Research ArticleOn the Kutta Condition in Potential Flow over Airfoil

Farzad Mohebbi and Mathieu Sellier

Department of Mechanical Engineering The University of Canterbury Private Bag Box 4800Christchurch 8140 New Zealand

Correspondence should be addressed to Farzad Mohebbi farzadmohebbiyahoocom

Received 30 October 2013 Revised 17 February 2014 Accepted 17 February 2014 Published 1 April 2014

Academic Editor Ujjwal K Saha

Copyright copy 2014 F Mohebbi and M Sellier This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper proposes a novel method to implement the Kutta condition in irrotational inviscid incompressible flow (potential flow)over an airfoil In contrast to common practice this method is not based on the panel method It is based on a finite differencescheme formulated on a boundary-fitted grid using an O-type elliptic grid generation technique The proposed algorithm usesa novel and fast procedure to implement the Kutta condition by calculating the stream function over the airfoil surface throughthe derived expression for the airfoils with both finite trailing edge angle and cusped trailing edge The results obtained show theexcellent agreement with the results from analytical and panel methods thereby confirming the accuracy and correctness of theproposed method

1 Introduction

The advent of high speed digital computers has revolution-ized the numerical treatment of fluid dynamics problemsNumerical methods nowadays have become a routine toolto investigate fluid flows over the bodies such as airfoilAmongst such fluid flows incompressible potential flows areof crucial importance in studying the low-speed aerodynam-ics problems The limitations associated with the exact (ana-lytical) solutions with complex variablesmethods (conformalmapping) motivated fluid dynamicists to develop numericaltechniques to solve incompressible potential flow problems(the Laplacersquos equation) over an airfoil Since the late 1960sthe panel methods have become the standard aerodynamictools to numerically treat such flows [1] Panel methodsare applicable to any fluid-dynamic problem governed byLaplacersquos equation In these methods the airfoil surface isdivided into piecewise straight line segments or panels andsingularities such as source doublet and vortex of unknownstrength are distributed on each panel Panel method usedfor the simulation of an incompressible potential flow pastan airfoil is concerned with the vortex panel strength andcirculation quantities and the evaluation of such quantitiesresults in the calculation of the velocity distribution over the

airfoil surface and hence the determination of the pressurecoefficients These methods have been extensively investi-gated in the aerodynamics literature [2ndash6] so these will notbe discussed any further hereThe interested reader can referto the above references for further information Howeverdealing with the panels and their attributes is numericallymuch more complex than the method proposed in this paperand of high programming effort The Kutta condition shouldbe introduced into the computational loop in order to solvethe derived systemof equations for the vortex panel strengthsIn this paper we propose a novel method to numericallysolve the incompressible potential flow over an airfoil whichis exempt from considering the quantities such as the vortexpanel strength and circulation This method takes advantageof an O-type elliptic grid generation technique to generatethe grid over the flow domain and approximate the flow fieldquantities such as stream function velocity and pressure atthe grid points The Kutta condition is implemented intothe computational loop by an exact derived expression Anexpression is derived for the finite-angle and cusped trailingedges Finally the obtained results from the proposedmethodare compared to those from the standard literature (bothanalytical and numerical) through several test cases

Hindawi Publishing CorporationJournal of AerodynamicsVolume 2014 Article ID 676912 10 pageshttpdxdoiorg1011552014676912

2 Journal of Aerodynamics

y

x

Vinfin

Vinfin

Vinfin

Vinfin120595 = constant

Figure 1 Boundary conditions at infinity and on the airfoil surface(no penetration)

2 Governing Equation for IrrotationalIncompressible Flow Laplace Equation

Consider the irrotational incompressible flow over an airfoil(Figure 1) The flow is governed by the Laplacersquos equationnabla2120595 = 0 (120595 is the stream function)The boundary conditions

are as shown in Figure 1

21 Conditions at Infinity Far away from the airfoil surface(toward infinity) in all directions the flow approaches theuniform free stream conditions If the angle of attack (AOA)is 120572 and the free stream velocity is 119881

infin then the components

of the flow velocity can be written as

119906 =120597120595

120597119910= 119881infincos120572 (1)

V = minus120597120595

120597119909= 119881infinsin120572 (2)

where 119906 and V are components of velocity vector V that isV = 119906i + Vj (i and j are the unit vectors in 119909 and 119910 directionsresp)

22 Condition on the Airfoil Surface For inviscid flow flowcannot penetrate the airfoil surface Thus the velocity vectormust be tangent to the surface This wall boundary conditioncan be expressed by

120597120595

120597119904= 0 or 120595 = constant (3)

where 119904 is tangent to the surface In the problem of the flowover an airfoil if the free stream velocity 119881

infinand the angle of

attack 120572 are known from the boundary conditions at infinity(see (1) and (2)) and the wall boundary condition (see (3))one can compute the stream function 120595 at any point of thephysical domain (flow region) Then by knowing 120595 one cancompute the velocity of all points in the physical domainSince for an incompressible flow the pressure coefficient is

a function of velocity only one can obtain the pressure at anypoint in the flow region as will be shown

23 Pressure Coefficient The pressure coefficient 119862119901

isdefined as

119862119901equiv

119901 minus 119901infin

(12) 120588infin1198812infin

= 1 minus (119881

119881infin

)

2

(4)

At standard sea level conditions

120588infin

= 123 kgm3 119901infin

= 101 times 105Nm2 (5)

3 Grid Generation

We have now presented all relations needed to obtain thepressure distribution in an incompressible irrotational invis-cid flow over an airfoil To calculate the pressure at anypoint in the flow region a grid should be generated over theregion The elliptic grid generation proposed by Thompsonet al [7] is based on solving a system of elliptic partialdifferential equations to distribute nodes in the interior of thephysical domain by mapping the irregular physical domainfrom the 119909 and 119910 physical plane (Figure 2) onto the 120585 and 120578

computational plane (Figure 3) which is a regular region Itis based on solving the Poisson equations as follows

120585119909119909

+ 120585119910119910

= 119875 (120585 120578)

120578119909119909

+ 120578119910119910

= 119876 (120585 120578)

(6)

where 120585 and 120578 are the computational coordinates correspond-ing to 119909 and 119910 in the physical coordinate respectively 119875 and119876 are grid control functionswhich control the density of gridstowards a specified coordinate line or about a specific gridpoint To find an explicit relation for 119909 and 119910 in terms ofgrid points 120585

119894(119894 isin [1119872]) and 120578

119895(119895 isin [1119873]) the following

relations may be used

120572119909120585120585

minus 2120573119909120585120578

+ 120574119909120578120578

= minus1198692(119875 (120585 120578) 119909

120585+ 119876 (120585 120578) 119909

120578)

120572119910120585120585

minus 2120573119910120585120578

+ 120574119910120578120578

= minus1198692(119875 (120585 120578) 119910

120585+ 119876 (120585 120578) 119910

120578)

(7)

where

120572 = 1199092

120578+ 1199102

120578

120573 = 119909120585119909120578+ 119910120585119910120578

120574 = 1199092

120585+ 1199102

120585

119869 = 119909120585119910120578minus 119909120578119910120585(Jacobian of transformation)

(8)

The solution of the above equations (using the finite dif-ference method to discretize the terms) gives 119909 and 119910

coordinates (in the physical domain) of coordinate (119894 119895) inthe computational domain

The O-type elliptic grid generation is employed herewhich results in a smooth and orthogonal grid over the airfoilsurfaceTheO-type elliptic grid generation technique has the

Journal of Aerodynamics 3

MM

A B

CD

E F

GH

x

y

(1 N)

(1 1) (M 1)(MN)

(MN1 + 2N2 + N3

N3

+ 5)

N=2N1 + 2N2 + N3 + 6

(MN1 + N2 + N3 + 4)

(MN1 + N2 N2

N2

+ 3) (MN1

N1

N1

+ 2)

Figure 2 The physical domain O-type scheme and discretizationof the boundaries

B

C

D

A

H G

F

E

M

M

N

(1 N)

(1 1) (M 1)

(MN)

(MN1 + 2N2 + N3 + 5)

N = 2N1 + 2N2 + N3 + 6

(MN1 + N2 + N3 + 4)

(MN1 + N2 + 3)

(MN1 + 2)

120578

120585

Figure 3 The computational domain showing the discretization ofthe physical domain boundaries

advantage that the grid around the airfoil is orthogonal Thediscretization of the physical domain and the correspondingcomputational domain are shown in Figures 2 and 3 respec-tively In the computational domain119872 and119873 = 2119873

1+21198732+

1198733+ 6 are the number of nodes in the 120585 and 120578 directions

respectively The resulting O-type grid scheme over an airfoilfor the case 119873

2= 1198731and 119873

3= 21198731minus 1 or 119873 = 6119873

1+ 5 is

shown in Figure 4The initial guess for the elliptic grid generation is per-

formed using the Transfinite Interpolation (TFI) methodSince the TFI method is an algebraic technique and does notneed much time to generate grids over the physical domainit will be an appropriate initial guess for the elliptic gridgeneration method and accelerate the convergence time forthe elliptic grid generation method Another advantage ofusing the TFI method as an initial guess is that it preventsthe grids generated by elliptic (O-type) method from folding

4 Solution Approach

Since119881infinand120572 are known the stream function120595 at any point

in the physical domain can be obtained from (1) and (2) asfollows

120595119894= 120595119895+ (119910119894minus 119910119895)119881infincos120572

120595119894= 120595119895minus (119909119894minus 119909119895)119881infinsin120572

(9)

where subscripts 119894 and 119895 refer to any two arbitrary grid pointsat the physical domain boundaries Equations (9) are appliedto vertical and horizontal boundaries of the physical domainrespectively By knowing the values of stream function 120595 onboundaries of the physical domain as well as on the airfoilsurface (from wall boundary condition) we can obtain thevalues of 120595 over the physical domain Since we deal withLaplacersquos equation it is necessary to find relationships for thetransformation of the first and second derivatives of the fieldvariable 120595 with respect to the position variables 119909 and 119910 Byusing the chain rule it can be concluded that

120597120595

120597119909=

120597120595

120597120585

120597120585

120597119909+

120597120595

120597120578

120597120578

120597119909=

120597120595

120597120585120585119909+

120597120595

120597120578120578119909

120597120595

120597119910=

120597120595

120597120585

120597120585

120597119910+

120597120595

120597120578

120597120578

120597119910=

120597120595

120597120585120585119910+

120597120595

120597120578120578119910

(10)

By interchanging 119909 and 120585 and 119910 and 120578 the followingrelationships can also be derived

120597120595

120597120585=

120597120595

120597119909

120597119909

120597120585+

120597120595

120597119910

120597119910

120597120585=

120597120595

120597119909119909120585+

120597120595

120597119910119910120585

120597120595

120597120578=

120597120595

120597119909

120597119909

120597120578+

120597120595

120597119910

120597119910

120597120578=

120597120595

120597119909119909120578+

120597120595

120597119910119910120578

(11)

By solving (11) for 120597120595120597119909 and 120597120595120597119910 we finally obtain

120597120595

120597119909=

1

119869(119910120578

120597120595

120597120585minus 119910120585

120597120595

120597120578) (12)

120597120595

120597119910=

1

119869(minus119909120578

120597120595

120597120585+ 119909120585

120597120595

120597120578) (13)

where 119869 = 119909120585119910120578minus 119909120578119910120585is Jacobian of the transformation To

transform terms in the Laplace equation the second orderderivatives are needed Therefore one has the following

In the physical domain (119909 119910)

nabla2120595 =

1205972120595

1205971199092+

1205972120595

1205971199102= 0 (14)

After transformation in the computational domain (120585 120578)

nabla2120595 =

1

1198692(120572120595120585120585

minus 2120573120595120585120578

+ 120574120595120578120578) + [(nabla

2120585) 120595120585+ (nabla2120578)120595120578]

(15)

4 Journal of Aerodynamics

minus06

minus04

minus02

0

02

04

06

0 05 1x

y

(a) Close-up view of O-type grid around the airfoil

minus015

015

minus01

minus005

0

005

01

minus01 0 01 02x

y

(b) Magnified view of grid around the leading edge

minus015

minus01

minus005

0

005

01

09 1 11x

y

(c) Magnified view of grid around the trailing edge

Figure 4 O-type grid (elliptic) around an airfoil The figure illustrates orthogonality and smoothness of the gridlines especially near airfoilsurface

where

120572 = 1199092

120578+ 1199102

120578

120573 = 119909120585119909120578+ 119910120585119910120578

120574 = 1199092

120585+ 1199102

120585

119869 = 119909120585119910120578minus 119909120578119910120585(Jacobian of transformation)

(16)

andnabla2120585 = 119875 andnabla

2120578 = 119876 are control functionswhichmay be

assumed to be zero in both the grid generation and the flowsolver sections (119875 = 119876 = 0) These assumptions lead to the

following equation to solve the above Laplacersquos equation andobtain 120595 at every grid point of the physical domain

((1199092

120578+ 1199102

120578) 120595120585120585

minus 2 (119909120585119909120578+ 119910120585119910120578) 120595120585120578

+ (1199092

120585+ 1199102

120585) 120595120578120578) = 0

(17)

To solve the above equation the finite differencemethodmaybe conveniently used

41 Kutta Condition TheKutta condition states that the flowleaves the sharp trailing edge of an airfoil smoothly [8] Toapply the Kutta condition in our calculation we need toconsider two possible configurations of the trailing edgeThe trailing edge can have a finite-angle or can be cusped(Figure 5)

Journal of Aerodynamics 5

a

Finite angle

Airfoil

V2

At point a V1 = V2 = 0

V1

(a)

Cusped

Airfoil

a

V2At point a V1 = V2 ne 0

V1

(b)

Figure 5 Different possible shapes of the trailing edge and theirrelation to the Kutta condition

1 N

1 2

2 11 1

x

y

AirfoilV1

1N minus 1

2N

VN

Figure 6 Grid notation of the trailing edge

Suppose that the velocities along the top surface andbottom surface are 119881

1and 119881

2 respectively For a finite-angle

trailing edge having two finite velocities in two differentdirections at the same point is physically impossible (Figure5(a)) and therefore the only possibility is that both velocitiesshould be zero (119881

1= 1198812= 0) For the cusped trailing edge

(Figure 5(b)) having two velocities in the same directions atpoint 119886 shows that both119881

1and119881

2can be finite However the

pressure at point 119886 is unique and Bernoulli equation statesthat [2]

119901119886+

1

21205881198812

1= 119901119886+

1

21205881198812

2(18)

or

1198811= 1198812 (19)

In order to obtain relationships for the Kutta condition interms of stream function 120595 consider the finite-angle trailingedge in the O-type grid scheme shown in Figure 6

From (1) we have

119906 = 120595119910 (20)

From the transformation relationship (see (13))

120595119910=

1

119869[minus (119909120578) (120595120585) + (119909

120585) (120595120578)] (21)

If 1198811and 119881

119873are the velocities of the grid points (1 1) and

(1119873) respectively the Kutta condition 1198811= 1198812= 0 gives

1198811= 119881119873

= 0 997904rArr 1199061= 119906119873

= 0

1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

100381610038161003816100381610038161003816100381610038161

=1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

10038161003816100381610038161003816100381610038161003816119873

= 0

minus119909120578120595120585+ 119909120585120595120578

100381610038161003816100381610038161= 0

(22)

By discretizing (22) in the computational domain we get

119909120578120595120585= 119909120585120595120578

[(11990912

minus 11990911

)] [(12059521

minus 12059511

)]

= [(11990921

minus 11990911

)] [(12059512

minus 12059511

)]

12059511

=12059521

(11990912

minus 11990911

) minus 12059512

(11990921

minus 11990911

)

11990912

minus 11990921

(23)

By considering the wall boundary condition (12059511

= 12059512) we

can simplify (23) to get

12059511

= 12059521

(24)

Since the grid points (1 1) and (1119873) are the same points inthe physical domain we have

12059511

= 1205951119873

= 12059521

(25)

This value is constant on the airfoil surface due to the wallboundary condition

The derivation of an equation for the cusped trailing edgeis more complicated Consider the cusped trailing edge andthe associated grid notation shown in Figure 7

Since for the cusped trailing edge both vectors 1198811and

119881119873are equal in the magnitude and direction from the Kutta

condition for the cusped trailing edge (1198811= 119881119873) we can write

1198811= 119881119873

997904rArr 1199061= 119906119873

[1

119869(minus119909120578120595120585+ 119909120585120595120578)]

1

= [1

119869(minus119909120578120595120585+ 119909120585120595120578)]

119873

(26)

But

119909120585

100381610038161003816100381610038161= 11990921

minus 11990911

119909120585

10038161003816100381610038161003816119873= 1199092119873

minus 1199091119873

(27)

Since 1199092119873

= 11990921

and 1199091119873

= 11990911

we have

119909120585

100381610038161003816100381610038161= 119909120585

10038161003816100381610038161003816119873 (28)

In similar approach we have

119910120585

100381610038161003816100381610038161= 119910120585

10038161003816100381610038161003816119873 (29)

6 Journal of Aerodynamics

Cusped

1 N

1 1

V1

1 N minus 1

VN

(2 1)

1 2

(2 N)

Figure 7 Cusped trailing edge and the associated grid notation

Furthermore120595120585|1= 12059521

minus12059511

and 120595120585|119873

= 1205952119873

minus1205951119873

Since12059511

= 1205951119873

and 12059521

= 1205952119873

120595120585

100381610038161003816100381610038161= 120595120585

10038161003816100381610038161003816119873 (30)

Moreover 120595120578|1= 12059512

minus 12059511

and 120595120578|119873

= 1205951119873

minus 1205951119873minus1

Since12059511

= 12059512

= 1205951119873minus1

= 1205951119873

(wall boundary condition) weobtain

120595120578

100381610038161003816100381610038161= 120595120578

10038161003816100381610038161003816119873= 0 (31)

By substituting (28) through (31) into (26) we have

1

(11990921

minus 11990911

) (11991012

minus 11991011

) minus (11990912

minus 11990911

) (11991021

minus 11991011

)

times [minus (11990912

minus 11990911

) (12059521

minus 12059511

) + 0]

=1

(11990921

minus 11990911

) (1199101119873

minus 1199101119873minus1

) minus (1199091119873

minus 1199091119873minus1

) (11991021

minus 11991011

)

times [minus (1199091119873

minus 1199091119873minus1

) (12059521

minus 12059511

) + 0]

(32)

By solving (32) for 12059511(using software Maple) we get

12059511

= 12059521

(33)

In addition 1205951119873

= 12059511

= 12059521 Equation (33) is the required

expression for the cusped trailing angleFigures 8 and 9 show the stream function 120595 for both the

finite-angle (NACA0012 airfoil with angle of attack of120572 = 40∘

and a free stream velocity of 119881infin

= 70ms) and the cusped(NACA64012with angle of attack of120572 = 40

∘ and a free streamvelocity of 119881

infin= 70ms) trailing edge respectively

42 Velocity Calculation There are three sections where thevelocity must be known

(1) the outer boundaries (four sides 119862119863119863119864 119864119865 and 119865119862

of the rectangle shown in Figure 2)(2) the airfoil surface (119860119867 in Figure 2)(3) the inside of the physical domain

0 2

minus2

minus15

minus1

minus05

0

05

1

15

2

NACA 0012 angle of attack =40

x

y

Figure 8 Stream function for a finite-angle trailing edgeThe figureshows the Kutta condition at the trailing edge

0 2x

minus2

minus2

minus1

0

1

2

y

NACA 64012 angle of attack =40

Figure 9 Stream function for a cusped trailing edge The figureshows the Kutta condition at the trailing edge

The velocity values on the outer boundaries are known fromthe conditions at infinity (using (1) and (2)) In other words119909-component of the velocity vector (119906) on all the outerboundaries is equal to 119881

infincos120572 and 119910-component of the

velocity vector (V) on all the outer boundaries is equal to119881infinsin120572 For the inside of the physical domain and the airfoil

surface we can use (12) and (13) as follows

119906119894119895

=120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895(120595120585)119894119895

+ (119909120585)119894119895(120595120578)119894119895]

V119894119895

= minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895(120595120585)119894119895

minus (119910120585)119894119895(120595120578)119894119895]

(34)

Journal of Aerodynamics 7

The central and one-sided difference schemes are used forthe inside of the physical domain and the airfoil surfacerespectively After obtaining the components of the velocityvector the total velocity can be computed by

119881119894119895

= radic1199062

119894119895+ V2119894119895 (35)

As stated before for an incompressible flow the pressurecoefficient can be expressed in terms of velocity only Thus(4) can be used to determine the pressure of any grid point inthe domain Therefore

119901119894119895

=1

2120588 (1198812

infinminus 1198812

119894119895) + 119901infin (36)

43 Kutta Condition in Terms of the Velocity Potential Theproposed method can be easily developed in terms of thevelocity potential 120601 The wall boundary condition may beexpressed in terms of either the velocity potential 120601 (120597120601120597119899 =

0) or the stream function 120595 (120597120595120597119904 = 0) where 119899 and119904 are the unit vector normal to the airfoil surface and thedistance along the body (airfoil) surface respectively Usingthe transformation relationships for mapping the physicaldomain onto the computational one we can write

120597120601

120597119899airfoil surface=

minus1

119869radic120572(120572120601120585minus 120573120601120578) = 0 (37)

where 119869120572 and120573 are defined in (16)The solution of the aboveequation for the airfoil surface using the finite differencemethod gives the value for 120601

1119895(119895 = 1 119873) From the

definition of the velocity potential

V = 120601119910 (38)

In a similar way to the derivation for Kutta condition in termsof the stream function given in (21) to (23) we get

120601119910=

1

119869[minus (119909120578) (120601120585) + (119909

120585) (120601120578)]

1198811= 119881119873

= 0 997904rArr V1= V119873

= 0

(39)

And finally

12060111

=12060121

(11990912

minus 11990911

) minus 12060112

(11990921

minus 11990911

)

11990912

minus 11990921

(40)

By including (37) and (40) into the solution loops we can findthe velocity potential over the domain The above procedurealso can be extended to the three dimension case

5 Results

51 Validation of the Results for the Pressure Distribution Theresults obtained here are compared with the results fromusing the panel methodThe results are obtained by a Fortrancompiler (PGI) and computations are run on a PC with IntelPentium Dual 173 and 1G RAM The tolerance used in theiterative loops (themesh generation and the stream function)is 10minus8

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 0012Angle of attack 9

Cp

xc

Figure 10 Comparison between the results from [2] and the resultsfrom our method for validation case 1The figure shows an excellentagreement between the results

52 Trailing Edge with Finite-Angle

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 0012 airfoil at an angle of attack of 120572 = 9∘ is

plotted The results are compared with the results from [2]The O-type grid size used in the computation is 155 times 155The computation time is 53 seconds (see Figure 10)

Validation Case 2 The pressure coefficient distribution (119862119901)

over a NACA 0024 airfoil at an angle of attack of 120572 = 0∘ is

plotted The results are compared with the results from [9]The O-type grid size used in the computation is 155 times 155The computation time is 41 seconds (see Figure 11)

Validation Case 3 The pressure coefficient distribution (119862119901)

over a NACA 4414 airfoil at an angle of attack of 120572 = 2∘

is plotted The results are compared with the results fromthe software Xfoil [10] The O-type grid size used in thecomputation is 155times155The computation time is 51 seconds(see Figure 12)

Validation Case 4 The pressure coefficient distribution (119862119901)

over a NACA 4412 airfoil at an angle of attack of 120572 = 10∘ is

plotted The results are compared with the results in [5] TheO-type grid size used in the computation is 155 times 155 Thecomputation time is 55 seconds (see Figure 13)

53 Cusped Trailing Edge

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 64012 airfoil at an angle of attack of 120572 = 6∘

8 Journal of Aerodynamics

0 02 04 06 08 1

0

05

1

minus05

minus1

Results from referenceResults from our methodAirfoil NACA 0024Angle of attack 0

Cp

xc

Figure 11 Comparison between the results from [9] and the results from our method for validation case 2 The figure shows an excellentagreement between the results

0 02 04 06 08 1

0

minus05

minus1

05

1

Results from XFoilResults from our methodAirfoil NACA 4414Angle of attack 2

Cp

xc

Figure 12 Comparison between the results from [10] and the results from our method for validation case 3 The figure shows an excellentagreement between the results

is plotted The results are compared with the results fromthe software XFLR5 [11] The O-type grid size used in thecomputation is 245times245 The computation time is 4 minutesand 15 seconds (see Figures 14 15 and 16)

Excellent agreement can be obtained by comparing theresults from ourmethod and the ones from the panel methodgiven in validation cases for both finite-angle and the cuspedtrailing edges As shown in the validation cases results the

Journal of Aerodynamics 9

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 4412Angle of attack 10

Cp

xc

Figure 13 Comparison between the results from [5] and the resultsfrom ourmethod for validation case 4The figure shows an excellentagreement between the results

minus05

minus05

0

0

05

05

1

1

15x

y

103000

102000

101000

100000

99000

98000

97000

96000

95000

94000

93000

92000

P(Pa)

Figure 14 Pressure distribution over the airfoil (NACA 64012) usedin validation case 1 (cusped trailing edge)

maximum value for 119862119901is exactly equal to 1 The pressure

coefficient at the trailing edge (TE) is equal to unity becausethe velocity is zero at this stagnation point Accordingly

119881TE = 0 119862119901= 1 minus

1198812

TE1198812infin

= 1 minus 0 = 1 (41)

For the cusped trailing edge119881TE = 0 Thus the value of 119862119901at

TE is not equal to 1 (119862119901

= 1) as shown in Figure 16

02 04 06 08 100

05 0

minus05minus10minus15minus20minus25minus30minus35

x

Cp

NACA 64012-Re = 100000-alpha = 600 inviscid

NACA 64012Thickness = 1200Max thick pos = 3740Max camber = minus000Max camber pos = 4420Number of panels = 245

Fixed speed polarReynolds = 100 000Mach = 0000NCrit = 9000Forced upper trans = 1000Forced lower trans = 1000Alpha = 600∘

C1 = 0619

Cd = 0000

Cm = 0009

Upper trans = 0000Lower trans = 0000

Figure 15 Pressure coefficient distribution over a NACA 64012airfoil at a 6∘ angle of attack obtained by XFLR5

0 02 04 06 08 1

minus1

minus2

minus3

minus4

0

1

Results from XFLR5 V610Results from our methodAirfoil NACA 64012 (using 245 points)Angle of attack 6

Cp

xc

Figure 16 Comparison between the results from the softwareXFLR5 and the results fromourmethod for validation case 1 (cuspedtrailing edge) The figure shows an excellent agreement between theresults

6 Conclusion

This paper presents a novel method to implement the Kuttacondition in the numerical solution of two-dimensionalincompressible potential flow over an airfoil The proposedmethod is based on solving the Laplacersquos equation for thestream function at each grid point generated by the ellipticgrid generation technique (O-type) Therefore it is exemptfrom considering the panels and the quantities such as thevortex panel strength and circulation used in the panel

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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Solid State PhysicsJournal of

 Computational  Methods in Physics

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Soft MatterJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

2 Journal of Aerodynamics

y

x

Vinfin

Vinfin

Vinfin

Vinfin120595 = constant

Figure 1 Boundary conditions at infinity and on the airfoil surface(no penetration)

2 Governing Equation for IrrotationalIncompressible Flow Laplace Equation

Consider the irrotational incompressible flow over an airfoil(Figure 1) The flow is governed by the Laplacersquos equationnabla2120595 = 0 (120595 is the stream function)The boundary conditions

are as shown in Figure 1

21 Conditions at Infinity Far away from the airfoil surface(toward infinity) in all directions the flow approaches theuniform free stream conditions If the angle of attack (AOA)is 120572 and the free stream velocity is 119881

infin then the components

of the flow velocity can be written as

119906 =120597120595

120597119910= 119881infincos120572 (1)

V = minus120597120595

120597119909= 119881infinsin120572 (2)

where 119906 and V are components of velocity vector V that isV = 119906i + Vj (i and j are the unit vectors in 119909 and 119910 directionsresp)

22 Condition on the Airfoil Surface For inviscid flow flowcannot penetrate the airfoil surface Thus the velocity vectormust be tangent to the surface This wall boundary conditioncan be expressed by

120597120595

120597119904= 0 or 120595 = constant (3)

where 119904 is tangent to the surface In the problem of the flowover an airfoil if the free stream velocity 119881

infinand the angle of

attack 120572 are known from the boundary conditions at infinity(see (1) and (2)) and the wall boundary condition (see (3))one can compute the stream function 120595 at any point of thephysical domain (flow region) Then by knowing 120595 one cancompute the velocity of all points in the physical domainSince for an incompressible flow the pressure coefficient is

a function of velocity only one can obtain the pressure at anypoint in the flow region as will be shown

23 Pressure Coefficient The pressure coefficient 119862119901

isdefined as

119862119901equiv

119901 minus 119901infin

(12) 120588infin1198812infin

= 1 minus (119881

119881infin

)

2

(4)

At standard sea level conditions

120588infin

= 123 kgm3 119901infin

= 101 times 105Nm2 (5)

3 Grid Generation

We have now presented all relations needed to obtain thepressure distribution in an incompressible irrotational invis-cid flow over an airfoil To calculate the pressure at anypoint in the flow region a grid should be generated over theregion The elliptic grid generation proposed by Thompsonet al [7] is based on solving a system of elliptic partialdifferential equations to distribute nodes in the interior of thephysical domain by mapping the irregular physical domainfrom the 119909 and 119910 physical plane (Figure 2) onto the 120585 and 120578

computational plane (Figure 3) which is a regular region Itis based on solving the Poisson equations as follows

120585119909119909

+ 120585119910119910

= 119875 (120585 120578)

120578119909119909

+ 120578119910119910

= 119876 (120585 120578)

(6)

where 120585 and 120578 are the computational coordinates correspond-ing to 119909 and 119910 in the physical coordinate respectively 119875 and119876 are grid control functionswhich control the density of gridstowards a specified coordinate line or about a specific gridpoint To find an explicit relation for 119909 and 119910 in terms ofgrid points 120585

119894(119894 isin [1119872]) and 120578

119895(119895 isin [1119873]) the following

relations may be used

120572119909120585120585

minus 2120573119909120585120578

+ 120574119909120578120578

= minus1198692(119875 (120585 120578) 119909

120585+ 119876 (120585 120578) 119909

120578)

120572119910120585120585

minus 2120573119910120585120578

+ 120574119910120578120578

= minus1198692(119875 (120585 120578) 119910

120585+ 119876 (120585 120578) 119910

120578)

(7)

where

120572 = 1199092

120578+ 1199102

120578

120573 = 119909120585119909120578+ 119910120585119910120578

120574 = 1199092

120585+ 1199102

120585

119869 = 119909120585119910120578minus 119909120578119910120585(Jacobian of transformation)

(8)

The solution of the above equations (using the finite dif-ference method to discretize the terms) gives 119909 and 119910

coordinates (in the physical domain) of coordinate (119894 119895) inthe computational domain

The O-type elliptic grid generation is employed herewhich results in a smooth and orthogonal grid over the airfoilsurfaceTheO-type elliptic grid generation technique has the

Journal of Aerodynamics 3

MM

A B

CD

E F

GH

x

y

(1 N)

(1 1) (M 1)(MN)

(MN1 + 2N2 + N3

N3

+ 5)

N=2N1 + 2N2 + N3 + 6

(MN1 + N2 + N3 + 4)

(MN1 + N2 N2

N2

+ 3) (MN1

N1

N1

+ 2)

Figure 2 The physical domain O-type scheme and discretizationof the boundaries

B

C

D

A

H G

F

E

M

M

N

(1 N)

(1 1) (M 1)

(MN)

(MN1 + 2N2 + N3 + 5)

N = 2N1 + 2N2 + N3 + 6

(MN1 + N2 + N3 + 4)

(MN1 + N2 + 3)

(MN1 + 2)

120578

120585

Figure 3 The computational domain showing the discretization ofthe physical domain boundaries

advantage that the grid around the airfoil is orthogonal Thediscretization of the physical domain and the correspondingcomputational domain are shown in Figures 2 and 3 respec-tively In the computational domain119872 and119873 = 2119873

1+21198732+

1198733+ 6 are the number of nodes in the 120585 and 120578 directions

respectively The resulting O-type grid scheme over an airfoilfor the case 119873

2= 1198731and 119873

3= 21198731minus 1 or 119873 = 6119873

1+ 5 is

shown in Figure 4The initial guess for the elliptic grid generation is per-

formed using the Transfinite Interpolation (TFI) methodSince the TFI method is an algebraic technique and does notneed much time to generate grids over the physical domainit will be an appropriate initial guess for the elliptic gridgeneration method and accelerate the convergence time forthe elliptic grid generation method Another advantage ofusing the TFI method as an initial guess is that it preventsthe grids generated by elliptic (O-type) method from folding

4 Solution Approach

Since119881infinand120572 are known the stream function120595 at any point

in the physical domain can be obtained from (1) and (2) asfollows

120595119894= 120595119895+ (119910119894minus 119910119895)119881infincos120572

120595119894= 120595119895minus (119909119894minus 119909119895)119881infinsin120572

(9)

where subscripts 119894 and 119895 refer to any two arbitrary grid pointsat the physical domain boundaries Equations (9) are appliedto vertical and horizontal boundaries of the physical domainrespectively By knowing the values of stream function 120595 onboundaries of the physical domain as well as on the airfoilsurface (from wall boundary condition) we can obtain thevalues of 120595 over the physical domain Since we deal withLaplacersquos equation it is necessary to find relationships for thetransformation of the first and second derivatives of the fieldvariable 120595 with respect to the position variables 119909 and 119910 Byusing the chain rule it can be concluded that

120597120595

120597119909=

120597120595

120597120585

120597120585

120597119909+

120597120595

120597120578

120597120578

120597119909=

120597120595

120597120585120585119909+

120597120595

120597120578120578119909

120597120595

120597119910=

120597120595

120597120585

120597120585

120597119910+

120597120595

120597120578

120597120578

120597119910=

120597120595

120597120585120585119910+

120597120595

120597120578120578119910

(10)

By interchanging 119909 and 120585 and 119910 and 120578 the followingrelationships can also be derived

120597120595

120597120585=

120597120595

120597119909

120597119909

120597120585+

120597120595

120597119910

120597119910

120597120585=

120597120595

120597119909119909120585+

120597120595

120597119910119910120585

120597120595

120597120578=

120597120595

120597119909

120597119909

120597120578+

120597120595

120597119910

120597119910

120597120578=

120597120595

120597119909119909120578+

120597120595

120597119910119910120578

(11)

By solving (11) for 120597120595120597119909 and 120597120595120597119910 we finally obtain

120597120595

120597119909=

1

119869(119910120578

120597120595

120597120585minus 119910120585

120597120595

120597120578) (12)

120597120595

120597119910=

1

119869(minus119909120578

120597120595

120597120585+ 119909120585

120597120595

120597120578) (13)

where 119869 = 119909120585119910120578minus 119909120578119910120585is Jacobian of the transformation To

transform terms in the Laplace equation the second orderderivatives are needed Therefore one has the following

In the physical domain (119909 119910)

nabla2120595 =

1205972120595

1205971199092+

1205972120595

1205971199102= 0 (14)

After transformation in the computational domain (120585 120578)

nabla2120595 =

1

1198692(120572120595120585120585

minus 2120573120595120585120578

+ 120574120595120578120578) + [(nabla

2120585) 120595120585+ (nabla2120578)120595120578]

(15)

4 Journal of Aerodynamics

minus06

minus04

minus02

0

02

04

06

0 05 1x

y

(a) Close-up view of O-type grid around the airfoil

minus015

015

minus01

minus005

0

005

01

minus01 0 01 02x

y

(b) Magnified view of grid around the leading edge

minus015

minus01

minus005

0

005

01

09 1 11x

y

(c) Magnified view of grid around the trailing edge

Figure 4 O-type grid (elliptic) around an airfoil The figure illustrates orthogonality and smoothness of the gridlines especially near airfoilsurface

where

120572 = 1199092

120578+ 1199102

120578

120573 = 119909120585119909120578+ 119910120585119910120578

120574 = 1199092

120585+ 1199102

120585

119869 = 119909120585119910120578minus 119909120578119910120585(Jacobian of transformation)

(16)

andnabla2120585 = 119875 andnabla

2120578 = 119876 are control functionswhichmay be

assumed to be zero in both the grid generation and the flowsolver sections (119875 = 119876 = 0) These assumptions lead to the

following equation to solve the above Laplacersquos equation andobtain 120595 at every grid point of the physical domain

((1199092

120578+ 1199102

120578) 120595120585120585

minus 2 (119909120585119909120578+ 119910120585119910120578) 120595120585120578

+ (1199092

120585+ 1199102

120585) 120595120578120578) = 0

(17)

To solve the above equation the finite differencemethodmaybe conveniently used

41 Kutta Condition TheKutta condition states that the flowleaves the sharp trailing edge of an airfoil smoothly [8] Toapply the Kutta condition in our calculation we need toconsider two possible configurations of the trailing edgeThe trailing edge can have a finite-angle or can be cusped(Figure 5)

Journal of Aerodynamics 5

a

Finite angle

Airfoil

V2

At point a V1 = V2 = 0

V1

(a)

Cusped

Airfoil

a

V2At point a V1 = V2 ne 0

V1

(b)

Figure 5 Different possible shapes of the trailing edge and theirrelation to the Kutta condition

1 N

1 2

2 11 1

x

y

AirfoilV1

1N minus 1

2N

VN

Figure 6 Grid notation of the trailing edge

Suppose that the velocities along the top surface andbottom surface are 119881

1and 119881

2 respectively For a finite-angle

trailing edge having two finite velocities in two differentdirections at the same point is physically impossible (Figure5(a)) and therefore the only possibility is that both velocitiesshould be zero (119881

1= 1198812= 0) For the cusped trailing edge

(Figure 5(b)) having two velocities in the same directions atpoint 119886 shows that both119881

1and119881

2can be finite However the

pressure at point 119886 is unique and Bernoulli equation statesthat [2]

119901119886+

1

21205881198812

1= 119901119886+

1

21205881198812

2(18)

or

1198811= 1198812 (19)

In order to obtain relationships for the Kutta condition interms of stream function 120595 consider the finite-angle trailingedge in the O-type grid scheme shown in Figure 6

From (1) we have

119906 = 120595119910 (20)

From the transformation relationship (see (13))

120595119910=

1

119869[minus (119909120578) (120595120585) + (119909

120585) (120595120578)] (21)

If 1198811and 119881

119873are the velocities of the grid points (1 1) and

(1119873) respectively the Kutta condition 1198811= 1198812= 0 gives

1198811= 119881119873

= 0 997904rArr 1199061= 119906119873

= 0

1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

100381610038161003816100381610038161003816100381610038161

=1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

10038161003816100381610038161003816100381610038161003816119873

= 0

minus119909120578120595120585+ 119909120585120595120578

100381610038161003816100381610038161= 0

(22)

By discretizing (22) in the computational domain we get

119909120578120595120585= 119909120585120595120578

[(11990912

minus 11990911

)] [(12059521

minus 12059511

)]

= [(11990921

minus 11990911

)] [(12059512

minus 12059511

)]

12059511

=12059521

(11990912

minus 11990911

) minus 12059512

(11990921

minus 11990911

)

11990912

minus 11990921

(23)

By considering the wall boundary condition (12059511

= 12059512) we

can simplify (23) to get

12059511

= 12059521

(24)

Since the grid points (1 1) and (1119873) are the same points inthe physical domain we have

12059511

= 1205951119873

= 12059521

(25)

This value is constant on the airfoil surface due to the wallboundary condition

The derivation of an equation for the cusped trailing edgeis more complicated Consider the cusped trailing edge andthe associated grid notation shown in Figure 7

Since for the cusped trailing edge both vectors 1198811and

119881119873are equal in the magnitude and direction from the Kutta

condition for the cusped trailing edge (1198811= 119881119873) we can write

1198811= 119881119873

997904rArr 1199061= 119906119873

[1

119869(minus119909120578120595120585+ 119909120585120595120578)]

1

= [1

119869(minus119909120578120595120585+ 119909120585120595120578)]

119873

(26)

But

119909120585

100381610038161003816100381610038161= 11990921

minus 11990911

119909120585

10038161003816100381610038161003816119873= 1199092119873

minus 1199091119873

(27)

Since 1199092119873

= 11990921

and 1199091119873

= 11990911

we have

119909120585

100381610038161003816100381610038161= 119909120585

10038161003816100381610038161003816119873 (28)

In similar approach we have

119910120585

100381610038161003816100381610038161= 119910120585

10038161003816100381610038161003816119873 (29)

6 Journal of Aerodynamics

Cusped

1 N

1 1

V1

1 N minus 1

VN

(2 1)

1 2

(2 N)

Figure 7 Cusped trailing edge and the associated grid notation

Furthermore120595120585|1= 12059521

minus12059511

and 120595120585|119873

= 1205952119873

minus1205951119873

Since12059511

= 1205951119873

and 12059521

= 1205952119873

120595120585

100381610038161003816100381610038161= 120595120585

10038161003816100381610038161003816119873 (30)

Moreover 120595120578|1= 12059512

minus 12059511

and 120595120578|119873

= 1205951119873

minus 1205951119873minus1

Since12059511

= 12059512

= 1205951119873minus1

= 1205951119873

(wall boundary condition) weobtain

120595120578

100381610038161003816100381610038161= 120595120578

10038161003816100381610038161003816119873= 0 (31)

By substituting (28) through (31) into (26) we have

1

(11990921

minus 11990911

) (11991012

minus 11991011

) minus (11990912

minus 11990911

) (11991021

minus 11991011

)

times [minus (11990912

minus 11990911

) (12059521

minus 12059511

) + 0]

=1

(11990921

minus 11990911

) (1199101119873

minus 1199101119873minus1

) minus (1199091119873

minus 1199091119873minus1

) (11991021

minus 11991011

)

times [minus (1199091119873

minus 1199091119873minus1

) (12059521

minus 12059511

) + 0]

(32)

By solving (32) for 12059511(using software Maple) we get

12059511

= 12059521

(33)

In addition 1205951119873

= 12059511

= 12059521 Equation (33) is the required

expression for the cusped trailing angleFigures 8 and 9 show the stream function 120595 for both the

finite-angle (NACA0012 airfoil with angle of attack of120572 = 40∘

and a free stream velocity of 119881infin

= 70ms) and the cusped(NACA64012with angle of attack of120572 = 40

∘ and a free streamvelocity of 119881

infin= 70ms) trailing edge respectively

42 Velocity Calculation There are three sections where thevelocity must be known

(1) the outer boundaries (four sides 119862119863119863119864 119864119865 and 119865119862

of the rectangle shown in Figure 2)(2) the airfoil surface (119860119867 in Figure 2)(3) the inside of the physical domain

0 2

minus2

minus15

minus1

minus05

0

05

1

15

2

NACA 0012 angle of attack =40

x

y

Figure 8 Stream function for a finite-angle trailing edgeThe figureshows the Kutta condition at the trailing edge

0 2x

minus2

minus2

minus1

0

1

2

y

NACA 64012 angle of attack =40

Figure 9 Stream function for a cusped trailing edge The figureshows the Kutta condition at the trailing edge

The velocity values on the outer boundaries are known fromthe conditions at infinity (using (1) and (2)) In other words119909-component of the velocity vector (119906) on all the outerboundaries is equal to 119881

infincos120572 and 119910-component of the

velocity vector (V) on all the outer boundaries is equal to119881infinsin120572 For the inside of the physical domain and the airfoil

surface we can use (12) and (13) as follows

119906119894119895

=120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895(120595120585)119894119895

+ (119909120585)119894119895(120595120578)119894119895]

V119894119895

= minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895(120595120585)119894119895

minus (119910120585)119894119895(120595120578)119894119895]

(34)

Journal of Aerodynamics 7

The central and one-sided difference schemes are used forthe inside of the physical domain and the airfoil surfacerespectively After obtaining the components of the velocityvector the total velocity can be computed by

119881119894119895

= radic1199062

119894119895+ V2119894119895 (35)

As stated before for an incompressible flow the pressurecoefficient can be expressed in terms of velocity only Thus(4) can be used to determine the pressure of any grid point inthe domain Therefore

119901119894119895

=1

2120588 (1198812

infinminus 1198812

119894119895) + 119901infin (36)

43 Kutta Condition in Terms of the Velocity Potential Theproposed method can be easily developed in terms of thevelocity potential 120601 The wall boundary condition may beexpressed in terms of either the velocity potential 120601 (120597120601120597119899 =

0) or the stream function 120595 (120597120595120597119904 = 0) where 119899 and119904 are the unit vector normal to the airfoil surface and thedistance along the body (airfoil) surface respectively Usingthe transformation relationships for mapping the physicaldomain onto the computational one we can write

120597120601

120597119899airfoil surface=

minus1

119869radic120572(120572120601120585minus 120573120601120578) = 0 (37)

where 119869120572 and120573 are defined in (16)The solution of the aboveequation for the airfoil surface using the finite differencemethod gives the value for 120601

1119895(119895 = 1 119873) From the

definition of the velocity potential

V = 120601119910 (38)

In a similar way to the derivation for Kutta condition in termsof the stream function given in (21) to (23) we get

120601119910=

1

119869[minus (119909120578) (120601120585) + (119909

120585) (120601120578)]

1198811= 119881119873

= 0 997904rArr V1= V119873

= 0

(39)

And finally

12060111

=12060121

(11990912

minus 11990911

) minus 12060112

(11990921

minus 11990911

)

11990912

minus 11990921

(40)

By including (37) and (40) into the solution loops we can findthe velocity potential over the domain The above procedurealso can be extended to the three dimension case

5 Results

51 Validation of the Results for the Pressure Distribution Theresults obtained here are compared with the results fromusing the panel methodThe results are obtained by a Fortrancompiler (PGI) and computations are run on a PC with IntelPentium Dual 173 and 1G RAM The tolerance used in theiterative loops (themesh generation and the stream function)is 10minus8

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 0012Angle of attack 9

Cp

xc

Figure 10 Comparison between the results from [2] and the resultsfrom our method for validation case 1The figure shows an excellentagreement between the results

52 Trailing Edge with Finite-Angle

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 0012 airfoil at an angle of attack of 120572 = 9∘ is

plotted The results are compared with the results from [2]The O-type grid size used in the computation is 155 times 155The computation time is 53 seconds (see Figure 10)

Validation Case 2 The pressure coefficient distribution (119862119901)

over a NACA 0024 airfoil at an angle of attack of 120572 = 0∘ is

plotted The results are compared with the results from [9]The O-type grid size used in the computation is 155 times 155The computation time is 41 seconds (see Figure 11)

Validation Case 3 The pressure coefficient distribution (119862119901)

over a NACA 4414 airfoil at an angle of attack of 120572 = 2∘

is plotted The results are compared with the results fromthe software Xfoil [10] The O-type grid size used in thecomputation is 155times155The computation time is 51 seconds(see Figure 12)

Validation Case 4 The pressure coefficient distribution (119862119901)

over a NACA 4412 airfoil at an angle of attack of 120572 = 10∘ is

plotted The results are compared with the results in [5] TheO-type grid size used in the computation is 155 times 155 Thecomputation time is 55 seconds (see Figure 13)

53 Cusped Trailing Edge

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 64012 airfoil at an angle of attack of 120572 = 6∘

8 Journal of Aerodynamics

0 02 04 06 08 1

0

05

1

minus05

minus1

Results from referenceResults from our methodAirfoil NACA 0024Angle of attack 0

Cp

xc

Figure 11 Comparison between the results from [9] and the results from our method for validation case 2 The figure shows an excellentagreement between the results

0 02 04 06 08 1

0

minus05

minus1

05

1

Results from XFoilResults from our methodAirfoil NACA 4414Angle of attack 2

Cp

xc

Figure 12 Comparison between the results from [10] and the results from our method for validation case 3 The figure shows an excellentagreement between the results

is plotted The results are compared with the results fromthe software XFLR5 [11] The O-type grid size used in thecomputation is 245times245 The computation time is 4 minutesand 15 seconds (see Figures 14 15 and 16)

Excellent agreement can be obtained by comparing theresults from ourmethod and the ones from the panel methodgiven in validation cases for both finite-angle and the cuspedtrailing edges As shown in the validation cases results the

Journal of Aerodynamics 9

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 4412Angle of attack 10

Cp

xc

Figure 13 Comparison between the results from [5] and the resultsfrom ourmethod for validation case 4The figure shows an excellentagreement between the results

minus05

minus05

0

0

05

05

1

1

15x

y

103000

102000

101000

100000

99000

98000

97000

96000

95000

94000

93000

92000

P(Pa)

Figure 14 Pressure distribution over the airfoil (NACA 64012) usedin validation case 1 (cusped trailing edge)

maximum value for 119862119901is exactly equal to 1 The pressure

coefficient at the trailing edge (TE) is equal to unity becausethe velocity is zero at this stagnation point Accordingly

119881TE = 0 119862119901= 1 minus

1198812

TE1198812infin

= 1 minus 0 = 1 (41)

For the cusped trailing edge119881TE = 0 Thus the value of 119862119901at

TE is not equal to 1 (119862119901

= 1) as shown in Figure 16

02 04 06 08 100

05 0

minus05minus10minus15minus20minus25minus30minus35

x

Cp

NACA 64012-Re = 100000-alpha = 600 inviscid

NACA 64012Thickness = 1200Max thick pos = 3740Max camber = minus000Max camber pos = 4420Number of panels = 245

Fixed speed polarReynolds = 100 000Mach = 0000NCrit = 9000Forced upper trans = 1000Forced lower trans = 1000Alpha = 600∘

C1 = 0619

Cd = 0000

Cm = 0009

Upper trans = 0000Lower trans = 0000

Figure 15 Pressure coefficient distribution over a NACA 64012airfoil at a 6∘ angle of attack obtained by XFLR5

0 02 04 06 08 1

minus1

minus2

minus3

minus4

0

1

Results from XFLR5 V610Results from our methodAirfoil NACA 64012 (using 245 points)Angle of attack 6

Cp

xc

Figure 16 Comparison between the results from the softwareXFLR5 and the results fromourmethod for validation case 1 (cuspedtrailing edge) The figure shows an excellent agreement between theresults

6 Conclusion

This paper presents a novel method to implement the Kuttacondition in the numerical solution of two-dimensionalincompressible potential flow over an airfoil The proposedmethod is based on solving the Laplacersquos equation for thestream function at each grid point generated by the ellipticgrid generation technique (O-type) Therefore it is exemptfrom considering the panels and the quantities such as thevortex panel strength and circulation used in the panel

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in Condensed Matter Physics

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AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Aerodynamics 3

MM

A B

CD

E F

GH

x

y

(1 N)

(1 1) (M 1)(MN)

(MN1 + 2N2 + N3

N3

+ 5)

N=2N1 + 2N2 + N3 + 6

(MN1 + N2 + N3 + 4)

(MN1 + N2 N2

N2

+ 3) (MN1

N1

N1

+ 2)

Figure 2 The physical domain O-type scheme and discretizationof the boundaries

B

C

D

A

H G

F

E

M

M

N

(1 N)

(1 1) (M 1)

(MN)

(MN1 + 2N2 + N3 + 5)

N = 2N1 + 2N2 + N3 + 6

(MN1 + N2 + N3 + 4)

(MN1 + N2 + 3)

(MN1 + 2)

120578

120585

Figure 3 The computational domain showing the discretization ofthe physical domain boundaries

advantage that the grid around the airfoil is orthogonal Thediscretization of the physical domain and the correspondingcomputational domain are shown in Figures 2 and 3 respec-tively In the computational domain119872 and119873 = 2119873

1+21198732+

1198733+ 6 are the number of nodes in the 120585 and 120578 directions

respectively The resulting O-type grid scheme over an airfoilfor the case 119873

2= 1198731and 119873

3= 21198731minus 1 or 119873 = 6119873

1+ 5 is

shown in Figure 4The initial guess for the elliptic grid generation is per-

formed using the Transfinite Interpolation (TFI) methodSince the TFI method is an algebraic technique and does notneed much time to generate grids over the physical domainit will be an appropriate initial guess for the elliptic gridgeneration method and accelerate the convergence time forthe elliptic grid generation method Another advantage ofusing the TFI method as an initial guess is that it preventsthe grids generated by elliptic (O-type) method from folding

4 Solution Approach

Since119881infinand120572 are known the stream function120595 at any point

in the physical domain can be obtained from (1) and (2) asfollows

120595119894= 120595119895+ (119910119894minus 119910119895)119881infincos120572

120595119894= 120595119895minus (119909119894minus 119909119895)119881infinsin120572

(9)

where subscripts 119894 and 119895 refer to any two arbitrary grid pointsat the physical domain boundaries Equations (9) are appliedto vertical and horizontal boundaries of the physical domainrespectively By knowing the values of stream function 120595 onboundaries of the physical domain as well as on the airfoilsurface (from wall boundary condition) we can obtain thevalues of 120595 over the physical domain Since we deal withLaplacersquos equation it is necessary to find relationships for thetransformation of the first and second derivatives of the fieldvariable 120595 with respect to the position variables 119909 and 119910 Byusing the chain rule it can be concluded that

120597120595

120597119909=

120597120595

120597120585

120597120585

120597119909+

120597120595

120597120578

120597120578

120597119909=

120597120595

120597120585120585119909+

120597120595

120597120578120578119909

120597120595

120597119910=

120597120595

120597120585

120597120585

120597119910+

120597120595

120597120578

120597120578

120597119910=

120597120595

120597120585120585119910+

120597120595

120597120578120578119910

(10)

By interchanging 119909 and 120585 and 119910 and 120578 the followingrelationships can also be derived

120597120595

120597120585=

120597120595

120597119909

120597119909

120597120585+

120597120595

120597119910

120597119910

120597120585=

120597120595

120597119909119909120585+

120597120595

120597119910119910120585

120597120595

120597120578=

120597120595

120597119909

120597119909

120597120578+

120597120595

120597119910

120597119910

120597120578=

120597120595

120597119909119909120578+

120597120595

120597119910119910120578

(11)

By solving (11) for 120597120595120597119909 and 120597120595120597119910 we finally obtain

120597120595

120597119909=

1

119869(119910120578

120597120595

120597120585minus 119910120585

120597120595

120597120578) (12)

120597120595

120597119910=

1

119869(minus119909120578

120597120595

120597120585+ 119909120585

120597120595

120597120578) (13)

where 119869 = 119909120585119910120578minus 119909120578119910120585is Jacobian of the transformation To

transform terms in the Laplace equation the second orderderivatives are needed Therefore one has the following

In the physical domain (119909 119910)

nabla2120595 =

1205972120595

1205971199092+

1205972120595

1205971199102= 0 (14)

After transformation in the computational domain (120585 120578)

nabla2120595 =

1

1198692(120572120595120585120585

minus 2120573120595120585120578

+ 120574120595120578120578) + [(nabla

2120585) 120595120585+ (nabla2120578)120595120578]

(15)

4 Journal of Aerodynamics

minus06

minus04

minus02

0

02

04

06

0 05 1x

y

(a) Close-up view of O-type grid around the airfoil

minus015

015

minus01

minus005

0

005

01

minus01 0 01 02x

y

(b) Magnified view of grid around the leading edge

minus015

minus01

minus005

0

005

01

09 1 11x

y

(c) Magnified view of grid around the trailing edge

Figure 4 O-type grid (elliptic) around an airfoil The figure illustrates orthogonality and smoothness of the gridlines especially near airfoilsurface

where

120572 = 1199092

120578+ 1199102

120578

120573 = 119909120585119909120578+ 119910120585119910120578

120574 = 1199092

120585+ 1199102

120585

119869 = 119909120585119910120578minus 119909120578119910120585(Jacobian of transformation)

(16)

andnabla2120585 = 119875 andnabla

2120578 = 119876 are control functionswhichmay be

assumed to be zero in both the grid generation and the flowsolver sections (119875 = 119876 = 0) These assumptions lead to the

following equation to solve the above Laplacersquos equation andobtain 120595 at every grid point of the physical domain

((1199092

120578+ 1199102

120578) 120595120585120585

minus 2 (119909120585119909120578+ 119910120585119910120578) 120595120585120578

+ (1199092

120585+ 1199102

120585) 120595120578120578) = 0

(17)

To solve the above equation the finite differencemethodmaybe conveniently used

41 Kutta Condition TheKutta condition states that the flowleaves the sharp trailing edge of an airfoil smoothly [8] Toapply the Kutta condition in our calculation we need toconsider two possible configurations of the trailing edgeThe trailing edge can have a finite-angle or can be cusped(Figure 5)

Journal of Aerodynamics 5

a

Finite angle

Airfoil

V2

At point a V1 = V2 = 0

V1

(a)

Cusped

Airfoil

a

V2At point a V1 = V2 ne 0

V1

(b)

Figure 5 Different possible shapes of the trailing edge and theirrelation to the Kutta condition

1 N

1 2

2 11 1

x

y

AirfoilV1

1N minus 1

2N

VN

Figure 6 Grid notation of the trailing edge

Suppose that the velocities along the top surface andbottom surface are 119881

1and 119881

2 respectively For a finite-angle

trailing edge having two finite velocities in two differentdirections at the same point is physically impossible (Figure5(a)) and therefore the only possibility is that both velocitiesshould be zero (119881

1= 1198812= 0) For the cusped trailing edge

(Figure 5(b)) having two velocities in the same directions atpoint 119886 shows that both119881

1and119881

2can be finite However the

pressure at point 119886 is unique and Bernoulli equation statesthat [2]

119901119886+

1

21205881198812

1= 119901119886+

1

21205881198812

2(18)

or

1198811= 1198812 (19)

In order to obtain relationships for the Kutta condition interms of stream function 120595 consider the finite-angle trailingedge in the O-type grid scheme shown in Figure 6

From (1) we have

119906 = 120595119910 (20)

From the transformation relationship (see (13))

120595119910=

1

119869[minus (119909120578) (120595120585) + (119909

120585) (120595120578)] (21)

If 1198811and 119881

119873are the velocities of the grid points (1 1) and

(1119873) respectively the Kutta condition 1198811= 1198812= 0 gives

1198811= 119881119873

= 0 997904rArr 1199061= 119906119873

= 0

1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

100381610038161003816100381610038161003816100381610038161

=1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

10038161003816100381610038161003816100381610038161003816119873

= 0

minus119909120578120595120585+ 119909120585120595120578

100381610038161003816100381610038161= 0

(22)

By discretizing (22) in the computational domain we get

119909120578120595120585= 119909120585120595120578

[(11990912

minus 11990911

)] [(12059521

minus 12059511

)]

= [(11990921

minus 11990911

)] [(12059512

minus 12059511

)]

12059511

=12059521

(11990912

minus 11990911

) minus 12059512

(11990921

minus 11990911

)

11990912

minus 11990921

(23)

By considering the wall boundary condition (12059511

= 12059512) we

can simplify (23) to get

12059511

= 12059521

(24)

Since the grid points (1 1) and (1119873) are the same points inthe physical domain we have

12059511

= 1205951119873

= 12059521

(25)

This value is constant on the airfoil surface due to the wallboundary condition

The derivation of an equation for the cusped trailing edgeis more complicated Consider the cusped trailing edge andthe associated grid notation shown in Figure 7

Since for the cusped trailing edge both vectors 1198811and

119881119873are equal in the magnitude and direction from the Kutta

condition for the cusped trailing edge (1198811= 119881119873) we can write

1198811= 119881119873

997904rArr 1199061= 119906119873

[1

119869(minus119909120578120595120585+ 119909120585120595120578)]

1

= [1

119869(minus119909120578120595120585+ 119909120585120595120578)]

119873

(26)

But

119909120585

100381610038161003816100381610038161= 11990921

minus 11990911

119909120585

10038161003816100381610038161003816119873= 1199092119873

minus 1199091119873

(27)

Since 1199092119873

= 11990921

and 1199091119873

= 11990911

we have

119909120585

100381610038161003816100381610038161= 119909120585

10038161003816100381610038161003816119873 (28)

In similar approach we have

119910120585

100381610038161003816100381610038161= 119910120585

10038161003816100381610038161003816119873 (29)

6 Journal of Aerodynamics

Cusped

1 N

1 1

V1

1 N minus 1

VN

(2 1)

1 2

(2 N)

Figure 7 Cusped trailing edge and the associated grid notation

Furthermore120595120585|1= 12059521

minus12059511

and 120595120585|119873

= 1205952119873

minus1205951119873

Since12059511

= 1205951119873

and 12059521

= 1205952119873

120595120585

100381610038161003816100381610038161= 120595120585

10038161003816100381610038161003816119873 (30)

Moreover 120595120578|1= 12059512

minus 12059511

and 120595120578|119873

= 1205951119873

minus 1205951119873minus1

Since12059511

= 12059512

= 1205951119873minus1

= 1205951119873

(wall boundary condition) weobtain

120595120578

100381610038161003816100381610038161= 120595120578

10038161003816100381610038161003816119873= 0 (31)

By substituting (28) through (31) into (26) we have

1

(11990921

minus 11990911

) (11991012

minus 11991011

) minus (11990912

minus 11990911

) (11991021

minus 11991011

)

times [minus (11990912

minus 11990911

) (12059521

minus 12059511

) + 0]

=1

(11990921

minus 11990911

) (1199101119873

minus 1199101119873minus1

) minus (1199091119873

minus 1199091119873minus1

) (11991021

minus 11991011

)

times [minus (1199091119873

minus 1199091119873minus1

) (12059521

minus 12059511

) + 0]

(32)

By solving (32) for 12059511(using software Maple) we get

12059511

= 12059521

(33)

In addition 1205951119873

= 12059511

= 12059521 Equation (33) is the required

expression for the cusped trailing angleFigures 8 and 9 show the stream function 120595 for both the

finite-angle (NACA0012 airfoil with angle of attack of120572 = 40∘

and a free stream velocity of 119881infin

= 70ms) and the cusped(NACA64012with angle of attack of120572 = 40

∘ and a free streamvelocity of 119881

infin= 70ms) trailing edge respectively

42 Velocity Calculation There are three sections where thevelocity must be known

(1) the outer boundaries (four sides 119862119863119863119864 119864119865 and 119865119862

of the rectangle shown in Figure 2)(2) the airfoil surface (119860119867 in Figure 2)(3) the inside of the physical domain

0 2

minus2

minus15

minus1

minus05

0

05

1

15

2

NACA 0012 angle of attack =40

x

y

Figure 8 Stream function for a finite-angle trailing edgeThe figureshows the Kutta condition at the trailing edge

0 2x

minus2

minus2

minus1

0

1

2

y

NACA 64012 angle of attack =40

Figure 9 Stream function for a cusped trailing edge The figureshows the Kutta condition at the trailing edge

The velocity values on the outer boundaries are known fromthe conditions at infinity (using (1) and (2)) In other words119909-component of the velocity vector (119906) on all the outerboundaries is equal to 119881

infincos120572 and 119910-component of the

velocity vector (V) on all the outer boundaries is equal to119881infinsin120572 For the inside of the physical domain and the airfoil

surface we can use (12) and (13) as follows

119906119894119895

=120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895(120595120585)119894119895

+ (119909120585)119894119895(120595120578)119894119895]

V119894119895

= minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895(120595120585)119894119895

minus (119910120585)119894119895(120595120578)119894119895]

(34)

Journal of Aerodynamics 7

The central and one-sided difference schemes are used forthe inside of the physical domain and the airfoil surfacerespectively After obtaining the components of the velocityvector the total velocity can be computed by

119881119894119895

= radic1199062

119894119895+ V2119894119895 (35)

As stated before for an incompressible flow the pressurecoefficient can be expressed in terms of velocity only Thus(4) can be used to determine the pressure of any grid point inthe domain Therefore

119901119894119895

=1

2120588 (1198812

infinminus 1198812

119894119895) + 119901infin (36)

43 Kutta Condition in Terms of the Velocity Potential Theproposed method can be easily developed in terms of thevelocity potential 120601 The wall boundary condition may beexpressed in terms of either the velocity potential 120601 (120597120601120597119899 =

0) or the stream function 120595 (120597120595120597119904 = 0) where 119899 and119904 are the unit vector normal to the airfoil surface and thedistance along the body (airfoil) surface respectively Usingthe transformation relationships for mapping the physicaldomain onto the computational one we can write

120597120601

120597119899airfoil surface=

minus1

119869radic120572(120572120601120585minus 120573120601120578) = 0 (37)

where 119869120572 and120573 are defined in (16)The solution of the aboveequation for the airfoil surface using the finite differencemethod gives the value for 120601

1119895(119895 = 1 119873) From the

definition of the velocity potential

V = 120601119910 (38)

In a similar way to the derivation for Kutta condition in termsof the stream function given in (21) to (23) we get

120601119910=

1

119869[minus (119909120578) (120601120585) + (119909

120585) (120601120578)]

1198811= 119881119873

= 0 997904rArr V1= V119873

= 0

(39)

And finally

12060111

=12060121

(11990912

minus 11990911

) minus 12060112

(11990921

minus 11990911

)

11990912

minus 11990921

(40)

By including (37) and (40) into the solution loops we can findthe velocity potential over the domain The above procedurealso can be extended to the three dimension case

5 Results

51 Validation of the Results for the Pressure Distribution Theresults obtained here are compared with the results fromusing the panel methodThe results are obtained by a Fortrancompiler (PGI) and computations are run on a PC with IntelPentium Dual 173 and 1G RAM The tolerance used in theiterative loops (themesh generation and the stream function)is 10minus8

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 0012Angle of attack 9

Cp

xc

Figure 10 Comparison between the results from [2] and the resultsfrom our method for validation case 1The figure shows an excellentagreement between the results

52 Trailing Edge with Finite-Angle

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 0012 airfoil at an angle of attack of 120572 = 9∘ is

plotted The results are compared with the results from [2]The O-type grid size used in the computation is 155 times 155The computation time is 53 seconds (see Figure 10)

Validation Case 2 The pressure coefficient distribution (119862119901)

over a NACA 0024 airfoil at an angle of attack of 120572 = 0∘ is

plotted The results are compared with the results from [9]The O-type grid size used in the computation is 155 times 155The computation time is 41 seconds (see Figure 11)

Validation Case 3 The pressure coefficient distribution (119862119901)

over a NACA 4414 airfoil at an angle of attack of 120572 = 2∘

is plotted The results are compared with the results fromthe software Xfoil [10] The O-type grid size used in thecomputation is 155times155The computation time is 51 seconds(see Figure 12)

Validation Case 4 The pressure coefficient distribution (119862119901)

over a NACA 4412 airfoil at an angle of attack of 120572 = 10∘ is

plotted The results are compared with the results in [5] TheO-type grid size used in the computation is 155 times 155 Thecomputation time is 55 seconds (see Figure 13)

53 Cusped Trailing Edge

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 64012 airfoil at an angle of attack of 120572 = 6∘

8 Journal of Aerodynamics

0 02 04 06 08 1

0

05

1

minus05

minus1

Results from referenceResults from our methodAirfoil NACA 0024Angle of attack 0

Cp

xc

Figure 11 Comparison between the results from [9] and the results from our method for validation case 2 The figure shows an excellentagreement between the results

0 02 04 06 08 1

0

minus05

minus1

05

1

Results from XFoilResults from our methodAirfoil NACA 4414Angle of attack 2

Cp

xc

Figure 12 Comparison between the results from [10] and the results from our method for validation case 3 The figure shows an excellentagreement between the results

is plotted The results are compared with the results fromthe software XFLR5 [11] The O-type grid size used in thecomputation is 245times245 The computation time is 4 minutesand 15 seconds (see Figures 14 15 and 16)

Excellent agreement can be obtained by comparing theresults from ourmethod and the ones from the panel methodgiven in validation cases for both finite-angle and the cuspedtrailing edges As shown in the validation cases results the

Journal of Aerodynamics 9

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 4412Angle of attack 10

Cp

xc

Figure 13 Comparison between the results from [5] and the resultsfrom ourmethod for validation case 4The figure shows an excellentagreement between the results

minus05

minus05

0

0

05

05

1

1

15x

y

103000

102000

101000

100000

99000

98000

97000

96000

95000

94000

93000

92000

P(Pa)

Figure 14 Pressure distribution over the airfoil (NACA 64012) usedin validation case 1 (cusped trailing edge)

maximum value for 119862119901is exactly equal to 1 The pressure

coefficient at the trailing edge (TE) is equal to unity becausethe velocity is zero at this stagnation point Accordingly

119881TE = 0 119862119901= 1 minus

1198812

TE1198812infin

= 1 minus 0 = 1 (41)

For the cusped trailing edge119881TE = 0 Thus the value of 119862119901at

TE is not equal to 1 (119862119901

= 1) as shown in Figure 16

02 04 06 08 100

05 0

minus05minus10minus15minus20minus25minus30minus35

x

Cp

NACA 64012-Re = 100000-alpha = 600 inviscid

NACA 64012Thickness = 1200Max thick pos = 3740Max camber = minus000Max camber pos = 4420Number of panels = 245

Fixed speed polarReynolds = 100 000Mach = 0000NCrit = 9000Forced upper trans = 1000Forced lower trans = 1000Alpha = 600∘

C1 = 0619

Cd = 0000

Cm = 0009

Upper trans = 0000Lower trans = 0000

Figure 15 Pressure coefficient distribution over a NACA 64012airfoil at a 6∘ angle of attack obtained by XFLR5

0 02 04 06 08 1

minus1

minus2

minus3

minus4

0

1

Results from XFLR5 V610Results from our methodAirfoil NACA 64012 (using 245 points)Angle of attack 6

Cp

xc

Figure 16 Comparison between the results from the softwareXFLR5 and the results fromourmethod for validation case 1 (cuspedtrailing edge) The figure shows an excellent agreement between theresults

6 Conclusion

This paper presents a novel method to implement the Kuttacondition in the numerical solution of two-dimensionalincompressible potential flow over an airfoil The proposedmethod is based on solving the Laplacersquos equation for thestream function at each grid point generated by the ellipticgrid generation technique (O-type) Therefore it is exemptfrom considering the panels and the quantities such as thevortex panel strength and circulation used in the panel

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Journal of Aerodynamics

minus06

minus04

minus02

0

02

04

06

0 05 1x

y

(a) Close-up view of O-type grid around the airfoil

minus015

015

minus01

minus005

0

005

01

minus01 0 01 02x

y

(b) Magnified view of grid around the leading edge

minus015

minus01

minus005

0

005

01

09 1 11x

y

(c) Magnified view of grid around the trailing edge

Figure 4 O-type grid (elliptic) around an airfoil The figure illustrates orthogonality and smoothness of the gridlines especially near airfoilsurface

where

120572 = 1199092

120578+ 1199102

120578

120573 = 119909120585119909120578+ 119910120585119910120578

120574 = 1199092

120585+ 1199102

120585

119869 = 119909120585119910120578minus 119909120578119910120585(Jacobian of transformation)

(16)

andnabla2120585 = 119875 andnabla

2120578 = 119876 are control functionswhichmay be

assumed to be zero in both the grid generation and the flowsolver sections (119875 = 119876 = 0) These assumptions lead to the

following equation to solve the above Laplacersquos equation andobtain 120595 at every grid point of the physical domain

((1199092

120578+ 1199102

120578) 120595120585120585

minus 2 (119909120585119909120578+ 119910120585119910120578) 120595120585120578

+ (1199092

120585+ 1199102

120585) 120595120578120578) = 0

(17)

To solve the above equation the finite differencemethodmaybe conveniently used

41 Kutta Condition TheKutta condition states that the flowleaves the sharp trailing edge of an airfoil smoothly [8] Toapply the Kutta condition in our calculation we need toconsider two possible configurations of the trailing edgeThe trailing edge can have a finite-angle or can be cusped(Figure 5)

Journal of Aerodynamics 5

a

Finite angle

Airfoil

V2

At point a V1 = V2 = 0

V1

(a)

Cusped

Airfoil

a

V2At point a V1 = V2 ne 0

V1

(b)

Figure 5 Different possible shapes of the trailing edge and theirrelation to the Kutta condition

1 N

1 2

2 11 1

x

y

AirfoilV1

1N minus 1

2N

VN

Figure 6 Grid notation of the trailing edge

Suppose that the velocities along the top surface andbottom surface are 119881

1and 119881

2 respectively For a finite-angle

trailing edge having two finite velocities in two differentdirections at the same point is physically impossible (Figure5(a)) and therefore the only possibility is that both velocitiesshould be zero (119881

1= 1198812= 0) For the cusped trailing edge

(Figure 5(b)) having two velocities in the same directions atpoint 119886 shows that both119881

1and119881

2can be finite However the

pressure at point 119886 is unique and Bernoulli equation statesthat [2]

119901119886+

1

21205881198812

1= 119901119886+

1

21205881198812

2(18)

or

1198811= 1198812 (19)

In order to obtain relationships for the Kutta condition interms of stream function 120595 consider the finite-angle trailingedge in the O-type grid scheme shown in Figure 6

From (1) we have

119906 = 120595119910 (20)

From the transformation relationship (see (13))

120595119910=

1

119869[minus (119909120578) (120595120585) + (119909

120585) (120595120578)] (21)

If 1198811and 119881

119873are the velocities of the grid points (1 1) and

(1119873) respectively the Kutta condition 1198811= 1198812= 0 gives

1198811= 119881119873

= 0 997904rArr 1199061= 119906119873

= 0

1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

100381610038161003816100381610038161003816100381610038161

=1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

10038161003816100381610038161003816100381610038161003816119873

= 0

minus119909120578120595120585+ 119909120585120595120578

100381610038161003816100381610038161= 0

(22)

By discretizing (22) in the computational domain we get

119909120578120595120585= 119909120585120595120578

[(11990912

minus 11990911

)] [(12059521

minus 12059511

)]

= [(11990921

minus 11990911

)] [(12059512

minus 12059511

)]

12059511

=12059521

(11990912

minus 11990911

) minus 12059512

(11990921

minus 11990911

)

11990912

minus 11990921

(23)

By considering the wall boundary condition (12059511

= 12059512) we

can simplify (23) to get

12059511

= 12059521

(24)

Since the grid points (1 1) and (1119873) are the same points inthe physical domain we have

12059511

= 1205951119873

= 12059521

(25)

This value is constant on the airfoil surface due to the wallboundary condition

The derivation of an equation for the cusped trailing edgeis more complicated Consider the cusped trailing edge andthe associated grid notation shown in Figure 7

Since for the cusped trailing edge both vectors 1198811and

119881119873are equal in the magnitude and direction from the Kutta

condition for the cusped trailing edge (1198811= 119881119873) we can write

1198811= 119881119873

997904rArr 1199061= 119906119873

[1

119869(minus119909120578120595120585+ 119909120585120595120578)]

1

= [1

119869(minus119909120578120595120585+ 119909120585120595120578)]

119873

(26)

But

119909120585

100381610038161003816100381610038161= 11990921

minus 11990911

119909120585

10038161003816100381610038161003816119873= 1199092119873

minus 1199091119873

(27)

Since 1199092119873

= 11990921

and 1199091119873

= 11990911

we have

119909120585

100381610038161003816100381610038161= 119909120585

10038161003816100381610038161003816119873 (28)

In similar approach we have

119910120585

100381610038161003816100381610038161= 119910120585

10038161003816100381610038161003816119873 (29)

6 Journal of Aerodynamics

Cusped

1 N

1 1

V1

1 N minus 1

VN

(2 1)

1 2

(2 N)

Figure 7 Cusped trailing edge and the associated grid notation

Furthermore120595120585|1= 12059521

minus12059511

and 120595120585|119873

= 1205952119873

minus1205951119873

Since12059511

= 1205951119873

and 12059521

= 1205952119873

120595120585

100381610038161003816100381610038161= 120595120585

10038161003816100381610038161003816119873 (30)

Moreover 120595120578|1= 12059512

minus 12059511

and 120595120578|119873

= 1205951119873

minus 1205951119873minus1

Since12059511

= 12059512

= 1205951119873minus1

= 1205951119873

(wall boundary condition) weobtain

120595120578

100381610038161003816100381610038161= 120595120578

10038161003816100381610038161003816119873= 0 (31)

By substituting (28) through (31) into (26) we have

1

(11990921

minus 11990911

) (11991012

minus 11991011

) minus (11990912

minus 11990911

) (11991021

minus 11991011

)

times [minus (11990912

minus 11990911

) (12059521

minus 12059511

) + 0]

=1

(11990921

minus 11990911

) (1199101119873

minus 1199101119873minus1

) minus (1199091119873

minus 1199091119873minus1

) (11991021

minus 11991011

)

times [minus (1199091119873

minus 1199091119873minus1

) (12059521

minus 12059511

) + 0]

(32)

By solving (32) for 12059511(using software Maple) we get

12059511

= 12059521

(33)

In addition 1205951119873

= 12059511

= 12059521 Equation (33) is the required

expression for the cusped trailing angleFigures 8 and 9 show the stream function 120595 for both the

finite-angle (NACA0012 airfoil with angle of attack of120572 = 40∘

and a free stream velocity of 119881infin

= 70ms) and the cusped(NACA64012with angle of attack of120572 = 40

∘ and a free streamvelocity of 119881

infin= 70ms) trailing edge respectively

42 Velocity Calculation There are three sections where thevelocity must be known

(1) the outer boundaries (four sides 119862119863119863119864 119864119865 and 119865119862

of the rectangle shown in Figure 2)(2) the airfoil surface (119860119867 in Figure 2)(3) the inside of the physical domain

0 2

minus2

minus15

minus1

minus05

0

05

1

15

2

NACA 0012 angle of attack =40

x

y

Figure 8 Stream function for a finite-angle trailing edgeThe figureshows the Kutta condition at the trailing edge

0 2x

minus2

minus2

minus1

0

1

2

y

NACA 64012 angle of attack =40

Figure 9 Stream function for a cusped trailing edge The figureshows the Kutta condition at the trailing edge

The velocity values on the outer boundaries are known fromthe conditions at infinity (using (1) and (2)) In other words119909-component of the velocity vector (119906) on all the outerboundaries is equal to 119881

infincos120572 and 119910-component of the

velocity vector (V) on all the outer boundaries is equal to119881infinsin120572 For the inside of the physical domain and the airfoil

surface we can use (12) and (13) as follows

119906119894119895

=120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895(120595120585)119894119895

+ (119909120585)119894119895(120595120578)119894119895]

V119894119895

= minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895(120595120585)119894119895

minus (119910120585)119894119895(120595120578)119894119895]

(34)

Journal of Aerodynamics 7

The central and one-sided difference schemes are used forthe inside of the physical domain and the airfoil surfacerespectively After obtaining the components of the velocityvector the total velocity can be computed by

119881119894119895

= radic1199062

119894119895+ V2119894119895 (35)

As stated before for an incompressible flow the pressurecoefficient can be expressed in terms of velocity only Thus(4) can be used to determine the pressure of any grid point inthe domain Therefore

119901119894119895

=1

2120588 (1198812

infinminus 1198812

119894119895) + 119901infin (36)

43 Kutta Condition in Terms of the Velocity Potential Theproposed method can be easily developed in terms of thevelocity potential 120601 The wall boundary condition may beexpressed in terms of either the velocity potential 120601 (120597120601120597119899 =

0) or the stream function 120595 (120597120595120597119904 = 0) where 119899 and119904 are the unit vector normal to the airfoil surface and thedistance along the body (airfoil) surface respectively Usingthe transformation relationships for mapping the physicaldomain onto the computational one we can write

120597120601

120597119899airfoil surface=

minus1

119869radic120572(120572120601120585minus 120573120601120578) = 0 (37)

where 119869120572 and120573 are defined in (16)The solution of the aboveequation for the airfoil surface using the finite differencemethod gives the value for 120601

1119895(119895 = 1 119873) From the

definition of the velocity potential

V = 120601119910 (38)

In a similar way to the derivation for Kutta condition in termsof the stream function given in (21) to (23) we get

120601119910=

1

119869[minus (119909120578) (120601120585) + (119909

120585) (120601120578)]

1198811= 119881119873

= 0 997904rArr V1= V119873

= 0

(39)

And finally

12060111

=12060121

(11990912

minus 11990911

) minus 12060112

(11990921

minus 11990911

)

11990912

minus 11990921

(40)

By including (37) and (40) into the solution loops we can findthe velocity potential over the domain The above procedurealso can be extended to the three dimension case

5 Results

51 Validation of the Results for the Pressure Distribution Theresults obtained here are compared with the results fromusing the panel methodThe results are obtained by a Fortrancompiler (PGI) and computations are run on a PC with IntelPentium Dual 173 and 1G RAM The tolerance used in theiterative loops (themesh generation and the stream function)is 10minus8

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 0012Angle of attack 9

Cp

xc

Figure 10 Comparison between the results from [2] and the resultsfrom our method for validation case 1The figure shows an excellentagreement between the results

52 Trailing Edge with Finite-Angle

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 0012 airfoil at an angle of attack of 120572 = 9∘ is

plotted The results are compared with the results from [2]The O-type grid size used in the computation is 155 times 155The computation time is 53 seconds (see Figure 10)

Validation Case 2 The pressure coefficient distribution (119862119901)

over a NACA 0024 airfoil at an angle of attack of 120572 = 0∘ is

plotted The results are compared with the results from [9]The O-type grid size used in the computation is 155 times 155The computation time is 41 seconds (see Figure 11)

Validation Case 3 The pressure coefficient distribution (119862119901)

over a NACA 4414 airfoil at an angle of attack of 120572 = 2∘

is plotted The results are compared with the results fromthe software Xfoil [10] The O-type grid size used in thecomputation is 155times155The computation time is 51 seconds(see Figure 12)

Validation Case 4 The pressure coefficient distribution (119862119901)

over a NACA 4412 airfoil at an angle of attack of 120572 = 10∘ is

plotted The results are compared with the results in [5] TheO-type grid size used in the computation is 155 times 155 Thecomputation time is 55 seconds (see Figure 13)

53 Cusped Trailing Edge

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 64012 airfoil at an angle of attack of 120572 = 6∘

8 Journal of Aerodynamics

0 02 04 06 08 1

0

05

1

minus05

minus1

Results from referenceResults from our methodAirfoil NACA 0024Angle of attack 0

Cp

xc

Figure 11 Comparison between the results from [9] and the results from our method for validation case 2 The figure shows an excellentagreement between the results

0 02 04 06 08 1

0

minus05

minus1

05

1

Results from XFoilResults from our methodAirfoil NACA 4414Angle of attack 2

Cp

xc

Figure 12 Comparison between the results from [10] and the results from our method for validation case 3 The figure shows an excellentagreement between the results

is plotted The results are compared with the results fromthe software XFLR5 [11] The O-type grid size used in thecomputation is 245times245 The computation time is 4 minutesand 15 seconds (see Figures 14 15 and 16)

Excellent agreement can be obtained by comparing theresults from ourmethod and the ones from the panel methodgiven in validation cases for both finite-angle and the cuspedtrailing edges As shown in the validation cases results the

Journal of Aerodynamics 9

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 4412Angle of attack 10

Cp

xc

Figure 13 Comparison between the results from [5] and the resultsfrom ourmethod for validation case 4The figure shows an excellentagreement between the results

minus05

minus05

0

0

05

05

1

1

15x

y

103000

102000

101000

100000

99000

98000

97000

96000

95000

94000

93000

92000

P(Pa)

Figure 14 Pressure distribution over the airfoil (NACA 64012) usedin validation case 1 (cusped trailing edge)

maximum value for 119862119901is exactly equal to 1 The pressure

coefficient at the trailing edge (TE) is equal to unity becausethe velocity is zero at this stagnation point Accordingly

119881TE = 0 119862119901= 1 minus

1198812

TE1198812infin

= 1 minus 0 = 1 (41)

For the cusped trailing edge119881TE = 0 Thus the value of 119862119901at

TE is not equal to 1 (119862119901

= 1) as shown in Figure 16

02 04 06 08 100

05 0

minus05minus10minus15minus20minus25minus30minus35

x

Cp

NACA 64012-Re = 100000-alpha = 600 inviscid

NACA 64012Thickness = 1200Max thick pos = 3740Max camber = minus000Max camber pos = 4420Number of panels = 245

Fixed speed polarReynolds = 100 000Mach = 0000NCrit = 9000Forced upper trans = 1000Forced lower trans = 1000Alpha = 600∘

C1 = 0619

Cd = 0000

Cm = 0009

Upper trans = 0000Lower trans = 0000

Figure 15 Pressure coefficient distribution over a NACA 64012airfoil at a 6∘ angle of attack obtained by XFLR5

0 02 04 06 08 1

minus1

minus2

minus3

minus4

0

1

Results from XFLR5 V610Results from our methodAirfoil NACA 64012 (using 245 points)Angle of attack 6

Cp

xc

Figure 16 Comparison between the results from the softwareXFLR5 and the results fromourmethod for validation case 1 (cuspedtrailing edge) The figure shows an excellent agreement between theresults

6 Conclusion

This paper presents a novel method to implement the Kuttacondition in the numerical solution of two-dimensionalincompressible potential flow over an airfoil The proposedmethod is based on solving the Laplacersquos equation for thestream function at each grid point generated by the ellipticgrid generation technique (O-type) Therefore it is exemptfrom considering the panels and the quantities such as thevortex panel strength and circulation used in the panel

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Aerodynamics 5

a

Finite angle

Airfoil

V2

At point a V1 = V2 = 0

V1

(a)

Cusped

Airfoil

a

V2At point a V1 = V2 ne 0

V1

(b)

Figure 5 Different possible shapes of the trailing edge and theirrelation to the Kutta condition

1 N

1 2

2 11 1

x

y

AirfoilV1

1N minus 1

2N

VN

Figure 6 Grid notation of the trailing edge

Suppose that the velocities along the top surface andbottom surface are 119881

1and 119881

2 respectively For a finite-angle

trailing edge having two finite velocities in two differentdirections at the same point is physically impossible (Figure5(a)) and therefore the only possibility is that both velocitiesshould be zero (119881

1= 1198812= 0) For the cusped trailing edge

(Figure 5(b)) having two velocities in the same directions atpoint 119886 shows that both119881

1and119881

2can be finite However the

pressure at point 119886 is unique and Bernoulli equation statesthat [2]

119901119886+

1

21205881198812

1= 119901119886+

1

21205881198812

2(18)

or

1198811= 1198812 (19)

In order to obtain relationships for the Kutta condition interms of stream function 120595 consider the finite-angle trailingedge in the O-type grid scheme shown in Figure 6

From (1) we have

119906 = 120595119910 (20)

From the transformation relationship (see (13))

120595119910=

1

119869[minus (119909120578) (120595120585) + (119909

120585) (120595120578)] (21)

If 1198811and 119881

119873are the velocities of the grid points (1 1) and

(1119873) respectively the Kutta condition 1198811= 1198812= 0 gives

1198811= 119881119873

= 0 997904rArr 1199061= 119906119873

= 0

1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

100381610038161003816100381610038161003816100381610038161

=1

119869[minus(119909120578)(120595120585) + (119909

120585)(120595120578)]

10038161003816100381610038161003816100381610038161003816119873

= 0

minus119909120578120595120585+ 119909120585120595120578

100381610038161003816100381610038161= 0

(22)

By discretizing (22) in the computational domain we get

119909120578120595120585= 119909120585120595120578

[(11990912

minus 11990911

)] [(12059521

minus 12059511

)]

= [(11990921

minus 11990911

)] [(12059512

minus 12059511

)]

12059511

=12059521

(11990912

minus 11990911

) minus 12059512

(11990921

minus 11990911

)

11990912

minus 11990921

(23)

By considering the wall boundary condition (12059511

= 12059512) we

can simplify (23) to get

12059511

= 12059521

(24)

Since the grid points (1 1) and (1119873) are the same points inthe physical domain we have

12059511

= 1205951119873

= 12059521

(25)

This value is constant on the airfoil surface due to the wallboundary condition

The derivation of an equation for the cusped trailing edgeis more complicated Consider the cusped trailing edge andthe associated grid notation shown in Figure 7

Since for the cusped trailing edge both vectors 1198811and

119881119873are equal in the magnitude and direction from the Kutta

condition for the cusped trailing edge (1198811= 119881119873) we can write

1198811= 119881119873

997904rArr 1199061= 119906119873

[1

119869(minus119909120578120595120585+ 119909120585120595120578)]

1

= [1

119869(minus119909120578120595120585+ 119909120585120595120578)]

119873

(26)

But

119909120585

100381610038161003816100381610038161= 11990921

minus 11990911

119909120585

10038161003816100381610038161003816119873= 1199092119873

minus 1199091119873

(27)

Since 1199092119873

= 11990921

and 1199091119873

= 11990911

we have

119909120585

100381610038161003816100381610038161= 119909120585

10038161003816100381610038161003816119873 (28)

In similar approach we have

119910120585

100381610038161003816100381610038161= 119910120585

10038161003816100381610038161003816119873 (29)

6 Journal of Aerodynamics

Cusped

1 N

1 1

V1

1 N minus 1

VN

(2 1)

1 2

(2 N)

Figure 7 Cusped trailing edge and the associated grid notation

Furthermore120595120585|1= 12059521

minus12059511

and 120595120585|119873

= 1205952119873

minus1205951119873

Since12059511

= 1205951119873

and 12059521

= 1205952119873

120595120585

100381610038161003816100381610038161= 120595120585

10038161003816100381610038161003816119873 (30)

Moreover 120595120578|1= 12059512

minus 12059511

and 120595120578|119873

= 1205951119873

minus 1205951119873minus1

Since12059511

= 12059512

= 1205951119873minus1

= 1205951119873

(wall boundary condition) weobtain

120595120578

100381610038161003816100381610038161= 120595120578

10038161003816100381610038161003816119873= 0 (31)

By substituting (28) through (31) into (26) we have

1

(11990921

minus 11990911

) (11991012

minus 11991011

) minus (11990912

minus 11990911

) (11991021

minus 11991011

)

times [minus (11990912

minus 11990911

) (12059521

minus 12059511

) + 0]

=1

(11990921

minus 11990911

) (1199101119873

minus 1199101119873minus1

) minus (1199091119873

minus 1199091119873minus1

) (11991021

minus 11991011

)

times [minus (1199091119873

minus 1199091119873minus1

) (12059521

minus 12059511

) + 0]

(32)

By solving (32) for 12059511(using software Maple) we get

12059511

= 12059521

(33)

In addition 1205951119873

= 12059511

= 12059521 Equation (33) is the required

expression for the cusped trailing angleFigures 8 and 9 show the stream function 120595 for both the

finite-angle (NACA0012 airfoil with angle of attack of120572 = 40∘

and a free stream velocity of 119881infin

= 70ms) and the cusped(NACA64012with angle of attack of120572 = 40

∘ and a free streamvelocity of 119881

infin= 70ms) trailing edge respectively

42 Velocity Calculation There are three sections where thevelocity must be known

(1) the outer boundaries (four sides 119862119863119863119864 119864119865 and 119865119862

of the rectangle shown in Figure 2)(2) the airfoil surface (119860119867 in Figure 2)(3) the inside of the physical domain

0 2

minus2

minus15

minus1

minus05

0

05

1

15

2

NACA 0012 angle of attack =40

x

y

Figure 8 Stream function for a finite-angle trailing edgeThe figureshows the Kutta condition at the trailing edge

0 2x

minus2

minus2

minus1

0

1

2

y

NACA 64012 angle of attack =40

Figure 9 Stream function for a cusped trailing edge The figureshows the Kutta condition at the trailing edge

The velocity values on the outer boundaries are known fromthe conditions at infinity (using (1) and (2)) In other words119909-component of the velocity vector (119906) on all the outerboundaries is equal to 119881

infincos120572 and 119910-component of the

velocity vector (V) on all the outer boundaries is equal to119881infinsin120572 For the inside of the physical domain and the airfoil

surface we can use (12) and (13) as follows

119906119894119895

=120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895(120595120585)119894119895

+ (119909120585)119894119895(120595120578)119894119895]

V119894119895

= minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895(120595120585)119894119895

minus (119910120585)119894119895(120595120578)119894119895]

(34)

Journal of Aerodynamics 7

The central and one-sided difference schemes are used forthe inside of the physical domain and the airfoil surfacerespectively After obtaining the components of the velocityvector the total velocity can be computed by

119881119894119895

= radic1199062

119894119895+ V2119894119895 (35)

As stated before for an incompressible flow the pressurecoefficient can be expressed in terms of velocity only Thus(4) can be used to determine the pressure of any grid point inthe domain Therefore

119901119894119895

=1

2120588 (1198812

infinminus 1198812

119894119895) + 119901infin (36)

43 Kutta Condition in Terms of the Velocity Potential Theproposed method can be easily developed in terms of thevelocity potential 120601 The wall boundary condition may beexpressed in terms of either the velocity potential 120601 (120597120601120597119899 =

0) or the stream function 120595 (120597120595120597119904 = 0) where 119899 and119904 are the unit vector normal to the airfoil surface and thedistance along the body (airfoil) surface respectively Usingthe transformation relationships for mapping the physicaldomain onto the computational one we can write

120597120601

120597119899airfoil surface=

minus1

119869radic120572(120572120601120585minus 120573120601120578) = 0 (37)

where 119869120572 and120573 are defined in (16)The solution of the aboveequation for the airfoil surface using the finite differencemethod gives the value for 120601

1119895(119895 = 1 119873) From the

definition of the velocity potential

V = 120601119910 (38)

In a similar way to the derivation for Kutta condition in termsof the stream function given in (21) to (23) we get

120601119910=

1

119869[minus (119909120578) (120601120585) + (119909

120585) (120601120578)]

1198811= 119881119873

= 0 997904rArr V1= V119873

= 0

(39)

And finally

12060111

=12060121

(11990912

minus 11990911

) minus 12060112

(11990921

minus 11990911

)

11990912

minus 11990921

(40)

By including (37) and (40) into the solution loops we can findthe velocity potential over the domain The above procedurealso can be extended to the three dimension case

5 Results

51 Validation of the Results for the Pressure Distribution Theresults obtained here are compared with the results fromusing the panel methodThe results are obtained by a Fortrancompiler (PGI) and computations are run on a PC with IntelPentium Dual 173 and 1G RAM The tolerance used in theiterative loops (themesh generation and the stream function)is 10minus8

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 0012Angle of attack 9

Cp

xc

Figure 10 Comparison between the results from [2] and the resultsfrom our method for validation case 1The figure shows an excellentagreement between the results

52 Trailing Edge with Finite-Angle

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 0012 airfoil at an angle of attack of 120572 = 9∘ is

plotted The results are compared with the results from [2]The O-type grid size used in the computation is 155 times 155The computation time is 53 seconds (see Figure 10)

Validation Case 2 The pressure coefficient distribution (119862119901)

over a NACA 0024 airfoil at an angle of attack of 120572 = 0∘ is

plotted The results are compared with the results from [9]The O-type grid size used in the computation is 155 times 155The computation time is 41 seconds (see Figure 11)

Validation Case 3 The pressure coefficient distribution (119862119901)

over a NACA 4414 airfoil at an angle of attack of 120572 = 2∘

is plotted The results are compared with the results fromthe software Xfoil [10] The O-type grid size used in thecomputation is 155times155The computation time is 51 seconds(see Figure 12)

Validation Case 4 The pressure coefficient distribution (119862119901)

over a NACA 4412 airfoil at an angle of attack of 120572 = 10∘ is

plotted The results are compared with the results in [5] TheO-type grid size used in the computation is 155 times 155 Thecomputation time is 55 seconds (see Figure 13)

53 Cusped Trailing Edge

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 64012 airfoil at an angle of attack of 120572 = 6∘

8 Journal of Aerodynamics

0 02 04 06 08 1

0

05

1

minus05

minus1

Results from referenceResults from our methodAirfoil NACA 0024Angle of attack 0

Cp

xc

Figure 11 Comparison between the results from [9] and the results from our method for validation case 2 The figure shows an excellentagreement between the results

0 02 04 06 08 1

0

minus05

minus1

05

1

Results from XFoilResults from our methodAirfoil NACA 4414Angle of attack 2

Cp

xc

Figure 12 Comparison between the results from [10] and the results from our method for validation case 3 The figure shows an excellentagreement between the results

is plotted The results are compared with the results fromthe software XFLR5 [11] The O-type grid size used in thecomputation is 245times245 The computation time is 4 minutesand 15 seconds (see Figures 14 15 and 16)

Excellent agreement can be obtained by comparing theresults from ourmethod and the ones from the panel methodgiven in validation cases for both finite-angle and the cuspedtrailing edges As shown in the validation cases results the

Journal of Aerodynamics 9

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 4412Angle of attack 10

Cp

xc

Figure 13 Comparison between the results from [5] and the resultsfrom ourmethod for validation case 4The figure shows an excellentagreement between the results

minus05

minus05

0

0

05

05

1

1

15x

y

103000

102000

101000

100000

99000

98000

97000

96000

95000

94000

93000

92000

P(Pa)

Figure 14 Pressure distribution over the airfoil (NACA 64012) usedin validation case 1 (cusped trailing edge)

maximum value for 119862119901is exactly equal to 1 The pressure

coefficient at the trailing edge (TE) is equal to unity becausethe velocity is zero at this stagnation point Accordingly

119881TE = 0 119862119901= 1 minus

1198812

TE1198812infin

= 1 minus 0 = 1 (41)

For the cusped trailing edge119881TE = 0 Thus the value of 119862119901at

TE is not equal to 1 (119862119901

= 1) as shown in Figure 16

02 04 06 08 100

05 0

minus05minus10minus15minus20minus25minus30minus35

x

Cp

NACA 64012-Re = 100000-alpha = 600 inviscid

NACA 64012Thickness = 1200Max thick pos = 3740Max camber = minus000Max camber pos = 4420Number of panels = 245

Fixed speed polarReynolds = 100 000Mach = 0000NCrit = 9000Forced upper trans = 1000Forced lower trans = 1000Alpha = 600∘

C1 = 0619

Cd = 0000

Cm = 0009

Upper trans = 0000Lower trans = 0000

Figure 15 Pressure coefficient distribution over a NACA 64012airfoil at a 6∘ angle of attack obtained by XFLR5

0 02 04 06 08 1

minus1

minus2

minus3

minus4

0

1

Results from XFLR5 V610Results from our methodAirfoil NACA 64012 (using 245 points)Angle of attack 6

Cp

xc

Figure 16 Comparison between the results from the softwareXFLR5 and the results fromourmethod for validation case 1 (cuspedtrailing edge) The figure shows an excellent agreement between theresults

6 Conclusion

This paper presents a novel method to implement the Kuttacondition in the numerical solution of two-dimensionalincompressible potential flow over an airfoil The proposedmethod is based on solving the Laplacersquos equation for thestream function at each grid point generated by the ellipticgrid generation technique (O-type) Therefore it is exemptfrom considering the panels and the quantities such as thevortex panel strength and circulation used in the panel

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Journal of Aerodynamics

Cusped

1 N

1 1

V1

1 N minus 1

VN

(2 1)

1 2

(2 N)

Figure 7 Cusped trailing edge and the associated grid notation

Furthermore120595120585|1= 12059521

minus12059511

and 120595120585|119873

= 1205952119873

minus1205951119873

Since12059511

= 1205951119873

and 12059521

= 1205952119873

120595120585

100381610038161003816100381610038161= 120595120585

10038161003816100381610038161003816119873 (30)

Moreover 120595120578|1= 12059512

minus 12059511

and 120595120578|119873

= 1205951119873

minus 1205951119873minus1

Since12059511

= 12059512

= 1205951119873minus1

= 1205951119873

(wall boundary condition) weobtain

120595120578

100381610038161003816100381610038161= 120595120578

10038161003816100381610038161003816119873= 0 (31)

By substituting (28) through (31) into (26) we have

1

(11990921

minus 11990911

) (11991012

minus 11991011

) minus (11990912

minus 11990911

) (11991021

minus 11991011

)

times [minus (11990912

minus 11990911

) (12059521

minus 12059511

) + 0]

=1

(11990921

minus 11990911

) (1199101119873

minus 1199101119873minus1

) minus (1199091119873

minus 1199091119873minus1

) (11991021

minus 11991011

)

times [minus (1199091119873

minus 1199091119873minus1

) (12059521

minus 12059511

) + 0]

(32)

By solving (32) for 12059511(using software Maple) we get

12059511

= 12059521

(33)

In addition 1205951119873

= 12059511

= 12059521 Equation (33) is the required

expression for the cusped trailing angleFigures 8 and 9 show the stream function 120595 for both the

finite-angle (NACA0012 airfoil with angle of attack of120572 = 40∘

and a free stream velocity of 119881infin

= 70ms) and the cusped(NACA64012with angle of attack of120572 = 40

∘ and a free streamvelocity of 119881

infin= 70ms) trailing edge respectively

42 Velocity Calculation There are three sections where thevelocity must be known

(1) the outer boundaries (four sides 119862119863119863119864 119864119865 and 119865119862

of the rectangle shown in Figure 2)(2) the airfoil surface (119860119867 in Figure 2)(3) the inside of the physical domain

0 2

minus2

minus15

minus1

minus05

0

05

1

15

2

NACA 0012 angle of attack =40

x

y

Figure 8 Stream function for a finite-angle trailing edgeThe figureshows the Kutta condition at the trailing edge

0 2x

minus2

minus2

minus1

0

1

2

y

NACA 64012 angle of attack =40

Figure 9 Stream function for a cusped trailing edge The figureshows the Kutta condition at the trailing edge

The velocity values on the outer boundaries are known fromthe conditions at infinity (using (1) and (2)) In other words119909-component of the velocity vector (119906) on all the outerboundaries is equal to 119881

infincos120572 and 119910-component of the

velocity vector (V) on all the outer boundaries is equal to119881infinsin120572 For the inside of the physical domain and the airfoil

surface we can use (12) and (13) as follows

119906119894119895

=120597120595

120597119910

10038161003816100381610038161003816100381610038161003816119894119895

=1

119869[minus(119909120578)119894119895(120595120585)119894119895

+ (119909120585)119894119895(120595120578)119894119895]

V119894119895

= minus120597120595

120597119909

10038161003816100381610038161003816100381610038161003816119894119895

= minus1

119869[(119910120578)119894119895(120595120585)119894119895

minus (119910120585)119894119895(120595120578)119894119895]

(34)

Journal of Aerodynamics 7

The central and one-sided difference schemes are used forthe inside of the physical domain and the airfoil surfacerespectively After obtaining the components of the velocityvector the total velocity can be computed by

119881119894119895

= radic1199062

119894119895+ V2119894119895 (35)

As stated before for an incompressible flow the pressurecoefficient can be expressed in terms of velocity only Thus(4) can be used to determine the pressure of any grid point inthe domain Therefore

119901119894119895

=1

2120588 (1198812

infinminus 1198812

119894119895) + 119901infin (36)

43 Kutta Condition in Terms of the Velocity Potential Theproposed method can be easily developed in terms of thevelocity potential 120601 The wall boundary condition may beexpressed in terms of either the velocity potential 120601 (120597120601120597119899 =

0) or the stream function 120595 (120597120595120597119904 = 0) where 119899 and119904 are the unit vector normal to the airfoil surface and thedistance along the body (airfoil) surface respectively Usingthe transformation relationships for mapping the physicaldomain onto the computational one we can write

120597120601

120597119899airfoil surface=

minus1

119869radic120572(120572120601120585minus 120573120601120578) = 0 (37)

where 119869120572 and120573 are defined in (16)The solution of the aboveequation for the airfoil surface using the finite differencemethod gives the value for 120601

1119895(119895 = 1 119873) From the

definition of the velocity potential

V = 120601119910 (38)

In a similar way to the derivation for Kutta condition in termsof the stream function given in (21) to (23) we get

120601119910=

1

119869[minus (119909120578) (120601120585) + (119909

120585) (120601120578)]

1198811= 119881119873

= 0 997904rArr V1= V119873

= 0

(39)

And finally

12060111

=12060121

(11990912

minus 11990911

) minus 12060112

(11990921

minus 11990911

)

11990912

minus 11990921

(40)

By including (37) and (40) into the solution loops we can findthe velocity potential over the domain The above procedurealso can be extended to the three dimension case

5 Results

51 Validation of the Results for the Pressure Distribution Theresults obtained here are compared with the results fromusing the panel methodThe results are obtained by a Fortrancompiler (PGI) and computations are run on a PC with IntelPentium Dual 173 and 1G RAM The tolerance used in theiterative loops (themesh generation and the stream function)is 10minus8

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 0012Angle of attack 9

Cp

xc

Figure 10 Comparison between the results from [2] and the resultsfrom our method for validation case 1The figure shows an excellentagreement between the results

52 Trailing Edge with Finite-Angle

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 0012 airfoil at an angle of attack of 120572 = 9∘ is

plotted The results are compared with the results from [2]The O-type grid size used in the computation is 155 times 155The computation time is 53 seconds (see Figure 10)

Validation Case 2 The pressure coefficient distribution (119862119901)

over a NACA 0024 airfoil at an angle of attack of 120572 = 0∘ is

plotted The results are compared with the results from [9]The O-type grid size used in the computation is 155 times 155The computation time is 41 seconds (see Figure 11)

Validation Case 3 The pressure coefficient distribution (119862119901)

over a NACA 4414 airfoil at an angle of attack of 120572 = 2∘

is plotted The results are compared with the results fromthe software Xfoil [10] The O-type grid size used in thecomputation is 155times155The computation time is 51 seconds(see Figure 12)

Validation Case 4 The pressure coefficient distribution (119862119901)

over a NACA 4412 airfoil at an angle of attack of 120572 = 10∘ is

plotted The results are compared with the results in [5] TheO-type grid size used in the computation is 155 times 155 Thecomputation time is 55 seconds (see Figure 13)

53 Cusped Trailing Edge

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 64012 airfoil at an angle of attack of 120572 = 6∘

8 Journal of Aerodynamics

0 02 04 06 08 1

0

05

1

minus05

minus1

Results from referenceResults from our methodAirfoil NACA 0024Angle of attack 0

Cp

xc

Figure 11 Comparison between the results from [9] and the results from our method for validation case 2 The figure shows an excellentagreement between the results

0 02 04 06 08 1

0

minus05

minus1

05

1

Results from XFoilResults from our methodAirfoil NACA 4414Angle of attack 2

Cp

xc

Figure 12 Comparison between the results from [10] and the results from our method for validation case 3 The figure shows an excellentagreement between the results

is plotted The results are compared with the results fromthe software XFLR5 [11] The O-type grid size used in thecomputation is 245times245 The computation time is 4 minutesand 15 seconds (see Figures 14 15 and 16)

Excellent agreement can be obtained by comparing theresults from ourmethod and the ones from the panel methodgiven in validation cases for both finite-angle and the cuspedtrailing edges As shown in the validation cases results the

Journal of Aerodynamics 9

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 4412Angle of attack 10

Cp

xc

Figure 13 Comparison between the results from [5] and the resultsfrom ourmethod for validation case 4The figure shows an excellentagreement between the results

minus05

minus05

0

0

05

05

1

1

15x

y

103000

102000

101000

100000

99000

98000

97000

96000

95000

94000

93000

92000

P(Pa)

Figure 14 Pressure distribution over the airfoil (NACA 64012) usedin validation case 1 (cusped trailing edge)

maximum value for 119862119901is exactly equal to 1 The pressure

coefficient at the trailing edge (TE) is equal to unity becausethe velocity is zero at this stagnation point Accordingly

119881TE = 0 119862119901= 1 minus

1198812

TE1198812infin

= 1 minus 0 = 1 (41)

For the cusped trailing edge119881TE = 0 Thus the value of 119862119901at

TE is not equal to 1 (119862119901

= 1) as shown in Figure 16

02 04 06 08 100

05 0

minus05minus10minus15minus20minus25minus30minus35

x

Cp

NACA 64012-Re = 100000-alpha = 600 inviscid

NACA 64012Thickness = 1200Max thick pos = 3740Max camber = minus000Max camber pos = 4420Number of panels = 245

Fixed speed polarReynolds = 100 000Mach = 0000NCrit = 9000Forced upper trans = 1000Forced lower trans = 1000Alpha = 600∘

C1 = 0619

Cd = 0000

Cm = 0009

Upper trans = 0000Lower trans = 0000

Figure 15 Pressure coefficient distribution over a NACA 64012airfoil at a 6∘ angle of attack obtained by XFLR5

0 02 04 06 08 1

minus1

minus2

minus3

minus4

0

1

Results from XFLR5 V610Results from our methodAirfoil NACA 64012 (using 245 points)Angle of attack 6

Cp

xc

Figure 16 Comparison between the results from the softwareXFLR5 and the results fromourmethod for validation case 1 (cuspedtrailing edge) The figure shows an excellent agreement between theresults

6 Conclusion

This paper presents a novel method to implement the Kuttacondition in the numerical solution of two-dimensionalincompressible potential flow over an airfoil The proposedmethod is based on solving the Laplacersquos equation for thestream function at each grid point generated by the ellipticgrid generation technique (O-type) Therefore it is exemptfrom considering the panels and the quantities such as thevortex panel strength and circulation used in the panel

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Aerodynamics 7

The central and one-sided difference schemes are used forthe inside of the physical domain and the airfoil surfacerespectively After obtaining the components of the velocityvector the total velocity can be computed by

119881119894119895

= radic1199062

119894119895+ V2119894119895 (35)

As stated before for an incompressible flow the pressurecoefficient can be expressed in terms of velocity only Thus(4) can be used to determine the pressure of any grid point inthe domain Therefore

119901119894119895

=1

2120588 (1198812

infinminus 1198812

119894119895) + 119901infin (36)

43 Kutta Condition in Terms of the Velocity Potential Theproposed method can be easily developed in terms of thevelocity potential 120601 The wall boundary condition may beexpressed in terms of either the velocity potential 120601 (120597120601120597119899 =

0) or the stream function 120595 (120597120595120597119904 = 0) where 119899 and119904 are the unit vector normal to the airfoil surface and thedistance along the body (airfoil) surface respectively Usingthe transformation relationships for mapping the physicaldomain onto the computational one we can write

120597120601

120597119899airfoil surface=

minus1

119869radic120572(120572120601120585minus 120573120601120578) = 0 (37)

where 119869120572 and120573 are defined in (16)The solution of the aboveequation for the airfoil surface using the finite differencemethod gives the value for 120601

1119895(119895 = 1 119873) From the

definition of the velocity potential

V = 120601119910 (38)

In a similar way to the derivation for Kutta condition in termsof the stream function given in (21) to (23) we get

120601119910=

1

119869[minus (119909120578) (120601120585) + (119909

120585) (120601120578)]

1198811= 119881119873

= 0 997904rArr V1= V119873

= 0

(39)

And finally

12060111

=12060121

(11990912

minus 11990911

) minus 12060112

(11990921

minus 11990911

)

11990912

minus 11990921

(40)

By including (37) and (40) into the solution loops we can findthe velocity potential over the domain The above procedurealso can be extended to the three dimension case

5 Results

51 Validation of the Results for the Pressure Distribution Theresults obtained here are compared with the results fromusing the panel methodThe results are obtained by a Fortrancompiler (PGI) and computations are run on a PC with IntelPentium Dual 173 and 1G RAM The tolerance used in theiterative loops (themesh generation and the stream function)is 10minus8

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 0012Angle of attack 9

Cp

xc

Figure 10 Comparison between the results from [2] and the resultsfrom our method for validation case 1The figure shows an excellentagreement between the results

52 Trailing Edge with Finite-Angle

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 0012 airfoil at an angle of attack of 120572 = 9∘ is

plotted The results are compared with the results from [2]The O-type grid size used in the computation is 155 times 155The computation time is 53 seconds (see Figure 10)

Validation Case 2 The pressure coefficient distribution (119862119901)

over a NACA 0024 airfoil at an angle of attack of 120572 = 0∘ is

plotted The results are compared with the results from [9]The O-type grid size used in the computation is 155 times 155The computation time is 41 seconds (see Figure 11)

Validation Case 3 The pressure coefficient distribution (119862119901)

over a NACA 4414 airfoil at an angle of attack of 120572 = 2∘

is plotted The results are compared with the results fromthe software Xfoil [10] The O-type grid size used in thecomputation is 155times155The computation time is 51 seconds(see Figure 12)

Validation Case 4 The pressure coefficient distribution (119862119901)

over a NACA 4412 airfoil at an angle of attack of 120572 = 10∘ is

plotted The results are compared with the results in [5] TheO-type grid size used in the computation is 155 times 155 Thecomputation time is 55 seconds (see Figure 13)

53 Cusped Trailing Edge

Validation Case 1 The pressure coefficient distribution (119862119901)

over a NACA 64012 airfoil at an angle of attack of 120572 = 6∘

8 Journal of Aerodynamics

0 02 04 06 08 1

0

05

1

minus05

minus1

Results from referenceResults from our methodAirfoil NACA 0024Angle of attack 0

Cp

xc

Figure 11 Comparison between the results from [9] and the results from our method for validation case 2 The figure shows an excellentagreement between the results

0 02 04 06 08 1

0

minus05

minus1

05

1

Results from XFoilResults from our methodAirfoil NACA 4414Angle of attack 2

Cp

xc

Figure 12 Comparison between the results from [10] and the results from our method for validation case 3 The figure shows an excellentagreement between the results

is plotted The results are compared with the results fromthe software XFLR5 [11] The O-type grid size used in thecomputation is 245times245 The computation time is 4 minutesand 15 seconds (see Figures 14 15 and 16)

Excellent agreement can be obtained by comparing theresults from ourmethod and the ones from the panel methodgiven in validation cases for both finite-angle and the cuspedtrailing edges As shown in the validation cases results the

Journal of Aerodynamics 9

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 4412Angle of attack 10

Cp

xc

Figure 13 Comparison between the results from [5] and the resultsfrom ourmethod for validation case 4The figure shows an excellentagreement between the results

minus05

minus05

0

0

05

05

1

1

15x

y

103000

102000

101000

100000

99000

98000

97000

96000

95000

94000

93000

92000

P(Pa)

Figure 14 Pressure distribution over the airfoil (NACA 64012) usedin validation case 1 (cusped trailing edge)

maximum value for 119862119901is exactly equal to 1 The pressure

coefficient at the trailing edge (TE) is equal to unity becausethe velocity is zero at this stagnation point Accordingly

119881TE = 0 119862119901= 1 minus

1198812

TE1198812infin

= 1 minus 0 = 1 (41)

For the cusped trailing edge119881TE = 0 Thus the value of 119862119901at

TE is not equal to 1 (119862119901

= 1) as shown in Figure 16

02 04 06 08 100

05 0

minus05minus10minus15minus20minus25minus30minus35

x

Cp

NACA 64012-Re = 100000-alpha = 600 inviscid

NACA 64012Thickness = 1200Max thick pos = 3740Max camber = minus000Max camber pos = 4420Number of panels = 245

Fixed speed polarReynolds = 100 000Mach = 0000NCrit = 9000Forced upper trans = 1000Forced lower trans = 1000Alpha = 600∘

C1 = 0619

Cd = 0000

Cm = 0009

Upper trans = 0000Lower trans = 0000

Figure 15 Pressure coefficient distribution over a NACA 64012airfoil at a 6∘ angle of attack obtained by XFLR5

0 02 04 06 08 1

minus1

minus2

minus3

minus4

0

1

Results from XFLR5 V610Results from our methodAirfoil NACA 64012 (using 245 points)Angle of attack 6

Cp

xc

Figure 16 Comparison between the results from the softwareXFLR5 and the results fromourmethod for validation case 1 (cuspedtrailing edge) The figure shows an excellent agreement between theresults

6 Conclusion

This paper presents a novel method to implement the Kuttacondition in the numerical solution of two-dimensionalincompressible potential flow over an airfoil The proposedmethod is based on solving the Laplacersquos equation for thestream function at each grid point generated by the ellipticgrid generation technique (O-type) Therefore it is exemptfrom considering the panels and the quantities such as thevortex panel strength and circulation used in the panel

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Journal of Aerodynamics

0 02 04 06 08 1

0

05

1

minus05

minus1

Results from referenceResults from our methodAirfoil NACA 0024Angle of attack 0

Cp

xc

Figure 11 Comparison between the results from [9] and the results from our method for validation case 2 The figure shows an excellentagreement between the results

0 02 04 06 08 1

0

minus05

minus1

05

1

Results from XFoilResults from our methodAirfoil NACA 4414Angle of attack 2

Cp

xc

Figure 12 Comparison between the results from [10] and the results from our method for validation case 3 The figure shows an excellentagreement between the results

is plotted The results are compared with the results fromthe software XFLR5 [11] The O-type grid size used in thecomputation is 245times245 The computation time is 4 minutesand 15 seconds (see Figures 14 15 and 16)

Excellent agreement can be obtained by comparing theresults from ourmethod and the ones from the panel methodgiven in validation cases for both finite-angle and the cuspedtrailing edges As shown in the validation cases results the

Journal of Aerodynamics 9

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 4412Angle of attack 10

Cp

xc

Figure 13 Comparison between the results from [5] and the resultsfrom ourmethod for validation case 4The figure shows an excellentagreement between the results

minus05

minus05

0

0

05

05

1

1

15x

y

103000

102000

101000

100000

99000

98000

97000

96000

95000

94000

93000

92000

P(Pa)

Figure 14 Pressure distribution over the airfoil (NACA 64012) usedin validation case 1 (cusped trailing edge)

maximum value for 119862119901is exactly equal to 1 The pressure

coefficient at the trailing edge (TE) is equal to unity becausethe velocity is zero at this stagnation point Accordingly

119881TE = 0 119862119901= 1 minus

1198812

TE1198812infin

= 1 minus 0 = 1 (41)

For the cusped trailing edge119881TE = 0 Thus the value of 119862119901at

TE is not equal to 1 (119862119901

= 1) as shown in Figure 16

02 04 06 08 100

05 0

minus05minus10minus15minus20minus25minus30minus35

x

Cp

NACA 64012-Re = 100000-alpha = 600 inviscid

NACA 64012Thickness = 1200Max thick pos = 3740Max camber = minus000Max camber pos = 4420Number of panels = 245

Fixed speed polarReynolds = 100 000Mach = 0000NCrit = 9000Forced upper trans = 1000Forced lower trans = 1000Alpha = 600∘

C1 = 0619

Cd = 0000

Cm = 0009

Upper trans = 0000Lower trans = 0000

Figure 15 Pressure coefficient distribution over a NACA 64012airfoil at a 6∘ angle of attack obtained by XFLR5

0 02 04 06 08 1

minus1

minus2

minus3

minus4

0

1

Results from XFLR5 V610Results from our methodAirfoil NACA 64012 (using 245 points)Angle of attack 6

Cp

xc

Figure 16 Comparison between the results from the softwareXFLR5 and the results fromourmethod for validation case 1 (cuspedtrailing edge) The figure shows an excellent agreement between theresults

6 Conclusion

This paper presents a novel method to implement the Kuttacondition in the numerical solution of two-dimensionalincompressible potential flow over an airfoil The proposedmethod is based on solving the Laplacersquos equation for thestream function at each grid point generated by the ellipticgrid generation technique (O-type) Therefore it is exemptfrom considering the panels and the quantities such as thevortex panel strength and circulation used in the panel

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Journal of Aerodynamics 9

0 02 04 06 08 1

minus1

minus2

minus3

minus4

minus5

minus6

0

1

Results from referenceResults from our methodAirfoil NACA 4412Angle of attack 10

Cp

xc

Figure 13 Comparison between the results from [5] and the resultsfrom ourmethod for validation case 4The figure shows an excellentagreement between the results

minus05

minus05

0

0

05

05

1

1

15x

y

103000

102000

101000

100000

99000

98000

97000

96000

95000

94000

93000

92000

P(Pa)

Figure 14 Pressure distribution over the airfoil (NACA 64012) usedin validation case 1 (cusped trailing edge)

maximum value for 119862119901is exactly equal to 1 The pressure

coefficient at the trailing edge (TE) is equal to unity becausethe velocity is zero at this stagnation point Accordingly

119881TE = 0 119862119901= 1 minus

1198812

TE1198812infin

= 1 minus 0 = 1 (41)

For the cusped trailing edge119881TE = 0 Thus the value of 119862119901at

TE is not equal to 1 (119862119901

= 1) as shown in Figure 16

02 04 06 08 100

05 0

minus05minus10minus15minus20minus25minus30minus35

x

Cp

NACA 64012-Re = 100000-alpha = 600 inviscid

NACA 64012Thickness = 1200Max thick pos = 3740Max camber = minus000Max camber pos = 4420Number of panels = 245

Fixed speed polarReynolds = 100 000Mach = 0000NCrit = 9000Forced upper trans = 1000Forced lower trans = 1000Alpha = 600∘

C1 = 0619

Cd = 0000

Cm = 0009

Upper trans = 0000Lower trans = 0000

Figure 15 Pressure coefficient distribution over a NACA 64012airfoil at a 6∘ angle of attack obtained by XFLR5

0 02 04 06 08 1

minus1

minus2

minus3

minus4

0

1

Results from XFLR5 V610Results from our methodAirfoil NACA 64012 (using 245 points)Angle of attack 6

Cp

xc

Figure 16 Comparison between the results from the softwareXFLR5 and the results fromourmethod for validation case 1 (cuspedtrailing edge) The figure shows an excellent agreement between theresults

6 Conclusion

This paper presents a novel method to implement the Kuttacondition in the numerical solution of two-dimensionalincompressible potential flow over an airfoil The proposedmethod is based on solving the Laplacersquos equation for thestream function at each grid point generated by the ellipticgrid generation technique (O-type) Therefore it is exemptfrom considering the panels and the quantities such as thevortex panel strength and circulation used in the panel

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Journal of Aerodynamics

method It applies for both finite-angle and cusped trailingedges A novel and very easy to implement expression for thestream function for the finite-angle and the cusped trailingedges is derived The accurate results obtained for both casesshow the correctness and accuracy of the numerical scheme

Conflict of Interests

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

References

[1] J L Hess and A M O Smith ldquoCalculation of potential flowabout arbitrary bodiesrdquo Progress in Aerospace Sciences vol 8pp 1ndash138 1967

[2] J D Anderson Fundamentals of Aerodynamics McGraw-HillNew York NY USA 2001

[3] J Katz and A Plotkin Low-Speed Aerodynamics CambridgeUniversity Press New York NY USA 2001

[4] J L Hess ldquoPanel methods in computational fluid dynamicsrdquoAnnual Review of Fluid Mechanics vol 22 no 1 pp 255ndash2741990

[5] R L Fearn ldquoAirfoil aerodynamics using panel methodsrdquo TheMathematica Journal vol 10 no 4 2008

[6] J Hess ldquoDevelopment and Application of Panel Methodsrdquo inAdvanced Boundary Element Methods T Cruse Ed pp 165ndash177 Springer Berlin Germany 1988

[7] J F Thompson F C Thames and C W Mastin ldquoAutomaticnumerical generation of body-fitted curvilinear coordinate sys-tem for field containing any number of arbitrary two-dimen-sional bodiesrdquo Journal of Computational Physics vol 15 no 3pp 299ndash319 1974

[8] M W Kutta Lifting Forces in Flowing Fluids 1902[9] E L Houghton and P W Carpenter Aerodynamics For Engi-

neering Students Elsevier Science Amsterdam The Nether-lands 2012

[10] M Drela ldquoXFOIL an analysis and design system for low reyn-olds number airfoilsrdquo in Low Reynolds Number Aerodynamicsvol 54 of Lecture Notes in Engineering pp 1ndash12 Springer BerlinGermany 1989

[11] XFLR5 an analysis tool for airfoils wings and planes httpwwwxflr5comxflr5htm

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of