An application of numerical optimization to the design of advanced airfoils for transonic aircraft showed that low-drag sections can be developed for a given design Mach number without an accompanying drag increase at lower Mach numbers. This is achieved by imposing a constraint on the drag coefficient at an off-design Mach number while minimizing the drag coefficient at the design Mach number. This multiple design-point numerical optimization has been implemented with the use of airfoil shape functions which permit a wide range of attainable profiles during the optimization process. Analytical data for the starting airfoil shape, a single design-point optimized shape, and a double design-point optimized shape are presented. Experimental data obtained in the NASA Ames two-by two-foot wind tunnel are also presented and discussed

Hicks, R. M.

Johnson, R. R.

English

NASA Technical Reports Server

• "00 49N 7'9 - ,,
_i_ APPLICATIONOF NUMERICALOPTIMIZATIONTO THE :_
i DESIGNOF ADVANCEDSUPERCRITICALAIRFOTLS
I RaymondR. Johnson
VoughtCorporation _
RaymondM. Hicks _..
NASAAmes ResearchCenter ,!_
SUMMARY '
A recenta_pllcationof numericaloptimizationto the designof advanced _!_
airfoilsfor transonicaircrafthas shown thatlow-dragsectionscan be
developedfor a givendesignMach numberwithoutan accompanyingdrag increase
at lowerMach numbers. This is achievedby imposinga constrainton the arag
coefficientat an off-designMech numberwhileminimizingthe dragcoefficient
at the designMach number. Thismultipledesign-pointnumericaloptimization
,, has beenimplementedwith the use of airfoilshapefunctlonswhich permita
wide rangeof _ttainableprofilesduringthe optimizationprocess. Analytical
datafor the startingairfoilshape,a singledesign-pointoptimizedshape.
and a doubledesign-polntoptimizedshape are presented. Experimenta_data
obtainedih the NASAAmes Two-byTwo-FootWind Tunnelare also presentedand
discussed.
INTRODUCTION
The designof supercriticalairfoilsfor advancedhigh speedaircrafthas
beenfacilitatedby the computerizedanalyticalmethodswhich have beendevel-
oped in recentyears. Althoughthesemethodsprovidegood performancepredic-
tionsfor each individualdesignpointwhich is considered,they do not allow
the designerto automaticallyconsideroff-designcharacteristicsduringthe
designprocess. A methodwhichdoes providemultiple,Jesign-poi_tcapability
is describedin reference I . It is basedon designby numericaloptimization.
An applicationof thatmethodto a singledesign-pointand a doubledesign-
pointairfoiloptimizationis addressedin the presentstudy. The double
design-pointoptimizationproduceda low drag supercrlticalairfoil_or a given ,:
Mach numbersubjectto a dragconstraintat a lowerMachnumber.
The treatmentof the supercriticalairfoildesignproblemby thismethod
has been facilitatedby the developmentof a set of airfoilshapefunctions
(referencel) which providea wide rangeof attainableprofilesduringthe
designprocess. The coefflclentsof th._seshapefunctionsare usedas design
varlablesin the numericaloptimizationtechniquewhich consistsof two exis-
ting computercodes: (a) an optimizationprogrambasedon the methodof
_I_
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feastble directions (reference 2) and (b) an aerodynamicanalysis program
basedon an iterative solution of the full potential equation for transonic
flow (reference 3).
SYMBOLS
4
ai shape function coefficients
c chord
CD section drag coefficient
C1 section lift coefficient
Cm section pitching momentcoefficient
Cp pressure coefficient
ft airfoil shape functions
M Machnumber
X airfoil abscissa
Y airfoilordinate
DESIGNMETHOD
Only a briefdescriptionof numericaloptimizationwill be given here.
A completediscussionof the techniquecan be found in reference4.
A schematicflowchartof the numericaloptimizationdesignprogramused
duringthisstudyis shown in figureI. A baselineairfoilis requiredto
starteach designproblem. The airfoilshape is representedin the program
by the followingequation:
Y = Ybasic + _ aifii
where Ybasicis the set of ordinatesof the baselineairfoiland fi are the shape
functions. The shapefunctionsare added linearlyto the baselineprofile
by the optimizationprogramto achievethe desireddesignimprovement.The
contributionof eachfunctionis determinedby the valueof the coefficient,
ai, associatedwith thatfunction These ai coefficientsare thereforethe
designvariables. Otherinputsto the programincludeMach number,angle of
attack,and any constraintsto be imposedon the design.
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1979011859-314
-'Yf
The hypotheticaldesignproblemrepresentedby the flowchart is drag
minimization at one Mach number, MI, with drag constrained to somespecified _
value at another _ch number, M2; The optimization program begins by changing _ ...
the designvariables,one by one, from the Inltlalvalueof zeroto 0.001. It :_
returns to the aerodynamics program for evaluation of the drag coefficient at
both_ch numbersM1 and M2 after each change. The value of O.OO1is somewhat
arbitrarybut has been foundto be an effectivestepchangein the design _
variablesto calculatethe requiredpartialderivatives,lhe partialderi- !_
i vativesof dragwith respectto eachdesignvariableform the gradientof drag, _
, VC_. The directionin which the designvariablesare changedto reducethe
dra_ coefficient at M1 ts -VC d (the steepest descent direction} if the drag I
constraintat M2 is not active. The optimizationprogramthen incrmnentsthe ,_
designvariablesin thlsdirectionuntilthe drag startsto increasebecause :!
of nonlinearityin the designspaceor the drag constraintat Mach numberM2 i
is encountered.If eitherof thesepossibilitiesoccurs,new gra_'ientsare
calculatedand a new directionis found thatwilldecreasedragwithoutvio- i
latingthe constraint.Whena minimumvalueof drag for Mach numberMI is :_
attainedwith a satisfieddrag constraintat M2, the requiredoptimizedalr- _i
foil has beenachieved.
t
AIRFOILSHAPEFUNCTIONS ;i
Supercrlticalairfoildesignby numericaloptimizationIs facilitated
by usinga set of geometricshapefunctions,each of whichaffectsa different
limitedregionof the profile. Generalclassesof suchfunctionswhich have
beenused successfullyto optimizesupercrltlcalairfoilsare describedin
referenceI. The shapefunctionsthatwere used in the presentstudywere
selectedfrom thosegeneralfunctionsand were applledto the airfoilupper
i surfaceonly. The exponentialdecayfunctionand the sine functionsare
presentedin figure2. The exponentialdecayfunction,fl providedvaria-
tionsin curvaturenear the airfoilleadingedge. In the'sinefunctions,the
exponentson the chordwisecoordinate,x, were assignedso thatthe maximum
perturbationsof f2, f3, f4 and f5 were at 20, 40, 60, and 80 percentof the
chordrespectively.The width of the regionaffectedby each sine function
was controlledby the locallzatlonpower,3. Previousstudies(referencel)
havefound thattheseshape functionsprovidea broadrangeof smoothairfoil
contourmodificationsduringthe optimizationprocess.
ANALYTICALDESIGNRESULTSAND DISCUSSION
The importanceof consideringoff-designperformanceof an airfoilduring
the designprocessw111 be illustratedby comparingthe resultsof a single
design-polntoptimizationwlth a doubledesign-pointoptimization,The first
involvesrecontouringthe upper surfaceof an existingsupercriticalairfoil
to reducethe wavedrag at a singledesignMach number. The secondconsists
of recontouringthe upper surfaceof the sameairfoilto reducethe wave drag
at the designMach numbersubjectto a drag constraintat a lower_lachnumber.
317
L
...................... L ' _ ' 1979011859 315
JThecalculatedwavedrag (reference3) for Machnumbersnear dragdlver-
gence for the starttng atrfot1 and the two opttm|zed atrfotls are presented
tn ftgure 3. All these data are for 0.40 C1, the destgn 11ft coefficient of
the starting airfoil. Machnumber0.78 was arbitrarily selected as the prtmary
design potnt, t.e., the Machnumberat whtch the drag would be minimized.
Results of the stngle destgn point optimization are Indicated as 412#41. The
drag atMach number0.78 ts significantly less than that of the starting air-
lot] and as a result the drag rtse occurs at a htgher l_lch number. ;:o;eYe_
the drag at lower Machnumbers, 0.76 and 0.77_ ts greate_ than that of the
starttng atrfot1. Thts local region of drag-creep could ltmtt the usefulness
of the improved drag rise characteristics of the optimized airfoil.
In order to avoid the drag-creep problem, the airfoil was optimized
a secondtime with an upper boundof .0005 tmposedon the drag coefficient
at Machnumber0.77. Results of thts double design-point optimization are
indicated tn figure 3 as 412142. The drag rise for thts atrfotl occurs at
a sltghtly lower Machnumberthan tt does for 412M1, but there is no drag-
creep over the range of Machnumbersfor which the atrfotls were analyzed.
Therefore, atrfotl 412M2is the more desirable design.
Chordwtse pressure distributions for the starttng airfoil and for airfoil
412M2at Machnumber0.77 are presented in figure 4. The reason for the lower
wave drag of the optimized atrfotl is obvious. The starttng airfoil has a
well deve]oped shock at approximately 40 percent of the chord, but airfoil
412M2does not. Instead, it exhibits a gradual recompression from approxi-
mately 16 percent to 50 percent of the chord. The geometric modification
which has producedthe pressure distribution change ts shownin figure 5.
This modification ts primarily a reduction tn surface curvature from 5 per-
cent to 40 percent of the _hord.
The aerodynamics code that was used in the optimization program is an
tnvtsctd, potential flow analysts method. In order to account for first
order viscous effects tn the flow field solution, a boundary layer displace-
ment thickness was added to the starting profile before the optimization
process. The displacement thickness was calculated for the pressure dts%rt-
button of the starting airfoil at a Machnumbernear its design condition,
0.78. I_ remained unchangedthroughout the optimization process, and each
of the optimized airfoils tncluded this samepassive displacement thickness.
Therefore, the analytical characteristics of the atrfotls did not reflect
potential changestn boundary layer behavior due to changes in the chordwtse
pressure distributions.
Another aerodynamic analysts code (reference 5) was used to evaluate
the acttve boundary layer characteristics of the starting atrfotl and opti-
mized atrfot] 412M2. That computer program ts also based on an tterattve
solutton of the full potential equation for transonic flow, and it tncludes
a momentum-integral calculation of the turbulent boundary layer parameters.
During the solution, the atrfotl geometry is regularly updated with the
boundary layer displacement thickness. The results of the viscous analyses
wlth thatcode for Mach numbersbetween0.76 and 0.81 indicatedthatthe
differencesin boundarylayercha'Rcteristicswould be small. The calculated
318
1979011859-316
wave drag for the startingairfoiland airfoil412M2 is presentedin figure6.
The relativeincreasein the dragriseMach numberis in good agreementwith
the resultsof the inviscidcode (figure3).
I EXPERIMENTALRESULTS
!
" Modelsof the startingairfoiland airfoil412M2 were testedin the NASA
A_s Two-byTwo-FootWindTunnel. Datawere obtainedat anglesof attack
from -4o to stallat Mach numbersfrom 0.20to 0.81. The testReynolds
numbervariedwlth Mach numberas presentedin Table I. Preliminarydata
fromthe test are presentedin figures7 and 8. The incrementalvaluesof
dragcoefficient,Cn, have beenreferencedto the minimumdragmeasuredfor
eitherof the alrfoITsat each llftcoefficient.Thereby,extraneouscom-
I ponentsin the absolutedrag levelhavebeen excludedfromthe comparison.
}
Dragcharacteristicsfor the startingairfoiland airfoil412M2 at lift
coefficient0.40 (figure7} indicatea differencein drag riseMach number
_: of 0.02 to 0.03. This improvementis greaterthan had beenpredictedby the
analyticalcodes (figures3 and 6). Dragcharacteristicsfor the two airfoils
at llft coefficient0.60 (figure8) also_ndicatesignificantlylessdrag for
the optimizedairfoil412M2 at all Mach numbers. Thereforethe airfoilper-
) fomance at this off-design condition has not been adversely affected by the
! desl;n improvementat 0.40 llft coefficient.
' The low speeddrag-creepwhichoccursbetweenMach numbers0.60 and 0.70
for both airfoilsis causedby the initialdevelopmentof supercritlcalveloc-
itiesoverthe uppersurfaceand the formationof a mild shockn_dr the leading
edge, Only Machnumbersgreaterthan0.76were consideredduringthe present
analyticaldeslgnstudy,but the numericaloptimizationtechniquecould also
be appliedto the minimizationof drag-creepat the lowerspeeds.
CONCLUDINGREMARKS
A techniquefor designinglow-dragsupercritlcalairfoilshas beendemon-
strated. The techniquewas usedto modifythe upper surfaceof an existing
12 percentthick supercrltlcalsectionto achievea substantialdrag reduction
at Mach number0.78withoutan accompanyingdrag increaseat lowerMach numbers.
The abilityto treatthis and othermultipledeslgn-pointproblemshas been
achievedby the use of a set of airfoilshapefunctionswhich providethe
necessaryflexibilityin the profilesthat are attainableP_ the numerical
optimizationdesigntechnique. Suchcapabilityis importantbecauseeach
designpointmightrequirethe modificationof a differentregionof the profile.
The two design-polntproblemconsideredin the presentstudyillustrates
the advantageof designby numericaloptimization.Aerodynamicrequirementsat
any numberof off-designconditionsare handledautomaticallywithoutmanual
interventionby the designer. Therefore.it providesa powerfultool for the
: 319
{_
1979011859-317
....................... -..,,L. ........
1
design, of airfoils to meet specified performance goals throughout a flight '
envelope.
REFERENCES
I. Hicks,R. M.; Vanderplaats,G. N.: Applicationof NumericalOptimization
to the Designof SupercrlticalAirfoilswithoutDrag-Creep.SAE Paper
770440, 1977.
2. Vanderplaats,G. N.: CONMIN-AFortranProgramfor ConstrainedFunction
Minimization.NASA TM X-62,282,1973.
3. Jameson,A.: IterativeSolutionof TransonicFlowsoverAirfoilsand Wings
IncludingFlowsat Mach I. Comm. PureAppl.Math.,Vol. 27, pp 283-309,
1974.
4. Vanderplaats,G. N.; Hicks,R. M.; Murman,E. M.: Applicationof Numerical
OptimizationTechniquesto AirfoilDesign. NASASP-347,Part II, 1975.
5. Bauer,F.; Garabedian,P.; Korn,D.; Jameson,A.: SupercriticalWing
SectionsII. LectureNotesin Economicsand MathematicalSystems,Vol.
I08,Springer-Verlag,1975.
320 'i
1979011859-318
| ,i
?
i s
j TABLEI. - HIND TUNNELTESTPARAHETERS
Angle of Attack
VazchNumber Range Reynolds Number
0.20 -40 to stall 1.9 x 106
0.40 3.0 x 106
0.60 4.0 x 106
0.70
0.75
0.76
0.77
0.78
0.79
0.80
0.81
321
L
1979011859-319
/ \
AERODYNAMICPROGRAM OPTIMIZATIONPROGRAMi i
INPUTINITIALAIRFOILAND
DRAGCONSTRAINTATM2 . PERTURBDESIGNVARIABLES -.m------
INITIALIZEDESIGNVARIABLES f
(al. • • an) /
" i / .......i;c;......ic;
I ...../ _FOR M1ANDM2
' f / FM j
CALCULATECdATM! ANDM2_ OR GRADIENTSVCd ATM1
ANDVCd ATM2
CALCULATECd ATM1ANDM2 L = SETa I .6.. an TONEW
VALUESASD.ETERMINEDBYGRADIENTS
F [ y. CDATM|ISMIN. CDATM],IS
l ANO NOTMIN.CDATM2 IS LESS AND/ORTHANSPECIFIED CDATM2 ISVALUE GREATERTHANx 1 SPECIFIEDVALUE
OUTPU_FINAL
PROFILEAND
C_, Cd. Cm
Figure i.- Flow chart of numerical optimization design program.
1.0_ f2 f3 f4 f5 Y = YBASIC+Zai fi
O.7.5- fl " x 11411- xl/e20x
f f2" SIN ('n'x 0.431)3
f3" SIN ('n'x0"75713
0,50 -
f4" SIN ('n'x1"35713
f5" SIN (w'x3.106)3
0'21 DESIGNVARIABLES:
a1, a2, a3, a4, a5
i:
_ O.20 O.40 O.60 O.80 1.O
! xi
i Figure 2.-Alrfoll shape functions.
!:
t 322 '_
rL
! e _'. e""'_"
1
_,, COMPUTERCODE: JAMESON2-D
.,..,
0.006 0 STARTINGAIRFOIL ___ ":
0 412M1 ' "
' D 412M2
:_: 0.005
!'; c.l- 0.40
0.004- _.
i C _;/ '
DINVISCID
0._-
,/
!,
, I I I I I
4_r {).75 O.76 O.77 O.78 O.79 O.80
_ MACHNUMBER
Figure 3.- Airfoil section optimization.
! i
i' M =0.77 C_ = 0.40
' SIARTINGAIRFOIL
412M2
t
" Cp
!.
i
L
! Figure 4 - Airfoil section pressure distributions.
r
3.!3
L,
I
....................................................._ ....... 197901 • _,,_, ,_---_Q_a_- 9_
f, STARTINGAIRFOIL (TIC)MAX --O.120
•- "-'"-"412M2 (TIC)MAX =O.119
Figure 5.- Airfoil geometry comparison.
COMPUTERCODE:GARABEDIAN-KORNWITHBOUNDARYlAYER
O.O_
0 STARTINGAIRFOIL
o 412M2
o._ 9.0.40 /
c, //i.
O.?5 O.76 O.77 O.78 O.79 O.80 O.81
MACH NUMBER
Figure 6.- Comparison of wave drag characteristics for the starting
airfoil and optimized airfoil 412M2.
324 #
lg7.clr) 11P,_q__99
_,_---.-___. / "_ • . . _, . _ ,
WINDTUNNELTESTRESULTS
REYNOLDSNUMBER2x106to4 x106
0 STARTINGAIRFOIL
[] 41_42
C_ - 0.400.006 ..:
Z_c° I
0.01_ c_
0""-"- 0
O 0 0
,,...,.,-J,--- I I I I I I
0.2 0.3 0.4 0.5 0.6 0.7 0.8
MACHNUMBER
Figure 7.- Experimental drag characteristics for the starting airfoil
and optimized airfoil 412M2 at C£ = 0.40.
WINDTUNNELTESTRESULTS
RE_'NOLDSNUMBER2x106to4 x106
0 STARTINGAIRFOIL
[] 412M2 I : l
!
cj .06o0.006
0.004
0.0_
I I I | I
0.2 0.3 0.4 0.5 0.6 Q7 0.8
MACHNUMBER
Figure 8.- Experimental drag characteristics for the starting airfoil
and optimized airfoil 412M2 at C£ - 0.60.
325
1979011859-323


Application of numerical optimization to the design of advanced supercritical airfoils

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