supercritical natural laminar flow airfoil optimization for regional aircraft wing design

7/25/2019 Supercritical Natural Laminar Flow Airfoil Optimization for Regional Aircraft Wing Design

1/13

Aerospace Science and Technology 43 (2015) 152164

Contents lists available at ScienceDirect

Aerospace

Science

and

Technology

www.elsevier.com/locate/aescte

Supercritical

natural

laminar

flow

airfoil

optimization

for

regional

aircraft

wing

design

Yufei Zhang a,1,Xiaoming Fang a,2,Haixin Chen a,,3,Song Fu a,3,Zhuoyi Duan b,4,Yanjun Zhang b,5

a TsinghuaUniversity,Beijing,100084,Chinab AVICTheFirstAircraftInstitute,Xian,710089,China

a

r

t

i

c

l

e

i

n

f

o a

b

s

t

r

a

c

t

Article

history:

Received29September2014

Receivedinrevisedform26February2015

Accepted27February2015

Availableonline4March2015

Keywords:

Airfoil

Optimization

Naturallaminarflow

Pressuregradientconstraint

Favorablepressuregradient

An

optimization

design

method

of

supercritical

natural

laminar

flow

airfoil

based

on

Genetic

Algorithm

and

Computational

Fluid

Dynamics

is

tested

in

this

paper.

Class

Shape

Transformation

method

is

adopted

as

geometry

parameterization

method.

Constraints

on

pressure

distribution

are

applied

to

gain

appropriate

flow

field

in

addition

to

the L/D performance. A fixed transition computation method

is

used

in

the

optimization

process

to

save

computation

time

while

giving the

reasonable

friction

drag

estimation

and

predicting the

influence

of

the

laminar

boundary

layer

on

airfoil

performances.

Specified

favorable

pressure

gradient

constraints

are

used

to

guarantee

the

expected

laminar

length.

Objective

of

optimization

is

set

to

weaken

the

shock

wave

and

minimize

the

pressure

drag.

Such

a

simplified

NLF

optimization

process

is

verified

by

natural

transition

computation.

The

optimal

setting

of

the

favorable

pressure

gradient

constraint,

which

is

important

for

the

trade-off

between

drag

reduction

and

laminar

stability,

is

then

studied

via

numerical

investigation.

Results

show

that

the

airfoil

optimized

by

constraining

a

favorable

pressure

gradient

larger

than

0.2

is

good

for

both

cruise

efficiency

and

robustness.

A

natural

laminar

wing

is

then

designed

based

on

the

optimized

airfoil.

Numerical

verifications

show

that

the

wing

has

good

natural

laminar

performance

and

low

speed

behavior.

2015ElsevierMassonSAS.All rights reserved.

1. Introduction

NaturalLaminarFlow(NLF)airfoilhasalreadybeenstudiedfor

severaldecades[15].However, itstilldrawshighattention inre-

cent years. As the friction drag is about half of the total drag

for modern civil aircrafts [10], laminar technology has great po-

tential to increase lift todragratio.Althougha lotofflight tests

had successfully validated the efficiency of NLF airfoil on mod-

ern large civil aircrafts [16,6], the technology is only realized on

wingsofseverallightbusinessaircraftsinthecommercialmarket

until

now,

such

as

the

Honda

Jet

[11,13] and

the

Aerion

Super-

SupportedbyNationalKeyBasicResearchProgramofChina(2014CB744801)

andNationalNaturalScienceFoundationofChina(11102098and11372160).

* Correspondingauthor.E-mailaddresses:[email protected](Y. Zhang),

[email protected](H. Chen).1 Assistantprofessor,SchoolofAerospaceEngineering.2 Masterstudent,SchoolofAerospaceEngineering.3 Professor,SchoolofAerospaceEngineering.4 Researchprofessor,GeneralDesignandAerodynamicDepartment.5 Seniorengineer,GeneralDesignandAerodynamicDepartment.

sonicBusinessJet[14,31].ParameteranalysisresultsofLammering

etal. [19] showed that,NLFdesignofaBoeing777-sizeairplane

could not show improvements on airplane direct operating costs

than conventional turbulent design unless the drag reduction is

morethan40counts.Inordertopreservelaminarregion,alower

leadingedgesweepangle isadopted in theNLFwing [19].Con-

sequently, thecruiseMachnumber isquite lower than turbulent

wing.IncreasingcruiseMachnumberisasimportantasreducing

skinfrictionforNLFwingdesign.Benefitandpenaltyof theNLF

technologyneedtobeclearlyquantified.

Airfoil

is

a

fundamental

element

of

a

wing.

Many

investigatorshavefocusedonthesupercriticalNLFairfoildesignbecauseofits

importanceforthehigh-subsonicNLFwing.BiberandTilmann[4]

developedasupercriticalNLFdesignmethodbasedon thepanel

and Euler codes coupled with boundary layer equation, and at-

temptedtoincreasethedragbucketoftheNLFairfoilinorderto

extend the operational speed range. Eggleston et al. [9] showed

that the peak Mach number, pressure gradient, and aft loading

werecriticalfactorsofafavorablepressuredistributionofanNLF

airfoil.Cellaetal. [5] successfullyused the ruleofcosine tode-

signahigh-subsonicNLFwingwithamulti-objectiveoptimization

method.Theyseparatelydesignedtheroot/kink/tipairfoilsandgot

http://dx.doi.org/10.1016/j.ast.2015.02.024

1270-9638/ 2015ElsevierMassonSAS.All rights reserved.
http://dx.doi.org/10.1016/j.ast.2015.02.024http://www.sciencedirect.com/http://www.elsevier.com/locate/aesctemailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.ast.2015.02.024http://crossmark.crossref.org/dialog/?doi=10.1016/j.ast.2015.02.024&domain=pdfhttp://dx.doi.org/10.1016/j.ast.2015.02.024mailto:[email protected]:[email protected]://www.elsevier.com/locate/aesctehttp://www.sciencedirect.com/http://dx.doi.org/10.1016/j.ast.2015.02.024


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Y. Zhang et al. / Aerospace Science and Technology 43 (2015) 152164 153

Nomenclature

TurbulenceintermittencyfactorRe Reynolds number based on boundary layer momen-

tumthickness

Ma Machnumber

Cp Pressurecoefficient

Cf

Frictioncoefficient

Cl Liftcoefficient

Cd Dragcoefficient

Cd,p Pressuredragcoefficient

Cd,f Frictiondragcoefficient

Cm Pitchingmoment

L/D Lifttodragratio

dCp/dX Pressurecoefficientgradientofairfoil

t/c Airfoilrelativethickness

c Chord length

QFPG quantityoffavorablepressuregradient,dCp/dX

anNLF wingwithgoodlaminarperformance.KhalidandJones[17]

showedthesupercriticalNLFairfoilswithdifferentthicknessesde-

signedattheNationalAeronauticalEstablishment,andtheexperi-

mentvalidatedgoodperformancesoftheairfoilsatReynoldsnum-

ber up to12.5million.Streit etal. [30] provideda new method

ofconvertingpressuredistributionoftwo-dimensionalNLFairfoil

tothree-dimensionalwingbyconsideringthesweepandtapered

effects. The pressure distribution of the Hondajet [12] provided

some new concepts of supercritical NLF airfoil design, on which

thetailingedgebubbleoftheuppersurfacewasadoptedtosup-

press

low

speed

flow

separation,

and

the

leading-edge

shape

was

carefullydesignedtocausetransitionathighanglesofattack(AoA)

to obtain higher maximum lift coefficient. Shockwave/boundary

layerinteractionisamajorcauseoftransonicdragrising.Aircraft

designerswouldhaveanopportunitytoraisecruiseMachnumber

iftheycoulddecreaseshockwavedrag.Thatisamainobjectiveof

supercriticalairfoildesign.Apparently,anotherobjectiveofsuper-

criticalNLFairfoildesignistodecreasethefrictiondrag.Inrealistic

high-subsonicdesignpractice,aerodynamicdesignerusuallyhasa

target(oranexpectation)oflaminarlengthforacertaincondition

based on experience or literature survey. Consequently, the po-

tentialof frictiondragreduction isapproximatelyconfirmed.The

designproblembecomeshowtoachievelaminarlengthandhow

toreduceshockwavedrag.Laminarflowlengthcouldbeachieved

through

maintaining

Favorable

Pressure

Gradient

(FPG,

or

negative

pressuregradient)[8].However,theFPGshouldnotbesogreatas

toavoidexcessiveshockstrength [8].Nevertheless, thequantita-

tive influenceofFPGondragofsupercriticalNLFairfoil isnotso

clear.Trade-offbetweenwavedragandfrictiondragisaproblem

ofNLFairfoildesign,whichiscloselyrelatedtotheFPG.

With the help of modern optimization methods, the applica-

tionofNLFtechnologycouldbepushedforward.Geneticalgorithm

[35,2] and adjoint method [21,22] are two kinds of widely used

optimization methodsonairfoildesign.Bothmethods have their

inherent problems. The latter is lack of global optimization abil-

ity and difficult to treat realistic design constraints. The former

hastheprobabilitytoachieveglobaloptimizationsolution,butre-

quires lots of computation costs. Computation cost of CFD must

be

carefully

controlled

in

genetic

algorithm

optimization.

Man-in-loopdesignprocessisapracticalcompromiseforengineering

applications, for example, introducing some pressure distribution

constraints in a design problem to guide optimization direction

[34,33]andartificiallyadjustingtheconstraintsandobjectivesdur-

ingdesigniteration.

Inthispaper,supercriticalNLFairfoilisoptimizedforthehigh-

subsonicNLFwingofaregionaljet.AReynoldsAveragedNavier

Stokes CFD solver is used as aerodynamic analysis tool. An in-

house developed optimization platform [27] based on the Non-

dominatedSortingGeneticAlgorithm-II(NSGA-II)[7]isadoptedas

backgroundschedulingsoftware.Classshapetransformation(CST)

methodisemployedasairfoilparameterizationmethod.Optimiza-

tion process is controlled by a series of realistic constraints. Su-

percritical

NLF

airfoil

is

obtained

through

optimization

based

on

RAE2822 supercritical airfoil. Effect of FPG on laminar character-

isticsisinvestigatedbysixairfoilswhichareoptimizedbydiffer-

entpressuredistributionconstraints.Agoodcompromisebetween

pressuredragand frictiondrag isachievedwhentheFPGon the

uppersurfacebefore50%chordislargerthan0.2.Ahigh-subsonic

NLFwingisgotbyassemblingandsimplymodifyingtheoptimized

airfoil.Numericalresultsdemonstratethatthewinghasbothlarge

laminarregionandgoodrobustness.

2.

Numerical

method

and

validation

2.1. Turbulencemodeling

AerodynamicanalysisinthispaperisbasedonaReynoldsAver-

agedNavierStokesCFDcode.Itisusedtocomputefixedtransition

flowfield in theoptimization, and tocalculatenatural transition

flowfieldafteroptimization.

InNLFwingdesign,theaccuracyoftransitionpredictionisan

important factor of design quality. Based on Shear Stress Trans-

port(SST)model[23],atransitionmodelhadbeendevelopedby

Menteretal. [24,25] through adding an intermittency factor ()equationandamomentumthicknessbasedReynoldsnumber(Re)

equationtotheturbulencemodels,calledasSSTRe model.Because

of

the

strong

source

terms

in

the

SST

Re model,

the

computation time of the SSTRe model is much longer thantheSSTmodel,as theCourantFriedrichsLewynumbermustbe

smaller.However, thecomputation time isacritical factorofge-

neticalgorithmoptimization.Analternativemethodisusedinthe

presentoptimizationprocesstoreducecomputationcost.Thepres-

suredistributionofairfoilispredictedbytheSSTturbulencemodel

with fixed transition when optimizing the airfoil shape and the

accurate transition location isvalidatedby theSSTRe modelafter optimization. The fixed transition location is located based

on thedesignexpectationof laminar length.Thefixed transition

computationcouldconsidertheinfluenceofthelaminarboundary

layerandaccuratelypredictthepressuredrag.Thelaminarlength

isachievedthroughmaintainingtheFPG.Inthenextsub-section,

the

code

is

validated

by

two

cases

with

experimental

data.

The

pressuredistributionsofthefixedtransitionandnaturaltransition

computationsarealsocompared.

2.2.

Validation

Inthispaper,wemainlyfocusontransitionpredictioncapabil-

ityoftheSSTRe transitionmodelforsupercriticalNLFairfoil.The transitionpredictionaccuracy isvalidatedby two testcases.

The first is a low speed NLF airfoil, NLF 0416. It is used to test

grid convergence, as well as fixed transition computation to en-

sure itasacheapersubstitution intheoptimizationprocess.The

secondcaseNLR7301isusedtovalidatethetransitionprediction

accuracy

of

transonic

flow.


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154 Y. Zhang et al. / Aerospace Science and Technology 43 (2015) 152164

Fig. 1. Computation grid of NLF 0416 airfoil (257 97 points).

2.2.1. NLF0416airfoil

NLF0416 isa lowspeednatural laminarflowairfoil.TheNa-

tionalAeronauticsandSpaceAdministrationprovidedalotofex-

perimentaldataoftheairfoil[29].Inthepresentstudy,flowcondi-

tionsarefreestreamMach number=0.1 andReynolds number=

1.0 106 .Freestreameddyviscosityissetas0.1timesofdynamic

viscosity,andturbulenceintensityis 0.1%.Fig. 1showsagridwith

25797 points.Thegrid isgeneratedbyan in-housedeveloped

gridcode.Generatedbysolvinganellipticequationwithasource

term,themeshisingoodorthogonalwiththesolidwall.Thefirst

layerinthenormaldirectionislessthan3.0E6toensureY+ less

than 1.0, and increasing rate of the boundary layer grid height

is 1.15.

Far-field

boundary

is

set

at

80

times

of

the

chord

length

awayfromtheairfoil.

Threemesheswithdifferentgridnumberareusedtotestgrid

convergence.Fig. 2showspressureandfrictiondistributionsofthe

three meshes at the same AoA by SST Re transition model,as well as the fixed transition result of 25797 points grid by

SSTmodel.Thepressuredistributionsof thedifferentgridscom-

puted by the SST Re transition model are almost the same,andthefrictiondistributionsalsoshowaconvergenttendency.The

fixedtransitionpositionsofupperand lowersurfacesarelocated

at40%and60%,respectively.Thepressuredistributionofthefixed

transition matches wellwith theothers. The friction distribution

isa littledifferentwith thenatural transitionbecauseof inaccu-

ratefixedtransitionlocation.Thefixedtransitioncomputationhas

quite

a

little

influence

on

the

drag

prediction,

as

in

Table 1.

The

fixed transition computation of the 257 97 grid costs about 2

minutesonanIntel2.8 GHzCPUcore.However,thenaturaltran-

sitioncomputationonthisgridneedsabout10minutes.Therefore

intheoptimization,thefixedtransitioncomputationcouldbeused

tominimizetheshockdraginordertoincreaseoptimizationeffi-

ciency.

Fig. 3 shows liftandlift-dragpolarcurvesofthecomputation

results and experimental data. The CFD results are computed by

the 257 97 points grid. Results of transition computations are

ingoodagreementwithexperimentaldata.However, thedragof

fullturbulencecomputationismuchhigherthanexperiment.Fig. 4

showstransitionlocationsofthecomputationcomparedwithex-

perimentaldata.Theexperimentcouldnotprovideexacttransition

locations. It employedmicrophones,whichconnected to the ori-

ficesontheairfoil,todeterminethetransitionlocation.Transition

locationsofpresentcomputationarealllocatedbetweenthelam-

inarand turbulenceorificesof theexperiment, whichshows the

Retransitionmodelsgoodcapabilityofcapturingnaturaltran-sition.

2.2.2.

NLR

7301

airfoil

NLR

7301

airfoil

is

a

typical

supercritical

airfoil

with

large

thick-ness.TheAdvisoryGroupforAerospaceResearchandDevelopment

providedaseriesoftransitionexperimentaldata[26].Computation

results are also available in publications [32,36]. In this section,

the airfoil is adopted to test the transition prediction accuracy

for transonic flow, especially with shock/boundary layer interac-

tion.Computationgrid isthesameasthe361137 gridofSec-

tion2.2.1.SeveralfreestreamMachnumbersarecalculatedtotest

dragrising.ReynoldsnumbersandAoAsintheexperimentareall

2.2106 and 0.85 . However, AoAs in the present computation

arecorrectedaccordingtocorrectionsintheexperiment,referring

toliterature[32].Freestreameddyviscosityissetas0.1timesof

dynamicviscosity,andturbulenceintensityis0.1%.

Fig. 5 shows the aerodynamic coefficients compared with ex-

periment,

including

lift,

drag

and

pitching

moment

coefficients.

All

of

the

curves

match

quite

well

before

Mach

number

0.75.

However,

theshockwave/boundarylayerinteractionbecomesverystrongaf-

terdragrising (Ma=0.75), leadingtodeviationof theRANSre-

sults. Nevertheless, the Mach number of drag divergence is well

captured.Fig. 6 showstransitionlocationscomparedwithexperi-

mentaldataatdifferentMachnumbers.Onbothupperandlower

surfaces, the computed transition locations match well with the

experimental data. Results of NLR 7301 validated the transition

predictioncapability for transonicflow,and theaccuracyofboth

pressureandfrictiondrag.Suchacapabilityprovidesthebasisfor

supercriticalNLFvalidation.

The 257 97 points grid is having a good accuracy together

withanacceptablecomputationcost.Inthefollowingsection,this

gridischosenasthedesigngridintheoptimizationprocess,and

the

361

137 points

grid

is

employed

as

a

natural

transition

vali-

dationgridafteroptimization.

3. Naturallaminarairfoiloptimization

3.1. DesignrequirementsofNLFairfoil

The NLF wing in this paper is designed for a high-subsonic

regionaljet.The objectivesandconstraintsof NLFairfoil arede-

rived from the needs of the wing. Cruise Mach number of the

regionaljet is 0.76, and flight altitude is 37000 ft. Fig. 7 shows

theplanformof thewing.Sweepanglesof leadingedgeand1/2

chord are 17.50 and 13.60 , respectively. Airfoil optimization is

focusedonprofilesoftheoutboardwing.Ruleofcosine[5]canbe

Fig. 2. Results of three different grids of NLF 0416 (Ma = 0.1, Re =1.0E6, AoA= 1 ).


4/13


Table 1

AerodynamiccoefficientsofdifferentcomputationsofNLF0416airfoil(Ma = 0.1,Re = 1.0E6,AoA =1).

187 69 grid

Natural transition

257 97 grid

Natural transition

361 137 grid

Natural transition

257 97 grid

Fixed transition

Lift coefficient, Cl 0.528 0.541 0.544 0.551

Drag coefficient, Cd 0.0089 0.0086 0.0087 0.0091

Pressure drag, Cd,p 0.0036 0.0034 0.0035 0.0036

Friction drag, Cd,f 0.0053 0.0052 0.0052 0.0055

Transition positions (upper/lower surfaces) 42%/62% 45%/62% 45%/63% 40%/60% (fixed)

Fig.3. Liftandlift/dragpolarcurvesofNLF0416airfoil(Ma = 0.1,Re =1.0E6,257

97 grid).

usedto transformwingparameter toairfoilrequirement. Ingen-

eralspeaking,anysweepanglefromleadingedgetotrailingedge

ofataperedwingcouldbeusedfor3Dto2Dparametertransfor-

mation.However,Streitetal.[30]suggestthatthesweepangleat

shockwavelocationcouldbeagoodchoicefortransonicairfoil.In

thispaper,theshockwavelocationisfoundtobenearthe1/2c,so

thesweepangleofthe1/2 chordisusedtodecidetheairfoilde-

signMachnumber.Relativethicknessoftheairfoilistransformed

from

the

kink

location.

The

parameters

of

airfoil

design

are

listed

inTable 2.

Therearethreetypesoftransitionmechanismsonsweepwing,

whichareTollmienSchlichting (TS) instability,crossflow instabil-ityandattachmentlineinstabilityorcontamination.TSinstability

could be suppressed by FPG. In this paper, the TS instability is

controlledbypressuredistributionconstraintsinoptimizationandpredicted by the SSTRe transition model after optimization.Crossflowinstabilityandattachmentlinecontaminationcouldnot

bedirectlycapturedin2-Dcomputation.Referringtoflighttestsof

AndersonandMeyer[1],thetransitionlocationisalmostthesame

whenthesweepangleislessthan20 .Therefore,crossflowinsta-

bilitycouldbeexcludedinthepresentplanform.Attachmentline

contaminationisalsoacauseofprematuretransition.Itisprimar-

ily influencedby leadingedgeradius.A leadingedgemomentum

Reynolds

number

is

often

used

to

estimate

the

leading

edge

con-

tamination [28,3]. In the present optimization, the leading edge

radiusisrestrictedtobelessthan0.035cbecausetheleadingedge

momentumReynoldsnumbershouldbelessthan100[28,3].

3.2. Class-shapetransformationmethod

Parameterization method should have fewer variables and

larger design space. In recent years, Class-Shape Transformation

(CST)[18,20] methodiswidelyusedinaerodynamicoptimization.

In themethod,anairfoilwithablunt trailingedge isdefinedas

theproductofaclass functionand ashape function,plusa lin-

eartermoftailthickness,asinEq. (1).The =Y/c,= X/c are

thenormalizedcoordinatesoftheairfoil.Thete isthetailthick-

ness.

The

CN1N2 () is

a

class

function,

as

in

Eq.(2).

Round

nose

and

Fig. 4. Transition locations of the computation compared with experiment.

pointedaftendairfoilscouldbedefinedbysetting the N1 = 0.5

and

N2=1.0.

In

this

paper,

the

shape

function

S() is

defined

byacombinationofBernsteinpolynomials B in(),as inEq. (3).The

bi isdesignvariablevector,whichrepresentsweightsofBernstein

polynomials.Upperandlowersurfacesofairfoilaregeneratedby

two independentpolynomials. Thus it is difficult to explicitly fix

theairfoilthicknessinthemethod.Whendesigninganairfoilwith

acertainthickness,Ycoordinateoftheairfoilisscaledtocontrol

thickness.

() =CN1N2()S() +

te

2 (1)

CN1N2()=N1(1 )N2 (2)

S() =

n

i=0

bi Bin()

whichB in() =Kin

i(1 )ni, Kin =n!

i!(n i)!(3)

Sixthorder(n=6)CSTpolynomialisusedforbothupperand

lowersurfacesinthispaper.Table 3showsrangesoftheCSTvari-

ables in the optimization. The first variables of both upper and

lowersurfaces aredifferent with theothers in order toobtaina

large leadingedge radius.Thesixthandseventhvariablesof the

lowersurface areadjusted to obtainaft loading.Eachvariable is

separatedby201stepsintheoptimization,thuspossibleparame-

tercombinationsareabout20114 1.76 1032.

3.3.

Airfoil

optimization

with

pressure

distribution

constraint

For

supercritical

NLF

airfoil,

wave

drag

is

primarily

determined

by theMachnumberbefore theshockwave,and frictiondrag is

controlledbythelengthoflaminarregion.SufficientFPGlengthin

the front part of NLF airfoil is needed to maintain laminar flow.

However,FPGwouldraise theMachnumberaheadofshockand

increasewavedrag.ThereforetheappropriateFPGisthekeyissue

ofNLFsupercriticalairfoildesign.

In the authors previous publications [34,33], pressure distri-

bution constraints can be applied on the airfoil or wing design

process to gain a good overall supercritical performance. Such a

pressuredistributionconstraintprovidesuswitha tool toobtain

airfoilswithdifferentpressuregradientsandcomparetheirperfor-

mances.

Inordertonotonlydesignanairfoilwithminimumdrag,but

also

separately

study

the

effects

of

the

FPG

on

pressure

drag

and


5/13


Fig. 5. Aerodynamic coefficients of the NLR 7301 airfoil varying with Mach number (Re = 2.2E6).

Fig. 6. Transition locationofNLR7301airfoilvaryingwithMachnumber(Re=

2.2E6).

Fig. 7. Planform of the natural laminar flow wing.

Table 2

NLFairfoildesignparameters.

Value

Cruise Mach number 0.739

Reynolds number 1.0E7

Lift coefficient 0.55

Relative thickness 12.35%

frictiondrag,astrategyisdesignedinthispaper.First,bysetting

different

pressure

gradient

constraints,

based

on

fixed

transition

computation

and

a

reasonable

transition

location,

corresponding

low pressure drag airfoils are designed by optimization [34,33].

Then the transitioncomputationsareconductedon theseairfoils

tovalidatethereallaminarflowlength,andtoinvestigatetheef-

fects of the pressure gradient, and at the same time, get overall

optimaldesign.

In this section, the fixed transition points of both upper and

lowersurfacesaresettobeatX= 0.52c,aswesupposethatthe

laminar region of the final design should be longer than 50%. It

alsoassumes that theshockwaveshould locateafter X= 0.52c,

because the computation would not be stable if flow after the

shock wave is still laminar. Such a setting is reasonable for su-

percriticalNLFairfoildesign.Objectiveofoptimization istomin-

imize

the

pressure

drag.

Shock

wave

drag

will

be

implicitly

mini-mizedwiththisobjective.NSGA-IIisemployedastheoptimization

method.The257 97 pointsgriddiscussedinSection2.2.1isused

intheoptimizationprocess.

Theoptimizationproblem isdefinedas inEq. (4).Thedesign

conditionsarefromSection3.1.LeadingedgeradiusRLE islimited

tobe less than0.035 in order to avoid leading edge contamina-

tion, and larger than 0.010 to ensure a good enough low speed

stallbehavior.PitchingmomentCm isexpectedtobe largerthan

0.10 to restrict trimdrag.Becauseoffixed transitioncomputa-

tion,thefrictiondragvariationofairfoilsisquitesmall.However,

ifflowseparationexists intheflow,frictiondragwouldbesmall

becauseof the reverseflowdirection.Consequently, frictiondrag

constraint Cd,f

> 0.0025 is employed to rule out designs with

flow

separation.

In

order

to

maintain

laminar

flow,

and

to

im-

proveoptimizationefficiency,thequantityoftheFPG,denotedas

QFPG = dCp/dX, are applied in the optimization. On the front

partofairfoil,whereX< 0.50,forbothupperandlowersurfaces

QFPG shouldbelargerthan0.0.dCp/dX islimitedtobelessthan

3.0 on the pressure recovery region (X>0.70) to avoid trailing

edgeseparation [8].

Table 3

ParameterrangesoftheCSTdesignvariables.

Upper surface parameter number 1 27

Parameter range [0.05,0.3] [0.0,0.3]

Lower surface parameter number 1 25 6 7

Parameter range [0.3,0.05] [0.3,0.0] [0.2,0.1] [0.0,0.3]


6/13


Fig. 8. IterationhistoryoftheNLFsupercriticalairfoiloptimization.(Forinterpretationofthereferencestocolorinthisfigure,thereaderisreferredtothewebversionof

thisarticle.)

Fig. 9. Airfoil shapes and pressure distributions of the airfoils.

min Cd,p

s.t. Ma=0.739; Re =1.0E7; Cl= 0.55

t/c=12.35%

RLE0.70< 3.0 (4)

Inthegeneticalgorithm,32individualsformthepopulationof

each

generation.

The

first

generation

contains

the

CST

parameters

of the RAE2822 airfoil and 31 random parameter combinations.

Optimizationisterminatedafter100generations.Totalnumberof

individualsgeneratedinthisprocessis3200.However,onlyabout

2100individualsarecomputedbyCFDbecauseduplicatedindivid-

uals could be generated due to the elite strategy of NSGA-II [7].

Total computation time is about 6 hours on a 32-core 2.8 GHz

workstation.

Fig. 8showstheiterationhistoryoftheoptimization.Eachair-

foil is given a Design ID, which is equal to generation pop-

ulation + individual number. Fig. 8(a) shows all results of the

optimizationprocess.Theredpointsareunfeasibledesignswhich

donotfullysatisfyallconstraintsofEq. (4).Theblackpointsare

feasibledesignsthatsatisfyalloftheconstraints.Theresultsshow

that

feasible

designs

first

appear

after

ID> 250,

and

a

lot

of

un-

feasibledesignshavelesspressuredragthanthefeasibledesigns.

Fig. 8(b)showsthehistoryoffeasibledesigns.Aclearconvergent

tendencycouldbeseenwhenID> 1500.Furthermore,feasiblede-signs become less and less when ID> 2500 because individuals

identicaltoalreadyexistdesignsareproducedmoreandmorefre-

quently,whichmeansconvergentsolutionisapproached.Threeairfoilshapesandpressuredistributionsintheoptimiza-

tion process are shown in Fig. 9. Corresponding Mach numbercontours are presented in Fig. 10. The original design (called as

ORD)inthefiguresisthescaledRAE2822airfoilofthefirstgen-

eration.TheORDisunfeasiblebecauseFPGonthelowersurfaceis

notsufficientforNLFdesign,andthepitchingmomentisalsotoo

large.Thefirstfeasibledesign(calledasFFD)istheairfoilwhichfirst

satisfies all constraints. The maximum thickness location of FFD

movestowardsthetrailingedgetoobtaina longerregionofFPG

onthelowersurface,asinFig. 9(b).However,Machnumberahead

ofshockexceeds 1.2,asinFig. 10(b),andtheshockwaveisverystrong.Theoptimizeddesign(calledasOPD) isthedesignwhich

hastheminimumpressuredraginallofthefeasibledesigns.Itis

fromthe97thgeneration.LowersurfacesoftheFFDandOPDare

similar. TheFPG region is longer than50%of the chordonboth

upperandlowersurface.Resultshowsthat,atthecurrentdesign

Machnumberand liftcoefficient,shockwavemightbeunavoid-

able.However,theshockwaveoftheOPDisobviouslyweakerthantheORDandFFD.TheMachnumberaheadofshockis 1.12,asin

Fig. 10(c),

which

is

an

acceptable

value

for

transonic

airfoil.


7/13


Fig. 10. Machnumbercontoursofthethreeairfoilsinoptimizationprocess.(Forinterpretationofthecolorsinthisfigure,thereaderisreferredtothewebversionofthis

article.)

Table 4

Aerodynamiccoefficientsofthethreeairfoils(Ma = 0.739,Re =10.0 million,Cl = 0.55).

Computation meth od Aerodyn amic coefficient Original design (ORD) First feasible design (FFD) Optimized design (OPD)

Fixed transition Pressure drag Cd,p 0.00314 0.00738 0.00195

Friction drag Cd,f 0.00303 0.00300 0.00315

Pitching moment Cm 0.117 0.0739 0.0894

Natural trans ition Pres sure dragCd,p 0.00314 0.00757 0.00212

Friction drag Cd,f 0.00316 0.00288 0.00299

Total drag Cd 0.00630 0.01045 0.00511

Lift to drag ratio L /D 87.36 52.63 107.52

Transition positions (upper/lower surfaces) 57%/44% 59%/55% 57%/54%

Fig. 11. Pressure distributions and friction coefficient distributions of the ORD and OPD.

Thethreeairfoilsarevalidatedbynatural transitioncomputa-

tion.Resultsofthefixedtransitionandnaturaltransitioncompu-

tationsarecomparedinTable 4.Itcouldbeseenthatthepressure

dragfrombothcomputationmethodsareclosetoeachother.After

optimization,the L/D isincreasedbyabout20.Mostofthedrag

reductioncomesfromthepressuredrag.Thefrictiondrag isalso

reducedbya little,althoughsuchareduction isnotexpectedby

usingsuchaprocess.

Fig. 11(a) shows pressure distributions of natural transition

computation and fixed transition computation. Pressure distribu-

tionsofthetwomethodsshowonlyalittledifference.Theresults

provethatthefixedtransitioncomputationcouldwellpredictthe

pressure

distribution

of

the

airfoils.

Fig. 11(b)

shows

friction

coef-

ficient distributions of the ORD and OPD airfoils which are both

computedby theSSTRe transitioncomputations.The transi-

tion location isclosely related to the lengthofFPGregion.As in

Fig. 11(b),laminarregionofthelowersurfaceisincreasedbyalot

afteroptimization.However,laminarregionontheuppersurfaceis

alittlebitreducedbecausetheshockwavelocationismovedback

towards the leading edge. However, it is necessary penalty paid

toweaken theshockwave.Nevertheless, laminarregion lengthof

theOPD ismore than55% for bothupperand lowersurfaces. It

isalittlelongerthanthepre-assumedtransitionlocationsforthe

fixedtransitioncomputation,whichmeansthattheFPGconstraint

isadequatetogeneratesuchalaminarflowregionatthisReynolds

number.

The

optimization

method

with

fixed

transition

computa-


8/13


Fig. 12. Design histories of drag coefficients of different FPG constraints.

Fig. 13. Shapes and pressure distributions of the airfoils with different QFPG constraints.

Fig. 14. Friction coefficient distributions of the six airfoils at different Reynolds numbers (Ma = 0.739, Cl = 0.55).

tionandpressuregradientconstraintcanproducereasonableand

satisfactorydesigns.

3.4. EffectofQFPG ontransition

Inlastsection,theexpectedpressuregradientcanbeobtained

bysettingaconstrainton QFPG throughtheoptimizationprocess,

and

the

expected

laminar

length

can

be

achieved

by

maintaining

a

reasonablyFPG.Aquestionisraised,whatshouldbeagood QFPGconstraint.Isitthelargerthebetter?IsitrelatedtoReynoldsnumbers?

Withthepressuregradientconstrainedoptimization,weare

abletodesignandcompareaseriesofairfoilswithdifferentpres-

sure distribution gradients, or different length of FPG. With the

pressuredragminimizedbytheoptimization,suchaprocesscould

provideaninsightintohowthepressuregradientisaffectingthe

transition performance, taking into account not only the friction

drag,

but

also

the

robustness

and

Reynolds

number

effects.


9/13


The optimization problem is defined as in Eq. (5). Different

pressuregradients from QFPG > 0.0 to QFPG> 1.0 aresetbefore

the50%chordofupper surface.The lengthofFPG on the lower

surfaceisexpectedtobemorethan 60%.Intheoptimizationpro-

cess,thefixedtransitionlocationsoftheupperandlowersurfaces

aresetat52%and62%,respectively.TheoptimizeddesigninSec-

tion3.3 isputintothefirstgenerationastheoriginaldesigninthe

followingoptimizations.

min Cd,p

s.t. Ma=0.739; Re =1.0E7; Cl= 0.55

t/c=12.35%

RLE0.70

0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 airfoils, respectively. Fig. 12 shows

thedragcoefficienthistoriesofthedifferentFPGconstraints.Only

thefeasibledesignsareplottedinthefigures.Allcasesareiterated

about90100 generations.WiththeincreaseofQFPG ,feasiblede-

signsarelessandless.ItisclearthatthedragincreaseswithQFPGincreasing, because the shock wave drag is increased, while the

frictiondragisalmostunaffectedinthefixedtransitioncomputa-

tion.

Fig. 13 compares theshapesand thepressuredistributionsof

fourairfoilsoutofthesixandtheoriginalairfoil.Shapesandlower

surface pressure distributions are similar. Lengths of the FPG on

the lowersurfacearemore than60%chord.Theshockwavebe-

comes stronger and stronger as the specified limitation of QFPGincreases.

Basedontheabovesixairfoils,fiveReynoldsnumbersranging

from10millionto20millionarecomputedbytransitionmodelto

testthesensitivityofpressuredistributiontotheFPGconstraints.

Different Reynolds numbers could represent flow conditions of

different types of aircrafts with different size, configuration and

speed. Reynolds number less than 1.0107 could be the flight

conditionof businessjet, andReynoldsnumber 1.5107 repre-

sentsthetypicalconditionofregionaljet,whileReynoldsnumber

higherthan2.0 107 mightcorrespondtothenarrowbodyairlin-

ers.Inthepresentcomputation,thefreestreameddyviscosity is

0.1timesofdynamicviscosity,andturbulenceintensityis 0.1%,as

theairfoilsaresupposedtobeinflightratherthaninwindtunnel.

Fig. 14 shows friction coefficient distributions of the six air-

foils

at

three

Reynolds

numbers

with

the

same

lift

coefficient.

As shown in Fig. 14, all of the airfoils have quite large areas of

laminarflow,whichindicatesthattheconstraintsof QFPG aread-

equatetoobtainlaminarflow.FromallthefiveReynoldsnumber

studied,wecouldfigureoutsometransitiontendenciesfromthe

results:

(1)WhentheReynoldsnumberis1.0 107 ,transitionlocations

ontheuppersurfacearealllaterthan50%chord.Rememberthat

theFPGregionisconstrainedtobelongerthan50%,actuallength

ofFPGontheuppersurfaceisdeterminedbyshockwavelocation,

so are the transition locations. Transition locations on the lower

surfaceskeepthesamefordifferentairfoils,noshock isdetected

on the lower surface. All these facts indicate that the transition

location is not sensitive to the quantity of FPG at this Reynolds

number

range,

but

only

to

the

shock

location.

Fig.15. FrictiondragandtotaldragofthesixairfoilsvaryingwithReynoldsnumber

(Ma = 0.739,Cl = 0.55).

(2)ForReynoldsnumberRe=1.5107,transition locationof

theuppersurfaceismostlybefore50%chord.Thedifferencesare

hence

not

produced

by

the

shock,

but

by

the

intensity

of

FPG.Transition locationsmovedownstreamwith the QFPG increasing.

TheonlyexceptionistheQFPG>1.0 airfoil.Theshockwaveofthis

airfoilisverystrongandinducesasmallseparationbubble,which

mightinfluencethetransitionlocation.FromairfoilsofQFPG> 0.0

toQFPG>0.8,transitionlocationsoftheuppersurfacesmovefrom

35%to53%chords.Forlowersurfaces,transitionlocationsarealso

all before 60% chord, which is also less than the assigned FPG

range. At this Reynoldsnumber, thepressure gradient should be

carefullydesignedtogetadequatelaminarlength.

(3)WhentheReynoldsnumberis2.0 107 ,theeffectofQFPGbecomessmall.Variationsoftheuppersurfacetransitionlocations

arelessthan6%chord.Thelongestlaminarregionislessthan30%

chord,whichisontheQFPG> 0.8 airfoil.ForsuchalargeReynolds

number,alargerangeofNLFisdifficulttorealize.HybridNLFand

laminar

control

measures

must

be

considered.

Fig. 15 showsfrictiondragandtotaldragofthesixairfoilsat

different Reynolds numbers. Friction drag of the QFPG > 0.0 air-

foil isthe largest.By increasingthe QFPG from0.0to 1.0,benefit

of friction drag achieved is about 4 drag counts with Reynolds

number less than 1.5107. However, the benefit drops to less

than 2 drag counts when Reynolds is 2.0107. Total drag of

the QFPG >1.0 airfoil is much larger than other airfoils, as the

shockwaveisquitestrong.Thetotaldragsofthe QFPG> 0.0 air-

foilandtheQFPG> 0.4 airfoilarealmostthesamewhenReynolds

numbers less than1.5107 ,whichmeansthevariationsoffric-

tion drag and pressure drag are balanced on these two points.

However, when Reynolds number is 2.0107 , the total drag of

QFPG> 0.0 airfoiland QFPG> 0.2 airfoilareaboutthesame.The

QFPG>0.2 airfoil

is

the

best

one

which

has

the

least

total

drag

for

allReynoldsnumbers. Itworth tobementioned thatconstraints

oftherecoveryregion(X> 0.70)andthepitchingmomentgreatly

limittheapplicationofaftloading,whichhasthepotentialoffur-

therreducingshockwavedrag.

Robustnessoftransitionlocationisalsoanimportantcriterion

fortheevaluationofNLFairfoil.Flightcouldbeunsafeifaerody-

namicforcesarenotstable.Flightcontrolproblemwouldarise if

theaerodynamicderivativeschangeincruiserange.Inthepresent

computation, the airfoils are computed at different AoAs to test

robustness.Fig. 16 showspolarcurvesofthe QFPG> 0.0,0.2and

0.4airfoils,whosecruisedragcoefficientsareclose.Fig. 17shows

thecorresponding transition locations.Cleardragbucketscanbe

seen on the curves of Fig. 16. The lower extent (Cl about 0.1

to 0.2)

of

the

drag

bucket

of

QFPG > 0.0 airfoil,

which

is

dom-


10/13


Fig. 16. Polar curves of three airfoils at Reynolds number 10.0M and 12.0M.

Fig. 17. Transition locations varying with angles of attack.

Fig. 18. Pressure distributions varying with angles of attack (Ma =0.739, Re = 10.0M).

inated by the lower surface transition, is larger than the other

twoairfoils;however, the liftcoefficientsmaynotappear in real

flight. The upper extents of the drag buckets are quite different.

Dragofthe QFPG> 0.0 airfoilshowsasuddenrisewhenCl > 0.4

(AoA> 0.75),whichdemonstrates thatpremature transitionoc-

cursontheuppersurface.AsshowninFig. 17,theuppersurface

transitionlocationsof QFPG> 0.0 airfoilatAoAsfrom0.5 to1.0

arenotstable,whichmeanspoorrobustness.Theothertwoairfoils

show better performance near the lift coefficient ofcruise range

(0.6

>Cl> 0.4).

The cause of the premature transition could be explained by

looking into thepressuredistribution.Fig. 18 shows thepressure

distributionsvaryingwithAoAs.WiththeAoAdeviatingfromthe

designpoint,thepressuredistributioncannotalwayskeepthefa-

vorablegradient. There willbean adverse regionon thesuction

roof-top. The adverse pressure gradient should be the cause of

premature transition.For the QFPG> 0.0 airfoil,adversepressure

gradient is strong when AoA is less than 1; However, for the

QFPG> 0.2 and0.4airfoils,theadversepressuregradientsonlyap-

pear

at

0.5 and

are

much

weaker.

Besides

the

increase

in

friction


11/13


Fig. 19. Lift coefficient curves of the QFPG> 0.0 and QFPG> 0.2 airfoils.

Fig. 20. Control airfoil locations of the NLF wing.

Fig. 21. Control airfoils of the NLF wing.

drag,theprematuretransitionalsochangestheslopeofliftcoef-

ficient,asshowninFig. 19.Fromtheaboveresults,wecouldsee

thatthe QFPG> 0.2 airfoilisthebestairfoilwithbothlowcruise

dragandgoodrobustness.

4.

High-subsonic

NLF

wing

design

Inthissection,theQFPG> 0.2 airfoilisusedasthebaselineair-

foil

of

high-subsonic

wing.

Based

on

the

conclusions

in

Section 3,

theFPG regionof the lowersurface isexpectedbeforeonly50%

chord.Thebaselineairfoilisoptimizedagainwiththenewdesign

constraints.Fig. 20showsthecontrolprofiles,whicharelocatedat

theroot,kinkandtiplocations.Therelativethicknessesoftheroot,

kinkandtiplocationsare15%,12%and 10%,respectively.Ruleof

cosine[5] isusedtoobtainthe2-Dairfoilthickness.Spline-based

surface is adopted to ensuresmoothness in thespan-wisedirec-

tion.Thebaselineairfoilisinstalledatthekinklocation.Thewing

isinstalledontoawing-bodyconfigurationtovalidatetheperfor-

mance.Cutandtrymethod isemployed toadjusttherootand

tip airfoils. At the beginning of wing design, the upper surfaces

oftherootandtipairfoilsarecopiedfromthekinkairfoil,while

thelowersurfacesarescaledfromthekinkairfoiltofitthethick-

ness

requirements.

The

kink

airfoil

is

kept

unchanged

during

the

designprocess.Afterabout30 iterationsofmanualmodifications

andCFDcomputations,anNLFwingisobtained.Fig. 21showsthe

controlairfoilsof theNLFwing.Theuppersurfaceshapesof the

airfoilsaresimilarwitheachother.Incontrast,thelowersurfaces

arequitedifferentbecauseofthedifferencesoftherelativethick-

ness.

Fig. 22 showssurface pressure contour and pressure distribu-

tions at the cruise condition Ma = 0.76, Cl = 0.42, Re= 1.6E7

(based on mean aerodynamic chord). The computation grid in-

volves15millionpoints,andtheY+ ofthefirstgridlayerisless

than 1.0. The isolines of the pressure in the span-wise direction

arealmoststraight,asshowninFig. 22(a).ClearFPGcanbeseen

in the stream-wise direction, as in Fig. 22(b). Fig. 23 shows the

skin

friction

contour

and

friction

contours

of

the

wing/body.

Both

theupperandlowersurfaceshavelargeareasofthelaminarflow.

Performancesof the cruise Mach number0.76 and low Mach

number0.2arealsovalidatedby theCFDmethod.Fig. 24 shows

lift and lift-drag ratio curves of the two Mach numbers. At the

cruiseMachnumber,theliftcoefficientremainslinearlyincreasing

untilAoA=3.5 ,onwhichtheliftcoefficientisabout 0.77,thatis

larger than1.3 timesof thecruise liftcoefficient 0.42.Moreover,

the lift-dragpolardoesnot haveunexpectedfluctuation.The lift

Fig. 22. Surfacepressurecontourandpressuredistributionsatfoursections(Ma=0.76,Cl = 0.42,Re=1.6E7).(Forinterpretationofthecolorsinthisfigure,thereaderis

referred

to

the

web

version

of

this

article.)


12/13


13/13


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