Further Studies of Airfoils Supporting Non-Unique Solutions in

17 Pages·2012·5.19 MB·English

AIAAJOURNAL
Vol.50,No.12,December2012
Further Studies of Airfoils Supporting Non-Unique
Solutions in Transonic Flow
AntonyJameson∗
StanfordUniversity,Stanford,California94305
JohnC.Vassberg†
TheBoeingCompany,HuntingtonBeach,California92647
and
KuiOu‡
StanfordUniversity,Stanford,California94305
DOI:10.2514/1.J051713
Non-uniquesolutionsoftheEulerequationswereoriginallydiscussedbyJamesonin1991(“AirfoilAdmitting
Non-UniqueSolutionstotheEulerEquations,”AIAAPaper1991-1625,June1991)forseveralhighlycambered
3 airfoils which were the result of aggressive shape optimization. In 1999 Hafez and Guo (“Nonuniqueness of
1
17 Transonic Flows,” Acta Mechanica, Vol. 138, Nos. 3–4, 1999, pp. 177–184) found non-unique solutions for a
5
4/1.J0 sFylomwmeAtrssicocpiaarteadllewlsiitdhedAamiraflogial,maantdiosnu/bSspelqitutienngtlyofKuSzu’pmerinsoannidcIRvaegnioovnas,(”“TThheeSortreuticctaulraanlIdnsCtaobmilpiutytaotifoTnraalnFsoluniidc
1
25 Dynamics,Vol.18,No.5,2004,pp.335–344)havediscoveredsomefullyconvexsymmetricairfoilsthatprovidenon-
10. uniquesolutions.Inthisarticlefournewsymmetricairfoils,allofwhichexhibitnon-uniquesolutionsinanarrow
OI: bandoftransonicMachnumbers,arestudied.Thefirst,NU4wastheresultofshapeoptimization.Thesecond,JF1is
D
g | anextremelysimpleparallelsidedairfoil.ThethirdJB1,isalsoparallelsidedbuthascontinuouscurvatureoverthe
aiaa.or zeenrtoireanpgrolefiloef.aTthtaecfko,utrhthe,PJ-C,6Z,-isacnodnNve-xbraanndcChe1scwointhtinpuoosuitsi.vCe,Lz(cid:1)er(cid:1)opalnodtsnoefgtahteisveealiifrtforielsspeexchtiibvietlyth.rAetesbormanecMheascaht
arc. numbersnostableZ-branchcouldbefound.WhentheP-branchiscontinuedtonegative(cid:1)insomecasesthereisa
p:// transitiontotheZ-branch,whileinothercasesthereisadirecttransitionfromtheP-totheN-branch.
htt
2 |
1
0
2
7,
2
ber I. Introduction question was further pursued in detail in a series of studies by
m
ece GIVEN that the equations governing steady inviscid com- Kuz’minandIvanova[7–11],whoconfirmedtheresultsofHafezand
n D pressibleflowarenonlinear,onecananticipatethepossibility Guoandalsoshowedthatairfoilswithpositivecurvatureeverywhere
o couldsupportnon-uniquesolutions.
2 ofnon-uniquesolutions.Afamiliarexampleisthecaseofsupersonic
3
0.2 flowpastawedgeatanangle(cid:1),inwhichtherearetwosolutionswith
6
4.1 different shock angles (cid:2) corresponding to the strong and weak II. ProblemStatement
171.6 bproatnenchtieasloflfotwhee(cid:2)qu(cid:1)at(cid:1)iodniawgrearme.dNisocno-vuenreiqdubeysoSltuetiinohnosfoffathnedtJraanmseosnoinc A. NU4Airfoil
by [1],whoobtainedliftingsolutionsforasymmetricJoukowskiairfoil The present authors have encountered another example as a
ded atzeroangleofattackinanarrowrangeofMachnumbersinthe consequenceofashapeoptimizationstudyforsymmetricairfoilsin
nloa neighborhood of Mach 0.85. This non-uniqueness could not be transonic flow [12], in which an attempt was made to find a 12-
ow duplicatedwiththeEulerequations,anditwasconjecturedbySalas percent-thick airfoil with ashock-free solution atMach0.84. The
D etal.[2]thatthenon-uniquenesswasaconsequenceoftheisentropic resultingairfoil,labelledNU4,hasanalmostshock-freesolutionat
flowapproximation.Subsequently,however,Jameson[3]discovered its designated Mach number, but it also allows a lifting and non-
several airfoils that supported non-unique solutions of the Euler lifting solution at zero angle of attack. To better understand this
equationsinanarrowMachband.Theseairfoilswerelifting. phenomenon,threeotherapproximately12-percent-thicksymmetric
Thequestionofnon-uniquetransonicflowswasre-examinedby airfoilshavebeenanalyzedindetail.
Hafez and Guo [4–6] who formed both lifting and non-lifting
solutionsfora12-percent-thicksymmetricairfoilwithparallelsides B. JF1Airfoil
from 25to75%chordinaMachrangefrom0.825to0.843.The Thefirst,theJF1airfoil,isevensimplerthantheshapeproposed
byHafezandGuo[4],consistingofaparallel-sidedslabclosedbya
semicircularnoseandtwoparabolicarcsattherear.Dependingon
Presented at the 29th AIAA Applied Aerodynamics Conference , theextentoftheparabolicarcs,aMachrangeexistsinwhichlifting
Honolulu,Hawaii,June27–30,2011;received1November2011;revision
solutionscanbefoundatzeroangleofattack.
received8March2012;acceptedforpublication29March2012.Copyright©
2012byAntonyJameson,JohnC.Vassberg,andKuiOu.Publishedbythe
AmericanInstituteofAeronauticsandAstronautics,Inc.,withpermission. C. JB1Airfoil
Copiesofthispapermaybemadeforpersonalorinternaluse,oncondition
Thesecond,theJB1airfoil,alsohasaparallelcentersectionbut
that the copier pay the $10.00 per-copy fee to the Copyright Clearance
the nose and tail are closed by higher-order curves that maintain
Center,Inc.,222RosewoodDrive,Danvers,MA01923;includethecode
continuityofthecurvatureatthejunctionpoints.Thenosesectionis
0001-1452/12and$10.00incorrespondencewiththeCCC.
∗ThomasV.JonesProfessor,AeronauticsandAstronauticsDepartment. definedbyaBeziercurvewiththefollowingfourcontrolpoints:
FellowAIAA.
†BoeingTechnicalFellow,PhantomWorks.FellowAIAA. x1(cid:2)0; y1(cid:2)0 x2(cid:2)0; y2(cid:2)1 x3(cid:2)1;
‡Ph.D. Candidate, Aeronautics and Astronautics Department, Student
y (cid:2)1 x (cid:2)1; y (cid:2)1
MemberAIAA. 3 4 4
2865
2866 JAMESON,VASSBERG,ANDOU
scaledtoalengthof0.125andaheightof0.0625.Theuppersurface 55%ofthechord,whichhaspositivecurvatureeverywhereandis
curveisdefinedby C continuous.
1
x(cid:2)0:125(cid:3)3t2(cid:1)2t3(cid:4) y(cid:2)0:0625(cid:3)3t(cid:1)3t2(cid:5)t3(cid:4); E. Overview
0(cid:6)t(cid:6)1 These four airfoils have very different characteristics, but they
allsharethepropertythatinnon-liftingtransonicflowtheyexhibit
a transition from a solution with two supersonic zones on each
Thetrailingcurvefromx(cid:2)0:625to1is
surface below a certain critical Mach number to a solution with
onesupersoniczoneoneachsurfaceabovethecriticalMachnumber.
(cid:1) (cid:2) (cid:3)1(cid:1)x(cid:4)(cid:5)3(cid:6) In the region of instability, solutions with positive lift are found
y(cid:2)0:0625 1(cid:1) 1(cid:1)
0:375 in which there is a single supersonic zone on the top surface and
two supersonic zones on the lower surface, and also solutions
with negative lift that are the mirror images of the solutions with
D. JC6Airfoil
positivelift. Thereresults are presentedinmore detailinthe next
Thethirdairfoil,JC6,isafullyconvexairfoildefinedbyasimple
section.
algebraicformula:
y(cid:3)x(cid:4)(cid:2)Cxn1(cid:3)1(cid:1)xn(cid:4); 0(cid:6)x(cid:6)1 III. Results
Thecalculationsforthefirstthreeairfoilswereperformedusing
3 wheretheconstantC,withavalueof0.06817,isadjustedtogivethe theauthors’SYN83code,whichimplementstheJameson–Schmidt–
171 specifiedmaximumthickness,12%ofthechord.Thechoicen(cid:2)6 Turkelscheme[13]onameshwithC-topology.Themeshcontained
5
1.J0 resultsinaveryblunt-nosedairfoilwithmaximumthicknessatabout 384 cells in the clockwise direction and 64 cells in the normal
4/
1
5
2
0.
1
232 on December 27, 2012 | http://arc.aiaa.org | DOI: Cp -0.4 -0.8 -1.2 -1.6 -2.0 1.2 0.8 0.4 0.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Nsup
64.160. NMUAC4 HA I R0.F83O7I0L0 ALPHA 0.00000 -12.0 0.00 500.00 1000.00 15W00o.r0k0 2000.00 2500.00 3000.0 -0.2 0
1. CL 0.000000 CD 0.008592 CM 0.000000 NU4 AIRFOIL
d by 17 GRID 384X 64 NDES 0 RES0.301E-07 GMAX 0.100E-05 MRWGERAOSICRDIDHK 1 03 .80134.918X9836E 7 . 0+060004 2 AR RLAEPTSHEIA D 2 000...9308009106E0-007
de a) Z-branch Cp distribution and mach contours b) Z-branch convergence history
a
o
nl
w
o
D
4.0 1.4
Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
NU4 AIRFOIL -12.0 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.0 -0.2 0
MACH 0.83700 ALPHA 0.00000 Work
CL 0.112434 CD 0.007120 CM -0.025056 NU4 AIRFOIL
GRID 384X 64 NDES 0 RES0.179E-07 GMAX 0.100E-05 MACH 0.83700 ALPHA 0.00000
RESID1 0.318E+02 RESID2 0.181E-07
WORK 1996.00 RATE 0.9894
GRID 384X 64
c) P-branch Cp distribution and mach contours d) P-branch convergence history
Fig.1 NU4atMach(cid:2)0:837showingthesymmetric solution(a)andconvergencehistory(b), andthe asymmetric solution(c)andconvergence
history(d).
JAMESON,VASSBERG,ANDOU 2867
4.0 1.4
Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -8.0 -6.0 -4.0 -2.0 -10.0 0.0 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
NMUAC4 HA I R0.F83O9I0L0 ALPHA 0.00000 -12.0 0.00 200.00 400.00 6W00o.r0k0 800.00 1000.00 1200.0 -0.2 0
CL 0.000001 CD 0.007383 CM 0.000000 NU4 AIRFOIL
GRID 384X 64 NDES 0 RES0.189E-03 GMAX 0.100E-05 MACH 0.83900 ALPHA 0.00000
RESID1 0.319E+02 RESID2 0.190E-03
WORK 998.00 RATE 0.9880
GRID 384X 64
a) Z-branch Cp distribution and mach contours b) Z-branch convergencehistory
3
05171 4.0 1.4
n December 27, 2012 | http://arc.aiaa.org | DOI: 10.2514/1.J CpNMCG 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 LRUA I C4D H0 A . 03 I 9R8074.F83X+++++++++++++++++++38+O++++++++++ ++93++++ ++I++ +0+ ++6L++ +++0+ ++4+ +++ C++ + +++ ++ +D+ ++ A++N+++ +++ L++D+ ++ ++P+0++E+++H.+++0S++++A0++ ++ +7+ + ++ +9+ +0+ ++2+ .++0+06++++ 0+ + + + +0+ +R+ +0+C+++E+0++M++S++++++0+ ++ +.+ ++1+-+++06+++++.7++0+++E++2++++-+4++0+++0++8+++0+++ ++ 9+++ +++G++++++++M++++++++++A++++++++X+++++++++ +++0++++++.+++1++++++0++++++0++++++E-05 Log (Error) -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 0.00 NMRW EUAO5S4CRI0 DAHK01I R .0 F 0 .013O.9108I993L6E9.0+0000 1 2 0AR RLA0EPT0SHEI.A D 0 2 0 000...91087082012E05-00W800o.0r0k2000.00 2500.00 3000.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 Nsup
2 o GRID 384X 64
3 c) P-branch Cp distribution and mach contours d) P-branch convergence history
2
60. Fig.2 NU4atMach(cid:2)0:839showingthesymmetric solution(a)andconvergencehistory(b), andthe asymmetric solution(c)andconvergence
1
4. history(d).
6
1.
7
1
y
b
d
e
ad direction.Inallcases,aliftingsolutionatzeroangleofattackwas WhenitbecomestoonarrowaboveMach0.835ortoowidebelow
o
nl obtainedbystartingatanangleofattackof0.005degandswitching Mach 0.825, the initial lifting solution at 0.005 deg decays when
w
Do to 0 deg after 500 iterations. The corresponding negative lifting theangleofattackisreducedtozero.Itmayalsobeobservedthat
solutioncanbeobtainedbystartingat(cid:1)0:005 degandistheexact the symmetric solutions at Mach 0.825 and 0.830 started to
mirrorimage. diverge after reaching extremely low values of the residual of the
order of 10(cid:1)12 and 10(cid:1)11, respectively, at around 1500 cycles,
suggesting that at these Mach numbers the symmetric solution
A. NU4Airfoil
may not be a stable equilibrium point. SYN83 uses a multigrid
ExaminingfirsttheNU4airfoil,stableasymmetricsolutionswere
solution procedure, in combination with variable local time steps
foundintherangeofMachnumbersfrom0.837to0.841.Asthe
andresidualaveraging.Theseprocedurescouldpossibilystabilize
Mach number is increased, there is a progressively decreasing
an unstable solution, and so time-accurate simulations will be
distancebetweenthetwosupersoniczonesonthelowersurface(in
needed to assess the true stability of the symmetric solutions.
the case of positivelift). This is illustrated in Figs. 1–3. At Mach
0.841,thesymmetricsolutionisclosetoshock-free,withasingle
relativelyweakshock. C. JB1Airfoil
TheJB1airfoildiffersfromtheJF1airfoilprimarilyinmaintaining
continuity of the curvature over the entire profile and having a
B. JF1Airfoil
slightly larger thickness of 12.5%. Nevertheless, the results dis-
Examining next the JF1 airfoil, which is closed at the rear by playedinFigs.7–9haveaverysimilarcharacter,withnon-unique
parabolic arcs from 76% chord, non-unique solutions were found
solutions in a somewhat narrower range of Mach numbers from
in the range of Mach numbers from 0.825 to 0.835, illustrated in
0.823to0.827.
Figs. 4–6. Figures 4a and 4c show the symmetric and positive
lifting solutions at Mach 0.825, respectively, together with their
convergence histories. Figures 5 and 6 show the corresponding D. JC6Airfoil
results at Mach 0.830 and 0.835, respectively. As the Mach CorrespondingresultsfortheJC6airfoilareshowninFigs.10–12.
number is increased, the double shock zone becomes narrower. Inthesecases,stableasymmetricsolutionswereformedintherange
2868 JAMESON,VASSBERG,ANDOU
4.0 1.4
Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -8.0 -6.0 -4.0 -2.0 -10.0 0.0 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
NMUAC4 HA I R0.F84O1I0L0 ALPHA 0.00000 -12.0 0.00 500.00 1000.00 15W00o.r0k0 2000.00 2500.00 3000.0 -0.2 0
CL 0.000000 CD 0.002708 CM 0.000000 NU4 AIRFOIL
GRID 384X 64 NDES 0 RES0.304E-07 GMAX 0.100E-05 MACH 0.84100 ALPHA 0.00000
RESID1 0.320E+02 RESID2 0.310E-07
WORK 1996.00 RATE 0.9897
GRID 384X 64
a) Z-branch Cp distribution and mach contours b) Z-branch convergence history
3
1
7
n December 27, 2012 | http://arc.aiaa.org | DOI: 10.2514/1.J051 CpNMCG 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 LRUA I C4D H0 A . 03 I 7R8064.F89X++++++++++++++++++4+5+O++++++++++ ++15++++ ++I++ 0++ ++6L++ ++0++ ++4 ++++ C++ + +++ ++ +D+ ++ A++N+++ +++ L++D +++ ++P+0++E+++H.+++0S++++A0++ ++ +8+ + ++ +3+ +0+ ++7+ .++00+1++++ 0+ + + + +0 ++R+ +0+C+++E+0++M++S+++++0++ ++ +.+ ++2-++++07++++.+8++0+++E++2++++-+1++0+++3++8+++5+++ ++ 7+++ +++G++++++++M++++++++++A++++++++X++++++++ ++++0++++++.+++1++++++0++++++0++++++E-05 Log (Error) -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 0.00 NM UA54C 0AH0I R . F 00O.80I4L100 1 A0L0P0HA.0 0 0.0001005W00o.0rk0 2000.00 2500.00 3000.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 Nsup
232 o RWGEROSIRDIDK 1 03 .813492X906E . +06004 2 R RAETSEI D 2 00..9288865E-08
60. c) P-branch Cp distribution and mach contours d) P-branch convergence history
1
64. Fig.3 NU4atMach(cid:2)0:841showingthesymmetricsolution(a)andconvergencehistory(b),andtheasymmetricsolutions(c)andconvergence
71. history(d).
1
y
b
d
e
d
a
o
wnl ofMachnumbersfrom0.844to0.848.Thesecalculationswerealso Inthefirstregime,thereappearstobenostablesolutionatzeroangle
o
D performedwithSYN83,butbecauseoftheextremebluntnoseshape ofattack,althoughitmaybepossibletoforceconvergencetovery
of this airfoil, a finer mesh with 768(cid:7)128 cells was used. The smallvaluesoftheerrorresidual.InthiscasethereisnoZ-branch.
generalcharacterofthesolutionsissimilartothatoftheJF1airfoils, TheP-branchcanbecontinuedtonegativeanglesofattackbefore
confirmingthatnon-uniquesolutionscanbefoundforanextremely thereisatransitiondirectlytotheN-branch.Inthesecondregime,
smoothconvexprofilewithpositivecurvatureeverywhere.Toverify thereisaZ-branchinwhichstablesolutionsarefoundoverasmall
that these solutions are not a consequence of the mesh or the rangeofangleofattack.However,whentheP-branchiscontinuedto
discretizationscheme,thisairfoilwasalsoanalyzedonanO-mesh negative(cid:3)thereiseventuallyadirecttransitiontotheN-branch.In
with512(cid:7)512cellswithanuniformaspectratioofunity.Figure13 thethirdregimethereisalargerZ-branch,andwhentheP-branchis
illustratesacorrespondingmeshwith128(cid:7)128cells.Calculations continuedtonegative(cid:3),thereisatransitiontotheZ-branch,andthen
wereperformedusingaversionoftheauthors’FLO82code,which as(cid:3)isfurtherreducedtoamorenegativevalue,thereisasecond
implementstheH-CUSPscheme[14],andwithNASA’sOverflow transitiontotheN-branch.Theseregimescanbeseenforthedifferent
code[15].Theresultsconfirmedthesametrend.Figure14showsthe airfoils in Figs. 15–18. Figure 15 shows the NU4 airfoil at Mach
three solutions for Mach 0.847 at zero angle of attack calculated 0.840.Itexhibitsthethirdbehavior,withatransitionfromtheP-to
usingFLO82. theN-branchviatheZ-branch.Figure16showsthattheJF1airfoil
hasasimilarbehavioratMach0.835.Thelowerfigureisazoomof
theupperpicture.Figure17showstheJB1airfoilatMach0.827.
E. PlotsofCL(cid:1)(cid:1)Sweeps Thisshowsthesecondregimewherethereisadirecttransitionfrom
To further elucidate the behavior, C versus (cid:3) curves were theP-totheN-branch.AtMach0.825,nostableZ-branchcouldbe
L
calculatedforallfourairfoilsattheMachnumberwheretheyexhibit foundforthisairfoil,indicatingthatitisinthefirstregime.Figure18
non-unique solutions. Overall, three different regimes can be showsthattheJC6airfoilisinthesecondregimeatMach0.847.In
identified as the Mach number is varied over a narrow band. this case the C (cid:1)(cid:3) sweep was calculated on 512(cid:7)512 O-mesh
L
Typically,theC (cid:1)(cid:3)curvehasthreebrancheswithzero,positive, usingbothFLO82-HCUSPandOverflow,toverifythebehavior.To
L
andnegativeliftatzeroangleofattack(theZ-,P-,andN-branches). better understand the stability of these solutions, time-accurate
JAMESON,VASSBERG,ANDOU 2869
8.0 1.4
CpJF 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 1 AIRFO+++++++++++++++++++++++++++++I+++++++++L+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -24.0 -20.0 -16.0 -12.0 -8.0 -4.0 0.0 4.0 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 Nsup
MACH 0.82500 ALPHA 0.00000 Work
CL 0.000000 CD 0.056972 CM 0.000000 JF1 AIRFOIL
GRID 384X 64 NDES 0 RES0.666E-10 GMAX 0.100E-05 MACH 0.82500 ALPHA 0.00000
RESID1 0.438E+02 RESID2 0.658E-10
WORK 1996.00 RATE 0.9865
GRID 384X 64
a) Z-branch Cp distribution and mach contours b) Z-branch convergencehistory
3
1
7
1
5
0
December 27, 2012 | http://arc.aiaa.org | DOI: 10.2514/1.J CpJF 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 1 AIRFO++++++++++++++++++++++++++++++I++++++++L+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 Nsup
32 on MCGLRA I CD H0 . 23 58084.88X22 56 0 6 0 4 C D AN LD P0EH.0SA5 6 7 0 2 .007 0 0 R 0CE0MS0 . 3-02.90E7-50386 8 GMAX 0.100E-05 JMRFEA1S CAIDHIR1 F 0 O .04I.38L82E50+00 2 A RLEPSHIAD 2 00..304050E0-0W08ork
0.2 WGROIRDK 3 8149X96 . 0604 RATE 0.9884
6
1
64. c) P-branch Cp distribution and mach contours d) P-branch convergencehistory
1.
7
1
y
b
d
Downloade CpJF -0.4 -0.8 -1.2 -1.6 -2.0 1.2 0.8 0.4 0.0 1 AIRFO++++++++++++++++++++++++++++++I++++++++L+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0 pNsu
MACH 0.82500 ALPHA 0.00000 Work
CL -0.258826 CD 0.056727 CM 0.075368 JF1 AIRFOIL
GRID 384X 64 NDES 0 RES0.329E-08 GMAX 0.100E-05 MACH 0.82500 ALPHA 0.00000
RESID1 0.438E+02 RESID2 0.345E-08
WORK 1996.00 RATE 0.9884
GRID 384X 64
e) N-branch Cp distribution and mach contours f) N-branch convergencehistory
Fig.4 JF1atMach(cid:2)0:825showingthesymmetricsolution(a)andconvergencehistory(b),andthepairofasymmetricsolutions(cande)and
convergencehistories(dandf).
2870 JAMESON,VASSBERG,ANDOU
4.0 1.4
Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -8.0 -6.0 -4.0 -2.0 -10.0 0.0 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
JF1 AIRFOIL -12.0 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.00 -0.2
MACH 0.83000 ALPHA 0.00000 Work
CL 0.000000 CD 0.054135 CM 0.000000 JF1 AIRFOIL
GRID 384X 64 NDES 0 RES0.588E-08 GMAX 0.100E-05 MACH 0.83000 ALPHA 0.00000
RESID1 0.441E+02 RESID2 0.578E-08
WORK 1996.00 RATE 0.9887
GRID 384X 64
3 a) Z-branch Cp distribution and mach contours b) Z-branch convergencehistory
1
7
1
5
0
14/1.J 4.0 1.4
mber 27, 2012 | http://arc.aiaa.org | DOI: 10.25 Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
on Dece JMFA1C AHI R 0F.O83I0L00 ALPHA 0.00000 -12.0 0.00 500.00 1000.00 15W00o.r0k0 2000.00 2500.00 3000.0 -0.2 0
2 CL 0.179321 CD 0.056586 CM -0.059885 JF1 AIRFOIL
60.23 GRID 384X 64 NDES 0 RES0.153E-09 GMAX 0.100E-05 MRWEAOSCRIDHK1 0 .014.9489136E0.0+0000 2 AR RLAEPTSHEIA D 2 000...9108606109E0-009
1 GRID 384X 64
4.
1.6 c) P-branch Cp distribution and mach contours d) P-branch convergence history
7
1
y
b
d
Downloade Cp -0.4 -0.8 -1.2 -1.6 -2.0 1.2 0.8 0.4 0.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Nsup
JMFA1C AHI R 0F.O83I0L00 ALPHA 0.00000 -12.0 0.00 500.00 1000.00 1W50o0.r0k0 2000.00 2500.00 3000.00 -0.2
CL -0.179321 CD 0.056586 CM 0.059885 JF1 AIRFOIL
GRID 384X 64 NDES 0 RES0.153E-09 GMAX 0.100E-05 MACH 0.83000 ALPHA 0.00000
RESID1 0.441E+02 RESID2 0.161E-09
WORK 1996.00 RATE 0.9869
GRID 384X 64
e) N-branch Cp distribution and mach contours f) N-branch convergence history
Fig.5 JF1atMach(cid:2)0:830showingthesymmetricsolution(a)andconvergencehistory(b),andthepairofasymmetricsolutions(cande)and
convergencehistories(dandf).
JAMESON,VASSBERG,ANDOU 2871
8.0 1.4
Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -20.0 -16.0 -12.0 -8.0 -4.0 0.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
JF1 AIRFOIL -24.0 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.00 -0.2
MACH 0.83500 ALPHA 0.00000 Work
CL 0.000000 CD 0.046047 CM 0.000000 JF1 AIRFOIL
GRID 384X 64 NDES 0 RES0.568E-13 GMAX 0.100E-05 MACH 0.83500 ALPHA 0.00000
RESID1 0.444E+02 RESID2 0.579E-13
WORK 1996.00 RATE 0.9830
GRID 384X 64
a) Z-branch Cp distribution and mach contours b) Z-branch convergenc history
3
1
7
1
5
0
14/1.J 4.0 1.4
on December 27, 2012 | http://arc.aiaa.org | DOI: 10.25 CpJMF 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 A1C AHI R 0F.O8+++++++++++++++++++++3+++++++++I++++++5++L++++++0++++++0+++++++ +++ +++ +++ +++A++++++++L++++++P++++++H+++++++A+++++++ ++ +++0++++.+++0++++0+++++0++++0+++++0++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0 2.0 0.00 500.00 1000.00 15W00o.r0k0 2000.00 2500.00 3000.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 Nsup
2 CL 0.056998 CD 0.051628 CM -0.024147 JF1 AIRFOIL
160.23 GRID 384X 64 NDES 0 RES0.210E-05 GMAX 0.100E-05 MRWGERAOSICRDIDHK 1 03 .80144.948X9436E 5 . 0+060004 2 AR RLAEPTSHEIA D 2 000...9209101406E0-005
4.
1.6 c) P-branch Cp distribution and mach contours d) P-branch convergencehistory
7
1
y
b
d
oade 4.0 1.4
Downl -2.0 2.0 1.2
Cp -0.4 -0.8 -1.2 -1.6 1.2 0.8 0.4 0.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -10.0 -8.0 -6.0 -4.0 -2.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Nsup
JF1 AIRFOIL -12.0 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.0 -0.2 0
MACH 0.83500 ALPHA 0.00000 Work
CL -0.056998 CD 0.051628 CM 0.024147 JF1 AIRFOIL
GRID 384X 64 NDES 0 RES0.210E-05 GMAX 0.100E-05 MACH 0.83500 ALPHA 0.00000
RESID1 0.444E+02 RESID2 0.214E-05
WORK 1996.00 RATE 0.9916
GRID 384X 64
e) N-branch Cp distribution and mach contours f) N-branch convergence history
Fig.6 JF1atMach(cid:2)0:835showingthesymmetricsolution(a)andconvergencehistory(b),andthepairofasymmetricsolutions(cande)and
convergencehistories(dandf).
2872 JAMESON,VASSBERG,ANDOU
8.0 1.4
-2.0 4.0 1.2
Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -20.0 -16.0 -12.0 -8.0 -4.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Nsup
JMBA1C AHI R 0F.8O2I3L00 ALPHA 0.00000 -24.0 0.00 500.00 1000.00 15W00o.0rk0 2000.00 2500.00 3000.0 -0.2 0
CL 0.000000 CD 0.024915 CM 0.000000 JB1 AIRFOIL
GRID 384X 64 NDES 0 RES0.155E-10 GMAX 0.100E-05 MACH 0.82300 ALPHA 0.00000
RESID1 0.419E+02 RESID2 0.155E-10
WORK 1299.00 RATE 0.9782
GRID 384X 64
3 a) Z-branch Cp distribution and mach contours b) Z-branch convergencehistory
1
7
1
5
0
514/1.J 8.0 1.4
ember 27, 2012 | http://arc.aiaa.org | DOI: 10.2 Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -20.0 -16.0 -12.0 -8.0 -4.0 0.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
232 on Dec JMCGBLRA1 I CD AH0 .I 13R 780F04.80OX29I 31 L 0 6 0 4 C D AN LD P0EH.0SA2 5 8 0 7 .009 0 0 R 0CE0MS0 . 3-08.50E4-81412 5 GMAX 0.100E-05 -24.0 0.00 JMB A51 CA0HI0R F. 0O0.I08L2300 1 0AL0P0H.A0 0 0.0001050W00o.0r0k 2000.00 2500.00 3000.00 -0.2
0. RESID1 0.419E+02 RESID2 0.404E-11
6 WORK 1996.00 RATE 0.9851
4.1 GRID 384X 64
1.6 c) P-branch Cp distribution and mach contours d) P-branch convergencehistory
7
1
y
b
d
Downloade Cp -0.4 -0.8 -1.2 -1.6 -2.0 1.2 0.8 0.4 0.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -20.0 -16.0 -12.0 -8.0 -4.0 0.0 4.0 8.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Nsup
JMBA1C AHI R 0F.8O2I3L00 ALPHA 0.00000 -24.0 0.00 500.00 1000.00 15W00o.0r0k2000.00 2500.00 3000.00 -0.2
CL -0.170091 CD 0.025879 CM 0.048425 JB1 AIRFOIL
GRID 384X 64 NDES 0 RES0.387E-11 GMAX 0.100E-05 MACH 0.82300 ALPHA 0.00000
RESID1 0.419E+02 RESID2 0.407E-11
WORK 1996.00 RATE 0.9851
GRID 384X 64
e) N-branch Cp distribution and mach contours f) N-branch convergencehistory
Fig.7 JB1atMach(cid:2)0:823showingthesymmetricsolution(a)andconvergencehistory(b),andthepairofasymmetricsolutions(cande)and
convergencehistories(dandf).
JAMESON,VASSBERG,ANDOU 2873
4.0 1.4
Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -8.0 -6.0 -4.0 -2.0 -10.0 0.0 2.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
JMBA1C AHI R 0F.8O2I5L00 ALPHA 0.00000 -12.0 0.00 200.00 400.00 6W00o.r0k0 800.00 1000.00 1200.0 -0.2 0
CL 0.000000 CD 0.023295 CM 0.000000 JB1 AIRFOIL
GRID 384X 64 NDES 0 RES0.182E-08 GMAX 0.100E-05 MACH 0.82500 ALPHA 0.00000
RESID1 0.420E+02 RESID2 0.188E-08
WORK 1197.00 RATE 0.9803
GRID 384X 64
a) Z-branch Cp distribution and mach contours b) Z-branch convergence history
3
1
7
1
5
0
14/1.J 8.0 1.4
ember 27, 2012 | http://arc.aiaa.org | DOI: 10.25 Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -20.0 -16.0 -12.0 -8.0 -4.0 0.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
n Dec JB1 AIRFOIL -24.0 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.00 -0.2
232 o MCGLRA I CD H0 . 13 48004.80X26 54 0 6 0 4 C D AN LD P0EH.0SA2 5 6 0 8 .009 0 0 R 0CE0MS0 . 5-05.00E4-21529 1 GMAX 0.100E-05 JMBA1 CAHIR F O0.I8L2500 ALPHA 0.0000W0 ork
60. RWEOSRIDK1 0 .1492906E.+000 2 R RAETSEI D 2 00..9587441E-12
4.1 GRID 384X 64
1.6 c) P-branch Cp Distribution and Mach Contours d) P-branch convergence history
7
1
y
b
d
Downloade Cp -0.4 -0.8 -1.2 -1.6 -2.0 1.2 0.8 0.4 0.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -20.0 -16.0 -12.0 -8.0 -4.0 0.0 4.0 8.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Nsup
JB1 AIRFOIL -24.0 0.00 500.00 1000.00 1500.00 2000.00 2500.00 3000.0 -0.2 0
MACH 0.82500 ALPHA 0.00000 Work
CL -0.140064 CD 0.025689 CM 0.042591 JB1 AIRFOIL
GRID 384X 64 NDES 0 RES0.574E-12 GMAX 0.100E-05 MACH 0.82500 ALPHA 0.00000
RESID1 0.420E+02 RESID2 0.605E-12
WORK 1996.00 RATE 0.9842
GRID 384X 64
e) N-branch Cp distribution and mach contours f) N-branch convergencehistory
Fig.8 JB1atMach(cid:2)0:825showingthesymmetricsolution(a)andconvergencehistory(b),andthepairofasymmetricsolutions(cande)and
convergencehistories(dandf).
2874 JAMESON,VASSBERG,ANDOU
4.0 1.4
CpJMB 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 A1C AHI R 0F.8O+++++++++++++++++++2++++++++++++I+++7++++L++++0++++++++0+++++++ +++ +++ +++ +++A++++++++L++++++P++++++H+++++++A+++++++ ++ +++0++++.+++0++++0+++++0++++0+++++0++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -8.0 -6.0 -4.0 -2.0 -12.0 -10.0 0.0 2.0 0.00 500.001000.0015W00o.r0k02000.002500.003000.0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0 Nsup
CL 0.000000 CD 0.020558 CM 0.000000 JB1 AIRFOIL
GRID 384X 64 NDES 0 RES0.109E-06 GMAX 0.100E-05 MACH 0.82700 ALPHA 0.00000
RESID1 0.421E+02 RESID2 0.109E-06
WORK 1299.00 RATE 0.9849
GRID 384X 64
3 a) Z-branch Cp distribution and mach contours b) Z-branch convergence history
1
7
1
5
0
14/1.J 8.0 1.4
ember 27, 2012 | http://arc.aiaa.org | DOI: 10.25 Cp 1.2 0.8 0.4 0.0 -0.4 -0.8 -1.2 -1.6 -2.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -20.0 -16.0 -12.0 -8.0 -4.0 0.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
on Dec JMBA1C AHI R 0F.8O2I7L00 ALPHA 0.00000 -24.0 0.00 500.00 1000.00 1W50o0.r0k0 2000.00 2500.00 3000.00 -0.2
2 CL 0.101583 CD 0.024709 CM -0.033702 JB1 AIRFOIL
160.23 GRID 384X 64 NDES 0 RES0.291E-10 GMAX 0.100E-05 MRWGERAOSICRDIDHK 1 03 .80144.928X9126E 7 . 0+060004 2 AR RLAEPTSHEIA D 2 000...9308006301E0-010
4.
1.6 c) P-branch Cp distribution and mach contours d) P-branch convergence history
7
1
y
b
d
oade 8.0 1.4
Downl Cp -0.4 -0.8 -1.2 -1.6 -2.0 1.2 0.8 0.4 0.0 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Log (Error) -20.0 -16.0 -12.0 -8.0 -4.0 0.0 4.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Nsup
JMBA1C AHI R 0F.8O2I7L00 ALPHA 0.00000 -24.0 0.00 500.00 1000.00 1W50o0.r0k0 2000.00 2500.00 3000.00 -0.2
CL -0.101583 CD 0.024709 CM 0.033702 JB1 AIRFOIL
GRID 384X 64 NDES 0 RES0.291E-10 GMAX 0.100E-05 MACH 0.82700 ALPHA 0.00000
RESID1 0.421E+02 RESID2 0.303E-10
WORK 1996.00 RATE 0.9861
GRID 384X 64
e) N-branch Cp distribution and mach contours f) N-branch convergence history
Fig.9 JB1atMach(cid:2)0:827showingthesymmetricsolution(a)andconvergencehistory(b),andthepairofasymmetricsolutions(cande)and
convergencehistories(dandf).