Influence of Wing Shape on Airfoil Performance: a Comparative Study

HOCINE HARES1 , GHAZALI MEBARKI2

1Department of Mechanical Engineering, LICEGS Laboratory,

University of Batna 2 Mostefa Ben Boulaid,

53, Route de Constantine. Fésdis, Batna,

ALGERIA

2Department of Mechanical Engineering, LESEI Laboratory,

University of Batna 2 Mostefa Ben Boulaid,

53, Route de Constantine. Fésdis, Batna,

ALGERIA

Abstract: - The aerodynamic performance of an aircraft mainly depends on the lift force, drag force, and the lift

to drag ratio. The geometric shapes of aircraft wings are considered crucial for this aerodynamic performance.

The purpose of this study is to determine the most efficient wing shape that improves the aerodynamic

performance of the airfoil. For that purpose, a numerical comparative study was carried out between the

rectangular and tapered wing shapes of the NACA 4412 airfoil for a wide range of angles of attack in the

subsonic regime. ANSYS Fluent software, based on the finite volume method, was used for the numerical

resolution of the governing equations. The Realizable k-ε model was chosen for the turbulence modeling. The

numerical procedure was validated based on experimental results obtained from the literature. The results show

an improvement in the lift coefficient and a reduction in the drag coefficient of the Tapered shape compared to

the rectangular shape at all angles of attack. However, a gain was achieved in the lift-to-drag coefficient ratio of

the Tapered shape.

Key-Words: - Rectangular wing, Tapered wing, Aerodynamic performance, Drag, Lift, Airfoil.

Received: October 18, 2022. Revised: August 8, 2023. Accepted: September 19, 2023. Published: October 6, 2023.

1 Introduction

The aerodynamic performance of the airfoil has an

important influence during the development of

airplanes. The main role of the wing shape is to

generate a lift force greater than the force of gravity

and to minimize the drag force. Indeed, the

performance of an airfoil depends on its

aerodynamic characteristics, which are influenced

by the shape and size of the wings. For that purpose,

several researches have been devoted to the

optimization of lift and drag forces by modifying the

aircraft wing’s structure. The study, [1], studied the

effect of a new vortex generator configuration, delta

wing shape, placed in the suction surface of a

rectangular profile NACA 4412. The experimental

study carried out in a wind tunnel showed an

improvement in the lift coefficient with a 20%

increase and a one-degree delay in the incidence

stall. The improvement of the aerodynamic

performance of the NACA 4415 airfoil by flow

control using a passive technique was investigated

[2]. This was achieved by attaching gothic-shaped

vortex generators to the surface of the wing. The

results of the parametric study show an increase in

the lift coefficient due to vortex generators at high

angles of attack. The study, [3], experimentally

investigated the effect of surface roughness features

on a tapered NACA 4412 wing. They showed that

the best-located wing roughness features

contributing to minimum drag and maximum lift are

located between 75 % and 95 % of the mean leading

edge chord compared to the other locations. A

comparative analysis of the aerodynamic

characteristics of rectangular and curved leading

edge wing planforms of the NACA 2412 airfoil was

carried out, [4]. A rectangular shape with straight

leading and trailing edges and a curved leading edge

with a straight trailing edge were tested. Lift and

drag forces were determined over a wide range of

angles of attack. The results show that the curved

leading edge wing planform has a higher coefficient

of lift and a lower coefficient of drag. Experimental

investigations have been conducted on the

performance of the NACA 4412 wing with a curved

leading edge, [5]. A rectangular model with straight

leading and trailing edges is compared to another

model with curved leading and straight trailing

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DOI: 10.37394/232013.2023.18.5

Hocine Hares, Ghazali Mebarki

E-ISSN: 2224-347X

Volume 18, 2023

edges. It was found that the curved leading edge

wing has a higher lift coefficient and lower drag

coefficient than the rectangular leading edge wing.

The study, [6], studied, numerically and

experimentally, the improvement of the

aerodynamic performance of the NACA 4412 airfoil

through the modification of the leading edge. The

results showed an improvement in aerodynamic

performance with the curved leading edge. The

study, [7], examined the morphing wings with

upward and downward deflections of the leading

edge at different frequencies. The numerical results

show that the deflection of the leading edge has the

most significant effect on the stall characteristics

and the stall angle of attack increases due to the

downward deflection of the leading edge. For

upward deflection, the results are reversed. The

study, [8], presents the experimental study on

rough-wing and smooth-wing models using PIV and

force measurements in wind tunnels. The rough

model was based on the actual dimensions of the

fully extended swift wing and compared with a

smooth one. The experimental results showed that

the aerodynamic performance of the roughened

swift wing can be improved. An experimental

analysis of the static aerodynamic stability of

different wing planform types of the NACA 0016

airfoil was presented by, [9]. Rectangular,

rectangular with tip curved, Tapered, and Tapered

with tip curved wing were chosen for this analysis.

All wings have been tested in the wind tunnel at

various low speeds and different angles of attack.

The tapered wing with a curved tip was found to be

the most stable wing planform. The study, [10],

studied the aerodynamic characteristics and the

static stability of the wing-in-ground effect aircraft.

The effect of geometric characteristics, namely twist

angle, dihedral angle, sweep angle, and taper ratio,

was investigated. The numerical results show that

the lift coefficient increases and the maximum drag

coefficient depends on the decrease in the ground

clearance torsion angle, the dihedral angle, the angle

of attack, and the torsion angle. To reduce fuel

consumption, [11], presents a fuel-saving double-

channel wing configuration. The main objective of

this work is to improve the lift-to-drag ratio of the

wing by taking advantage of the beneficial influence

of the propeller on the wing. The numerical results

show that the proposed wing configuration increases

the lift-to-drag ratio by 13.29 % and reduces wing

drag by 10.41 %, resulting in a fuel saving of

20.15%. The study, [12], developed a new airfoil

design for an unmanned aerial vehicle by using CFD

to analyze the performance of six combined wing

designs. The results show that the best blended-

winglet configuration is the 0.3 taper ratio

combination, as it improves the average lift-to-drag

ratio by 9.84 % while reducing the average wingtip

vortex by 17 %. The study, [13], studied the

improvement of the aerodynamic performance of

fixed-wing unmanned aerial vehicles operating in

the low-speed subsonic regime. Based on the results

of the aerodynamic investigations, an in-depth

review of drag reduction technologies is carried out

based on the existing literature, and the most

promising technologies are proposed. The study,

[14], used the 'wing smarting' approach to study the

effects of twist angle variation on aerodynamic

coefficients and the flow field. A specific range of

angles of attack and twist angles was investigated.

The results show that the aerodynamic efficiency is

relatively related to the increase in twist angle and

improves over a wide range of angles of attack.

In the present study, a numerical study is carried

out to investigate the effect of wing shape on the

performance of the NACA 4412 airfoil. For this

purpose, two wing shape models (rectangular and

conical) were tested. The aim was to compare the

effectiveness of the deformation structure on the

aerodynamic profiles.

2 Mathematical Formulation

In this study, the fluid was assumed to be

Newtonian and incompressible. The fluid flow has

been considered stationary, three-dimensional, and

turbulent. The governing equations are given by:





(1)

()

j i j

j i j j

U U U

X X X X







   

   

(2)

The Reynolds stress equation is given by

()



   



i j t i j

U U k

   

(3)





 

(4)

For turbulence modeling, the Realizable k-ε

turbulence model was chosen. It is characterized by

the following equations:

( ) ( ) ( )

j j K j

k b M k

k kU

t X X X

P P Y S



   





   

  



   





    

(5)

1 2 1 3

( ) ( ) ( )

j j j

t X X X

C S C C C P S



   



   



 



   

  



   





   



(6)

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The constant values are given by:



=1.44,

=1.9,



=1,





=1.2

The Aerodynamic coefficients are given by:

( ) / ( / 2)

C P P U





(7)

/ ( / 2)

C F SU





(8)

/ ( / 2)

C F SU





(9)

For the boundary conditions, at the inlet; the

velocity is assumed to be equal to 34 m/s,

corresponding to a 2.17×105 Reynolds number.

At the outlet, according to the fully developed flow

regime, the following parameters are imposed:

0, 0, 0

U P T

y y y

  

  

  

(10)

At the top and bottom of the domain symmetrical

conditions are imposed.

3 Profile Shape Design Overview

SolidWorks software was used to design the

NACA4412 profile for both configurations

(Rectangular and Tapered shapes). The chord length

for the rectangular wing is 0.1 m, while for the

Tapered wing, two different chords are used, 0.1 m

and 0.025 m. Both shapes have a wingspan of 0.15m

(Figure 1 and Figure 2).

Fig. 1: Rectangular planform design

Fig. 2: Tapered planform design

4 Numerical Resolution Procedure

Fluent software, based on the finite volume method,

was used to numerically solve the above

mathematical equations. The SIMPLE algorithm

was used to resolve the pressure-velocity coupling.

In contrast, the pressure-based solver, the standard

pressure interpolation scheme, and the implicit

formulation method were chosen. The second-order

upwind scheme was adopted in the momentum

equation discretization to obtain more accurate

results.

4.1 Mesh Independence Study

A mesh-independence study was performed to select

an optimal number of elements to ensure that the

solution obtained is mesh-independent. The

independence of the mesh size has been evaluated

by varying the lift, drag, and lift-to-drag ratio

coefficients. For this purpose, two different types of

mesh (tetrahedral and polyhedral) with different

degrees of refinement have been tested. The details

of the tetrahedral mesh are given in Table 1. The

conversion of tetrahedral meshes to polyhedral

meshes was evaluated to reduce the number of

elements in the tetrahedral mesh. Polyhedral mesh

details are given in Table 2.

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Table 1. Tetrahedral mesh details

Types

Elements

numbers

Nodes

numbers

Tetrahedral 1

1466189

256514

Tetrahedral 2

2170055

378944

Tetrahedral 3

3492362

603809

Tetrahedral 4

9092704

1552911

Table 2. Conversion of tetrahedral to polyhedral

meshes

Types

Elements

numbers

Nodes

numbers

Polyhedral 1

248969

1406517

Polyhedral 2

393796

2259657

Polyhedral 3

619195

3596170

Polyhedral 4

1569187

9240635

Figure 3, Figure 4, and Figure 5 show the lift, drag,

and lift-to-drag ratio coefficient variations with the

angle of attack obtained by numerical simulation

compared to those obtained by, [1].

Fig. 3: Lift coefficient for different mesh sizes

compared with the results of, [1]

Fig. 4: Drag coefficient for different mesh sizes

compared with the results of, [1]

Fig. 5: Lift-to-drag coefficient for different mesh

sizes compared with the results of, [1]

A good agreement is observed between the

experimental results of, [1], and the refined

polyhedral meshes of 2 mm and 3 mm. However,

the 3 mm refined mesh is adopted because it

contains fewer elements than the 2 mm refined mesh

for optimum computation time. The domain used in

this study is shown in Figure 6 and the chosen mesh

is shown in Figure 7 and Figure 8.

Fig. 6: Control domain dimension in (mm)

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Fig. 7: The used mesh

(a)

(b)

Fig. 8: Polyhedral mesh: (a) Rectangular planform,

(b) Tapered planform

4.2 Turbulence Modeling

Two well-known turbulence models (k-ε Realisable

and Spalart-Allmaras) were tested to determine the

best turbulence model. The numerical results were

compared with those obtained by, [1].

It is clear from Figure 9, Figure 10 and Figure 11

that a good agreement was obtained with the k-ε

Realizable turbulence model. Consequently, our

numerical procedure was validated and the

Realizable k-ε turbulence model was adopted in the

present study.

Fig. 9: Lift coefficient for different turbulence

models compared to, [1] results

Fig. 10: Drag coefficient for different turbulence

models compared to, [1] results

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Fig. 11: Lift-to-drag coefficient for different

turbulence models compared to, [1] results

5 Results and Discussion

The lift and drag coefficients as a function of angle

of attack for the rectangular and tapered planform

wings are shown respectively in Figure 12 and

Figure 13. It is clear that both shapes have the same

drag coefficient. However, the lift coefficient of the

tapered wing shape is higher than that of the

rectangular shape, especially at high angles of attack

(from 20° to 25°). The aerodynamic performances

have been improved by the tapered wing shape at all

angles of attack (0° to 25°) compared to the

rectangular wing shape, as shown by the lift-to-drag

coefficient ratio (Figure 14).

Fig. 12: Lift coefficient for rectangular and tapered

wing planforms

Fig. 13: Drag coefficient for rectangular and tapered

wing planforms

Fig. 14: Lift-to-drag coefficient ratio for rectangular

and tapered wing planforms

Velocity contours around tapered and

rectangular wings at different angles of attack and

for two wing chord positions (20 % and 80 %) are

shown in Figure 15. It can be seen that at zero

angles of attack, the same velocity contours are

observed for both positions, with a very small

variation. As the angle of attack increases, the

recirculation zone (flow separation) at the trailing

edge of the wing becomes significant, especially at

large positions (80%). At low positions (20%), the

recirculation zone is reduced for tapered wings.

Pressure contours around tapered and

rectangular wings at different angles of attack and

for two wing chord positions (20 % and 80 %) are

shown in Figure 16. At zero angle of attack the

same tendency is observed for both wing shapes.

However, the pressure distribution is strongly

affected at high angles of attack. In this case, lower

pressures on the upper wing surface and higher

pressures on the bottom wing surface are obtained

for tapered wings.

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Fig. 15: Velocity contours around tapered and

rectangular wings for different angles of attack and

at two wing chord positions (20 % and 80 %).

Fig. 16: Pressure contours around tapered and

rectangular wings for different angles of attack and

at two wing chord positions (20 % and 80 %).

Tapered wing at 20 % of the chord.

Tapered wing at 80 % of the chord.

Rectangular wing at 80 % of the chord.

Rectangular wing at 20 % of the chord.

Angle of attack = 0°

[m/s]

Tapered wing at 20 % of the chord.

Tapered wing at 80 % of the chord.

Rectangular wing at 80 % of the chord.

Rectangular wing at 20 % of the chord.

Angle of attack = 8°

[m/s]

Angle of attack = 16°

Tapered wing at 80 % of the chord.

Tapered wing at 20 % of the chord.

Rectangular wing at 80 % of the chord.

Rectangular wing at 20 % of the chord.

[m/s]

Angle of attack = 20°

Tapered wing at 80 % of the chord.

Tapered wing at 20 % of the chord.

Rectangular wing at 80 % of the chord.

Rectangular wing at 20 % of the chord.

[m/s]

Angle of attack = 24°

Tapered wing at 80 % of the chord.

Tapered wing at 20 % of the chord.

[m/s]

Rectangular wing at 20 % of the chord.

Rectangular wing at 80 % of the chord.

Angle of attack = 0°

Tapered wing at 80 % of the chord.

Tapered wing at 20 % of the chord.

[Pa]

Rectangular wing at 20 % of the chord.

Rectangular wing at 80 % of the chord.

Angle of attack = 8°

[Pa]

Tapered wing at 20 % of the chord.

Tapered wing at 80 % of the chord.

Rectangular wing at 20 % of the chord.

Rectangular wing at 80 % of the chord.

Angle of attack = 16°

Tapered wing at 20 % of the chord.

Tapered wing at 80 % of the chord.

Rectangular wing at 20 % of the chord.

Rectangular wing at 80 % of the chord.

[Pa]

Rectangular wing at 80 % of the chord.

Rectangular wing at 0 % of the chord.

Tapered wing at 20 % of the chord.

Tapered wing at 80 % of the chord.

Angle of attack = 20°

[Pa]

Rectangular wing at 80 % of the chord.

Rectangular wing at 0 % of the chord.

Tapered wing at 20 % of the chord.

Tapered wing at 80 % of the chord.

Angle of attack = 24°

[Pa]

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6 Conclusion

In this paper, a numerical comparative study of the

shape geometry was performed of the NACA 4412

airfoil for two configurations, the rectangular and

the Tapered planforms. The numerical procedure

using Fluent software has been validated by the

experimental results. The numerical results show

that the tapered wing improves the lift coefficient

and reduces the drag coefficient at high angles of

attack (from 20° to 25°) compared to the rectangular

wing. In addition, the improvements in aerodynamic

performance at all angles of attack (from 0° to 25°)

are shown by the lift-to-drag ratio for the tapered

wing shape. In addition, alternative aircraft design

techniques such as roughened surfaces, ailerons, and

multi-element wings could be explored to improve

the aerodynamic performance of airfoils. It is also

worth investigating these techniques to determine

the most cost-effective and size-appropriate method.

Alternatively, vortex generators on aircraft wings

are suggested to improve their aerodynamic

performance.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Ghazali Mebarki, Methodology and initial design.

- Hocine Hares, Simulation and interpretation.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflict of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

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