Research Article An Integrated Method for Designing...

Research ArticleAn Integrated Method for Designing Airfoils Shapes

Wang Xudong12 Wang Licun12 and Xia Hongjun1

1Chongqing Key Laboratory of Manufacturing Equipment Mechanism Design and ControlChongqing Technology and Business University Chongqing 400067 China2Research Center of System Health Maintenance Chongqing Technology and Business University Chongqing 400067 China

Correspondence should be addressed to Wang Xudong wangxudong916163com

Received 31 July 2015 Accepted 22 October 2015

Academic Editor Mustafa Tutar

Copyright copy 2015 Wang Xudong et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new method for designing wind turbine airfoils is presented in this paper As a main component in the design method airfoilprofiles are expressed in a trigonometric series form using conformal transformations and series of polynomial equations Thecharacteristics of the coefficient parameters in the trigonometric expression for airfoils profiles are first studied As a directconsequence three generic airfoil profiles are obtained from the expression To validate and show the generality of the trigonometricexpression the profiles of the NACA 64418 and S809 airfoils are expressed by the present expression Using the trigonometricexpression for airfoil profiles a so-called integrated designmethod is developed for designingwind turbine airfoils As airfoil shapesare expressed with analytical functions the airfoil surface can be kept smooth in a high degree In the optimization step drag andlift force coefficients are calculated using the XFOIL code Three new airfoils CQ-A15 CQ-A18 and CQ-A21 with a thickness of15 18 and 21 respectively are designed with the new integrated design method

1 Introduction

Design of airfoils for wind turbine blades is a very basic andimportant task for designing wind turbine rotors [1] Fromthe seventies of last century NASA began on working withairfoil design using a code developed by Eppler and Somers[2 3] The philosophy of the design method is that a lift-dragpolar was first defined according to the requirements of adesigned airfoil and then a pressure coefficient distributionalong the airfoil was deduced Based on the pressure distri-bution a potential velocity distribution was obtained Usingthe conformal mapping method and the prescribed velocitydistribution the shape of the candidate airfoil was designedCompared with other inverse methods the conformal map-ping method was used in the Eppler code and it allowed thevelocity distribution to be specified along the airfoil surfaceat different angles of attack [4ndash8]

In the past 20 years various airfoils have been designedespecially for wind turbinesThe first examples are the NRELnine airfoil families from the National Renewable EnergyLaboratory (NREL) [9 10] that were designed in 1995 for

various rotors of horizontal-axis wind turbines (HAWTs)using the EpplerAirfoil Design andAnalysis CodeGenerallythe new airfoil familieswere designed such that themaximumlift coefficient is relatively insensitive to wall roughnessTheseairfoil families have been successfully used in stall-regulatedvariable-pitch and variable-rpm wind turbines The secondexample is the DU airfoils that were designed at DUT (DelftUniversity of Technology) and tested in theDUTwind tunnel[11]TheDUairfoils are referred to asDUyy-W-xxx inwhichDU stands for Delft University yy is the year in which theairfoil was designed W denotes the wind energy applicationand the last three digits give 10 times the airfoil maximumthickness in percent of the chord The third example is theRISOslash airfoils from RISOslash National Laboratory in DenmarkThe development of these airfoils started in the mid-1990sand until now three airfoil families have been developedRISOslash-A1 RISOslash-P and RISOslash-B1 [12] Another type of windturbine airfoils is the FFA airfoils from the AeronauticalResearch Institute of Sweden [13]

In this paper a new airfoil design method is describedBased on the common characteristics of the existing airfoils

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 838674 12 pageshttpdxdoiorg1011552015838674

2 Mathematical Problems in Engineering

a general expression for airfoils is developed The airfoilshape is controlled by the coefficients in the series Theairfoil performance is calculated by using the XFOIL code[14] The accuracy of XFOIL for airfoil flow predictions wasinvestigated in [15] Since the code is based on solving theviscous integral boundary layer equation and the inviscidEuler equation it can be run fast on a computer and thereforeit is a favorable solver to be used in the design model As theairfoil equation is directly related to the lift and the drag of theairfoil it is easier to design any expected airfoils Comparedto the traditional inverse design method the new designmethod is more direct and accurate

2 Integrated Design Method forAirfoil Profiles

21 Trigonometric Expression of Airfoil Profiles From the ear-lier Joukowsky transformation investigations on airfoils (eg[16 17]) it is known that any airfoil profile can be expressedwith a conformal mapping and an analytical function of afinite series of Fourier expansions Following this idea theshape expression of an airfoil will be constructed similarly byusing a conformal mapping and a Taylor series

In general any airfoil profile can be mapped to a nearcircle by the relation

119911 = 1199111015840+

1198862

1199111015840 (1)

where 119886 = 1198884 119888 is airfoil chord 119911 is the complex variable inthe airfoil plane and 1199111015840 is the one in the near circle planeThecoordinates of 119911 are defined as

119911 = 119909 + 119894119910 (2)

Using polar coordinates (119903 120579) in the near circle plane thecoordinates of the airfoil can be expressed as

119909 = (119903 +

1198862

119903

) cos 120579

119910 = (119903 minus

1198862

119903

) sin 120579

(3)

On the other hand the coordinates of 1199111015840 in the near circleplane can also be expressed as

1199111015840= 119903 exp (120593 + 119894120579) (4)

where the real part of the exponential argument 120593 is a func-tion of 120579 as120593 = 120593(120579)The function120593(120579) is expressed in a seriesof sine and cosine functions as

120593 (120579) = 1198861(1 minus cos 120579) + 119887

1sin 120579 + 119886

2(1 minus cos 120579)2

+ 1198872sin2120579 + sdot sdot sdot + 119886

119896(1 minus cos 120579)119896 + 119887

119896sin119896120579

+ sdot sdot sdot 119896 = 1 2 3 119899

(5)

where 1198861 1198871 1198862 1198872 119886

119896 119887119896are the unknown coefficients

which are determining the airfoil shape From (5) it is noticed

minus04 minus02 0 02 04

minus02

0

02

04

xc

yc

Figure 1 First type of airfoil profiles

that 120593(0) = 0 This point corresponds to the sharp trailingedge of the airfoilThe shape of the airfoil is determined com-bining (1) and (5)

22 Characteristics of the Trigonometric Expression In orderto analyse the characteristics of an airfoil desired by using theintroduced expression three special cases will be consideredin this section These cases correspond to three types of air-foils For convenience the airfoil coordinates are normalizedby the airfoil chord

Case 1 Letting the first two coefficients be 1198861= 01 and 119887

1=

005 putting all other coefficients equal to 0 the function120593(120579)becomes

120593 (120579) = 01 times (1 minus cos 120579) + 005 times sin 120579 (6)

From (1) (4) and (6) an airfoil shape is obtained asshown in Figure 1

Case 2 Letting the third and fourth coefficients be 1198862= 005

and 1198872= 005 respectively setting all other coefficients equal

to 0 the function 120593(120579) becomes

120593 (120579) = 005 times (1 minus cos 120579)2 + 005 times sin2120579 (7)

From (1) (4) and (7) a second airfoil shape is obtainedas shown in Figure 2

Case 3 Letting the fifth and sixth coefficients be 1198863= 003




From (1) (4) and (8) the shape of this airfoil shape isobtained as shown in Figure 3

Mathematical Problems in Engineering 3

minus02 0 02 04minus04

minus02

0

02

04

xc

yc

minus04

Figure 2 Second type of airfoil profiles


minus02

0

02

04

xcminus04

yc

Figure 3 Third type of airfoil profiles

From Figures 1ndash3 it is seen that the three shapes obtainedfrom the integrated expression have the basic characteristicsof an airfoil

23 Generality of the Trigonometric Expression In order toshow the universality of the trigonometric expression twopopular airfoils the NACA 64418 and the S809 airfoils areconstructed using the present analytical expression

Combining (2) and (4)120595 and 120579 can be expressed in termsof 119909 and 119910 as

cosh120593 =

119909

2119886 cos 120579

sinh120593 =

119910

2119886 sin 120579

2sin2120579 = 119901 + radic1199012+ (

119910

119886

)

2

(9)

Table 1 Coefficients for determining the NACA 64418 and S809airfoils

Airfoils NACA 64418 S8091198861

089482 1474861198871

004960 0047331198862

minus035665 minus0556411198872

minus035445 minus0657731198863

minus002444 minus0074361198873

minus001001 minus005212

where 119901 = 1 minus (1199092119886)2minus (1199102119886)

2 If the coordinates 119909 and119910 of an existing airfoil are known the coefficients of (5) canbe determined from (9) If we know 119899 points on an airfoilsurface the airfoil shape can be determined in principle withthe first 119899 coefficients (5) When the number of coefficientsis big the computing time will be long This is not veryconvenient for optimizations In order to use a small numberof coefficients in the expression and represent airfoil shapeswith a sufficient accuracy only the first six coefficients of theequation are chosen

120593 (120579) = 1198861(1 minus cos 120579) + 119887

1sin 120579 + 119886

2(1 minus cos 120579)2

+ 1198872sin2120579 + 119886

3(1 minus cos 120579)3 + 119887

3sin3120579

(10)

Using the profile data of the NACA 64418 or the S809airfoils we choose six key data points that approximately candetermine the shape of the airfoil Putting the coordinates(119909 119910) of these six points into (9) the values of 120593 canbe determined The six coefficients [119886

1 1198871 1198862 1198872 1198863 1198873] are

obtained as shown in Table 1 In order to analyse the qualityof the airfoil derived from the reduced expression the NACA64418 and S809 airfoils are reproduced using (1) (2) (4) and(10) Figure 4 shows the reproduced and the original NACA64418 and S809 airfoils From the figure it is seen that theintegrated expression can express the two airfoils with anacceptable accuracy It is worth noting that a different choiceof the six key points results in different six coefficients and adifferent airfoil shape If a higher accuracy is required morecoefficients are needed Thus we can conclude that a generalairfoil can be represented by the introduced expression usinga relatively limited number of coefficients

3 Integrated Design Method

The integrated design method presented in this paper is anoptimization design method which includes an optimizationprocess using the shape expression to represent the profilesof the airfoil Six coefficients are chosen as design variables torepresent the shape of the airfoilThe lift and drag coefficientswhich are themain design objectives are calculated using thefast and robust XFOIL code by Drela [14]

31 Design Objective An important element during theairfoil design procedure is the criteria for a high lift and a lowdrag which can increase the energy capture and reduce the


minus05 0 05

0

02

04

xc

yc

Integration equationNACA 64418

minus02

(a)

minus05 0 05

minus02

0

02

xc

yc

Integration equationS809

(b)

Figure 4 Plots of the airfoil shapes obtained with the integrated expression (a) NACA 64418 airfoil (b) S809 airfoil

cost of energy In most cases it is desirable to obtain a highlift and drag ratio in the design 120572 rangeTherefore the designobjective in the study is the maximum ratio of lift and dragcoefficients 119888

119897119888119889

119891 = max(119888119897

119888119889

) (11)

32 Design Variables and Constraints As it is known thathigh roughness on an airfoil can cause earlier transition toturbulence keeping the airfoil shape smooth is essential inthe optimization From the previous sections it was shownthat the shape of an airfoil can be expressed analyticallyusing the trigonometric expression This also implies thatanalytical expression results in a smooth airfoil shape In ausual optimization procedure for airfoils the design variablesare chosen to be a spline that can control the shape of airfoilIn the present study the coefficients of the shape expressionare chosen to be the design variables

Since the analytical expression can express airfoil shapeswith the first six coefficients and the optimization with asmall number of design variables can run fast the first sixcoefficients are used to design airfoils It means that only thefollowing coefficients are active

119883 = [1198861 1198871 1198862 1198872 1198863 1198873] (12)

In order to design airfoils the basic structural features ofthe airfoil shape need to be satisfied The airfoil thickness-to-chord ratio is one of the most important parametersto determine the basic structure Besides the location ofthe maximum thickness is also important The location ofthe maximum thickness is always controlled to be locatedbetween 20 and 40of the airfoil chordmeasured from the

leading edge [18] Therefore the constraint of the location ofthe maximum thickness is applied as

02 le

119909

119888

le 04 (13)

4 Results and Discussion

The optimization design of airfoil profiles is achieved bysolving the function expression model using MATLAB Andin this section three new airfoils CQ-A15 CQ-A18 and CQ-A21 with a thickness of 15 18 and 21 respectively aredesigned by the integrated design method The aerodynamicperformance of the designed airfoils is calculated by theXFOIL code and compared to that of a few existing windturbine airfoils such as the RISOslash DU FFA and NACAairfoils

41 Characteristics of the New Designed Airfoils The threenew airfoils are designed to have a high lift-drag ratio 119888

119897119888119889

for an attack angle 120572 isin [2∘ 10∘] For airfoils with 015 018

and 021 thickness-chord ratios used for constructing theoutboard part of a wind turbine blade and play an importantrole for the output power Table 2 lists the characteristics ofthe three airfoils where 119905 is airfoil thickness 119905119888 is thickness-to-chord ratio and Re is Reynolds number

Figure 5 shows the shape of the CQ-A15 airfoil with athickness-chord ratio of 015 The location of the maximumthickness of this airfoil is at 025 chords from the leadingedge The airfoil has a maximum lift coefficient of 186 anda maximum lift-drag ratio of 14392 at a Reynolds numberRe = 16 times 10

6 The maximum lift coefficient is found at anattack angle of about 18∘ and the maximum lift-drag ratio islocated at an attack angle of about 65∘ Figure 6 shows the


Table 2 Geometric parameters of the designed airfoils

Designed airfoil119905119888 119909119888 at max 119905119888 Re times 10

6 max 119888119897

max(119888119897119888119889)

015 025 16 186 14392018 025 16 187 15009021 023 16 196 13010

0

02

04

yc

minus02

05 10xc

Figure 5 The new designed CQ-A15 airfoil with a thickness-chordratio of 015

shape of the CQ-A18 airfoil with thickness-chord ratio of 018The location of themaximum thickness of this airfoil is at 025chords from the leading edge The airfoil has a maximum liftcoefficient of 187 and a maximum lift-drag ratio of 15009at a Reynolds number Re = 16 times 10

6 The maximumlift coefficient is found at an attack angle of about 18∘ andthe maximum lift-drag ratio is located at an attack angle ofabout 55∘ The shape of the designed airfoil with thickness-chord ratio of 021 is shown in Figure 7 The location of themaximum thickness of this airfoil is at 023 chordsThe airfoilhas a maximum lift coefficient of 196 and a maximum lift-drag ratio of 13010 at Re = 16 times 10

6 The maximum liftcoefficient of the CQ-A21 is found at an attack angle of 18∘and the maximum lift-drag ratio appears at an attack angle of6∘

42 Aerodynamic Performance of the New Airfoils and Com-parisons to Existing Airfoils All results shown here arecarried out using the viscous-inviscid interactive XFOILcode In order to analyse the sensitivity of the new airfoils toturbulent inflow and wall roughness computations for bothfree transitional and fully turbulent flows at Re = 16 times 10

6

are carried out Figure 8 shows the lift coefficient 119888119897and the

lift-drag relation of the new CQ-A15 airfoil From the figureit is seen that the lift coefficient is not very sensitive to theinflow turbulence level but the lift-drag ratio is decreased forturbulent flow The maximum 119888

119897and 119888119897119888119889for fully turbulent

0 05 1

0

02

04

xc

minus02

yc


0 05 1

0

02

04

minus02

yc

xc


flow are estimated to be 183 and 11676 Figure 9 shows thelift coefficient 119888

119897and the lift-drag relation of the new CQ-

A18 airfoil From the figure it is seen that the lift coefficientis slightly more sensitive to inflow turbulence The changesin lift-drag ratio are very similar to the CQ-A15 airfoil Themaximum 119888

119897and 119888119897119888119889for fully turbulent flow are reduced to

182 and 12767 Figure 10 shows the lift and drag coefficients 119888119897

and 119888119889of the new airfoil CQ-A21 From the figure very similar

features are seenThemaximum 119888119897and 119888119897119888119889for turbulent flow

are reduced to 189 and 11310 compared to free transitionalflow

In order to demonstrate the performance of the designedairfoils a comparison is made between the new airfoils andsome existing wind turbine airfoils such as RISOslashDUNACAand FFA airfoils A data base of force characteristics on theexisting airfoils is presented byBertagnolio et al [19 20] using


0 5 10 15 200

05

1

15

2

Attack angle (deg)

Free transitionTurbulent flow

c l

(a)


0 5 10 15 200

50

100

150

Attack angle (deg)

c lc

d

(b)

Figure 8 Lift coefficient 119888119897(a) and lift-drag ratio (b) for the new designed airfoil CQ-A15 at Re = 16 times 10

6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 5 10 15 200

50

100

150

200

Attack angle (deg)


c lc

d

(b)


6

the XFOIL code Figure 11 shows the lift and drag coefficients119888119897and 119888

119897119888119889for the new designed CQ-A15 airfoil and the

NACA 63215 airfoil at Re = 109 times 106 From the figure it

is seen that the lift coefficients for the CQ-A15 airfoil increasemonotonously for attack angle ranging between 0∘ and 20∘but the lift coefficients of the NACA 63215 airfoil start todecrease at an attack angle of 17∘ and also the 119888

119897values are

much smaller than that of the new design airfoil The dragcoefficients are similar for both airfoils Due to the higher liftcoefficient 119888

119897of the new airfoil the lift-drag ratio 119888

119897119888119889is also

much bigger than that of the NACA 63215 airfoil

Figure 12 shows the lift and drag coefficients 119888119897and 119888119897119888119889

for the CQ-A18 airfoil and the NACA 64418 airfoil at Re =

16 times 106 From the figure similar features are seen where

the lift coefficients for both airfoils increasemonotonously forattack angle ranging between 0∘ and 20∘ but the lift coefficientof the new designed airfoil is much higher than that of theNACA 64418 airfoil The drag coefficients are also similar forboth airfoils As it is in the previous comparison the lift-drag ratio 119888

119897119888119889is also much bigger than that of the NACA

64418 airfoil Figure 13 shows the lift coefficient 119888119897and lift-

drag ratio 119888119897119888119889at Re = 16times10

6 for the new designed CQ-A18


0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


c l

(a)

c lc

d


0 5 10 15 200

50

100

150

Attack angle (deg)

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)

CQ-A15 airfoilNACA 63-215 airfoil

c l

(a)

0 005 01 015 020

05

1

15

2


c l

cd

(b)

Figure 11 Lift coefficient 119888119897(a) and lift-drag chart (b) for flows past the newCQ-A15 airfoil and past the NACA 63215 airfoil at Re = 109times10

6

airfoil and theRISOslash-A1-18 airfoil which have the same relativethickness From the figure it is seen that the new designedairfoil produces a higher lift coefficient 119888

119897for an attack angle

ranging between 0∘ and 20∘ It is worth noting that the liftcoefficient 119888

119897of the RISOslash-A1-18 airfoil decreases at an attack

angle of 12∘Similar comparisons for the 21 thickness airfoil CQ-

A21 are shown below Figure 14 shows the aerodynamicperformance of the new CQ-A21 airfoil and the RISOslash-A1-21airfoil with the same thickness-chord ratio at Re = 16 times 10

6TheRISOslash-A1-21 airfoil stalls at120572 = 12

∘ where the lift suddenly

decreases and the drag increases The lift coefficient 119888119897of the

new airfoil is seen to reach a value of about 20 When the liftcoefficient 119888

119897reaches 2 the drag coefficient 119888

119889starts to increase

quickly Figure 15 shows the aerodynamic performance of theCQ-A21 airfoil and the DU93-W-210 airfoil at Re = 10 times 10

6From the figure it is obvious that the designed airfoil attainsa much bigger 119888

119897during the whole attack angle range between

0∘ and 20∘ but the slopes of the lift coefficient 119888119897against the

angle of attack 120572 are similar The drag coefficient 119888119889for the

two airfoils is very similar at angle of attack up to stall Sincea bigger 119888

119897is obtained for the new airfoil the lift-drag ratio


c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)

CQ-A18 airfoilNACA 64418 airfoil

(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)

Figure 12 Lift coefficient 119888119897(a) and lift-drag chart (b) for flows past the new CQ-A18 airfoil and past the NACA 64418 airfoil at Re = 16times10

6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)

CQ-A18 airfoilRISOslash-A1-18 airfoil

(a)

c l

0 005 01 015 020

05

1

15

2


cd

(b)

Figure 13 Lift coefficient 119888119897(a) and lift-drag chart (b) for flows past the new CQ-A18 airfoil and past the RISOslash-A1-18 airfoil at Re = 16 times 10

6

119888119897119888119889of the new designed airfoil is also bigger than that of

the DU93-W-210 airfoil Figure 16 shows the aerodynamicperformance between theCQ-A21 airfoil and the FFA-W3-211airfoil at Re = 18times10

6 Compared to the FFA-W3-211 airfoilthe lift coefficient 119888

119897of the new designed airfoil is much bigger

whereas the drag coefficients 119888119889for both airfoils are similar It

means that the lift-drag ratio 119888119897119888119889of the designed airfoil is

much bigger than that of the FFA-W3-211 airfoil

43 Roughness Sensitivity Study for the New Airfoils Rough-ness in the region near the airfoil leading edge is formed

by accumulation of dust dirt and bugs which can lead topremature transition in the laminar boundary layer and resultin earlier separation To simulate the influence of roughnesson the performance of an airfoil the fixed-transition onthe upper and lower surfaces is usually used In the RISOslashexperiments [21 22] transition was fixed at 5 and 10 onthe upper and lower surfaces respectively

In order to test the sensitivity of the new airfoils tran-sition was fixed at the same locations 5 and 10 on theupper and lower surfaces respectively The lift and dragcoefficients of the new three airfoils were calculated using


c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

0 005 01 015 020

05

1

15

2

25


cd

(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)

CQ-A21 airfoilDU93-W-210 airfoil

(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)

Figure 15 Comparison of the lift coefficient 119888119897(a) and the lift-drag ratio 119888

119897119888119889(b) obtained for flows past the CQ-A21 airfoil and past the

DU93-W-210 airfoil at Re = 10 times 106

the XFOIL code In Figure 17 the performances of the CQ-A15 airfoil with both clean and rough walls are plotted It isseen that for the rough airfoil the lift coefficient is slightlysmaller than that of the clean airfoil but themaximum valuesare almost the same Figure 18 shows the comparison of theperformances of the CQ-A18 airfoil with clean and roughwalls From the figure it is seen that the lift coefficient ofthe rough airfoil is significantly smaller than the clean airfoilespecially at angle of attack between 5∘ and 15∘ It should benoted that the drag coefficient at fixed transition is bigger than

the free transition case Similarly Figure 19 shows the resultsfor CQ-A21 airfoil From the figure it is seen that the liftcoefficient is decreased and the drag coefficient is increasedfor the rough airfoil comparing to the clean airfoil

5 Conclusions

In this paper a new integrated design method for windturbines airfoils using an analytical expression of series andconformal transformations has been developed Using this


c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)

CQ-A21 airfoilFFA-W3-211 airfoil

(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)


119897119888119889(b) for flows past the CQ-A21 airfoil and past the FFA-W3-211

airfoil at Re = 18 times 106

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)

Clean airfoilRough airfoil

(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)

Figure 17 Lift coefficient 119888119897(a) and lift-drag chart (b) for the new CQ-A15 airfoil with clean and rough walls at Re = 16 times 10

6

method three new airfoils namedCQ-A15 CQ-A18 andCQ-A21 with a thickness-chord ratio of 015 018 and 021 respec-tively are designed The performance of the new airfoils (liftcoefficient 119888

119897and lift-drag ratio 119888

119897119888119889) is calculated using the

XFOIL code for both free transition and fully turbulent flowsat a Reynolds number of Re = 16 times 10

6 The sensitivityon wall roughness is simulated with fixed transition onairfoil wall Computations on the new airfoils with roughwallshow that the CQ-A15 airfoil is not very sensitive on wall

roughness whereas the other two airfoils are sensitive Theforce characteristics of the new airfoils are also compared tothe common wind turbine airfoils such as RISOslash DU NACAand FFA airfoils The results show that the new method isfeasible for designing wind turbines airfoils

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment

This work was supported by the National Natural ScienceFoundation of China (Grant no 51205430) Natural ScienceFoundation of Chongqing (Grant no cstc2011jjA70002)and China Postdoctoral Science Foundation (Grant no2013T60842)

References

[1] O Turhan and G Bulut ldquoOn nonlinear vibrations of a rotatingbeamrdquo Journal of Sound and Vibration vol 322 no 1-2 pp 314ndash335 2009

[2] R Eppler and D M Somers ldquoLow speed airfoil design andanalysis Advanced technology airfoil researchmdashvolume IrdquoNASA CP-2045 Part 1 1979

[3] R Eppler and D M Somers ldquoA computer program for thedesign and analysis of low-speed airfoilsrdquo NASA TM-802101980

[4] S Sarkar andH Bijl ldquoNonlinear aeroelastic behavior of an oscil-lating airfoil during stall-induced vibrationrdquo Journal of Fluidsand Structures vol 24 no 6 pp 757ndash777 2008

[5] DN Srinath and SMittal ldquoOptimal aerodynamic design of air-foils in unsteady viscous flowsrdquo Computer Methods in AppliedMechanics and Engineering vol 199 no 29ndash32 pp 1976ndash19912010


[6] A Filippone ldquoAirfoil inverse design and optimization bymeansof viscous-inviscid techniquesrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 56 no 2-3 pp 123ndash136 1995

[7] K Y Maalawi and M A Badr ldquoA practical approach for select-ing optimum wind rotorsrdquo Renewable Energy vol 28 no 5 pp803ndash822 2003

[8] C Bak and P Fuglsang ldquoModification of the NACA 632-415leading edge for better aerodynamic performancerdquo Journal ofSolar Energy EngineeringmdashTransactions of the ASME vol 124no 4 pp 327ndash334 2002

[9] J L Tangler and D M Somers ldquoStatus of the special-purposeairfoil familiesrdquo SERI TP-217-3246 1987

[10] J L Tangler and D M Somers ldquoNREL airfoil families forHAWTrsquosrdquo in Proceedings of the WINDPOWER pp 117ndash123Washington DC USA 1995

[11] W A Timmer and R P J O M Van Rooij ldquoSummary of theDelft university wind turbine dedicated airfoilsrdquo Journal of SolarEnergy Engineering vol 125 no 4 pp 488ndash496 2003

[12] P Fuglsang C Bak M Gaunaa and I Antoniou ldquoDesign andverification of the Risoslash-B1 airfoil family for wind turbinesrdquoJournal of Solar Energy Engineering vol 126 no 4 pp 1002ndash1010 2004

[13] A Bjork Coordinates and Calculations for the FFA-W1-xxxFFA-W2-xxx and FFA-w3-xxx Series of Airfoils for HorizontalAxis Wind Turbines FFA TN Stockholm Sweden 1990

[14] M Drela ldquoXFOIL an analysis and design system for lowReynolds number airfoilsrdquo in Low Reynolds Number Aero-dynamics vol 54 of Lecture Notes in Engineering pp 1ndash12Springer 1989

[15] W Z Shen and J N Soslashrensen ldquoQuasi-3D Navier-Stokes modelfor a rotating airfoilrdquo Journal of Computational Physics vol 150no 2 pp 518ndash548 1999

[16] I H Abbott and A E Von Doenhoff Theory of Wing SectionsDover Publications New York NY USA 1959

[17] VHMorcos ldquoAerodynamic performance analysis of horizontalaxis wind turbinesrdquo Renewable Energy vol 4 no 5 pp 505ndash5181994

[18] P Thokala and J R R A Martins ldquoVariable-complexity opti-mization applied to airfoil designrdquo Engineering Optimizationvol 39 no 3 pp 271ndash286 2007

[19] F Bertagnolio N Soslashrensen J Johansen and P Fuglsang ldquoWindturbine airfoil cataloguerdquo Risoslash-R 1280(EN) Risoslash NationalLaboratory Roskilde Denmark 2001

[20] F Bertagnolio N N Soslashrensen and F Rasmussen ldquoNew insightinto the flow around a wind turbine airfoil sectionrdquo Journal ofSolar Energy EngineeringmdashTransactions of the ASME vol 127no 2 pp 214ndash222 2005

[21] P Fuglsang and C Bak ldquoWind tunel tests of the Risoslash-A1-18Risoslash-A1-21 and Risoslash-A1-24 airfoilsrdquo Report Risoslash-R-1112(EN)Risoslash National Laboratory Roskilde Denmark 1999

[22] P Fuglsang andC Bak ldquoDevelopment of the RISOslashwind turbineairfoilsrdquoWind Energy vol 7 no 2 pp 145ndash162 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a general expression for airfoils is developed The airfoilshape is controlled by the coefficients in the series Theairfoil performance is calculated by using the XFOIL code[14] The accuracy of XFOIL for airfoil flow predictions wasinvestigated in [15] Since the code is based on solving theviscous integral boundary layer equation and the inviscidEuler equation it can be run fast on a computer and thereforeit is a favorable solver to be used in the design model As theairfoil equation is directly related to the lift and the drag of theairfoil it is easier to design any expected airfoils Comparedto the traditional inverse design method the new designmethod is more direct and accurate

2 Integrated Design Method forAirfoil Profiles

21 Trigonometric Expression of Airfoil Profiles From the ear-lier Joukowsky transformation investigations on airfoils (eg[16 17]) it is known that any airfoil profile can be expressedwith a conformal mapping and an analytical function of afinite series of Fourier expansions Following this idea theshape expression of an airfoil will be constructed similarly byusing a conformal mapping and a Taylor series

In general any airfoil profile can be mapped to a nearcircle by the relation

119911 = 1199111015840+

1198862

1199111015840 (1)

where 119886 = 1198884 119888 is airfoil chord 119911 is the complex variable inthe airfoil plane and 1199111015840 is the one in the near circle planeThecoordinates of 119911 are defined as

119911 = 119909 + 119894119910 (2)

Using polar coordinates (119903 120579) in the near circle plane thecoordinates of the airfoil can be expressed as

119909 = (119903 +

1198862

119903

) cos 120579

119910 = (119903 minus

1198862

119903

) sin 120579

(3)

On the other hand the coordinates of 1199111015840 in the near circleplane can also be expressed as

1199111015840= 119903 exp (120593 + 119894120579) (4)

where the real part of the exponential argument 120593 is a func-tion of 120579 as120593 = 120593(120579)The function120593(120579) is expressed in a seriesof sine and cosine functions as

120593 (120579) = 1198861(1 minus cos 120579) + 119887

1sin 120579 + 119886

2(1 minus cos 120579)2

+ 1198872sin2120579 + sdot sdot sdot + 119886

119896(1 minus cos 120579)119896 + 119887

119896sin119896120579

+ sdot sdot sdot 119896 = 1 2 3 119899

(5)

where 1198861 1198871 1198862 1198872 119886

119896 119887119896are the unknown coefficients

which are determining the airfoil shape From (5) it is noticed

minus04 minus02 0 02 04

minus02

0

02

04

xc

yc

Figure 1 First type of airfoil profiles

that 120593(0) = 0 This point corresponds to the sharp trailingedge of the airfoilThe shape of the airfoil is determined com-bining (1) and (5)

22 Characteristics of the Trigonometric Expression In orderto analyse the characteristics of an airfoil desired by using theintroduced expression three special cases will be consideredin this section These cases correspond to three types of air-foils For convenience the airfoil coordinates are normalizedby the airfoil chord

Case 1 Letting the first two coefficients be 1198861= 01 and 119887

1=

005 putting all other coefficients equal to 0 the function120593(120579)becomes

120593 (120579) = 01 times (1 minus cos 120579) + 005 times sin 120579 (6)

From (1) (4) and (6) an airfoil shape is obtained asshown in Figure 1

Case 2 Letting the third and fourth coefficients be 1198862= 005




From (1) (4) and (7) a second airfoil shape is obtainedas shown in Figure 2

Case 3 Letting the fifth and sixth coefficients be 1198863= 003




From (1) (4) and (8) the shape of this airfoil shape isobtained as shown in Figure 3



minus02

0

02

04

xc

yc

minus04



minus02

0

02

04

xcminus04

yc





cosh120593 =

119909

2119886 cos 120579

sinh120593 =

119910

2119886 sin 120579

2sin2120579 = 119901 + radic1199012+ (

119910

119886

)

2

(9)



089482 1474861198871

004960 0047331198862

minus035665 minus0556411198872

minus035445 minus0657731198863

minus002444 minus0074361198873




120593 (120579) = 1198861(1 minus cos 120579) + 119887

1sin 120579 + 119886

2(1 minus cos 120579)2

+ 1198872sin2120579 + 119886

3(1 minus cos 120579)3 + 119887

3sin3120579

(10)


1 1198871 1198862 1198872 1198863 1198873] are






minus05 0 05

0

02

04

xc

yc


minus02

(a)

minus05 0 05

minus02

0

02

xc

yc


(b)



119897119888119889

119891 = max(119888119897

119888119889

) (11)



119883 = [1198861 1198871 1198862 1198872 1198863 1198873] (12)



02 le

119909

119888

le 04 (13)




119897119888119889








6 max 119888119897

max(119888119897119888119889)

015 025 16 186 14392018 025 16 187 15009021 023 16 196 13010

0

02

04

yc

minus02

05 10xc






6




0 05 1

0

02

04

xc

minus02

yc


0 05 1

0

02

04

minus02

yc

xc












0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)


0 5 10 15 200

50

100

150

Attack angle (deg)

c lc

d

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 5 10 15 200

50

100

150

200

Attack angle (deg)


c lc

d

(b)


6





119897values are



119897119888119889is also











0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


c l

(a)

c lc

d


0 5 10 15 200

50

100

150

Attack angle (deg)

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 005 01 015 020

05

1

15

2


c l

cd

(b)


6





















c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

c l

0 005 01 015 020

05

1

15

2


cd

(b)


6












c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

0 005 01 015 020

05

1

15

2

25


cd

(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)






5 Conclusions



c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)




c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)


6










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus02

0

02

04

xc

yc

minus04



minus02

0

02

04

xcminus04

yc





cosh120593 =

119909

2119886 cos 120579

sinh120593 =

119910

2119886 sin 120579

2sin2120579 = 119901 + radic1199012+ (

119910

119886

)

2

(9)



089482 1474861198871

004960 0047331198862

minus035665 minus0556411198872

minus035445 minus0657731198863

minus002444 minus0074361198873




120593 (120579) = 1198861(1 minus cos 120579) + 119887

1sin 120579 + 119886

2(1 minus cos 120579)2

+ 1198872sin2120579 + 119886

3(1 minus cos 120579)3 + 119887

3sin3120579

(10)


1 1198871 1198862 1198872 1198863 1198873] are






minus05 0 05

0

02

04

xc

yc


minus02

(a)

minus05 0 05

minus02

0

02

xc

yc


(b)



119897119888119889

119891 = max(119888119897

119888119889

) (11)



119883 = [1198861 1198871 1198862 1198872 1198863 1198873] (12)



02 le

119909

119888

le 04 (13)




119897119888119889








6 max 119888119897

max(119888119897119888119889)

015 025 16 186 14392018 025 16 187 15009021 023 16 196 13010

0

02

04

yc

minus02

05 10xc






6




0 05 1

0

02

04

xc

minus02

yc


0 05 1

0

02

04

minus02

yc

xc












0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)


0 5 10 15 200

50

100

150

Attack angle (deg)

c lc

d

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 5 10 15 200

50

100

150

200

Attack angle (deg)


c lc

d

(b)


6





119897values are



119897119888119889is also











0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


c l

(a)

c lc

d


0 5 10 15 200

50

100

150

Attack angle (deg)

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 005 01 015 020

05

1

15

2


c l

cd

(b)


6





















c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

c l

0 005 01 015 020

05

1

15

2


cd

(b)


6












c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

0 005 01 015 020

05

1

15

2

25


cd

(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)






5 Conclusions



c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)




c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)


6










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus05 0 05

0

02

04

xc

yc


minus02

(a)

minus05 0 05

minus02

0

02

xc

yc


(b)



119897119888119889

119891 = max(119888119897

119888119889

) (11)



119883 = [1198861 1198871 1198862 1198872 1198863 1198873] (12)



02 le

119909

119888

le 04 (13)




119897119888119889








6 max 119888119897

max(119888119897119888119889)

015 025 16 186 14392018 025 16 187 15009021 023 16 196 13010

0

02

04

yc

minus02

05 10xc






6




0 05 1

0

02

04

xc

minus02

yc


0 05 1

0

02

04

minus02

yc

xc












0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)


0 5 10 15 200

50

100

150

Attack angle (deg)

c lc

d

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 5 10 15 200

50

100

150

200

Attack angle (deg)


c lc

d

(b)


6





119897values are



119897119888119889is also











0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


c l

(a)

c lc

d


0 5 10 15 200

50

100

150

Attack angle (deg)

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 005 01 015 020

05

1

15

2


c l

cd

(b)


6





















c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

c l

0 005 01 015 020

05

1

15

2


cd

(b)


6












c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

0 005 01 015 020

05

1

15

2

25


cd

(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)






5 Conclusions



c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)




c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)


6










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












6 max 119888119897

max(119888119897119888119889)

015 025 16 186 14392018 025 16 187 15009021 023 16 196 13010

0

02

04

yc

minus02

05 10xc






6




0 05 1

0

02

04

xc

minus02

yc


0 05 1

0

02

04

minus02

yc

xc












0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)


0 5 10 15 200

50

100

150

Attack angle (deg)

c lc

d

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 5 10 15 200

50

100

150

200

Attack angle (deg)


c lc

d

(b)


6





119897values are



119897119888119889is also











0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


c l

(a)

c lc

d


0 5 10 15 200

50

100

150

Attack angle (deg)

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 005 01 015 020

05

1

15

2


c l

cd

(b)


6





















c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

c l

0 005 01 015 020

05

1

15

2


cd

(b)


6












c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

0 005 01 015 020

05

1

15

2

25


cd

(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)






5 Conclusions



c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)




c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)


6










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)


0 5 10 15 200

50

100

150

Attack angle (deg)

c lc

d

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 5 10 15 200

50

100

150

200

Attack angle (deg)


c lc

d

(b)


6





119897values are



119897119888119889is also











0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


c l

(a)

c lc

d


0 5 10 15 200

50

100

150

Attack angle (deg)

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 005 01 015 020

05

1

15

2


c l

cd

(b)


6





















c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

c l

0 005 01 015 020

05

1

15

2


cd

(b)


6












c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

0 005 01 015 020

05

1

15

2

25


cd

(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)






5 Conclusions



c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)




c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)


6










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


c l

(a)

c lc

d


0 5 10 15 200

50

100

150

Attack angle (deg)

(b)


6

0 5 10 15 200

05

1

15

2

Attack angle (deg)


c l

(a)

0 005 01 015 020

05

1

15

2


c l

cd

(b)


6





















c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

c l

0 005 01 015 020

05

1

15

2


cd

(b)


6












c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

0 005 01 015 020

05

1

15

2

25


cd

(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)






5 Conclusions



c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)




c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)


6










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

c l

0 005 01 015 020

05

1

15

2


cd

(b)


6












c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

0 005 01 015 020

05

1

15

2

25


cd

(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)






5 Conclusions



c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)




c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)


6










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

0 005 01 015 020

05

1

15

2

25


cd

(b)


6

c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)






5 Conclusions



c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)




c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)


6










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)c l

cd

0 005 01 015 020

05

1

15

2

25


(b)




c l

0 5 10 15 200

05

1

15

2

Attack angle (deg)


(a)

cd

c l


0 005 01 015 020

05

1

15

2

(b)


6










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 200

05

1

15

2c l

Attack angle (deg)


(a)

cd

c l

0 005 01 015 020

05

1

15

2


(b)


6

c l

0 5 10 15 200

05

1

15

2

25

Attack angle (deg)


(a)

c l


0 005 01 015 020

05

1

15

2

25

cd

(b)


6

Acknowledgment


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article An Integrated Method for Designing...

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