panel method for mixed configurations with finite thickness with zero thickness

8/20/2019 Panel Method for Mixed Configurations With Finite Thickness With Zero Thickness

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Panel method for mixed congurations with nite thicknessand zero thickness

José M. Ezquerro, Victoria Lapuerta n, Ana Laverón-Simavilla, José M. García, Taisir Avilés

Universidad Politécnica de Madrid, Plaza de Cardenal Cisneros, 3, 28040 Madrid, Spain

a r t i c l e i n f o

Article history:

Received 23 July 2013

Received in revised form10 January 2014

Accepted 10 April 2014Available online 8 May 2014

Keywords:

Aerodynamics

Panel method

Mixed conguration

Potential ow

a b s t r a c t

Panel methods are well-known methods for solving potential uid ow problems. However, mixed

congurations of obstacles with nite thickness and zero thickness have not been solved with these

methods. Such congurations arise naturally in delta wings, sailing boats, and even in complete aircraftaerodynamics. In this work, a new numerical approach is proposed for solving 2D mixed congurations

of obstacles with nite thickness and zero thickness. The method is based on the Dirichlet and Neumann

formulations and is checked by comparison with analytical results.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Methods for solving potential uid ows are frequently used in

engineering practices, mainly for preliminary design. Although non-

potential models like CFDs (Computational Fluid Dynamics) are wide-

spread, they require long calculation times and the results are notalways reliable so inputs from potential methods are often needed.

One of the most developed potential methods is the panel

method [1–3]. The main advantage of this method is that it

reduces the dimension of the problem by one order, so the

numerical cost is very low compared with non-potential methods.

The panel methods allow one to calculate numerically the

solution of any given problem as long as the velocity potential

satises the Laplace equation. There has been much work and

many numerical codes based on panel methods [4–11] since the

pioneering work of Hess and Smith [4].

The panel method based on Green's formula was rst intro-

duced in the work of Morino and Kuo [5], in which the primary

unknown was the velocity potential. There are two main formula-

tions of panel methods based on Green's formula: Neumann andDirichlet [3]. The Dirichlet formulation solves the Laplace equation

numerically and the velocity potential is obtained. However,

with the Neumann formulation only differences of potential are

obtained. Dirichlet formulation is more stable, more suitable to

numerical computation than Neumann formulation and leads to

numerical errors of smaller order of magnitude.

However, these formulations cannot be directly applied to

mixed congurations of both nite and zero thickness, such as

the mast and the sail of a sailing boat, traf c signs [12], delta wings

or congurations like Gurney aps [13]. In [14] a distribution of

vortices is used to model the surface of a mast and sail and sources

and sinks are used to represent the ow separation, which is justan empirical t to data from wind tunnel and is not related to the

subject of this paper. Nevertheless, in this work we show that

methods based on discrete vortex do not recover correctly the ow

around mixed two-dimensional congurations.

In this paper a numerical scheme for mixed two-dimensional

(2D) congurations of nite and zero-thickness bodies is pre-

sented. This method does not introduce spurious singularities in

the numerical resolution and gives very good precision results,

even for very thin airfoils or airfoils with cusped trailing edge.

The paper is organized as follows: in Section 2 Dirichlet and

Neumann formulations are reviewed. In Section 3 a new numerical

scheme for mixed two-dimensional (2D) congurations of nite

thickness and zero-thickness bodies is presented. This formulation

is also applicable to bodies without thickness, providing moreprecise results than the discrete vortex solution. In Section 4 an

analytical solution is calculated in order to check the results

and the precision of the numerical method. Section 5 is dedicated

to the analysis of the results and, nally, in Section 6 the main

conclusions are extracted.

2. Review of Dirichlet and Neumann formulations

In this section Dirichlet and Neumann formulations are reviewed

and reformulated.

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/enganabound

Engineering Analysis with Boundary Elements

http://dx.doi.org/10.1016/j.enganabound.2014.04.011

0955-7997/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.

E-mail addresses: [email protected] (J.M. Ezquerro),

[email protected] (V. Lapuerta),

[email protected] (A. Laverón-Simavilla), [email protected] (T. Avilés).

Engineering Analysis with Boundary Elements 44 (2014) 28–35

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2.1. Dirichlet formulation for nite-thickness bodies

The formulation is based on Green's integral equation [3],

Φð x pÞ ¼

Z Σ B

Φ∇Φm n ds

Z Σ W

ðΦþ Φ Þ∇Φm n ds þΦ1 ð1Þ

where Φ is the velocity potential at a generic point P , of

coordinates x p, outside the body (see Fig. 1), n is the normalvector dened such that it always points into the uid region, Φ1is the potential of the stationary uid far enough from the body,

Φþ is the velocity potential on the upper side of the discontinuitysurface, Σ w, and Φ

on the lower one. Φm is the potential at x pdue to sources located at the body or at the discontinuity surface

with unit strength and ∇Φm n is the potential at x p due todoublets located at the body or at the discontinuity surface with

their axes perpendicular to the Σ B or Σ w surfaces. The internalpotential inside the body has been considered zero. The velocity

potential dened by Eq. (1) already fullls the zero normal velocity

at the body boundary (∂Φ=∂n ¼ 0).Eq. (1) represents the velocity potential Φ of a distribution of

doublets on both the surface of the body and the discontinuity

surface, with axis n and intensities Φ and Φþ Φ , respectively.The solution of this integral equation is obtained by letting x ptend to Σ B.

The basic idea of this method consists in solving Green's

integral equation by discretization of the body: the body is

replaced by N straight panels, see Fig. 2, and it is assumed that

the velocity potential Φ is constant on each panel, Φ j, whichcorrespond to a constant distribution of doublets along the panel.

Each panel is dened by two limiting nodes placed on the body

surface, whose coordinates are ð x j; z jÞ. In the middle point of each

panel a collocation point is placed, ð xcp j ; z cp j Þ. The numbering of the

panels is in a clockwise sense, panel N ¼1 being the rst panel of

the lower surface starting from the trailing edge of the obstacle.

The discontinuity surface is modeled as a single panel of innite

length (it is numbered panel N þ 1).

In this formulation the unknown variables of the problem arethe velocity potential values on each panel, Φ j. The equationswhich have to be solved are obtained by particularizing Eq. (1) on

the collocation points. The problem is reduced to an algebraic

system of equations:

Φk ¼ Φ1k ∑N

j ¼ 1

Φ j2π

Z panel j

x x cpk j x x cpk j

2

n ds ΦN Φ1

2π

Z panel N þ 1

x x cpk j x x cpk j

2 n ds ð2Þ

where Φ1k is the potential of the stationary uid far enough fromthe body calculated in the collocation points. Using an angular

reference parallel to the discontinuity surface panel and a local

frame attached to each panel (see Fig. 3), Eq. (2) can be reformu-

lated as

Φk ¼ Φ1k þ ∑N

j ¼ 1

Φ j2π

ðθ jkF θ

jkI Þ þ

ΦN Φ12π

θ 1k ð3Þ

where θ kF j , θ kI

j and θ 1k are dened in Fig. 3. Note that the velocity

potential of each panel, given by the term ðΦ j=2π Þðθ jkF θ

jkI Þ in

Eq. (3), does not introduce any discontinuity in the uid ow.

It introduces a discontinuity segment, the straight line between

the two nodes j and j þ1, which models the surface of the body.

The velocity potential value changes from Φ j=2 in the uppersurface to Φ j=2 in the lower surface of the panel since θ

jkF θ

jkI

jumps from π to π . As mentioned above, in order to solve Eq. (1),the point x p, in the ow outside the body, tends to the collocation

points over the body, therefore the potential on the upper surface

of the panel, Φ j=2, which is the potential of the outer ow around

the discretized body, is selected hereafter. The only discontinuityline in the uid domain is the discontinuity surface which starts in

the trailing edge of the prole and ends at innity.

2.2. Neumann formulation for zero-thickness bodies

Nowadays, the potential methods most widely used to solve

zero-thickness bodies are the vortex-lattice method for 3D con-gurations and the discrete vortex method for 2D congurations

[3]. These methods can be derived from the Neumann formulation,

but some information of Green's integral is lost in the derivation

and the errors are larger than those obtained using the Dirichlet

formulation.

Here Neumann equations are reformulated by modeling the

zero-thickness bodies by panels with two wet faces (see Fig. 4),as a degenerate boundary [15].

Eq. (3) leads to

2πΦk ¼ 2πΦ1k þ ∑ j ¼ N

j ¼ 1

Φ jðθ jkF θ

jkI Þ þðΦN Φ1Þθ

1

k ð4Þ

and expanding the summation and the discontinuity term,

∑ j ¼ N

j ¼ 1

Φ jðθ jkF θ


1

k ¼Φ1ðθ 1kF θ

1kI θ

1

k Þ

þΦN ðθ N kF θ

N kI þθ

1k Þ þ ∑

j ¼ N =2

j ¼ 2

Φ jðθ jkF θ

jkI Þ þ ∑

j ¼ N 1

j ¼ N =2 þ 1

Φ jðθ jkF θ

jkI Þ

ð5Þ

Fig. 1. Fluid domain in Green’s integral equation.

Fig. 2. Discretization of geometry for a nite-thickness obstacle.

Fig. 3. Denition of θ kF

j , θ kI

j and θ 1k

.

J.M. Ezquerro et al. / Engineering Analysis with Boundary Elements 44 (2014) 28– 35 29


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By using,

θ 1kI þθ 1k ¼ 0 ð6Þ

θ N kF þθ 1

k ¼ 0 ð7Þ

θ jkI ¼ θ N þ 1 jkF ð8Þ

and being γ j the potential jump between upper and lower surfacesof the same panel,

γ j ¼ Φ j ΦN þ 1 j ð9Þ

then, Eq. (5) can be rewritten as

∑ j ¼ N

j ¼ 1

Φ jðθ jkF θ


1k ¼ γ N θ

N kI þ ∑

j ¼ N 1

j ¼ N =2 þ 1

γ jðθ jkF θ

jkI Þ

ð10Þ

By introducing Eq. (10) in Eq. (4) and imposing the boundary

condition over the body at the collocation point k,

∇Φk nk ¼ 0 ð11Þ

this leads to

0 ¼ 2π ∇Φ1k nk γ N θ 0N kI þ ∑

j ¼ N 1

j ¼ N =2 þ 1

γ jðθ 0 jkF θ

0 jkI Þ ð12Þ

where

∇ x k θ j

k nk ¼ θ 0 j

k ð13Þ

and the Kutta condition,

γ N þ 1 γ N ¼ 0 ð14Þ

has been imposed.

Renaming the panels in the sum in order to begin from the

leading edge of the airfoil, reordering terms in Eq. (12) and

changing the dummy index j to l ¼ j N =2 we nally have the

algebraic system of equations that allows one to obtain M values of

the jump of the potential. It is remarkable that with Neumann

conditions the N unknowns of the potential cannot be obtained,

but only M ¼N /2 unknowns: the jump of potential between the

faces of the panels, γ l:

γ M θ 0M kI ∑

l ¼ M 1

l ¼ 1γ lðθ

0lkF θ

0lkI Þ ¼ 2π ∇Φ1k nk ð15Þ

3. Method for mixed nite-thickness and zero-thickness

bodies

In this section a new formulation is developed to solve zero-

thickness bodies and mixed congurations with both nite-

thickness and zero-thickness bodies.

3.1. Mixed Dirichlet –Neumann formulation for zero-thickness bodies

Here we obtain a new formulation for zero-thickness bodies.

First, Eq. (3) is duplicated and applied at the collocation points k

situated on the lower surface, krN =2, and upper surface, k4N =2:

Φk ¼ Φ1k þ ∑N =2

j ¼ 1 ja k

Φ j2π

ðθ jkF θ

jkI Þ þ ∑

N

j ¼ N =2 þ 1 jaN þ 1 þ k

Φ j2π

ðθ jkF θ

jkI Þ

þΦk2

þΦN þ 1 k

2 þ

ΦN Φ12π

θ 1k ; krN

2 ð16Þ

Φk ¼ Φ1k þ ∑N =2

j ¼ 1 jaN þ 1 þ k

Φ j

2π

ðθ jkF θ

jkI Þ þ ∑

N

j ¼ N =2 þ 1 ja k

Φ j

2π

ðθ jkF θ

jkI Þ

þΦk2

þΦN þ 1 k

2 þ

ΦN Φ12π

θ 1k ; k4N

2 ð17Þ

setting,

l ¼ N þ 1 j ð18Þ

and introducing it into the rst sum of Eqs. (16) and (17) we have

Φk ¼ Φ1k þ ∑N =2 þ 1

l ¼ N la k

Φl2π

ðθ lkF θ

lkI Þ þ ∑

N

j ¼ N =2 þ 1 jaN þ 1 k

Φ j2π

ðθ jkF θ

jkI Þ

þΦk2

þΦN þ 1 k

2 þ

ΦN Φ12π

θ 1k ; krN

2 ð19Þ

Φk ¼ Φ1k þ ∑

N =2 þ 1

l ¼ N la k

Φl2π ðθ

l

kF θ l

kI Þ þ ∑

N

j ¼ N =2 þ 1 ja k

Φ j2π ðθ

j

kF θ j

kI Þ

þΦk2

þΦN þ 1 k

2 þ

ΦN Φ12π

θ 1k ; k4N

2: ð20Þ

Reversing the order in the rst sum of Eqs. (19) and (20), taking

into account

θ jkF θ jkI ¼ ðθ

N þ 1 jkF θ

N þ 1 jkI Þ; ð21Þ

noting that k is the same point on the lower and upper surfaces

and identifying the corresponding potentials on the upper surface

(Φþk ) and lower surface (Φk ), then become

Φk ¼Φ1k þ ∑N

j ¼ N =2 þ 1 ja k

Φþ j Φ

j

2π ðθ

jkF θ

jkI Þ

þΦk

2 þ

Φþk2

þΦN Φ1

2π θ

1

k ð22Þ

Φþk ¼Φ1k þ ∑N

j ¼ N =2 þ 1 ja k

Φþ j Φ

j

2π ðθ jkF θ

jkI Þ

þΦþk

2 þ

Φk2

þΦN Φ1

2π θ

1

k ð23Þ

Finally, using γ j ¼ Φþ

j Φ

j , as dened in the previous section,

we have

Φk ¼Φ1k þ ∑M

j ¼ 1 jak

γ j2π

ðθ jkF θ

jkI Þ

γ j2

þγ M 2π

θ 1k ð24Þ

Φþk ¼Φ1k þ ∑M

j ¼ 1 jak

γ j2π

ðθ jkF θ jkI Þ þ

γ j2

þγ M 2π

θ 1k ð25Þ

This system needs the Neumann formulation for zero-thick-

ness bodies, formulated in the previous section, so the combined

equation system is composed by Eqs. (15), (24) and (25).

The velocity on each side of the body can be calculated using

V 7i ¼Φ7i 1 Φ

7

i

c i; i ¼ 2;…N ; ð26Þ

where c i is the length of each panel. The pressure coef cient can

be calculated on the lower and upper surfaces with

c 7 pi ¼ 1 V 7i

V 1 2

: ð27Þ

Fig. 4. Discretization of geometry for a zero thickness obstacle.

J.M. Ezquerro et al. / Engineering Analysis with Boundary Elements 44 (2014) 28– 3530


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3.2. Formulation for mixed nite-thickness and zero-thickness bodies

The geometry of the body is shown in Fig. 5. The zero-thickness

part of the body is modeled by M panels numbered from the

trailing edge of the nite-thickness airfoil towards the trailing

edge of the airfoil itself and with normal vectors pointing

upwards. The nite-thickness part is divided into N panels

numbered clockwise and starting from the lower surface trailing

edge. The normal vector for the panels in the zero-thickness part

points towards the upper surface. The discontinuity surface is

taken as panel N þM þ1 and its normal vector is oriented upwards.

As in the previous case, a collocation point is placed at the center

of each panel.

The solution of the coupled problem is achieved by calculating

the potential jump at each of the panels with Eq. (15) and equating

the potential of the inner surface of the nite-thickness panels to

the internal potential, that is, zero. Finally, a set of algebraic

equations is obtained,

γ N þ M θ 0N þ M kI ∑

j ¼ N þ M 1

j ¼ 1

γ jðθ 0 jkF θ

0 jkI Þ ¼ 2π ∇Φ1k nk ;

k ¼ N þ 1;…N þ M ð28Þ

∑ j ¼ N þ M

j ¼ 1 ja k

γ j2π

ðθ jkF θ

jkI Þ

γ k2

þγ N þ M

2π θ 1k ¼ Φ1k ; k ¼ 1;…N : ð29Þ

In matrix form,

½ A jk

fγ jg ¼ fbg ð30Þ

The potential of each panel is easily obtained with Eq. (25), andtaking into account the denition of the jump of the potential,

Eq. (9), we have

for k ¼ 1;…; N

Φþk ¼ γ k ð31Þ

Φk ¼Φi ¼ 0 ð32Þ

and for k ¼ N þ1;…; N þM ;

Φþk ¼Φ1k þ ∑ j ¼ N þ M

j ¼ 1 ja k

γ j2π

ðθ jkF θ

jkI Þ þ

γ k2

þγ N þ M

2π θ

1

k ð33Þ

Φk ¼Φþk γ k ð34Þ

4. Analytical solution

In order to validate the numerical scheme proposed for mixed

zero-thickness and nite-thickness congurations an analytical

expression is obtained for the conguration presented at the top of

Fig. 6, which consists of a curved airfoil and a camber line. Next,

the complete transformation outlined in Fig. 6 is described.

First, to obtain curved airfoils, a generalized Karman–Trefftz

transformation is used to map a symmetric conguration formed

by a circular body and a at plate (t -plane) into a curved airfoil and

a camber line (s-plane).

The whole conguration in the t -plane is rotated an angle, β , about the origin,

β ¼ arctan aδ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

R2 ðaδ Þ2q

0B@

1CA ð35Þ

and translated to point t 0 ¼ að λþiδ Þ, where δ is the camberparameter of Karman–Trefftz transformation, λ is the thickness

parameter of Karman–Trefftz transformation, R is the radius of the

circular body and a ¼ R=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ λÞ2 þδ 2

q . Then, the Karman–Trefftz

transformation leads to

s¼ ka1þc

1c ð36Þ

where

c ¼ ðte i β þt 0Þ a

ðte i β þt 0Þ þa

!kð37Þ

and 1rkr2.

The second transformation, a Joukowski transformation from

the t -plane to the τ -plane, maps the circle and zero-thickness plateonto a single plate (see Fig. 6),

τ ¼ t þR2

t

l2

2ðRþlÞ ð38Þ

The length of the plate in the τ -plane is

L ¼4RðRþlÞ þl2

Rþl ð39Þ

The third transformation is a Joukowski transformation from

the τ -plane to the Ω-plane, the inverse of which is

τ ¼ Ωþ ρ2

Ω ð40Þ

with ρ ¼ L=4.The circle theorem is used and an appropriate vortex intensity

is included to obtain the complex velocity potential in Ω plane,

g ð ΩÞ ¼ U 1 Ωe iα þ

ρ2

Ω eiα

þ

iΓ

2π log Ω ð41Þ

Γ ¼ 4πρU 1 sin α ð42Þ

Finally, the conjugate velocity is

dg

ds¼

dg

d Ωd Ω

dτ dτ

dt

dt

ds¼

dg

d Ω1

dτ

d Ω

dτ

dt

1

ds

dt

ð43Þ

where

dτ

d Ω ¼ 1

ρ2

Ω2 ð44Þ

dτ

dt ¼ 1

R2

t 2 ð45Þ

ds

dt ¼ 2ka

ððte i β þt 0Þ aÞk 1

ððte i β þt 0Þ þ aÞ3

e i β ð46Þ

5. Results and discussion

Fig. 7 compares the pressure coef cient from the analytical

solution described in Section 4 with the pressure coef cient

obtained with the numerical scheme proposed in Section 3. As

this gure shows, the agreement is very good even with only 59

panels for the body and 31 for the plate. The amount of panels in

Fig. 5. Mixed conguration (thick and non-thick) panel discretization.



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the surfaces of the thick prole and the camber line is such that

their lengths are approximately the same.To check the convergence of the method, the error in the

circulation, Γ , has been calculated. Note that the lift is propor-tional to the circulation. Fig. 8 shows the relative error of the

numerical method as a function of N , being Γ num ¼ ΦN Φ1 . Theresults are very good. For example, for N ¼100, the relative error is

0.4% and for N ¼500 the error is 0.07%.



obtained with the numerical scheme proposed in Section 3 and

with the result obtained using the discrete vortex method. All

the calculations of the discrete vortex method presented in the

paper have been performed using the standard method [3] (vortex

placed at a quarter of the panel chord and collocation point

at three quarters of the panel chord). The error in the pressure

coef cient is calculated as error¼jc p;numerical c p;analyticalj and is

plotted in Fig. 10 computed with each numerical method for anincreasing number of panels. These gures show that:

(i) The error in our method is signicantly smaller than the one

in discrete vortex method, and the difference is more impor-

tant for a small number of panels. Increasing the number of

panels in the discrete vortex method does not reduce the

error to the level found in our method. For example, in the

case of Fig. 10(c), the mean relative error with respect to

the analytical solution is: 1% in the upper surface and 0.2% in

the lower surface calculated with our method, and 16% in the

upper surface and 10% in the lower surface calculated with

the discrete vortex method.

(ii) When the number of panels increases, the error of the discrete

vortex method decreases, but this method never recovers

Fig. 6. Complete conformal mapping transformation.



6/8

correctly the behavior in the junction between the nite

thickness and zero-thickness elements. The maximum error in

the junction obtained with our method in case (a) is 0.4559 for

C þ

p and 0.8388 for C

p , in case (b) is 0.3641 for C þ

p and 0.8354

for C p , and in case (c) is 0.3131 for C þ

p and 0.6121 for C

p . The

maximum error in the junction obtained with the discrete

vortex method in case (a) is 3.143 for C þ p and 4.367 for C

p , in

case (b) is 3.423 for C þ p and 5.069 for C

p , and in case (c) is 2.406

for C þ p and 3.819 for C

p . Our method does signicantly better in

the junction even with a small number of panels.

Fig. 7. Comparison of the pressure coef cient obtained with the analytical solution

of Section 4 and the mixed Dirichlet–Neumann numerical approach proposed in

Section 3 (D–N) for N ¼59, M ¼32. The parameters of the conguration are α ¼121

and l/R ¼7.

Fig. 8. Relative error in the circulation for the numerical approach proposed in

Section 3. The parameters of the conguration are α ¼121 and l/R¼7.

Fig. 9. Pressure coef cient obtained with the analytical solution of Section 4, the

mixed Dirichlet–Neumann numerical approach proposed in Section 3 (D–N) and

the discrete vortex method for N ¼49, M ¼17. The data of the conguration are

α β ¼ 21, l/R¼3, k ¼1.8, λ¼0.05, δ ¼0.3.

Fig. 10. Error in the pressure coef cient calculated with the mixed Dirichlet–

Neumann numerical approach proposed in Section 3 (D–N) and the discrete vortex

method for (a) N ¼49, M ¼17, (b) N ¼94, M ¼ 32 and (c) N ¼146, M ¼49. The data of

the conguration are α β ¼ 21, l/R¼3, k ¼1.8, λ¼0.05, δ ¼0.3.



7/8

(iii) Our method ts much better than the discrete vortex method

with the analytical solution in the nite thickness part. This is

because the order of magnitude of the error in the discrete

vortex method is the same as the order of magnitude in

the Neumann method error, which is higher than the order of

magnitude in the Dirichlet method error.

Fig. 11 shows the pressure coef cient obtained using thenumerical approach of Section 3 for a Karman–Trefftz body with

a cosine-shaped camber line. Notice that in the lower surface the

angle in the junction is greater than 1801, and therefore the

velocity is innite, whereas in the upper surface the angle is

smaller than 1801 and the velocity is zero. The method recovers

correctly this behavior.



obtained with the numerical scheme proposed in Section 3 for a

very thin airfoil. The prole used is a Karman–Trefftz, with

k ¼1.915, which gives an airfoil with 10.8% of maximum thickness

and a trailing edge angle of 15.31. The pressure coef cient is

calculated for an angle of the incident ow of 51. We have used 60

panels in the airfoil and 35 panels in the tail. As can be seen the

numerical solution ts extremely well with the analytical exact

solution even for this very thin airfoil. It is well known that the

discrete vortex method is not applicable to very thin congura-

tions, as this one.

The method here described is also capable of providing good

results even for congurations with cusped trailing edges as can

be seen in Fig. 13. In this gure the pressure coef cient computed

Fig. 11. Pressure coef cient calculated with the mixed Dirichlet–Neumann numer-

ical approach proposed in Section 3 (D–N) for N ¼59, M ¼46. The data of the

conguration are α β ¼ 21, k ¼1.8, λ¼0.19, δ ¼0.2.

Fig.12. Pressure coef cient calculated with the analytical solution of Section 4 and

the mixed Dirichlet–Neumann numerical approach proposed in Section 3 (D–N) for

N ¼60, M ¼35. The data of the conguration are α β ¼ 51, l/R¼5, k ¼1.915, λ¼0.04,

δ ¼0.

Fig.13. Pressure coef cient calculated with the analytical solution of Section 4, the

mixed Dirichlet–Neumann numerical approach proposed in Section 3 ( D–N) and

the discrete vortex method for (a) N ¼31, M ¼11, (b) N ¼80, M ¼ 26 and (c) N ¼153,

M ¼ 48. The data of the conguration are α β ¼ 21

, l/R¼3, k ¼2, λ ¼0.2, δ ¼0.3.



8/8

with the discrete vortex method and our method is compared to

the pressure coef cient obtained with the analytical solution, for

an increasing number of panels. The error in the calculation of the

pressure coef cient obtained with the numerical scheme and the

discrete vortex method is shown in Fig. 14, for an increasing

number of panels. These gures show that:

(i) Our method ts much better than the discrete vortex method

with the analytical solution. In fact, in the discrete vortex

method the error increases when the number of panels

increases and this method does not converge to the analytical

solution. The maximum error in the junction obtained with

our method in case (a) is 0.1787 for C þ

p and 0.0970 for C

p , incase (b) is 0.2313 for C þ p and 0.1256 for C

p , and in case (c) is

0.2368 for C þ p and 0.1457 for C

p . The maximum error in the

junction obtained with the discrete vortex method in case

(a) is 2.732 for C þ p and 6.603 for C

p , in case (b) is 4.183 for C þ

p

and 2.456 for C p , and in case (c) is 13.367 for C þ

p and 5.583

for C p .

(ii) Our method ts very well with the analytical solution even for

a low number of panels. For increasing number of panels our

method converges to the analytical solution, and the method

performs well even near the junction.

6. Conclusions

In this work a new formulation for combined nite-thickness

and zero-thickness bodies has been developed. This formulation

has been tested by comparison with analytical solutions and gives

very good agreement even for very thin airfoils and airfoils with

cusped trailing edge. The convergence of this new formulation has

also been tested.

The method can be very useful for preliminary design in all

kinds of problems that combine both nite-thickness and zero-

thickness bodies; these include sailing boats, Gurney ap cong-

urations, and the study of realistic aircraft aerodynamics.

References

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Fig. 14. Error in the pressure coef cient obtained with the mixed Dirichlet–

Neumann numerical approach proposed in Section 3 (D–N) and the discrete vortex

method (a) N ¼ 31, M ¼11, (b) N ¼80, M ¼ 26 and (c) N ¼153, M ¼48. The data of the

conguration are α β ¼ 21, l/R¼ 3, k ¼2, λ¼0.2, δ ¼0.3.

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panel method for mixed configurations with finite thickness with zero thickness

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