Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 2 Year 2008

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NUMERICAL SOLUTIONS FOR POTENTIAL FLOW PASTS
2 – D LIFTING AIRFOILS

Hayder Aziz Neema
Kufa University, Faculty of Engineering

Abstract
The ability of the panel methods in solving the potential flow problem around 2-D lifting

airfoils is tested in this study. The approximate solutions for potential flow pasts thick, thin,
symmetrical, and non-symmetrical airfoils have been calculated by using panel methods and then
compared with either the exact analytical solution or the numerical solution obtained by using a
perturbation method. As a results, the linear-varying strength vortex method is seemed to be the
better one in precision in solving the all four problems.

األبعادحلول عددية لجريان جهدي حول سطوح رفع هوائية ثنائية

حيدر عزيز نعمة
جامعة الكوفة-الهندسة آلية تدريسي في

الخالصة

الحلول . األبعاد رفع هوائية ثنائية أسطحهذا البحث يفحص قابلية طرق الصفيحة في حل مشكلة الجريان الجهدي حول
حسبت باستخدام طرق الصفيحة و من و غير متناظرة ةمتناظر هوائية سميكة و نحيفة و أسطحالتقريبية للجريان الجهدي حول

آنتيجة نهائية طريقة الدوامة ذات الشدة . رابما مع الحل التحليلي الحقيقي او مع حل عددي باستخدام طريقة االضط ثم قورنت إ
.األربعةالخطية بدت األحسن بالدقة في حل آل المسائل

Nomenclature

PC pressure coefficient
L.E leading edge
T.E trailing edge
c airfoil chord
x horizontal coordinate
α angle of attack
ε thickness ratio

Subscripts
i collocation point
j singularity element
n number of panels
∞ free stream

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181

Introduction
Due to the applicability of the results of the potential flow problem solution in airfoil design,

this problem has received a good deal of attention. Also, due to the phenomenal growth in the use of
computers, a method that is valid for completely arbitrary geometry is needed. Thus, panel methods
have come to stay. In the panel methods the airfoil contour has been approximated by inscribed
polygon. That is, the contour has been approximated by a number of straight-line segments. Each
segment is distributed by singularities of certain kind that have an undetermined strength. These
singularities deflect the oncoming stream so that it will flow around the airfoil. The requirement the
oncoming flow be tangent to every segment at a particular location ( collocation point ) gives a set
of equations which is used to compute the singularities strength. Thus the flow consisting of a
uniform flow and the flow induced by the singularities on a finite number of segments becomes
determined and the velocity and the pressure at any point in the flow field can be calculated. The
requirement of the flow must be tangent to the segments at the collocation points leads to the
normal component of the flow must be zero at these collocation points. This is the physical
boundary condition (that is, the normal component of the fluid velocity must be zero on the solid
boundaries). Therefore, two boundary conditions can be used to solve the Laplace's equation and
they are; the Neumann B. C. which utilizes the physical B. C. directly, and the Dirichlet B. C.
which specifies the velocity potential on the boundaries so that indirectly zero normal flow will be
met.

Hess and Smith [1] is a review of such methods. Both source and vortex distributions have
been used. The two-dimensional surface source method [1] approximates the body contour by an
inscribed polygon. The number and distribution of the surface elements determine the accuracy of
the numerical solution. The surface source strength is assumed to be constant on each element, and
the zero normal velocity boundary condition is applied at the midpoint of each element (collocation
point) to produce a set of linear algebraic equations for the values of the source strength on the
elements. For lift problem an auxiliary constant-strength vortex singularity element has been
distributed. The total strength of this distribution is adjusted to specify the Kutta condition, which
requires the equality of the surface pressures at the collocation points adjacent to the trailing edge.
Some difficulties have been encountered for airfoils that have very thin trailing edges. Moreover,
the difficulties have been made more severe by the use of the relatively small element numbers that
are appropriate for the three-dimensional cases. Even in the two-dimensional cases reduced element
numbers are becoming more desirable to conserve computing time in multielement airfoil cases.
One technique for increasing the computational accuracy is the so-called higher-order formulation,
which is described in [2] for non-lifting case. While higher-order surface singularity distributions
for lifting cases is described in [3].

Formulation of The Problem and its Numerical Solution
The general problem of calculating the incompressible, 2-D potential flow requires the
solution of Laplace's equation ( 02 =Φ∇ ) for the velocity potential Φ , with boundary conditions of
zero normal velocity component on all solid surfaces (Vn=0), and zero velocity perturbation at large
distances from the airfoil (V∞=Constant). Auxiliary conditions (Kutta conditions) are also imposed
to fix the value of the circulation about the airfoils.

The numerical solution can be constructed by six ordinary steps, and they are:
1-Selection of the Singularity Element. The first and one of the most important decisions is the
type of singularity element or elements that will be used. This includes the selection of source,
doublet, or vortex representation and the method of discretizing these distributions (constant, linear,
or quadratic). Six singularity distributions have been used in this study and they are; constant-
strength doublet, constant-strength vortex, combined constant-strength source and doublet, linear-
varying strength doublet, linear-varying strength vortex, and quadratic-varying strength doublet and
for more details it is preferred to return to [4].

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2-Descritization of Geometry. Once the basic solution element is selected, the geometry of the
problem has to be subdivided such that it will consist of those basic solution elements. In this
generation process, the elements’ end points and the collocation points are defined. The collocation
points are points where the boundary conditions, such as the zero normal flow to a solid surface,
will be enforced.
3- Influence Coefficients. The influence coefficient ija is the normal component of the flow
velocity induced by a unit strength element j at a collocation point i. In this phase, for each of the
elements an algebraic equation (based on the boundary condition) is derived at the collocation
point.
4- Establish the Right- Hand Side (RHS). The RHS of the matrix equation is the known portion
of the free-stream velocity or the potential and requires mainly the computation of geometric
quantities.
5- Solve Linear set of Equations. The influence coefficients and the RHS equations are obtained in
the previous steps and now the equations are solved by a standard matrix technique (Gauss-
elimination technique).
6- Secondary Computations. The solution of the matrix equation results in the singularity
strengths and the velocity field and any secondary information can be computed. The pressure will
be computed by the Bernoulli’s equation, and the loads and the aerodynamic coefficients will be
computed by adding up the contributions of the elements.
Actually the results of the first four steps mentioned above is the following system:

⎥
⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

⎥
⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

⎥
⎥
⎥
⎥
⎥
⎥
⎥

⎦

⎤

⎢
⎢
⎢
⎢
⎢
⎢
⎢

⎣

⎡

nnn3n2n1

2n232221

1n131211

RHS
.
.

RHS

σ
.
.

...aaaa
.
.

...aaaa

....aaaa

Where aij is the influence coefficient

j
σ is the singularity unknown strength

The above system of equation have been solved by using the Guassian elimination method
for

j
σ

Fortran 77 programming language have been used to solved the problem, just few second
have been consumed to get upon the final results.

The Airfoils Used in the Study

Four types of airfoils were used in this study in solving the thickness, thinness, symmetry,
and non-symmetry problems and they are:
A-Van de Vooren airfoil with non-cusped trailing edge: To illustrate the effectiveness of the
methods used in solving the problem for ordinary shapes of airfoils with finite trailing edge
angles.

Non-cusped T.E
L.E

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183

B- Van de Vooren airfoil with cusped trailing edge: To illustrate the effectiveness of the methods
used in solving the problem for ordinary shapes of airfoils with zero trailing edge angles.

C- Symmetric Joukowski airfoil: To illustrate the effectiveness of the methods used in solving the
problem for thin airfoils with cusped trailing edge.

D- Cambered Joukowski airfoil: To illustrate the effectiveness of the methods used in solving the
problem of camber for thin airfoils with cusped trailing edge.

Numerical Results and Discussion
The pressure distribution for the seven surface singularity methods on a thick Van de Voorn
airfoil with non-cusped trailing edge and with cusped trailing edge are shown in Fig. (1) and
Fig. (2) respectively. The agreement between the analytical solution [5] and the numerical solution
of the panel methods for the first case is excellent as shown in Fig. (1) espcialy for (n=60). That is
due to the collocation points position be near the surface area of the airfoil where the analytical
solution have been calculated (i.e, in the panal method, the collocation points are some where inside
the airfoil counter and they will be nearest to the airfoil counter as the number of the used panel
have been increased). Some deviation of the numerical solutions from the analytical solution
appears near the trailing edge as shown in Fig. (2), and that is due to the very thin region there
(cusped) and a very high interaction of adjacent singularities will appear.
Depending on the earlier results, the pressure distributions for two surface singularity
methods on thin symmetrical and non-symmetrical (cambered) airfoils are shown in Fig. (3) and
Fig. n(4) respectively. Generally, very good agreement between the solutions obtained by these
methods and the solutions obtained by method of [6].

cusped T.E

Non-cusped T.E

cusped T.E

Camber line

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184

CONCLUSIONS
1-The methods depending upon the Dirichlet B.C. are better than the methods depending upon the
Neumann B.C. in the solution of the potential flow around thick airfoils without cusped trailing
edges, especially the constant-strength doublet method because of the ease construction, low
computational effort, and good results even for low density of panels.
2-The linear-varying strength vortex method with the Neumann B.C. and the constant-strength
doublet, combined constant-strength source and doublet, and quadratic-varying strength doublet
methods with Dirichlet B.C. can be used to solve the potential flow problem around symmetric
thick and thin airfoils with cusped trailing edges. But only the linear-varying strength vortex
method with Neumann B.C. and the combined constant-strength source and doublet method with
Dirichlet B.C. can be used to solve the potential flow problem around asymmetric and thin airfoils
with cusped trailing edges.
3-From the above conclusions, the decision of the type of the elementary solution distribution and
the boundary condition selection depends upon the case under testing.

EFERENCESR

1. J. L. Hess and A. M. O. Smith, “ Calculation of Potential Flow about Arbitrary
Bodies,”in: Prog. in Aeronautical Sciences 8(Pergamon, New York, 1966).

2. J. L. Hess, “ Higher-Order Numerical Solution of the Integral Equation for 2-D
Neumann Problem,” Computer Methods in Applied Mechanics and Engineering 2 (1973) 1-15.

3. J. L. Hess, “ The Use of Higher-Order Surface Singularity Distribution to Obtain
Improved Potential Flow Solutions for 2-D Lifting Airfoils,” Computer Methods in Applied
Mechanics and Engineering 5 (1975) 11-35.

4. Aimen R. Noor, “ The Use of Panel Method in the Prediction of the Potential Flow
Around 2-D Configurations,” M.Sc. thesis, Babylon University, 2001.

5. J. Katz, and A. Poltkin, “Low-Speed Aerodynamics from Wing Theory to Panel
Methods,” Mc Graw Hill, Inc. 1998.

6. T. C. Wong, C. H. Liu, and J. Geer,“ Comparison of Uniform Perturbation and Numerical
Solutions for Some Potential Flows Past Slender Bodies,” Computers & Fluids, Vol. 13, No. 3, pp.
271-283, 1985.

Al-Qadisiya Journal For Engineering Sciences Vol. 1 No. 2 Year 2008

185

(g)

Fig. 1 Pressure coefficient distribution for a 15% thick Van de Vooren airfoil with non-
cusped trailing edge at an angle of attack (α) of 5ο.

(c)

Linear-varying strength doublet
method with Dirichlet B.C.

Analytical solution

10 panels

30 panels

60 panels

0.0 0.5 1.0
-1.8
-1.5
-1.2
-0.9
-0.6
-0.3
0.0
0.3
0.6
0.9
1.2

0.0 0.5 1.0
x/c

-1.8
-1.5
-1.2
-0.9
-0.6
-0.3
0.0
0.3
0.6
0.9
1.2

x/c x/c

0.0 0.5 1.0
x/c

-1.8
-1.5
-1.2
-0.9
-0.6
-0.3
0.0
0.3
0.6
0.9
1.2

Constant-strength doublet
method with Neumann B.C.

Constant-strength vortex
method with Neumann B.C.

Linear-varying strength vortex
method with Neumann B.C.

(a)

Combined constant-strength source and
doublet method with Dirichlet B.C.

(b)

Constant-strength doublet
method with Dirichlet B.C.

(d) (e) (f)

Quadratic-varying strength
doublet method with Dirichlet B.C.

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186

(g)

Fig. 2 Pressure coefficient distribution for a 15 % thick Van de Vooren airfoil with cusped
trailing edge at an angle of attack (α) of 5ο.

0.0 0.5 1.0
x/c

-1.9
-1.6
-1.3
-1.0
-0.7
-0.4
-0.1
0.2
0.5
0.8
1.1

0.0 0.5 1.0
x/c

-1.9
-1.6
-1.3
-1.0
-0.7
-0.4
-0.1
0.2
0.5
0.8
1.1

0.0 0.5 1.0
x/c

-1.9
-1.6
-1.3
-1.0
-0.7
-0.4
-0.1
0.2
0.5
0.8
1.1

Analytical solution

40 Panels

60 Panels

Constant-strength doublet
method with Neumann B.C.

Constant-strength vortex
method with Neumann

B.C.

Linear-varying strength vortex
method with Neumann B.C.

(a)
Combined constant-strength source and

doublet method with Dirichlet B.C.

(b)
Constant-strength doublet
method with Dirichlet B.C.

method with Dirichlet B.C.

(d) (e) (f)
Quadratic-varying strength
doublet method with Dirichlet B.C.

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187

(a) (b)

Fig. 3 Pressure coefficient distribution for a thin (ε = 0.01624) and symmetric Joukowski
airfoil with cusped trailing edge at α = 6ο.

(a) (b)
Fig. 4 Pressure coefficient distribution for a thin (ε = 0.02688) and cambered Joukowski

airfoil with cusped trailing edge at α = 0ο.

0.0 0.5 1.0
x/c

-7.0

-5.0

-3.0

-1.0

1.0

0.0 0.5 1.0
x/c

-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0

0.0 0.5 1.0
x/c

-7.0
-6.0
-5.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0

0.0 0.5 1.0
x/c

-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0

Perturbation method

40 Panels

50 Panels

60 PanelsLinear-varying strength vortex
method with Neumann B.C.

Combined constant-strength
source and doublet method with
Dirichlet B.C.

Quadratic-varying strength doublet
method with Dirichlet B.C.

Constant-strength doublet method
with Dirichlet B.C

Perturbation method

40 Panels

50 Panels

60 Panels
Linear-varying strength vortex
method with Neumann B.C.

Combined constant-strength
source and doublet method with
Dirichlet B.C.

Quadratic-varying strength doublet
method with Dirichlet B.C.

Constant-strength doublet method
with Dirichlet B.C