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Journal of Environmental Science and Engineering, 5 (2011) 1175-1182

Aerodynamic Performances of Wind Turbine Airfoils

Using a Panel Method

M.M. Oueslati1, A.W. Dahmouni

1, M. Ben Salah

2, F. Askri

3, C. Kerkeni

1and S. Ben Nasrallah

3

1. Laboratory of Wind Energy Management and Waste Energy Recovery, Research and Technology Center of Energy, Hammam Lif

2050, Tunisia

2. Laboratory of Thermal Process, Research and Technology Center of Energy, Hammam Lif 2050, Tunisia

3. Laboratory of Studies of Thermal and Energy Systems, National Engineering School of Monastir, Monastir 5019, Tunisia

Received: January 13, 2011 / Accepted: May 6, 2011 / Published: September 20, 2011.

Abstract: One of the key features of Laplaces Equation is the property that allows the equation governing the flow field to be

converted from a 3D problem throughout the field to a 2D problem for finding the potential on the surface. The solution is then found

using this property by distributing singularities of unknown strength over discretized portions of the surface: panels. Hence the flow

field solution is found by representing the surface by a number of panels, and solving a linear set of algebraic equations to determine the

unknown strengths of the singularities. In this paper a Hess-Smith Panel Method is then used to examine the aerodynamics of NACA

4412 and NACA 23015 wind turbine airfoils. The lift coefficient and the pressure distribution are predicted and compared with

experimental result for low Reynolds number. Results show a good agreement with experimental data.

Key words: Panel method, wind turbine airfoils, incompressible potential flow, pressure distribution.

1. Introduction

Knowledge of wind power technology has increased

over the years. Lanchester and Betz were the first to

predict the maximum power output of an ideal wind

turbine. The major break-through was achieved by

Glauert who formulated the Blade Element Momentum

(BEM) method in 1935 [1].

Almost design codes of today are still based on the

Blade Element Momentum method. However, this

method needs knowledge of the blade aerodynamic

which depends on the airfoil nature and his intrinsiccharacteristic. Therefore, the aerodynamic research is

today shifting toward a more fundamental approach

since the basic aerodynamic mechanisms are not fully

understood and the importance of accurate design

A.W. Dahmouni, Ph.D., research fields: fluid mechanics,

aerodynamic, wind turbine. E-mail:[email protected].

Corresponding author: M.M. Oueslati, Ph.D., researchfields: fluid mechanics, aerodynamic, wind turbine. E-mail:

[email protected].

models increases as the turbines are becoming larger.

To evaluate wind turbine performance its necessaryto study on airfoil aerodynamic characteristic such as

pressure distribution, moment coefficients, lift, and

drag forces which must required an expensive process

of testing in a wind tunnel.

To minimize the cost of experiments, many

researchers have used theoretical method to predict

airfoil performances such as the panel method.

This method provides an elegant methodology for

solving a class of flows past arbitrarily shaped bodies

in both two and three dimensions.

Panel methods were initially developed as lower

order methods for incompressible and subsonic flows.

The first successful panel method for supersonic flow

became available in the mid-1960s developed by

Woodward-Carmichael. Hess and Smith developed

together the Hess-Smith code in 1962 based on flat

constant source Panels. Woodward-Carmichael has

used into the series of computer programs known as

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Aerodynamic Performances of Wind Turbine Air foil s Using a Panel Method 1176

USSAERO in 1973. Many other numerical programs

such as SOUSSA, PAN AIR, VSAERO and

QUADPAN are developed with different boundary

condition configurations, and different numerical

techniques of resolution [2].

In the recently work, this method was used to predict

wind turbine performance. This method was used to

predict performance of the NACA 63(2)215 and the

FFA-W3-241 airfoils [3]. The results show a good

agreement with experimental data. The method has

used also to calculate the unsteady loading and radiate

noise from airfoils in incompressible turbulent flow [4].

This method is used also in the design of wind turbine

blades in order to increase the energy produced by theaero-generators. A related method has used that

imposes circulation instead of the blade load [5]. The

current design method imposes indirectly the

circulation by prescribing the pressure difference

between both sides of the blades and hence the lift.

Furthermore, the method hasused in the direct analysis

of fast panel method for blade cascades [6]. The

approach starts by assuming an initial geometry and

calculates the flow field caused by it. The differences

between the calculated flow field and the desired one

are used to modify the original geometry. In the

following, the authors will present the theoretical Hess

and Smith Panel method with constant source strength

and comparison of the theoretical result with these

obtained by experiment of the lift coefficient and the

pressure distribution over the NACA 4412, and the

NACA 23015 most airfoils types used in wind turbine.

2. Theoretical Method

The basic idea of the Panel method is to:

y Discretize the body in terms of a singularity

distribution on the body surface;

y Satisfy the necessary boundary conditions;

y Find the resulting distribution of singularity on the

surface thereby obtain fluid dynamic properties of the

flow.

The body geometry is represented in terms of

smaller subunits called panels, hence the name panel

method.

Each panel is constructed to have some kind of

singularity distribution. The singularities used can be

sources, doublets, or vortices. Depending on the

accuracy, computational speed and other factors one

can use constant, linear, parabolic, or even higher

orders of distribution of the singularity on each panel.

The number of panels that represent the body can also

be varied.

The actual singularity distribution is initially

unknown, but by enforcing the boundary conditions on

the body, it is possible to solve for them. The boundaryconditions can be represented in terms of the velocity

field, called the Neumann condition, or in terms of the

potential inside the body, called the Dirichlet

condition.

The formulation of the panel method consists in the

resolution of Laplaces equation Eq. (1) through the

superposition of simpler solutions of elementary flows

distributed throughout the body. This characteristic

makes the method fast, because it is not necessary the

discretization of all flow domains. The Laplace

equation is written as:

(1)

The total potential can be written as follow:

(2)

Where, the total potential function, the

potential of free stream, S the source distribution, and

V

the vortex distribution.

These last two distributions have potentially locally

varying strengths q (s) and (s), where s is an

arc-length coordinates which spans the complete

surface of the airfoil in any way you want.

The potentials created by the distribution of

sources/sinks and vortices are given by:

(3)

2 2

2 20

x y

+ =

S V = + +

( )ln

2S

S

q sr ds

=

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(4)

where the various quantities are defined in the Fig. 1.

The total potential can be written as:

(5)

Hess and Smith made the following valid

simplification: Take the vortex strength to be constant

over the whole airfoil and use the Kutta condition to fix

its value, while allowing the source strength to vary

from panel to panel so that, together with the constant

vortex distribution, the flow tangency boundary

condition is satisfied everywhere.

Fig. 2 illustrates the representation of a smooth

surface by a series of line segments. The numbering

system starts at the lower surface trailing edge and

proceeds forward, around the leading surface and aft tothe upper surface trailing edge. N+1 points define N

panels.

The authors can discretize Eq. (5) in the following

way:

(6)

With q (s) taken to be constant on each panel,

allowing us to write q (s) = qi, i = 1, ...N.

The flow tangency boundary condition is given by

= 0V n , and is written using the relations given here

as:

(7)

The velocity components at any point i are given by

contributions from the velocities induced by the source

and vortex distributions over each panel. The

mathematical statement is:

Fig. 1 Airfoil analysis nomenclature for panel methods.

Fig. 2 Representation of a smooth air foil with straight line

segments.

(8)

Where, qi and

are the singularity strengths, and

the usij, vsij, uvij, and vvij are the influence coefficients.

To find usij, vsij, uvij, and vvij the authors need to work

in a local panel coordinate system x*, y* which leads to

a straightforward means of integrating source and

vortex distributions along a straight line segment.

In general, if the authors locate the sources along the

x-axis at a point x = t, and integrate over a length l, the

velocities induced by the source distributions are

obtained from:

(9)

To obtain the influence coefficients, write down this

equation in the ( )* coordinate system, with q (t) = 1

(unit source strength):

( )

2V

S

sds

=

{

uniform onset flowcos sin

i s t he 2D t hi s i s a vo rt ex si ng ula ri tysource strength of strength (s)

( ) ( )ln

2 2

V x V y

q

S

q s sr ds

= +

= +

142 43 123

1 1

1 1

cos

sin

ij ij

ij ij

N N

i j s v

j j

N N

i j s v

j j

u V q u u

v V q v v

= =

= =

= + +

= + +

2 20

2 20

( )

2 ( )

( )

2 ( )

t l

st

t l

st

q t x t u dt

x t y

q t yv dt

x t y

=

=

=

=

=

+

= +

( )1panel

( )cos sin ln

2 2

N

j j

q sV x y r dS

=

= + +

( ) ( )sin cos 0i i i iu v + + =i j i j

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(10)

These integrals can be found (from tables) in closed

form:

(11)

To interpret these expressions, examine Fig. 3. The

notation adopted and illustrated in the sketch makes iteasy to translate the results back to global coordinates.

Note that the formulas for the integrals given in Eq.

(14) can be interpreted as a radius and an angle.

Substituting the limits into the expressions and

evaluating results in the final formulas for the influence

coefficients due to the sources:

(12)

Here, rij is the distance from the jth

node to the point i,

which is taken to be the control point location of the ith

panel. The angle is the angle subtended at the middle of

the ith

panel by the jth

panel.

Using the same analysis used for source singularities

for vortex singularities the equivalent vortex

distribution results can be obtained. Summing over the

panel with vortex strength of unity the authors get the

formulas for the influence coefficients due to the vortex

distribution:

(13)

Then the authors obtain a system of equations of the

form:

y*

x*lj

jj + 1

x*,y*i i

rr

ij

i,j+1

ij

l0

Fig. 3 Relations between the point x*, y* and a panel.

(14)

Which are solved for the unknown source and vortex

strengths. After manipulation and substituting

equations the authors obtain the final result as:

(15)

The remaining relation is found from the Kutta

condition. This condition states that the flow must

leave the trailing edge smoothly. In practice this

implies that at the trailing edge the pressures on the

upper and lower surface are equal. Here the authors

satisfy the Kutta condition approximately by equating

velocity components tangential to the panels adjacent

to the trailing edge on the upper and lower surface as

illustrated in Fig. 4.

Equating the magnitude of the tangential velocities

on the upper and lower surface:

(16)

This is expanded to obtain the final relation:

Fig. 4 Trailing edge panel nomenclature.

**

* 2 *20

**

* 2 *20

1

2 ( )

1

2 ( )

j

ij

j

ij

li

s

i i

li

s

i i

x tu dt

x t y

yv dt

x t y

=

+

=

+

( )1

2 2* * *2

0

** 1

*

0

1ln

2

1tan

2

j

ij

j

ij

t l

s i i

t

t l

i

s

i t

u x t y

yv

x t

=

=

=

=

= +

=

, 1*

* 0

1ln

2

2 2

ij

ij

i j

s

ij

ijls

ru

r

v

+

=

= =

**

* 2 *20

*, 1*

* 2 *20

1

2 ( ) 2

1 1ln

2 ( ) 2

j

ij

j

ij

liji

v

i i

li ji

v

i i ij

yu dt

x t y

rx tv dt

x t y r

+

=+ = +

= = +

( ), 1,

, 1

, 1

1 ,

1 1sin( )ln cos

2 2

1cos( )ln sin( )

2

sin( )

i j

ij i j i j ij

i j

Ni j

i N i j i j ij

j i j

i i

rA

r

rA

r

b V

+

++

=

= +

=

=

, 1

1

1,...N

ij j i N i

j

A q A b i N+=

+ = =

1 Nt tu u=

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(17)

To complete the system of N + 1equations, the

authors use the Kutta condition and substitute into this

expression the formulas for the velocities due to thefree stream and singularities given above.

In this case they are written as:

(18)

Substituting into the Kutta condition equation the

authors obtain:

(19)

Then the authors manipulate this expression into the

form:

(20)

which is the N + 1st equation which completes thesystem for the N + 1 unknowns.

The final equations associated with the Kutta

condition are:

(21)

(22)

(23)

The coefficients derived above provide the required

coefficients to solve a system of linear algebraic

equations for the N+1 unknowns, qi, i = 1,...,N :

(24)

To determine the pressure distribution over the

airfoil the authors must calculate the surface pressure

coefficient Cp.

At each control point, the authors substitute v.n = 0

and find ut the tangential velocity by:

(25)

Using the ( )* values of the influence coefficients,

and some trigonometric identities, the authors obtain

the final result:

(26)

The surface pressure coefficient can be found from:

(27)

3. Results and Discussion

In this section ,the authors will present some result

obtained by the developed code over the NACA 4412,

and NACA 23015 (Fig. 5). To examine the code the

1 1 1 1cos sin cos sinN N N Nu v u v + = +

1 1

1 1

1

1 1

1

1 1

1 1

1 1

cos

sin

cos

sin

j j

j j

Nj Nj

Nj Nj

N N

j s v

j j

N N

j s v

j j

N N

N j s v

j j

N N

N j s v

j j

u V q u u

v V q v v

u V q u u

v V q v v

= =

= =

= =

= =

= + +

= + +

= + +

= + +

1, 1, 1 1

1

N

N j j N N N

j

A q A b+ + + +=

+ =

1 1, ,

1, 1, 1 , 1

1

1, ,

sin( ) sin( )

1

2 cos( )ln cos( )ln

j j N j N j

N j j N j

j N j

j N j

A r r

r r

+ + +

+

=

( ) ( )1, 1 , 11, ,1, 1

1

1 1, ,

sin ln sin ln1

2cos( ) cos( )

j N jN

j N j

i j N jN N

j

j j N j N j

r r

r rA

+ +

+ +=

+ =

+ +

1 1cos( ) cos( )N Nb V V + =

, 1

1

1, 1, 1 1

1

1,...N

ij j i N i

j

N

N j j N N N

j

A q A b i N

A q A b

+=

+ + + +=

+ = =

+ =

1 1

1 1

cos sin

cos cos

sin sin

i

ij ij

ij ij

t i i i i

N N

s j v i

j j

N N

s j v ij j

u u v

V u q u

V v q v

= =

= =

= +

= + +

+ + +

( ) , 11 ,

, 1

1 ,

cos sin( ) cos( )ln2

sin( )ln cos( )2

i

Ni ji

t i i j ij i j

j i j

Ni j

i j i j ij

j i j

rqu V

r

r

r

+

=

+

=

= +

+ +

2

1 ii

t

P

uC

V

=

1 1

1 1

1

1 1

1

1 1

1 1

1 1

cos cos

sin sin

cos cos

sin sin 0

j j

j j

Nj Nj

Nj Nj

N N

j s v

j j

N N

j s v

j j

N N

j s v N

j j

N N

j s v N

j j

V q u u

V q v v

V q u u

V q v v

= =

= =

= =

= =

+ +

+ + +

+ + +

+ + + =

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authors will check the sensitivity of the solution to the

number of panels by comparing force results and

pressure distributions with increasing numbers of

panels.

Figs. 6 and 7 represent the change of lift with the

number of panels. They show that the lift becoming

Fig. 5 (a) NACA 4412; (b) NACA 23015.

Fig. 6 Change of lift with number of panels of the NACA

4412, incidence angle 4.

Fig. 7 Change of lift with number of panels of the NACA

23015, incidence angle 4 .

constant as the number of panels increase, and it

indicates that 80-100 panels (40 upper, 40 lower for

example) should be enough panels.

The sensitivity of the pressure distributions to

changes in panel density has been also investigated.

Fig. 8 shows the pressure distribution over the

NACA 4412 for 4 of incidence angle with 20 panels.

Its clear that more panels are required to define the

details of the pressure distribution. The stagnation

pressure region on the lower surface of the leading

edge is not yet distinct. The expansion peak and trailing

edge recovery pressure are also not resolved clearly.

Fig. 9 contains a comparison between 20 and 60

panel cases. In this case it appears that the pressuredistribution is well defined with 60 panels. This is

confirmed in Fig. 10, which demonstrates that it is

almost impossible to identify the differences between

the 60 and 80 panel cases.

After the examination of the convergence of the

Fig. 8 Pressure distribution for NACA 4412-4-20 panels

Fig. 9 Pressure distribution for NACA 4412-4 comparing

results using 20 and 60 panels.

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Fig. 10 Pressure distribution for NACA 4412-4

comparing results using 60 and 80 panels.

program and the dependency of the result on the panelnumber the authors will investigate the agreement with

experimental data.

Figs. 11 and 12 compare the lift coefficients from the

inviscid solutions obtained from panel with

experimental data obtained from Refs. [7, 8].

Agreement is good at low angles of attack, where the

flow is fully attached. The agreement deteriorates as

the angle of attack increases, and viscous effects start to

show up as a reduction in lift with increasing angle of

attack, until, finally, the airfoil stalls. The inviscid

solutions from panel cannot capture this part of the

physics.

The authors need to compare the pressure

distributions predicted with panel to experimental data.

Figs. 13 and 14 show comparison of experimental

pressure distribution of NACA 4412 with panel

predictions at two different angles of attack (-4) and

(1.875) for Reynolds number equal to 3,000,000 and

720,000 respectively. In general there are good

agreements between predicted and experimental data

of pressure distribution.

The primary area of disagreement is at the trailing

edge. Here viscous effects act to prevent the recovery

of the experimental pressure to the levels predicted by

the inviscid solution. The disagreement on the lower

surface is surprising, and suggests that the angle of

attack from the experiment is not precise.

Fig. 11 Comparison of panel lift predictions with

experimental data (Ref. [7]) of NACA 4412).

Fig. 12 Comparison of panel lift predictions with

experimental data (Ref. [8]) of NACA 23015).

Fig. 13 Comparison of pressure distribution predictions

with experimental data (Ref. [9]) of NACA 4412 incidence

angle (-4).

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Fig. 14 Comparison of pressure distribution predictions

with experimental data (Ref. [10]) of NACA 4412 incidence

angle (1.875).

4. Conclusion

The panel method is used to develop a numerical

code to predict airfoil performances. A comparison

between experimental data and numerical prediction of

lift coefficient and pressure distribution over NACA

4412 and NACA 23015 is made. A good agreement in

some cases for low Reynolds number is observed.

The future work consists to compute the viscous

effects on the airfoil and compare results of forces and

moment with experimental data to model the

aerodynamic stall zone.

References

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Elforsk Report, 2009.

[2] L.L. Erickson, Panel methodsAn introduction: Technical,

Report No. 2995, NASA, 1990.

[3] B. Kamoun, D. Afungchui, A. Chauvin, A wind turbine

blade profile analysis code based on the singularities

method, Journal of Renewable Energy 30 (2005) 339-352 .

[4] S.A.L. Glegg, W.J. Devenport, Panel methods for airfoils

in turbulent flow, Journal of Sound and Vibration 18 (2010)

3709-3720.

[5] B. Kamoun, D. Afungchui, M. Abid, The inverse design of

the wind turbine blade sections by the singularities method,

Journal of Renewable Energy 31 (2006) 2091-2107.

[6] J.C.C. Henriques, F. Marques da Silva, A.I. Estanqueiro,

L.M.C. Gato, Desgin of a new urban wind turbine airfoil

using a pressure-load inverse method, Journal of

Renewable Energy 34 (2009) 2728-2734.

[7] I.H. Abbott, A.E. von Doenhoff, Theory of Wing Sections,

Dover, New York, 1959.

[8] L.A. Tavares de Vargas, P.H.I. Andrade de Oliveira, R.L.

U. de Freitas Pinto, M.V. Bortoulus, M. da S. e Souza,

Comparison between modern procedures for aerodynamic

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aircraft designs, in: Proceedings of COBEM 18th

International Congress of Mechanical Engineering, 2005.

[9] R.M. Pinkerton, Calculated and measured pressure

distributions over the midsapan section of the NACA 4412

airfoil, NACA Report No. 563, 1937, p. 372.

[10] J. Stack, W.F. Lindsey, R.E. Littell, The compressibility

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forces acting on an airfoil, NACA Report No. 646, 1939, p.

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