AerodynamicDesignOptimizationofTransonic Natural-Laminar

Research ArticleAerodynamic Design Optimization of TransonicNatural-Laminar-Flow Airfoil at Low Reynolds Number

Jing Li Cong Wang and Huiting Bian

School of Mechanics and Safety Engineering Zhengzhou University Zhengzhou 450001 China

Correspondence should be addressed to Huiting Bian bianhtzzueducn

Received 7 January 2020 Revised 3 May 2020 Accepted 16 June 2020 Published 16 July 2020

Academic Editor Haopeng Zhang

Copyright copy 2020 Jing Li et al is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e position and size of laminar separation bubble on airfoil surfaces exert a profound impact on the efficiency of transonicnatural-laminar-flow airfoil at low Reynolds number Based on the particle swarm algorithm an optimization methodology in thecurrent work would be established with the aim of designing a high and robust performance transonic natural-laminar-flow airfoilat low Reynolds number is methodology primarily includes two design processes a traditional deterministic optimization aton-design point and a multi-objective of uncertainty-based optimization First a multigroup cooperative particle swarm op-timization was used to obtain the optimal deterministic solution e crowing distance multi-objective particle swarm opti-mization and the non-intrusive polynomial chaos expansion method were then adopted to determinate the Pareto-optimal frontof uncertainty-based optimization Additionally the c minus Reθt transition model was employed to predict the laminar-turbulenttransition Regarding to the established optimization methodology a propeller tip airfoil of solar energy unmanned aerial vehiclewas finally designed During optimization processes the minimized pressure drag was particularly chosen as the optimizationobjective while the friction drag increment served as a constraint condition e deterministic results indicate that the optimizedairfoil has a good ability to control the separation and reattachment positions and the pressure drag can be greatly reduced whenthe laminar separation bubble is weakenedemulti-objective results show that the uncertainty-based optimized airfoil possessesa significant robust performance by considering the uncertainty of Mach number e findings evidently demonstrate that theproposed optimization methodology and mathematical model are valuable tools to design a high-efficiency airfoil for thepropeller tip

1 Introduction

e solar powered unmanned aerial vehicle (UAV) generallyflying at an altitude of 10ndash40 km where it is with low at-mospheric density has attracted wide attention due to itssignificant practical engineering application in recent years[1 2] Nevertheless note that the low atmospheric densityexisting in high altitude environment could result in theReynolds number range of the propeller airfoils falling within105ndash106 [3] Subsequently the aerodynamic performance ofairfoil could be deteriorated at such low Reynolds number(LRN) and then the propeller efficiency would be plum-meted [4 5] As such inspired by the importance and urgentneed of low-speed flight at LRN comprehensive studies havebeen dedicated to improve the aerodynamic efficiency of itspropeller [6 7] Zhang et al carried out the optimization

design of E387 and PSU 94ndash097 airfoils at Re 20 times 105 andRe 40 times 105 respectively [8] e designed airfoils showedbetter aerodynamic characteristics predicted through theXfoil code than the original ones Ma et al established ahierarchical multi-objective optimization platform based onXfoil code [9] eir results indicated that three new airfoilsoptimized from LRN high-lift airfoils have excellent aero-dynamic characteristics in both on-design and off-designstates Wang et al accomplished the multi-objective opti-mization design for SD7073 airfoil and proposed a significantdesign concept that is to change boundary layer properties viacontrolling the transition positions [10] Chen et al discov-ered the low lift-to-drag ratio for the obtained tandem airfoilat Re 220000 that could be overcome under the lower liftcoefficient and it particularly demonstrates a manifestation ata high angle of attack [11]

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 5812129 22 pageshttpsdoiorg10115520205812129

ese above new optimized natural-laminar-flow (NLF)airfoils presented the outstanding performance in differentaspects for low-speed flight at LRN but they can scarcely beimplemented in the propeller tip e reason for this is thatthe velocity nearby the blade tip would be high subsonic ortransonic while the velocity of its root region is rather low[12 13] Up to now there are very few transonic LRN airfoilsthat are available for the blade tip of solar powered UAVUndoubtedly it gives rise to more difficulties in designingtransonic LRN airfoil compared with low-speed NLF oneMoreover LSB will inevitably appear on one or both surfacesof the airfoil which are mainly responsible for increasingsensitivity of aerodynamic performance as the Reynoldsnumber decreases to and even below 500000 [14ndash16] Forthis it is necessary to employ the active flow control methodsto completely eliminate the laminar separation [17 18]Furthermore it has a high probability of developing shockwaves on the upper airfoil surface at transonic speed flightese phenomena would subsequently produce the strongadverse pressure gradients and compressibility effect thatgreatly influence the size and position of LSB and enhancethe design complexity [19 20] For another the fluctuationsof the flight conditions leading to the instabilities of the sizesand locations of separation bubbles also cause more diffi-culties [21]

According to these previous studies [12ndash21] it is badly inneed of putting forward an effective optimization meth-odology to design the transonic NLF airfoil at LRN towardsachieving direct application in the blade to enhance thepropellerrsquos efficiency e particle swarm optimization(PSO) algorithm has been widely used in aerodynamicoptimization because of its excellent optimal-searchingability [22ndash24] Khurana et al built an adaptive mutation-PSO (AM-PSO) method in use for redesigning airfoil atflight envelopes encompassing low-to-high Mach numbers[25] eir results illustrated that the AM-PSO can seek outthe optimal airfoil with minimal computational resourcesTao et al proposed an improved PSO algorithm on the basisof centroidal Voronoi tessellations (CVTs) and fitness dis-tance ratio (FDR) [26] e optimization of single point androbust showed that the optimized airfoil and wing both haveoutstanding characteristics at on-design point Masoumiet al developed a hybrid algorithm combining the big bang-big crunch and PSO which can be applied for reverse anddirect optimization approaches [27] And this proposedalgorithm has proper accuracy and convergence speed thatfacilitates to reach the desired airfoil geometry However itshould be mentioned that most of the above PSO algorithmswould fall into a local optimum e main reason is theevolution information for individual guidance only comingfrom the single group Under such conditions the evolu-tionary ability of algorithm could be declined and thereupongives rise to an algorithm premature in local optimum[28ndash30] Of particular interest is the high-order nonlinearfeature of the aerodynamic design space that certainly re-quires the algorithm to have a strong global optimizationcapability Moreover massive computational consumptionof aerodynamic characteristics needs the algorithm to have afaster convergence speed [31 32]

With respect to the significant role of single-objectivePSO algorithms in deterministic optimization the highrobustness airfoil should be taken into consideration inwhich the performances have less sensitivity in response toflight conditionsrsquo uncertainty Usually the mean and vari-ance of the characteristics were treated as the indicators forthe uncertainty design However the mean and variancealways were converted into a single-objective optimizationby a linear-weight combination [33 34] So the optimizationresults would be influenced by the treatment of weightcoefficients In this study the mean and variance are treatedas two objectives to get over the disadvantage of the weightmethod In line with previous study the multi-objectiveparticle swarm optimization (MOPSO) employed in manymulti-objective aerodynamic optimization tried to get thePareto-optimal front [35] Azabi et al proposed an inter-active MOPSO with the benefits of reducing computationaltime and avoiding the complexity of postdata analysis [36]Huang et al constructed an improved MOPSO with prin-cipal component analysis methodology enhancing theconvergence of Pareto front [37] e level of non-domi-nanted Pareto front was effectively improved and the multi-objective requirements of a wide-body aircraft were satisfiedHowever the number of the Pareto-optimal front will growdramatically for multi-objective aerodynamic design ereasonable dominant ranking and selecting for Pareto-op-timal sets are essential factors in enhancing the capability ofthe MOPSO [38]

In present work an optimization methodology based onPSO algorithm was proposed It aims to design a high androbust performance transonic NLF airfoil at LRN isarticle is organized as follows in Section 21 the PSOmethodology is built up and the algorithm performances arevalidated by the aerodynamic design of a supercritical airfoilthe numerical simulation and airfoil parameterizationmethods are given in Sections 22 and 23 respectively inSection 3 deterministic optimization and multi-objectiverobust design are conducted to get an optimal airfoil to meetthe design requirements and in Section 4 several importantdesign conclusions are summarized by the comparison oforiginal and newly optimized airfoil

2 Methodology

21 Optimization Methodology Based on the PSO Algorithm

211 Multigroups Cooperative Particle Swarm OptimizationAlgorithm (MCPSO) To avoid algorithm premature aMCPSO algorithm based on the biological systems symbi-osis is put forward in this section e populations are di-vided into several subgroups Every subgroup obtains theevolution information from the group itself and the othersymbiotic subgroups e symbiotic relationships amongsubgroups are described by a master-slave structure eslave subgroups and the master one separately undertake theglobal and local searches

e basic principles of this algorithm are conducted asfollows the populations are divided into four subgroupsmarked as the 1st 2nd 3rd and 4th subgroups e 1st

2 Mathematical Problems in Engineering

subgroup is chosen as the master subgroup which is mainlyin charge of local search and the others are specified as theslave subgroups for the global search e 1st subgroup is toadopt the standard PSO while the others are to adopt theimproved PSO algorithms accomplished in our prior studies[39 40] e speed and position of the 1st subgroup isupdated according to the following equation

vrarr

i(t + 1) ω vrarr

i(t) + c1r1 prarr1

i (t) minus xrarr

i(t)1113874 1113875 + c2r2 prarr1

g(t) minus xrarr

i(t)1113874 1113875 + fc3r3 prarr2middotmiddotmiddot

g (t) minus xrarr

i(t)1113874 1113875

xrarr

i(t + 1) xrarr

i(t) + vrarr

i(t + 1)

⎧⎪⎨

⎪⎩(1)

where vrarr

i and xrarr

i are the speed and position of the particlerespectively p

rarr1i and p

rarr1g are the local and global best par-

ticles of the 1st subgroup respectively prarr2middotmiddotmiddot

g is the global bestparticle of the other subgroups c1 and c2 are the learningfactors and f isin (0 1) is the migration factor

e ability to search the optimum is verified by designingthe supercritical airfoil NASA-SC (2) 0712 e on-designpoint is Mainfin 076 CL 070 and Re 30 times 106 and theobjective is minimizing the total drag at this cruise state efollowing optimized mathematical model is built up for thisdesign problem

objective min CD

stCL 07

thicknessmax ge 0121113896

(2)

where thicknessmax is the maximum thickness of theairfoil

In this testing aerodynamic design the characteristicsare obtained by solving the Reynolds-averaged NavierndashStokes (RANS) equations [41] e airfoil parameterizationmethod adopted in this design will be displayed in Section23 ere are five design variables for the upper or lowersurfaces ten design variables in all for each airfoil e cellnumber of the grids is 29000

e standard PSO is marked as GPSO e populationsize of the two algorithms (GPSO and MCPSO) counts 60particles and the convergence condition is the maximumiteration which is set as 50 e damping strategy is utilizedfor the boundary treatment method [42]

Aerodynamic characteristics for the original and opti-mized airfoils at on-design point are shown in Table 1 edrag coefficient of MCPSO is smallest and decreases 328counts by comparing with the original one e lift-to-dragratio increases by 245

e optimization results of GPSO and MCPSO areshown in Figure 1 e convergent curves indicate thatMCPSO has less probability of falling into the local op-timum than GPSO One thing should be mentioned thatthe fitness of MCPSO could attain 013505 in the 25thiteration while such value only could be achieved byGPSO in its 40th iteration Furthermore the GPSO fallsinto the local optimum at the iteration end comparedwith the better fitness (0013394) of the MCPSO Seen

from the surface pressure distributions presented inFigure 1(b) the strong shocks disappear on the uppersurface of both two optimized airfoils Notice that the suckpeak of optimized surface pressure distribution is higherthan that of the original one e results indicate that theconvergence speed of MCPSO is faster than that of GPSOand the better optimal solution could be found in thedesign space quickly and accurately

212 MOPSO Algorithm Based on Crowding-DistanceSorting eMOPSO proposed in the current study is basedon the crowding-distance sorting and the update andcontrol of Pareto sets are as follows

(1) Assume that the maximum size of the Pareto setsnamed as At isM and the size of At in the t iterationis m1(m1 leM) ere will be m2 new Pareto solu-tions in the following iteration which puts togetherwith At into another temporary Pareto sets named asAtprime

(2) If the two solutions have the same objectives ran-domly eliminate one of the duplicate solutions in thetemporary Pareto sets At

prime(3) e size of At

prime will be m3(m3 lem1 + m2) when de-leting the dominated solutions in the temporaryPareto sets At

prime(4) Sort the populations of At

prime in the ascending orderaccording to the crowing distance marked as At

Primeecrowing-distance computation is as follows

idistance 1113944

Nobj

m11113944

Nset minus 1

i2

fmi+1 minus fm

iminus 1fmmax minus fm

min (3)

where idistance refers to the crowing distance of the ith

population fmi+1 refers to the mth objective function

value of the i+ 1th individual fmmax and fm

min are themaximum and minimum values of the mth objectivefunctions respectively and Nobj and Nset are thenumbers of objectives and set At

prime Herein thecrowing-distance computation requires sorting thepopulation corresponding to each objective functionvalue in ascending order

Table 1 Aerodynamic characteristics for the original and opti-mized airfoils at on-design point

Characteristics NASA SC (2)-0712 GPSO MCPSODrag coefficient 0016678 0013505 0013394Lift-to-drag ratio 4197 5183 5226Maximum thickness 12 12 12

Mathematical Problems in Engineering 3

(5) e sorting sets AtPrime will be chosen as the new At

prime innext iteration if m3 leM Otherwise the exact top Mpopulation members from the At

Prime will be chosen asthe new At

prime

In general the optimization results of MOPSO can beverified by generational distance (GD) and spacing metric(SP) e GD can represent the distance between the non-dominated set and true Pareto-optimal front of which theformula is as follows

GD

1113936

ni1 d2

i

1113969

n (4)

where n is the size of nondomination set and di is thesmallest distance between the ith individual and true Pareto-optimal front Notice that the smaller the value of GD is thebetter the convergence is

e SP is used to calculate the distribution uniformityand its formula is as follows

SP

1n minus 1

1113944

n

i1

1113936nminus 1i1 di

n minus 1minus di1113888 1113889

211139741113972

(5)

A smaller SP denotes a better uniformity of the setsus three typical multi-objective functions labeled byZDT1 ZDT2 and ZDT3 are to verify the performance

ZDT1

min f1(x) x1

min f2(x) g(x) 1 minus

x1

g(x)

1113970

1113888 1113889

g(x) 1 + 91113936

ni2 xi

n minus 1

n 30 x isin [0 1]

(6)

ZDT2

min f1(x) x1

min f2(x) g(x) 1 minusx1

g(x)1113874 1113875

21113888 1113889

g(x) 1 + 91113936

ni2 xi( 1113857

n minus 1

n 30 x isin [0 1]

(7)

Iterations

Fitn

ess

10 20 30 40 50

0135

014

0145

015

GPSOMCPSO

(a)

xc

Cp y

0 02 04 06 08 1

ndash08

0

08

160

015

03

045

OriginalGPSOMCPSO

(b)

Figure 1 Optimization results (a) convergent curves (b) comparisons of pressure distributions and airfoils


ZDT3

min f1(x) x1


x1

g(x)

1113970

minusx1 sin 10πx1( 1113857

g(x)1113888 1113889

g(x) 1 + 91113936

ni2 xi

n minus 1

n 30 x isin [0 1]

(8)Figure 2 shows the Pareto-optimal fronts of these three

functions obtained from MOPSO e newly designed frontsare extremely consistent with the real ones Moreover as shownin Table 2 theGDs ofMOPSO are better than those of NSGA-II[43] for all three functions As for SPs it is hard to distinguishbetween the two algorithms to find which one performs better

213 Uncertainty Quantification by Polynomial Chaos Ex-pansion (PCE) e main difficulties for uncertainty quanti-fication are the excessive computational cost and unsatisfactoryaccuracies of the mean and variance For instance the widelyused Monte Carlo simulation (MCS) [44] method basicallyrequires at least 104 samples to get sufficiently accurate estimateof the mean and variance which certainly causes a much highcost of computation In the aspect of efficient and accurateuncertainty quantification the PCE is a promising method foraccurate estimation of mean and variance of the objectives[45 46] A non-instrusive PCE method is adopted in this partwhich is a question-answer system and unnecessary to modifythe original governing equations [47]

is formula assumes that the random variable θ obeys astandard normal distribution and thus p polynomial order of ageneral random process X(ξ(θ)) can be expressed as follows

X(ξ(θ)) a0H0 + 1113944infin

i11ai1

H1 ξi1(θ)1113872 1113873

+ 1113944infin

i111113944

i1

i21ai1 i2

H2 ξi1(θ) ξi2

(θ)1113872 1113873 + middot middot middot

1113944

NP+1

j1bjψj(ξ)

(9)

where Hn is the Hermite polynomial (ξi1 ξin

) are thevectors consisting of n Gaussian variables NP + 1

(p + n)pn and n is the number of random variablesen coefficients bj can be solved by the following

formula

bj langX(ξ)ψj(ξ)ranglangψj(ξ)ψj(ξ)rang

1113946 X(ξ)ψj(ξ)ω(ξ)dξ

langψj(ξ)ψj(ξ)rang

1113936

ki11 middot middot middot 1113936

kiN1 ωi1

middot middot middotωiNX xi1

xiN1113872 1113873ψj xi1

xiN1113872 1113873


(10)

where the number of collocation points is indicated by M

kn and ωi and xi are the GaussianndashHermite quadraturecoefficient and point respectively

According to the orthogonality of GaussianndashHermitepolynomials the mean and variance of X(ξ(θ)) are pre-sented as follows

μ(X(ξ(θ))) b1

σ2(X(ξ(θ))) 1113936NP+1

j2b2jlangψ2

jrang

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)

e accuracy and efficiency for PCE and MCS ap-proaches are compared through the following function

f(X) 20eminus (x+27)22( )

2π

radic +eminus (xminus 27)2 2 22( )( )( )

2π

radic2

1113888 1113889 (12)

where X is assumed to be normally distributed with μ 20and σ 08 e random variables are assumed asξi isin N(0 12) and xi is expressed as xi 20 + 08ξi

e real mean and variance of this function would becalculated by the following equations

μY E[f(X)] 3520985

σY

E f(X) minus μY1113858 1113859

1113969

0507313(13)

Figure 3 plots the comparisons of estimated mean (μY)and variance (σY) between PCE and MCS methods esample numbers for MCS fetch two values ie 100 and 500that are utilized to appraise its estimation accuracy of thestochastic processe results demonstrate that the mean andvariance will become convergent with the increasing order PAnd the estimation accuracy of mean is slightly higher thanthat of variance particularly for the P value within 2ndash4Additionally the estimated μY and σY by the PCE methodboth fall within 05 percent error bands when the polynomialorder P is up to 4 Meanwhile the estimated accuracy of μY

agrees well with that of MCS using 500 samples But theestimated accuracy of σY becomes closer to that of MCS using500 samples with the polynomial order P increasing from 4ereupon our results demonstrate that the PCE method isreasonable for the uncertain quantification with respect to itsreliable accuracy and efficiency And the same conclusion wasalso derived by the former study that verified the estimatedresults of aerodynamic characteristics by PCE [48]Figure 3(c) shows the time cost of PCE and MCS approachesvarying with the error of variances When the error of var-iance is at the same level the time cost of theMCS approach ismuch more than that of the PCE approach erefore thePCE method is chosen to estimate the mean and variance ofthe objectives in present work for its efficiency

22 Numerical Simulation Method for AerodynamicCharacteristics e c minus Reθt transition model [49 50] iscombined with the local variable of SST k minus ω turbulencemodel to predict transition e proposed model incorpo-rated intermittent function with transition experience rela-tionship So it can control the distribution of intermittentfunctions in boundary layers through the relevance functions


of transition momentum thickness Reynolds number andafterward dominates turbulence through the intermittentfunctions Moreover since the momentum thickness

Reynolds number is related to the strain rate the calculationof momentum thickness can be prevented thereby enhancingthe prediction accuracy of separation transition [50]

e transitionmodel validation is performed onNLR7301airfoil applying the same boundary conditions adopted in theexperiments e simulation condition isMa 0747 andRe 2 times 106 e airfoil was tested at theNLR pilot closed-circuit wind tunnel [51]ere are 760 times 179structured grid for the transition model solver as shown inFigure 4 As displayed in Figure 5 the transition position (TP)and surface pressure distribution obtained from the transitionmodel and experiment data illustrate good consistency

Fitness_1

Fitn

ess_

2

0 02 04 06 08 1

02

04

06

08

1

MOPSOReal front

(a)Fi

tnes

s_2

MOPSOReal front

Fitness_10 02 04 06 08 1

0

02

04

06

08

1

(b)

Fitn

ess_

2

MOPSOReal front

Fitness_10 02 04 06 08

ndash05

0

05

1

(c)

Figure 2 e comparison of the real and Pareto-optimal fronts obtained from MOPSO (a) ZDT1 (b) ZDT2 (c) ZDT3

Table 2 e GD and SP obtained from MOPSO and NAGA-II interms of three typical functions

Test functionsGD SP

NSGA-II MOPSO NSGA-II MOPSOZDT1 8146e minus 2 3396e minus 4 9539e minus 2 7539e minus 2ZDT2 2334e minus 2 2127e minus 3 6722e minus 2 1269e minus 1ZDT3 1037e minus 1 5957e minus 3 4613e minus 1 5613e minus 2


Also the validation of capturing LSB is accomplished onE387 airfoil at low speed and Re 2 times 105 e airfoilperformances predicted by transition model are comparedto University of Illinois Urbana-Campaign (UIUC) wind-tunnel data [52] in Figure 6 e results displayed inFigure 6(a) demonstrate that the lift-drag curve is in goodagreement with the experimental one As presented inFigure 6(b) both the location and size of LSB obtained fromCFD are coincident with the experiment data

Consequently these phenomena make us to concludethat the numerical simulation technology of transition

boundary layer is accurate and reliable for the aerodynamicdesign optimization of transonic NLF airfoil at LRN

23 Airfoil Parameterization Method e disturbing basisfunction based on the classshape function transformation(CST) [53] is introduced in airfoil parameterization eparameterization method constructed here can not onlyminimize the errors caused by fitting the initial parametersbut also preserve the advantages of the original CST pa-rameterization method

P

microY

2 3 4 5 6 7 8

348

35

352

354

356

PCEMCS-100MCS-500

05 error bands

(a)

σY

PCEMCS-100MCS-500

P2 3 4 5 6 7 8

048

049

05

051

052

053

05 error bands

(b)

Error of variance

Tim

e of M

CS (s

)

Tim

e of P

CE (s

)

10ndash4 10ndash3 10ndash2

102

103

04

06

08

1

PCEMCS

(c)

Figure 3 Comparison of estimated mean and variance between PCE and MCS methods (a) mean μY (b) variance σY (c) time cost


(a) (b)

Figure 4 e computation structured grid (a) grid around the airfoil (b) leading edge grid

α

TP o

f upp

er su

rface

ndash2 ndash1 0 1 2 30

02

04

06

08

CFDEXP

(a)

αndash2 ndash1 0 1 2 3

0

02

04

06

08

TP o

f low

er su

rface

CFDEXP

(b)

Figure 5 Continued


e original CST method is shown in the followingequation

ζ(ψ) ψ05(1 minus ψ) 1113944

n

i0AiKiψ

i(1 minus ψ)

nminus i+ ψζT (14)

where ψ and ζ(ψ) are the x-coordinates and y-coordinates ofthe airfoil respectively ζT is the y-coordinate of the trailingedge point Ai is the design variables andKi Ci

n ni(n minus i)e disturbance ΔAi is added to the initial design var-

iable Ai and then equation (14) can be written as

CD

CL

001 002 003 004 005 006

02

04

06

08

1

12

CFDEXP

(a)

α

xc

0 1 2 3 4 5 6ndash02

0

02

04

06

08

1

SP_CFDRP_CFD

SP_EXPRP_EXP

(b)

Figure 6 e aerodynamic characteristics of the E387 airfoil obtained from CFD and experiment (a) lift-drag curves (b) locations of theupper surface flow features

xc

Cp

0 02 04 06 08 1

ndash2

ndash1

0

1

α = ndash2

CFDEXP

(c)

Cp

xc0 02 04 06 08 1

ndash2

ndash1

0

1

α = 2

CFDEXP

(d)

Figure 5 e aerodynamic characteristics of the NLR 7301 obtained from CFD and experiment (a) AoA effect on TP of upper surface (b)AoA effect on TP of lower surface (c) surface pressure distribution when AoA is equal to minus 2 (d) surface pressure distribution when AoA isequal to 2


ζ(ψ)prime ψ05(1 minus ψ) 1113944

n

i0Ai + ΔAi( 1113857Kiψ

i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)

where ζ(ψ)prime and ζ(ψ) are the separable y-coordinates of newand original airfoils and Δζ(ψ) are the disturbances of y-coordinates which can be denoted as

Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)

According to the aforementioned equations the new y-coordinates can be treated as two parts ie the original y-coordinates (ζ(ψ)) and disturbances (Δζ(ψ)) us equa-tion (16) can be denoted as the disturbing basis functionbased on CST

3 Aerodynamic Design Problem

31 Optimization Design Definition As mentioned abovethe size and position of LSB and transition location have agreat impact on aerodynamic characteristics of the tran-sonic NLF airfoil at LRN So these two factors should beconsidered during the design process e classical laminarbubble separation theory [54ndash56] indicates that the adversepressure gradient of the upper surfaces will promote thelaminar separation and the transition will take place in freeshear layer above the airfoil surface of which the location isdownstream of the SP Consequently the LSB would be-come smaller with the forward SP and RP and the pressuredrag would decrease Conversely the friction drag wouldincrease by the forward TP It is of particular interest topoint out that the friction drag decreases dramatically bythe backward TP Nevertheless an extra length of laminarflow region would result in the excessive adverse pressuregradient at the pressure recovery section thus yielding alarger LSB on upper surface Accordingly the pressure dragsharply increases because of the larger LSB Regarding tothe above analysis it should keep a balance between thesetwo factors in the aerodynamic design of transonic NLFairfoil at LRN

e flight condition of the propeller tip airfoil utilizedin a solar energy UAV is transonic at LRN As for this theon-design point is assumed as Ma 072 andRe

11 times 105 For the original airfoil SD-8000 its CFD resultsshow that there is a large LSB on the upper surface and theproportion of pressure drag is 75 erefore the decreasein pressure drag would correspondingly lead to a reductionin total drag And it can be achieved by weakening the LSBHowever the friction drag would increase when the TPmoves forward

32 Deterministic Optimization Results and Discussione following optimized mathematical model is built up forthe deterministic design problem

objective min CDp

st

CL 05


CoptimizedDv minus C

originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)

where CL is the lift coefficient CDp is the pressure dragcoefficient thicknessmax is the maximum thickness of airfoiland C

optimizedDv and C

originalDv are the friction drag coefficients of

the optimized and original airfoils respectively e ob-jective of this model is to reduce the pressure drag coeffi-cient ere are some constraints including the growingproportion of friction drag coefficient less than 10 themaximum thickness of airfoil greater than 892 and the liftcoefficient being 05

ere are five design variables for the upper or lowersurfaces a total of ten design variables for each airfoil ecell number of the grids is 760 times 179 [57] of which thedistributions for the leading edge and trailing edge aredemonstrated in Figures 7(a) and 7(b)With respect to the gridconvergence for the original airfoil presented in Figure 7(c)when the circumferential grid number is more than 600 thevariations of the total and pressure drags are within 00008

In the deterministic optimization the algorithm pop-ulation in each iteration is set as 32 particles and themaximum iteration step is set as 60 Table 3 lists theaerodynamic characteristics for both the original and op-timized airfoils It can be summarized that the total drag andpitching moment coefficients separately decrease by 48counts and 266 while the friction drag coefficient and lift-to-drag ratio increase by 74 and 287 at on-design pointrespectively e friction drag increases because of the TPmoving forward and the pressure drag reduces because ofthe smaller size of LSB as displayed in Figure 8(b) From thegeometric characteristics enumerated in Table 4 it can benoticed that there is a backward location of the maximumthickness on optimized airfoil Its maximum camber issmaller than that of the original one And the same situationhappens to the leading edge radius of which the value foroptimized airfoil is also smaller

Figure 8 displayed two airfoils and surface pressuredistributions at on-design point Compared with the originalairfoil the optimized one with a flat upper surface has asmaller camber before the 60 chord and larger camberafter the 60 chord ere is a stronger inverse pressuregradient roughly existing in 5 to 20 chord for the op-timized pressure distribution and then this gradient be-comes weaker for 50 to 75 chord Moreover the suctionpeak of the optimized airfoil obviously becomes larger Asmarked in Figure 8 the SP and RP of the optimized airfoilare moved forward at the position of 15 and 60 chordrespectively Figure 9 shows the Mach number distribution


contours and the streamlines of the original and optimizedairfoils at on-design point e separation bubble on uppersurface moves forward and becomes smaller compared withthat of the original one We assume that the transition willtake place in free shear layer above the airfoil surface of

which the location is the downstream of SP e forward TPresults in a slight increase of friction drag Simultaneouslythe small LSB results in a considerable decrease of pressuredrag Moreover the height of the bubble is also smaller Itcan be concluded from Figures 8 and 9 that the stronginverse pressure gradient on the leading edge could lead tothe forward SP and the early recover pressure could result inthe forward RPe forward SP and RP minimize the LSB toreduce the pressure drag Nevertheless the forward LSB maybe conducive to the formation of a forward transition po-sition TP subsequently causing the increase of friction drage lift-to-drag ratio curves of the original and optimizedairfoils are shown in Figure 10 For the optimized airfoilthere is a low drag region over the range of lift coefficient

(a) (b)

Circumferential grid number

Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)

Figure 7 (a) e grid distributions of the leading edge (b) the grid distributions of the trailing edge (c) variation of drag with cir-cumferential grid number


CL CD KCLCD CDp CDv Cm Xtri

Originalairfoil 05 00222 225 00168 00054 minus 00785 0598

Optimizedairfoil 05 00174 287 00116 00058 minus 00576 0488


xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01

Optimized airfoilOriginal airfoil

(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position

Reattachment position

(b)

Figure 8 (a) e original and optimized airfoils (b) surface pressure distributions of original and optimized airfoils at on-design point

Table 4 Geometric characteristics for the original and optimized airfoils

Maximumthickness

Location of maximumthickness

Maximumcamber

Location of maximumcamber

Leading edgeradius

Original airfoil 0089 0274 00197 0491 00078Optimizedairfoil 0089 0306 00168 0697 00035

Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)

Figure 9 e Mach number contours and streamlines at on-design point (a) original airfoil (b) optimized airfoil


from 03 to 052 It is caused by the sharp decrease in thepressure drag as clearly presented in Figure 10

It can be seen from Figure 10 that the drag of optimizedairfoil is larger than that for the original one when CL = 02By comparing the streamline distributions of the off-designpoint (CL = 02) for original airfoil in Figure 11(a) it isobviously noticed that there is a large flow separation inwake region of the optimized airfoil which will increase thepressure drag in Figure 11(b) Furthermore its skin frictioncoefficient increases in the trailing edge of the optimizedairfoil in Figure 11(c) which will significantly give rise to thegrowth of the friction drag erefore both the increases ofpressure and friction drags result in increasing the total drag

Figure 12 shows the comparison of aerodynamic char-acteristics between the original and optimized ones epressure drag of optimized airfoil shows a considerabledecrease over the Mach number ranging from 070 to 076e friction drag of optimized airfoil shows a slight increasewhich meets the constraints at on-design point Note thatthe maximum increase of the friction drag is only up to 5counts It can be summarized from the drag divergencecurves that the total drag becomes decreased significantlywith the substantial decrease of the pressure drag eoptimized airfoil has the smaller drag coefficients over therange of Mach numbers from 070 to 076 e SP and RPvarying with the Mach numbers are displayed inFigure 12(d) Over the Mach number ranging from 070 to076 the LSB of the optimized airfoil is smaller than that ofthe original one and thus leads to a larger reduction of thepressure drag However when Magt 075 the RP cannot bedetected on the original airfoil leading to the sharp rise ofthe pressure drag erefore the varying trend of thepressure drag is mostly influenced by the size and position ofLSB over the studied Mach number range

Figure 13(a) shows the pressure distributions of originaland optimized airfoils varying with the Mach number It canbe seen that the obvious flat pressure distribution appearsaround the position of separation bubble e results alsopresent that the strong inverse pressure gradient on theleading edge leads to a forward SP on the optimized airfoiland an inverse pressure gradient around the RP esecharacteristics indicate that the laminar separation of op-timized airfoil coincides with the classical LSB theorye SPof original airfoil moves forward when the Mach numberbecomes larger but the RP is difficult to be detected on theupper surface ese characteristics are consistent with thetrailing edge separation bubble theory

33 Multi-objective Robust Results and DiscussionAlthough the optimized airfoil of deterministic has anoutstanding performance at on-design point the pressureand total drags both have the sharp increases whenMagt 075 erefore the fluctuation of flight conditionswould cause a sudden drop in the airfoilrsquos efficiency Amultiobjective robust design is carried out in this sectionaiming to reduce the performancesrsquo sensitivity to the un-certainties of flight conditionse PCEmethod is employedto obtain the mean and variance of the objectives e four

GaussianndashHermite integration points are introduced in thenonintrusive PCE method to calculate the PCE coefficientsand then a 3-order PCE model is set up for the robustdesign e population of MOPSO is set as 40 and themaximum iteration is 20

e Pareto optimal solution is chosen to deal with themean and variance as two objectives rather than convertingthem into a single objective We assume Mach number to benormally distributed Mainfin isin N(072 0022) e followingoptimized mathematical model is built up for the multi-objective design problem

objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

where μCDpand σCDp

are the mean and the variance of thepressure drag respectively μCDv

and μCoriginalDv

are the separatemean friction drags of the optimized and original ones

We focus on the marked individuals of the Pareto-op-timal front named as optimization_2 as shown in Figure 14e deterministic optimized airfoil named as optimiza-tion_1 also plots in Figure 14 e optimization_2 has asmaller camber before the 40 chord compared with op-timization_1 However the camber distributions for thesetwo airfoils are aligned with each other after 40 chord

Table 5 lists the aerodynamic force coefficients of theoptimization ones at on-design point It can be summarizedthat the total and pressure drag coefficients of optimization_2are smaller than those of optimization_1e lift-to-drag ratioof optimization_2 is better than that of optimization_1 Ta-ble 6 displays the geometric characteristics for two optimizedairfoils Compared with these of optimization_1 the positionsof both the maximum thickness and camber of optimiza-tion_2 move forward slightly And its leading edge radiusbecomes smaller demonstrated in Table 6

Figure 15 shows the comparisons of aerodynamic char-acteristics between deterministic and robust optimizationsWhen Malt 072 optimization_2 has the larger pressure dragcoefficients than the optimization_1 However the contra-dictory trend is found for them when Magt 072 With atradeoff of the pressure drag the varying trend of pressuredrag becomes gentler over the range of Mach number from070 to 076 e friction drag coefficients of optimization_2are smaller than the those for optimization_1 From the dragdivergence curves of the optimized airfoils we can see that therapid change of drag calms down through the robust designFor the performance of optimization_2 although it has atrade-off of the pressure drag its friction drag acquires a


X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05

Original airfoilOptimized airfoil

(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)

Figure 10e lift-to-drag ratio curves of the original and optimized airfoil (Re 11 times 105 CL 05) (a) lift to total drag (b) lift to pressuredrag


xc

Cf

0 02 04 06 08 1

0

005

01

Original_upperOriginal_lower

Optimized_upperOptimized_lower

(c)

Figure 11 Off-design point (CL 02) (a) the streamlines of original airfoil (b) the streamlines of optimized airfoil (c) the skin frictiondistributions

Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued


slighter drop erefore the airfoil has a wide range of lowerdrag e SP and RP varying with the Mach numbers aredisplayed in Figure 15(d) Two optimizations have the similarLSB length however the SP and RP of optimization_2 areboth forward compared with those of optimization_1

Figure 16 shows the aerodynamic characteristics of opti-mized airfoils at on-design point ere is a stronger inverse

pressure gradient occurring in 5 to 20 chord on optimi-zation_2 and then a weaker inverse pressure gradient can befound from 40 to 60 chord Furthermore the suction peakof the optimization_2 becomes larger than that of optimiza-tion_1 For clarity the SPs andRPs are alsomarked in Figure 16e SP and RP of optimization_2 would move forward at thepositions of 11 and 58 chord respectively As mentioned

Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150

SPOriginalRPOriginal

SPOptimizedRPOptimized

(d)

Figure 12e comparison of aerodynamic characteristics between the original and optimized airfoils (a) pressure drag divergence curves(b) friction drag divergence curves (c) drag divergence curves (d) SP and RP

xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)

Figure 13 e surface pressure distribution varying with Mach number (a) original airfoil (b) optimized airfoil


Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005

Optimization_1Optimization_2

(b)

Figure 14 (a) Pareto-optimal front (b) optimized airfoils

Table 5 Aerodynamic force coefficients for the optimized airfoils at on-design point


Optimization_1 05 00174 287 00116 00058 minus 00576 0488Optimization_2 05 00171 292 00115 00056 minus 00552 0476

Table 6 Geometric characteristics for the optimized airfoils

Maximumthickness


Maximumcamber


Leading edgeradius

Optimization_1 0089 0306 00168 0697 000351Optimization_2 0089 0304 00170 0696 000345

072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued


before the identical circumstance also happens here that is thestrong inverse pressure gradient on the leading edge and theforward SP both lead to a forward TP e position and lengthof LSB also can be seen from surface friction distributions asshown in Figure 16e separation bubble of optimization_2 onupper surface moves forward and becomes smaller than that ofoptimization_1 Figure 17 displays the aerodynamic

characteristics of optimized airfoils at Re 11 times 105 andCL

05 ere is an obvious lower drag region of optimization_2over the range of lift coefficient from 01 to 055 Additionallywith the Mach number increasing the suction peak becomessmaller and the pressure recovery is postponed

e pitching moment curves of the original and opti-mized airfoils are shown in Figure 18 It can be seen that the

072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076

SPOptimization_1vRPOptimization_1

SPOptimization_2RPOptimization_2

(d)

Figure 15 e comparison of aerodynamic characteristics between the deterministic and robust optimizations (a) pressure drag di-vergence curves (b) friction drag divergence curves (c) drag divergence curves (d) SP and RP

CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)

Figure 16 e aerodynamic characteristics of optimized airfoils at on-design point (a) surface pressure distributions (b) surface frictiondistributions


pitching moments of the optimized airfoils are smaller thanthat of the original one Additionally as shown in Figures 814 and 16 both the maximum camber and after load of the

optimized airfoils are also smaller than those of the originalone and thereby they all will result in the decrease in thepitching moment

CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)

Figure 17 e aerodynamic characteristics of optimized airfoils at Re 11 times 105 andCL 05 (a) lift-to-drag coefficient curves (b) thesurface pressure distribution of optimization_2 varying with Mach number

Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm

Original airfoilOptimization_1Optimization_2

(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)

Figure 18 e comparison of moment coefficient between the original airfoil and optimized airfoils (a) varying with Mach number (b)varying with AOA


Regarding to all above discussions the mulit-objectiverobust design could produce an excellent airfoil e opti-mized surface pressure distribution not only keeps the lowdrag at on-design point but also catches the robustness oflow drag to overcome the fluctuation of flight condition

4 Conclusions

In this paper we carry out the aerodynamic design optimi-zation of a propeller tip airfoil in a solar energy UAV eflight condition of this airfoil is treated as transonic at LRNUnder such conditions we develop an optimization meth-odology based on the PSO algorithm applied to the design of atransonic NLF airfoil at LRN is new proposed method-ology briefly contains two optimization design processesey are traditional deterministic optimization at on-designpoint on the basis of MCPSO and multi-objective robustdesign based on MOPSO e non-intrusive PCE method isadopted to get sufficiently accurate estimations of the meanand variance It facilitates to the fact that the optimal de-terministic solution and Pareto-optimal front of aerodynamicdesign can be captured in the space quickly and accuratelyFrom the findings of deterministic optimizations and multi-objective robust design we obtained some important con-clusions for transonic NLF airfoil at LRN as follows

(1) We construct a new optimization model that is theminimized pressure drag is typically treated as theoptimization objective while the friction drag in-crement is a constraint condition According to thedeterministic optimization and robust design resultsthis proposed model is reliable and efficacious todesign a high-performance airfoil of the propellertip

(2) e pressure drag is mostly affected by the size andposition of LSB e deterministic optimized pres-sure distribution can lead to a small and forwardLSB e SP and RP of the airfoil move forward byvirtue of the deterministic optimized pressuredistribution

(3) e multi-objective robust design can weaken thesensitivity of airfoil performance to the flight fluc-tuation e proper inverse pressure gradient at theleading edge and early pressure recover show thegood abilities to control the LSB

(4) e proposed optimization methodology is an effi-cient and reasonable approach for the aerodynamicdesign of transonic NLF airfoil at LRN

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors gratefully acknowledge financial support fromthe China Scholarship Council e authors are also gratefulfor computing support by the Supercomputing Center ofZhengzhou University is research was sponsored by theNational Science Foundation of China (NSFC) (grantnumber 11602226) University Science Project of HenanProvince (grant number 16A130001) and China Postdoc-toral Science Foundation (grant number 2019M652579)

References

[1] S Morton L Scharber and N Papanikolopoulos ldquoSolarpowered unmanned aerial vehicle for continuous flightconceptual overview and optimizationrdquo in Proceedings of the2013 IEEE International Conference on Robotics and Auto-mation pp 766ndash771 Karlsruhe Germany May 2013

[2] S Leutenegger M Jabas and R Y Siegwart ldquoSolar airplaneconceptual design and performance estimationrdquo UnmannedAerial Vehicles vol 61 no 1ndash4 pp 545ndash561 2010

[3] J Brandt and M Selig ldquoPropeller performance data at lowReynolds numbersrdquo in Proceedings of the 49th AIAA Aero-space Sciences Meeting including the New Horizons Forum andAerospace Exposition Orlando FL USA 2011

[4] W Chen and L Bernal ldquoDesign and performance of lowReynolds number airfoils for solar-powered flightrdquo in Pro-ceedings of the 46th AIAA Aerospace Sciences Meeting andExhibit Reno NV USA 2008

[5] R W Deters G K A Krishnan and M S Selig ldquoReynoldsnumber effects on the performance of small-scale propellersrdquoin Proceedings of the 32nd AIAA Applied AerodynamicsConference Atlanta GA USA 2014

[6] K Wang Z Zhou X Zhu and X Xu ldquoAerodynamic design ofmulti-propellerwing integration at low reynolds numbersrdquoAerospace Science and Technology vol 84 pp 1ndash17 2019

[7] B R Rakshith S M Deshpande R Narasimha andC Praveen ldquoOptimal low-drag wing planforms for tractor-configuration propeller-driven aircraftrdquo Journal of Aircraftvol 52 no 6 pp 1791ndash1801 2015

[8] S Zhang H Li and A A Abbasi ldquoDesign methodology usingcharacteristic parameters control for low Reynolds numberairfoilsrdquo Aerospace Science and Technology vol 86 pp 143ndash152 2019

[9] R Ma B Zhong and P Liu ldquoOptimization design study oflow-reynolds-number high-lift airfoils for the high-efficiencypropeller of low-dynamic vehicles in stratosphererdquo ScienceChina Technological Sciences vol 53 no 10 pp 2792ndash28072010

[10] K Wang X Zhu Z Zhou and X Xu ldquoStudying optimizationdesign of low reynolds number airfoil using transitionmodelrdquoJournal of Northwestern Polytechnical University vol 33pp 580ndash587 2015

[11] F Chen J Yu and YMei ldquoAerodynamic design optimizationfor low reynolds tandem airfoilrdquo Proceedings of the Institutionof Mechanical Engineers Part G Journal of Aerospace Engi-neering vol 232 no 6 pp 1047ndash1062 2017

[12] Z Q Zhu X L Wang Z C Wu and Z M Chen ldquoAero-dynamic characteristics of smallmicro unmanned aerialvehicles and their shape designrdquo Acta Aeronauticaet Astro-nautica Sinica vol 27 pp 353ndash364 2006

[13] F Li and P Bai Aircraft Aerodynamic at Low ReynoldsNumber China Aerospace Press Beijing China 2017


[14] R M Hicks and S E Cliff An Evaluation of Hree Two-Dimensional Computational Fluid Dynamics Codes IncludingLow Reynolds Numbersand Transonic Mach Numbers NASASilicon Valley CA USA 1991

[15] R Hain C J Kahler and R Radespiel ldquoDynamics of laminarseparation bubbles at low-reynolds-number aerofoilsrdquo Jour-nal of Fluid Mechanics vol 630 pp 129ndash153 2009

[16] M Jahanmiri Laminar Separation Bubble Its StructureDynamics and Control Chalmers University of TechnologyGothenburg Sweden 2011

[17] D Kamari M Tadjfar and A Madadi ldquoOptimization ofSD7003 airfoil performance using TBL and CBL at lowreynolds numbersrdquoAerospace Science and Technology vol 79pp 199ndash211 2018

[18] C Chen R Seele and I Wygnanski ldquoFlow control on a thickairfoil using suction compared to blowingrdquo AIAA Journalvol 51 no 6 pp 1462ndash1472 2013

[19] M Drela ldquoTransonic low-reynolds number airfoilsrdquo Journalof Aircraft vol 29 no 6 pp 1106ndash1113 1992

[20] R Placek and P Ruchała ldquoe flow separation developmentanalysis in subsonic and transonic flow regime of the laminarairfoilrdquo Transportation Research Procedia vol 29 pp 323ndash329 2018

[21] H Zhao Z Gao Y Gao and C Wang ldquoEffective robustdesign of high lift NLF airfoil under multi-parameter un-certaintyrdquo Aerospace Science and Technology vol 68pp 530ndash542 2017

[22] A Banks J Vincent and C Anyakoha ldquoA review of particleswarm optimization Part II hybridisation combinatorialmulticriteria and constrained optimization and indicativeapplicationsrdquo Natural Computing vol 7 no 1 pp 109ndash1242008

[23] S Sun and J Li ldquoA two-swarm cooperative particle swarmsoptimizationrdquo Swarm and Evolutionary Computation vol 15pp 1ndash18 2014

[24] Q Luo and D Yi ldquoA co-evolving framework for robustparticle swarm optimizationrdquo Applied Mathematics andComputation vol 199 no 2 pp 611ndash622 2008

[25] M Khurana and K Massey ldquoSwarm algorithm with adaptivemutation for airfoil aerodynamic designrdquo Swarm and Evo-lutionary Computation vol 20 pp 1ndash13 2015

[26] J Tao G Sun X Wang and L Guo ldquoRobust optimization fora wing at drag divergence Mach number based on an im-proved PSO algorithmrdquo Aerospace Science and Technologyvol 92 pp 653ndash667 2019

[27] H Masoumi and F Jalili ldquoInverse and direct optimizationshape of airfoil using hybrid algorithm big bang-big crunchand particle swarm optimizationrdquo Journal of Heoretical andApplied Mechanics vol 57 no 3 pp 697ndash711 2019

[28] M Kohler M M B R Vellasco and R Tanscheit ldquoPSO+ anew particle swarm optimization algorithm for constrainedproblemsrdquo Applied Soft Computing vol 85 Article ID105865 2019

[29] S Kiranyaz J Pulkkinen and M Gabbouj ldquoMulti-dimen-sional particle swarm optimization for dynamic environ-mentsrdquo in Proceedings fo the 2008 International Conference onInnovations in Information Technology pp 34ndash38 Al LinUnited Arab Emirates December 2008

[30] A P Engelbrecht ldquoParticle swarm optimization global bestor local bestrdquo in Proceedings of the BRICS Congress onComputational Intelligence and 11th Brazilian Congress onComputational Intelligence pp 124ndash135 Ipojuca BrazilSeptember 2013

[31] H Likeng G Zhenghong and Z Dehu ldquoResearch on multi-fidelity aerodynamic optimization methodsrdquo Chinese Journalof Aeronautics vol 26 no 2 pp 279ndash286 2013

[32] H Han L Liang Y Shujin and H Zhifeng ldquoParticle swarmoptimization with convergence speed controller for large-scale numerical optimizationrdquo Soft Computing vol 23 no 12pp 4421ndash4437 2018

[33] M Nemec D W Zingg and T H Pulliam ldquoMultipoint andmulti-objective aerodynamic shape optimizationrdquo AIAAJournal vol 42 no 6 pp 1057ndash1065 2004

[34] H Zhao and Z Gao ldquoUncertainty-based design optimizationof NLF airfoil for high altitude long endurance unmanned airvehiclesrdquo Engineering Computations vol 36 no 3pp 971ndash996 2019

[35] Y Naranjani and J-Q Sun ldquoMulti-objective optimal airfoildesign for cargo aircraftsrdquo in Proceedings of the ASME 2016International Mechanical Engineering Congress andExposition Portland OR USA 2016

[36] Y Azabi A Savvaris and T Kipouros ldquoe interactive designapproach for aerodynamic shape design optimisation of theaegis UAVrdquo Aerospace vol 6 no 4 2019

[37] J Huang Z Zhou Z Gao M Zhang and L Yu ldquoAerody-namic multi-objective integrated optimization based onprincipal component analysisrdquo Chinese Journal of Aeronau-tics vol 30 no 4 pp 1336ndash1348 2017

[38] A de Campos A T R Pozo and E P Duarte ldquoParallel multi-swarm PSO strategies for solving many objective optimizationproblemsrdquo Journal of Parallel and Distributed Computingvol 126 pp 13ndash33 2019

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[40] J Li Z Gao J Huang and K Zhao ldquoAerodynamic designoptimization of nacellepylon position on an aircraftrdquoChineseJournal of Aeronautics vol 26 no 4 pp 850ndash857 2013

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[42] J LI Research on the High Performance Aircraft AerodynamicDesign Northwestern Polytechnical University Xirsquoan China2014

[43] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2pp 182ndash197 2002

[44] L Zhang H Wu P Li Y Hu and C Song ldquoDesign analysisand experiment of multiring permanent magnet bearings bymeans of equally distributed sequences based Monte Carlomethodrdquo Mathematical Problems in Engineering vol 2019Article ID 4265698 17 pages 2019

[45] K Chikhaoui D Bitar N Kacem and N Bouhaddi ldquoRo-bustness analysis of the collective nonlinear dynamics of aperiodic coupled pendulums chainrdquo Applied Sciences vol 7no 7 p 684 2017

[46] H Zhao Z Gao F Xu Y Zhang and J Huang ldquoAn efficientadaptive forward-backward selection method for sparsepolynomial chaos expansionrdquo Computer Methods in AppliedMechanics and Engineering vol 355 pp 456ndash491 2019

[47] M Eldred and J Burkardt ldquoComparison of non-intrusivepolynomial chaos and stochastic collocation methods foruncertainty quantificationrdquo in Proceedings of the 47th AIAAAerospace Sciences Meeting including He New Horizons Fo-rum and Aerospace Exposition Orlando FL USA January2009


[48] K Zhao Z Gao J Huang and J Li ldquoUncertainty quantifi-cation and robust design of airfoil based on polynomial chaostechniquerdquo Chinese Journal of Heoretical and Applied Me-chanics vol 46 pp 10ndash19 2014

[49] R B Langtry and F R Menter ldquoCorrelation-based transitionmodeling for unstructured parallelized computational fluiddynamics codesrdquo AIAA Journal vol 47 no 12 pp 2894ndash2906 2009

[50] T H Nguyen and K Chang ldquoComparison of the point-collocation non-intrusive polynomial (NIPC) and non-in-trusive spectral projection (NISP) methods for the c-Rθtransition modelrdquo Applied Sciences vol 9 no 7 2019

[51] B Ernesto and P Rita ldquoLaminar to turbulent boundary layertransition investigation on a supercritical airfoil using the cndashθtransitional modelrdquo in Proceedings of the 40th Fluid DynamicsConference and Exhibit Chicago IL USA June 2010

[52] M S Selig and J J Guglielmo ldquoHigh-lift low reynolds numberairfoil designrdquo Journal of Aircraft vol 34 no 1 pp 72ndash791997

[53] B Kulfan ldquoUniversal parametric geometry representationmethodrdquo Journal of Aircraft vol 45 no 1 pp 142ndash158 2008

[54] M P Simens and A G Gungor ldquoInfluence of the transition ofa laminar separation bubble on the downstream evolution ofstrong adverse pressure gradient turbulent boundary layersrdquoEuropean Journal of Mechanics-BFluids vol 67 pp 70ndash782018

[55] S S Diwan and O N Ramesh ldquoLaminar separation bubblesdynamics and controlrdquo Sadhana vol 32 no 1-2 pp 103ndash1092007

[56] H P Horton Laminar Separation Bubbles in Two and HreeDimensional Incompressible Flow Queen Mary CollegeUniversity of London London UK 1968

[57] Z Huan Z Gao C Wang and Y Gao ldquoResearch on thecomputing grid of high speed laminar airfoilrdquo Chinese Journalof Applied Mechanics vol 2018 no 4 pp 351ndash357 2019


ese above new optimized natural-laminar-flow (NLF)airfoils presented the outstanding performance in differentaspects for low-speed flight at LRN but they can scarcely beimplemented in the propeller tip e reason for this is thatthe velocity nearby the blade tip would be high subsonic ortransonic while the velocity of its root region is rather low[12 13] Up to now there are very few transonic LRN airfoilsthat are available for the blade tip of solar powered UAVUndoubtedly it gives rise to more difficulties in designingtransonic LRN airfoil compared with low-speed NLF oneMoreover LSB will inevitably appear on one or both surfacesof the airfoil which are mainly responsible for increasingsensitivity of aerodynamic performance as the Reynoldsnumber decreases to and even below 500000 [14ndash16] Forthis it is necessary to employ the active flow control methodsto completely eliminate the laminar separation [17 18]Furthermore it has a high probability of developing shockwaves on the upper airfoil surface at transonic speed flightese phenomena would subsequently produce the strongadverse pressure gradients and compressibility effect thatgreatly influence the size and position of LSB and enhancethe design complexity [19 20] For another the fluctuationsof the flight conditions leading to the instabilities of the sizesand locations of separation bubbles also cause more diffi-culties [21]

According to these previous studies [12ndash21] it is badly inneed of putting forward an effective optimization meth-odology to design the transonic NLF airfoil at LRN towardsachieving direct application in the blade to enhance thepropellerrsquos efficiency e particle swarm optimization(PSO) algorithm has been widely used in aerodynamicoptimization because of its excellent optimal-searchingability [22ndash24] Khurana et al built an adaptive mutation-PSO (AM-PSO) method in use for redesigning airfoil atflight envelopes encompassing low-to-high Mach numbers[25] eir results illustrated that the AM-PSO can seek outthe optimal airfoil with minimal computational resourcesTao et al proposed an improved PSO algorithm on the basisof centroidal Voronoi tessellations (CVTs) and fitness dis-tance ratio (FDR) [26] e optimization of single point androbust showed that the optimized airfoil and wing both haveoutstanding characteristics at on-design point Masoumiet al developed a hybrid algorithm combining the big bang-big crunch and PSO which can be applied for reverse anddirect optimization approaches [27] And this proposedalgorithm has proper accuracy and convergence speed thatfacilitates to reach the desired airfoil geometry However itshould be mentioned that most of the above PSO algorithmswould fall into a local optimum e main reason is theevolution information for individual guidance only comingfrom the single group Under such conditions the evolu-tionary ability of algorithm could be declined and thereupongives rise to an algorithm premature in local optimum[28ndash30] Of particular interest is the high-order nonlinearfeature of the aerodynamic design space that certainly re-quires the algorithm to have a strong global optimizationcapability Moreover massive computational consumptionof aerodynamic characteristics needs the algorithm to have afaster convergence speed [31 32]

With respect to the significant role of single-objectivePSO algorithms in deterministic optimization the highrobustness airfoil should be taken into consideration inwhich the performances have less sensitivity in response toflight conditionsrsquo uncertainty Usually the mean and vari-ance of the characteristics were treated as the indicators forthe uncertainty design However the mean and variancealways were converted into a single-objective optimizationby a linear-weight combination [33 34] So the optimizationresults would be influenced by the treatment of weightcoefficients In this study the mean and variance are treatedas two objectives to get over the disadvantage of the weightmethod In line with previous study the multi-objectiveparticle swarm optimization (MOPSO) employed in manymulti-objective aerodynamic optimization tried to get thePareto-optimal front [35] Azabi et al proposed an inter-active MOPSO with the benefits of reducing computationaltime and avoiding the complexity of postdata analysis [36]Huang et al constructed an improved MOPSO with prin-cipal component analysis methodology enhancing theconvergence of Pareto front [37] e level of non-domi-nanted Pareto front was effectively improved and the multi-objective requirements of a wide-body aircraft were satisfiedHowever the number of the Pareto-optimal front will growdramatically for multi-objective aerodynamic design ereasonable dominant ranking and selecting for Pareto-op-timal sets are essential factors in enhancing the capability ofthe MOPSO [38]

In present work an optimization methodology based onPSO algorithm was proposed It aims to design a high androbust performance transonic NLF airfoil at LRN isarticle is organized as follows in Section 21 the PSOmethodology is built up and the algorithm performances arevalidated by the aerodynamic design of a supercritical airfoilthe numerical simulation and airfoil parameterizationmethods are given in Sections 22 and 23 respectively inSection 3 deterministic optimization and multi-objectiverobust design are conducted to get an optimal airfoil to meetthe design requirements and in Section 4 several importantdesign conclusions are summarized by the comparison oforiginal and newly optimized airfoil

2 Methodology

21 Optimization Methodology Based on the PSO Algorithm

211 Multigroups Cooperative Particle Swarm OptimizationAlgorithm (MCPSO) To avoid algorithm premature aMCPSO algorithm based on the biological systems symbi-osis is put forward in this section e populations are di-vided into several subgroups Every subgroup obtains theevolution information from the group itself and the othersymbiotic subgroups e symbiotic relationships amongsubgroups are described by a master-slave structure eslave subgroups and the master one separately undertake theglobal and local searches

e basic principles of this algorithm are conducted asfollows the populations are divided into four subgroupsmarked as the 1st 2nd 3rd and 4th subgroups e 1st



vrarr

i(t + 1) ω vrarr

i(t) + c1r1 prarr1

i (t) minus xrarr

i(t)1113874 1113875 + c2r2 prarr1

g(t) minus xrarr


g (t) minus xrarr

i(t)1113874 1113875

xrarr

i(t + 1) xrarr

i(t) + vrarr

i(t + 1)

⎧⎪⎨

⎪⎩(1)

where vrarr

i and xrarr


rarr1i and p





objective min CD

stCL 07


(2)















idistance 1113944

Nobj

m11113944

Nset minus 1

i2

fmi+1 minus fm


min (3)












prime


GD

1113936

ni1 d2

i

1113969

n (4)



SP

1n minus 1

1113944

n

i1



211139741113972

(5)


ZDT1

min f1(x) x1


x1

g(x)

1113970

1113888 1113889

g(x) 1 + 91113936

ni2 xi

n minus 1

n 30 x isin [0 1]

(6)

ZDT2

min f1(x) x1


g(x)1113874 1113875

21113888 1113889

g(x) 1 + 91113936

ni2 xi( 1113857

n minus 1

n 30 x isin [0 1]

(7)

Iterations

Fitn

ess

10 20 30 40 50

0135

014

0145

015

GPSOMCPSO

(a)

xc

Cp y

0 02 04 06 08 1

ndash08

0

08

160

015

03

045

OriginalGPSOMCPSO

(b)



ZDT3

min f1(x) x1


x1

g(x)

1113970


g(x)1113888 1113889

g(x) 1 + 91113936

ni2 xi

n minus 1

n 30 x isin [0 1]





X(ξ(θ)) a0H0 + 1113944infin

i11ai1

H1 ξi1(θ)1113872 1113873

+ 1113944infin

i111113944

i1

i21ai1 i2

H2 ξi1(θ) ξi2


1113944

NP+1

j1bjψj(ξ)

(9)




formula


1113946 X(ξ)ψj(ξ)ω(ξ)dξ


1113936


kiN1 ωi1


xiN1113872 1113873ψj xi1

xiN1113872 1113873


(10)




μ(X(ξ(θ))) b1

σ2(X(ξ(θ))) 1113936NP+1

j2b2jlangψ2

jrang

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)


f(X) 20eminus (x+27)22( )

2π


2π

radic2

1113888 1113889 (12)



μY E[f(X)] 3520985

σY

E f(X) minus μY1113858 1113859

1113969

0507313(13)








Fitness_1

Fitn

ess_

2

0 02 04 06 08 1

02

04

06

08

1

MOPSOReal front

(a)Fi

tnes

s_2

MOPSOReal front

Fitness_10 02 04 06 08 1

0

02

04

06

08

1

(b)

Fitn

ess_

2

MOPSOReal front

Fitness_10 02 04 06 08

ndash05

0

05

1

(c)



Test functionsGD SP







P

microY

2 3 4 5 6 7 8

348

35

352

354

356

PCEMCS-100MCS-500

05 error bands

(a)

σY

PCEMCS-100MCS-500

P2 3 4 5 6 7 8

048

049

05

051

052

053

05 error bands

(b)

Error of variance

Tim

e of M

CS (s

)

Tim

e of P

CE (s

)


102

103

04

06

08

1

PCEMCS

(c)



(a) (b)


α

TP o

f upp

er su

rface


02

04

06

08

CFDEXP

(a)


0

02

04

06

08

TP o

f low

er su

rface

CFDEXP

(b)

Figure 5 Continued



ζ(ψ) ψ05(1 minus ψ) 1113944

n

i0AiKiψ

i(1 minus ψ)





CD

CL

001 002 003 004 005 006

02

04

06

08

1

12

CFDEXP

(a)

α

xc

0 1 2 3 4 5 6ndash02

0

02

04

06

08

1

SP_CFDRP_CFD

SP_EXPRP_EXP

(b)


xc

Cp

0 02 04 06 08 1

ndash2

ndash1

0

1

α = ndash2

CFDEXP

(c)

Cp

xc0 02 04 06 08 1

ndash2

ndash1

0

1

α = 2

CFDEXP

(d)




n


i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)


Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)







objective min CDp

st

CL 05



originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


optimizedDv and C









(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References






























































vrarr

i(t + 1) ω vrarr

i(t) + c1r1 prarr1

i (t) minus xrarr

i(t)1113874 1113875 + c2r2 prarr1

g(t) minus xrarr


g (t) minus xrarr

i(t)1113874 1113875

xrarr

i(t + 1) xrarr

i(t) + vrarr

i(t + 1)

⎧⎪⎨

⎪⎩(1)

where vrarr

i and xrarr


rarr1i and p





objective min CD

stCL 07


(2)















idistance 1113944

Nobj

m11113944

Nset minus 1

i2

fmi+1 minus fm


min (3)












prime


GD

1113936

ni1 d2

i

1113969

n (4)



SP

1n minus 1

1113944

n

i1



211139741113972

(5)


ZDT1

min f1(x) x1


x1

g(x)

1113970

1113888 1113889

g(x) 1 + 91113936

ni2 xi

n minus 1

n 30 x isin [0 1]

(6)

ZDT2

min f1(x) x1


g(x)1113874 1113875

21113888 1113889

g(x) 1 + 91113936

ni2 xi( 1113857

n minus 1

n 30 x isin [0 1]

(7)

Iterations

Fitn

ess

10 20 30 40 50

0135

014

0145

015

GPSOMCPSO

(a)

xc

Cp y

0 02 04 06 08 1

ndash08

0

08

160

015

03

045

OriginalGPSOMCPSO

(b)



ZDT3

min f1(x) x1


x1

g(x)

1113970


g(x)1113888 1113889

g(x) 1 + 91113936

ni2 xi

n minus 1

n 30 x isin [0 1]





X(ξ(θ)) a0H0 + 1113944infin

i11ai1

H1 ξi1(θ)1113872 1113873

+ 1113944infin

i111113944

i1

i21ai1 i2

H2 ξi1(θ) ξi2


1113944

NP+1

j1bjψj(ξ)

(9)




formula


1113946 X(ξ)ψj(ξ)ω(ξ)dξ


1113936


kiN1 ωi1


xiN1113872 1113873ψj xi1

xiN1113872 1113873


(10)




μ(X(ξ(θ))) b1

σ2(X(ξ(θ))) 1113936NP+1

j2b2jlangψ2

jrang

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)


f(X) 20eminus (x+27)22( )

2π


2π

radic2

1113888 1113889 (12)



μY E[f(X)] 3520985

σY

E f(X) minus μY1113858 1113859

1113969

0507313(13)








Fitness_1

Fitn

ess_

2

0 02 04 06 08 1

02

04

06

08

1

MOPSOReal front

(a)Fi

tnes

s_2

MOPSOReal front

Fitness_10 02 04 06 08 1

0

02

04

06

08

1

(b)

Fitn

ess_

2

MOPSOReal front

Fitness_10 02 04 06 08

ndash05

0

05

1

(c)



Test functionsGD SP







P

microY

2 3 4 5 6 7 8

348

35

352

354

356

PCEMCS-100MCS-500

05 error bands

(a)

σY

PCEMCS-100MCS-500

P2 3 4 5 6 7 8

048

049

05

051

052

053

05 error bands

(b)

Error of variance

Tim

e of M

CS (s

)

Tim

e of P

CE (s

)


102

103

04

06

08

1

PCEMCS

(c)



(a) (b)


α

TP o

f upp

er su

rface


02

04

06

08

CFDEXP

(a)


0

02

04

06

08

TP o

f low

er su

rface

CFDEXP

(b)

Figure 5 Continued



ζ(ψ) ψ05(1 minus ψ) 1113944

n

i0AiKiψ

i(1 minus ψ)





CD

CL

001 002 003 004 005 006

02

04

06

08

1

12

CFDEXP

(a)

α

xc

0 1 2 3 4 5 6ndash02

0

02

04

06

08

1

SP_CFDRP_CFD

SP_EXPRP_EXP

(b)


xc

Cp

0 02 04 06 08 1

ndash2

ndash1

0

1

α = ndash2

CFDEXP

(c)

Cp

xc0 02 04 06 08 1

ndash2

ndash1

0

1

α = 2

CFDEXP

(d)




n


i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)


Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)







objective min CDp

st

CL 05



originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


optimizedDv and C









(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References
































































prime


GD

1113936

ni1 d2

i

1113969

n (4)



SP

1n minus 1

1113944

n

i1



211139741113972

(5)


ZDT1

min f1(x) x1


x1

g(x)

1113970

1113888 1113889

g(x) 1 + 91113936

ni2 xi

n minus 1

n 30 x isin [0 1]

(6)

ZDT2

min f1(x) x1


g(x)1113874 1113875

21113888 1113889

g(x) 1 + 91113936

ni2 xi( 1113857

n minus 1

n 30 x isin [0 1]

(7)

Iterations

Fitn

ess

10 20 30 40 50

0135

014

0145

015

GPSOMCPSO

(a)

xc

Cp y

0 02 04 06 08 1

ndash08

0

08

160

015

03

045

OriginalGPSOMCPSO

(b)



ZDT3

min f1(x) x1


x1

g(x)

1113970


g(x)1113888 1113889

g(x) 1 + 91113936

ni2 xi

n minus 1

n 30 x isin [0 1]





X(ξ(θ)) a0H0 + 1113944infin

i11ai1

H1 ξi1(θ)1113872 1113873

+ 1113944infin

i111113944

i1

i21ai1 i2

H2 ξi1(θ) ξi2


1113944

NP+1

j1bjψj(ξ)

(9)




formula


1113946 X(ξ)ψj(ξ)ω(ξ)dξ


1113936


kiN1 ωi1


xiN1113872 1113873ψj xi1

xiN1113872 1113873


(10)




μ(X(ξ(θ))) b1

σ2(X(ξ(θ))) 1113936NP+1

j2b2jlangψ2

jrang

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)


f(X) 20eminus (x+27)22( )

2π


2π

radic2

1113888 1113889 (12)



μY E[f(X)] 3520985

σY

E f(X) minus μY1113858 1113859

1113969

0507313(13)








Fitness_1

Fitn

ess_

2

0 02 04 06 08 1

02

04

06

08

1

MOPSOReal front

(a)Fi

tnes

s_2

MOPSOReal front

Fitness_10 02 04 06 08 1

0

02

04

06

08

1

(b)

Fitn

ess_

2

MOPSOReal front

Fitness_10 02 04 06 08

ndash05

0

05

1

(c)



Test functionsGD SP







P

microY

2 3 4 5 6 7 8

348

35

352

354

356

PCEMCS-100MCS-500

05 error bands

(a)

σY

PCEMCS-100MCS-500

P2 3 4 5 6 7 8

048

049

05

051

052

053

05 error bands

(b)

Error of variance

Tim

e of M

CS (s

)

Tim

e of P

CE (s

)


102

103

04

06

08

1

PCEMCS

(c)



(a) (b)


α

TP o

f upp

er su

rface


02

04

06

08

CFDEXP

(a)


0

02

04

06

08

TP o

f low

er su

rface

CFDEXP

(b)

Figure 5 Continued



ζ(ψ) ψ05(1 minus ψ) 1113944

n

i0AiKiψ

i(1 minus ψ)





CD

CL

001 002 003 004 005 006

02

04

06

08

1

12

CFDEXP

(a)

α

xc

0 1 2 3 4 5 6ndash02

0

02

04

06

08

1

SP_CFDRP_CFD

SP_EXPRP_EXP

(b)


xc

Cp

0 02 04 06 08 1

ndash2

ndash1

0

1

α = ndash2

CFDEXP

(c)

Cp

xc0 02 04 06 08 1

ndash2

ndash1

0

1

α = 2

CFDEXP

(d)




n


i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)


Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)







objective min CDp

st

CL 05



originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


optimizedDv and C









(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References





























































ZDT3

min f1(x) x1


x1

g(x)

1113970


g(x)1113888 1113889

g(x) 1 + 91113936

ni2 xi

n minus 1

n 30 x isin [0 1]





X(ξ(θ)) a0H0 + 1113944infin

i11ai1

H1 ξi1(θ)1113872 1113873

+ 1113944infin

i111113944

i1

i21ai1 i2

H2 ξi1(θ) ξi2


1113944

NP+1

j1bjψj(ξ)

(9)




formula


1113946 X(ξ)ψj(ξ)ω(ξ)dξ


1113936


kiN1 ωi1


xiN1113872 1113873ψj xi1

xiN1113872 1113873


(10)




μ(X(ξ(θ))) b1

σ2(X(ξ(θ))) 1113936NP+1

j2b2jlangψ2

jrang

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(11)


f(X) 20eminus (x+27)22( )

2π


2π

radic2

1113888 1113889 (12)



μY E[f(X)] 3520985

σY

E f(X) minus μY1113858 1113859

1113969

0507313(13)








Fitness_1

Fitn

ess_

2

0 02 04 06 08 1

02

04

06

08

1

MOPSOReal front

(a)Fi

tnes

s_2

MOPSOReal front

Fitness_10 02 04 06 08 1

0

02

04

06

08

1

(b)

Fitn

ess_

2

MOPSOReal front

Fitness_10 02 04 06 08

ndash05

0

05

1

(c)



Test functionsGD SP







P

microY

2 3 4 5 6 7 8

348

35

352

354

356

PCEMCS-100MCS-500

05 error bands

(a)

σY

PCEMCS-100MCS-500

P2 3 4 5 6 7 8

048

049

05

051

052

053

05 error bands

(b)

Error of variance

Tim

e of M

CS (s

)

Tim

e of P

CE (s

)


102

103

04

06

08

1

PCEMCS

(c)



(a) (b)


α

TP o

f upp

er su

rface


02

04

06

08

CFDEXP

(a)


0

02

04

06

08

TP o

f low

er su

rface

CFDEXP

(b)

Figure 5 Continued



ζ(ψ) ψ05(1 minus ψ) 1113944

n

i0AiKiψ

i(1 minus ψ)





CD

CL

001 002 003 004 005 006

02

04

06

08

1

12

CFDEXP

(a)

α

xc

0 1 2 3 4 5 6ndash02

0

02

04

06

08

1

SP_CFDRP_CFD

SP_EXPRP_EXP

(b)


xc

Cp

0 02 04 06 08 1

ndash2

ndash1

0

1

α = ndash2

CFDEXP

(c)

Cp

xc0 02 04 06 08 1

ndash2

ndash1

0

1

α = 2

CFDEXP

(d)




n


i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)


Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)







objective min CDp

st

CL 05



originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


optimizedDv and C









(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References
































































Fitness_1

Fitn

ess_

2

0 02 04 06 08 1

02

04

06

08

1

MOPSOReal front

(a)Fi

tnes

s_2

MOPSOReal front

Fitness_10 02 04 06 08 1

0

02

04

06

08

1

(b)

Fitn

ess_

2

MOPSOReal front

Fitness_10 02 04 06 08

ndash05

0

05

1

(c)



Test functionsGD SP







P

microY

2 3 4 5 6 7 8

348

35

352

354

356

PCEMCS-100MCS-500

05 error bands

(a)

σY

PCEMCS-100MCS-500

P2 3 4 5 6 7 8

048

049

05

051

052

053

05 error bands

(b)

Error of variance

Tim

e of M

CS (s

)

Tim

e of P

CE (s

)


102

103

04

06

08

1

PCEMCS

(c)



(a) (b)


α

TP o

f upp

er su

rface


02

04

06

08

CFDEXP

(a)


0

02

04

06

08

TP o

f low

er su

rface

CFDEXP

(b)

Figure 5 Continued



ζ(ψ) ψ05(1 minus ψ) 1113944

n

i0AiKiψ

i(1 minus ψ)





CD

CL

001 002 003 004 005 006

02

04

06

08

1

12

CFDEXP

(a)

α

xc

0 1 2 3 4 5 6ndash02

0

02

04

06

08

1

SP_CFDRP_CFD

SP_EXPRP_EXP

(b)


xc

Cp

0 02 04 06 08 1

ndash2

ndash1

0

1

α = ndash2

CFDEXP

(c)

Cp

xc0 02 04 06 08 1

ndash2

ndash1

0

1

α = 2

CFDEXP

(d)




n


i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)


Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)







objective min CDp

st

CL 05



originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


optimizedDv and C









(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References

































































P

microY

2 3 4 5 6 7 8

348

35

352

354

356

PCEMCS-100MCS-500

05 error bands

(a)

σY

PCEMCS-100MCS-500

P2 3 4 5 6 7 8

048

049

05

051

052

053

05 error bands

(b)

Error of variance

Tim

e of M

CS (s

)

Tim

e of P

CE (s

)


102

103

04

06

08

1

PCEMCS

(c)



(a) (b)


α

TP o

f upp

er su

rface


02

04

06

08

CFDEXP

(a)


0

02

04

06

08

TP o

f low

er su

rface

CFDEXP

(b)

Figure 5 Continued



ζ(ψ) ψ05(1 minus ψ) 1113944

n

i0AiKiψ

i(1 minus ψ)





CD

CL

001 002 003 004 005 006

02

04

06

08

1

12

CFDEXP

(a)

α

xc

0 1 2 3 4 5 6ndash02

0

02

04

06

08

1

SP_CFDRP_CFD

SP_EXPRP_EXP

(b)


xc

Cp

0 02 04 06 08 1

ndash2

ndash1

0

1

α = ndash2

CFDEXP

(c)

Cp

xc0 02 04 06 08 1

ndash2

ndash1

0

1

α = 2

CFDEXP

(d)




n


i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)


Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)







objective min CDp

st

CL 05



originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


optimizedDv and C









(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References





























































(a) (b)


α

TP o

f upp

er su

rface


02

04

06

08

CFDEXP

(a)


0

02

04

06

08

TP o

f low

er su

rface

CFDEXP

(b)

Figure 5 Continued



ζ(ψ) ψ05(1 minus ψ) 1113944

n

i0AiKiψ

i(1 minus ψ)





CD

CL

001 002 003 004 005 006

02

04

06

08

1

12

CFDEXP

(a)

α

xc

0 1 2 3 4 5 6ndash02

0

02

04

06

08

1

SP_CFDRP_CFD

SP_EXPRP_EXP

(b)


xc

Cp

0 02 04 06 08 1

ndash2

ndash1

0

1

α = ndash2

CFDEXP

(c)

Cp

xc0 02 04 06 08 1

ndash2

ndash1

0

1

α = 2

CFDEXP

(d)




n


i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)


Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)







objective min CDp

st

CL 05



originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


optimizedDv and C









(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References






























































ζ(ψ) ψ05(1 minus ψ) 1113944

n

i0AiKiψ

i(1 minus ψ)





CD

CL

001 002 003 004 005 006

02

04

06

08

1

12

CFDEXP

(a)

α

xc

0 1 2 3 4 5 6ndash02

0

02

04

06

08

1

SP_CFDRP_CFD

SP_EXPRP_EXP

(b)


xc

Cp

0 02 04 06 08 1

ndash2

ndash1

0

1

α = ndash2

CFDEXP

(c)

Cp

xc0 02 04 06 08 1

ndash2

ndash1

0

1

α = 2

CFDEXP

(d)




n


i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)


Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)







objective min CDp

st

CL 05



originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


optimizedDv and C









(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References






























































n


i(1 minus ψ)

nminus i+ ψζT

ζ(ψ) + Δζ(ψ)

(15)


Δζ(ψ) ψ05(1 minus ψ) 1113944

n

i0ΔAiKiψ

i(1 minus ψ)

nminus i (16)







objective min CDp

st

CL 05



originalDv

CoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(17)


optimizedDv and C









(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References































































(a) (b)


Dra

g co

effic

ient

400 600 800 10000014

0016

0018

002

0022

0024

0026

CDCDp

(c)







xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References





























































xc

y

00 02 04 06 08 10

ndash02

ndash01

00

01


(a)


xc

CP

00 02 04 06 08 10

ndash15

ndash10

ndash05

00

05

10

15

Seperation position


(b)



Maximumthickness


Maximumcamber


Leading edgeradius


Ma 002 014 026 038 05 062 074 086 098 11

(a)

Ma 005 02 035 05 065 08 095 11

(b)










objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References




































































objectives minμCDp

σCDp

⎧⎪⎪⎨

⎪⎪⎩

st

CL 05


μCDvminus μ

CoriginalDv

μCoriginalDv

le 01

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)



and μCoriginalDv






X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References





























































X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Original airfoil

(a)

X

Y

07 075 08 085 09 095 1 105 11

ndash01

0

01

Optimized airfoil

(b)

Figure 11 Continued

CD

CL

001 002 003 004 005

ndash05

00

05


(a)

CL

CDp

001 002 003 004 005

ndash050

000

050


(b)



xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References





























































xc

Cf

0 02 04 06 08 1

0

005

01



(c)


Ma

CDp

07 072 074 0760010

0015

0020

0025

0030


(a)

Ma

CDv

07 072 074 07600050

00055

00060

00065

00070


(b)

Figure 12 Continued





Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References
































































Ma

CD

07 072 074 0760015

0020

0025

0030

0035

0040


(c)

Ma

xc

07 072 074 076

050

100

150



(d)


xc

Cp

0 02 04 06 08 1

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(a)

Cp

Ma = 070Ma = 072

Ma = 074Ma = 076

xc0 02 04 06 08 1

ndash20

ndash10

00

10

20

(b)



Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References





























































Fitness_1

Fitn

ess_

2 (1

0ndash3)

0012 0013 0014 0015 0016

2

4

6

8 Optimization_2

(a)

xc

y

00 02 04 06 08 10ndash010

ndash005

000

005


(b)






Maximumthickness


Maximumcamber


Leading edgeradius


072 074070 076Ma

0010

0012

0014

0016

0018

CDp


(a)

Ma070 072 074 076


0005

0006

0007

0008

CDv

(b)

Figure 15 Continued






072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References

































































072 074070 076Ma


0015

0020

0025

0030

CD

(c)x

c

020

040

060

080

100

120

Ma070 072 074 076



(d)


CP

ndash15

ndash10

ndash05

00

05

10

15

Separation position


02 04 06 08 10xc


(a)

Cf

ndash004

ndash002

000

002

004


02 04 06 08 1000xc

(b)





CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References































































CL


003 004 005 006002CD

ndash05

00

05

(a)

Cp

xc00 02 04 06 08 10

ndash20

ndash10

00

10

20

Ma = 070Ma = 072

Ma = 074Ma = 076

(b)


Ma070 072 074 076

ndash0100

ndash0080

ndash0060

ndash0040

ndash0020

Cm


(a)

α000 200 400 600 800 1000

ndash016

ndash012

ndash008

ndash004

000

Cm


(b)




4 Conclusions






Data Availability




Acknowledgments


References






























































4 Conclusions






Data Availability




Acknowledgments


References






















































































































Documents

AerodynamicDesignOptimizationofTransonic Natural-Laminar