desktop application to plot and simulate single and

DESKTOP APPLICATION TO PLOT AND SIMULATE SINGLE AND MULTI-ELEMENT AEROFOILS FOR APPLICATION IN FORMULA VEHICLE AERODYNAMICS A project submitted by

DIPAK KUMAR SISODIYA (UR13AE021) in partial fulfilment for the award of the degree of

BACHELOR OF TECHNOLOGY in

AEROSPACE ENGINEERING under the supervision of Dr. PRADEEP KUMAR

SCHOOL OF CIVIL AND MECHANICAL SCIENCES KARUNYA UNIVERSITY (Karunya Institute of Technology and Sciences) (Declared as Deemed-to-be-under Sec-3 of the UGC Act, 1956) Karunya Nagar, Coimbatore - 641 114. INDIA

APRIL 2017

II

BONAFIDE CERTIFICATE Certified that this project report “DESKTOP APPLICATION TO PLOT AND SIMULATE SINGLE AND MULTI-ELEMENT AEROFOILS FOR APPLICATION IN FORMULA VEHICLE AERODYNAMICS” is the bonafide work of “DIPAK KUMAR SISODIYA (UR13AE021)”who carried out the project work under my supervision.

SIGNATURE

Dr. P.D. Arumairaj Professor and Head of the Department School of mechanical and civil sciences

SIGNATURE

Dr. Pradeep Kumar Professor of Aerospace engineering Department of aerospace engineering School of mechanical and civil sciences

Submitted for the (Phase I/Phase II/Full Semester/Half Semester)* Viva Voce held on ………………………. Internal Examiner

External Examiner

III

ABSTRACT In the current formula sports, front and rear wings alone amount for two third of the total downforce created by the vehicle. This downforce in turn increases the cornering speed of the car while decreasing the turn radius. The project aims to create a platform which can effectively plot aerofoils from a data file series for use primarily in formula cars. It will also calculate the flow properties around the wing.

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ACKNOWLEDGEMENT I thank all the men of science who since time immortal have contributed to the betterment of humanity by their continuous selfless work. I wish to thank the management of Karunya University and our Vice Chancellor, Dr. S. Sundar Manoharan, Ph.D and Registrar, Dr. C. Joseph Kennady , Ph.D. for extending all facilities. I would also like to thank Dr. P.D. Arumairaj, HOD, School of Civil and Mechanical Sciences for extending the academic facilities. I also take this opportunity to express my profound gratitude and deep regards to my guide Dr. Pradeep kumar, Professor, Department of Aerospace Engineering, for his guidance, monitoring and constant encouragement throughout the course of this thesis.

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Title

Page No.

BONAFIDE CERTIFCATE ABSTRACT TABLE OF CONTENTS LIST OF FIGURES LIST OF SYMBOLS AND ABBREVIATIONS 1. INTRODUCTION

1

1.1. Importance of wings in formula one

1

1.2. Basic definitions

2

1.3. Aim & Objective

3

2. 2D VORTEX PANEL METHOD

4

2.1. Introduction to panel method

4

2.2. Panel distribution

4

2.3. Boundary condition

5

2.4. Linear equation

6

3. LIFTING LINE THEORY

7

3.1. Finite and Infinite wings

7

3.2. Lifting line theory

7

3.2.1. Important points in lifting line theory

7

3.2.2. Trailing vortices and downwash

8

3.3. Input data

8

3.4. Summary

9

4. GROUND EFFECT

10

4.1. Definition

10

4.2. Ground effect calculations using python code

10

5. RESULTS - VORTEX PANEL METHOD OVER AN AIRFOIL

12

6. RESULTS – FINITE WING

16

VI

6.1. Introduction

16

6.2. Lift curve for finite wing

16

6.2.1. Input parameters

16

6.2.2. Lift curve

17

6.3. Lift curve with respect to aspect ratio

18

6.3.1. Input parameters

18

6.3.2. Coefficient of lift and drag at different aspect ratio

19

7. RESULTS – MULTI ELEMENT AIRFOIL

21

7.1. Introduction

21

7.2. Application of panel method

21

7.3. Lift curve

21

7.4. Variation of Cl with distance from trailing edge

22

8. DISCUSSION OF RESULTS

24

8.1. 2D Panel Method

24

8.2. Ground effect

24

8.3. Lifting Line Theory

24

8.4. Panel Method on Two Element Airfoil

25

9. CONCLUSION AND FUTURE WORK

26

9.1. Remarks

26

9.2. Future Work

26

REFERENCES

28

APPENDICES

29

VII

LIST OF FIGURES

Page No

Figure 2.1: Distribution of panels over an airfoil

4

Figure 2.2: Boundary conditions over the panels

5

Figure 2.3: Liner equation matrix

6

Figure 3.1: Vortex generation over a finite wing

7

Figure 4.1: Method of image using python code

11

Figure 5.1: Coefficient of lift for different number of panels at AOA 3

12

Figure 5.2: Coefficient of lift for different number of panels at AOA 9

13

Figure 5.3: Lift curve for different number of panels

14

Figure 5.4: Lift curve for different h/c

15

Figure 6.1: Lift curve for finite wing

17

Figure 6.2: Coefficient of drag along different angle of attacks

17

Figure 6.3: Lift curve for finite wing at aspect ratio 4

18

Figure 6.4: Lift curve for finite wing at different AR

19

Figure 6.5: Coefficient of drag along different angle of attacks

19

Figure 7.1: Lift curve for two element airfoil

22

Figure 7.2: Cl along w.r.t y distance from leading edge

22

VIII

LIST OF SYMBOLS AND ABBREVIATIONS ρ - Density in Kg/m3 ʎ - Taper ratio V – Velocity α – Angle of attack σ – Strength of source γ – Strength of vortex A – Source contribution coefficient B – Vortex contribution coefficient Г(y) – Strength of circulation at y L(y) – Lift at y Wy – Downwash velocity Dv – Drag due to vortex Cp - Co-efficient of pressure Cd - Coefficient of drag AOA - Angle of attack in degrees AR – Aspect ratio 2D – Two dimensional

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CHAPTER 1 INTRODUCTION 1.1 IMPORTANCE OF WINGS IN FORMULA ONE The difference between designing a championship challenging formula car and a tail ender in the sports of formula one is aerodynamics. In fact, aerodynamics plays such a vital role in motorsports that the FIA (Governing body of formula one) had to limit the amount of time to be spent on aerodynamic testing so as to ensure that the sports remain driver oriented. Although always important in race car design, aerodynamics became a truly serious proposition in the late 1960s when several teams started to experiment with the now familiar wings. Race car wings operate on exactly the same principle as aircraft wings, only in reverse. Planes use their wings to create lift while race cars use theirs to create negative lift or downforce. A modern Formula One car (such as the VJM10 by Force India) is capable of developing 3.5 g lateral cornering force (three and a half times its own weight) thanks to aerodynamic downforce. That means that, theoretically, at high speeds they could drive upside down. The most obvious aerodynamic devices on a Formula One car are the front and rear wings, which together account for around 60 percent of overall downforce (with the floor responsible for the majority of the rest). These wings are fitted with different profiles depending on the downforce requirements of a particular track. Tight, slow circuits like Monaco require very aggressive wing profiles to maximise downforce, whilst at high-speed circuits like Monza the amount of wing is minimised to reduce drag and increase speed on the long straights. By the mid-1970s, engineers had found out the effect of downforce on the performance of the formula car. This virtually gave teams such as Lotus a few seconds faster lap timing. Soon after rule changes followed to limit the benefits of ground effects. Sport regulators have time and again tightened the rules around the body work and wing placement limiting the overall effect of aerodynamics on the race results.

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As a result, today’s aerodynamicists have considerably less freedom than their counterparts from the past, with strict rules dictating the height, width and location of bodywork. However, with every additional kilogram of downforce equating to several milliseconds of lap time saved, the teams still invest considerable amounts of time and money into wind tunnel programmes and computational fluid dynamics (CFD) – the two main forms of aerodynamic research. This report basically concerns around the calculation of aerodynamics forces around a wing in ground effect using a rather simpler approach known as the panel method. 1.2 SOME BASIC DEFINITIONS 1. Lift: Lift is a mechanical aerodynamic force produced by the motion of the airplane through the air. The primary source of lift generation in an airplane is the wings. 2. Downforce: a downwards force created by the aerodynamic characteristics of a car. The purpose of downforce is to allow a car to travel faster through a corner by increasing the vertical force on the tires, thus creating more grip. 3. NACA airfoils: The NACA airfoils are airfoil shapes for aircraft wings developed by the National Advisory Committee for Aeronautics (NACA). The shape of the NACA airfoils is described using a series of digits following the word "NACA". The parameters in the numerical code can be entered into equations to precisely generate the cross-section of the airfoil and calculate its properties. 4. Ground Effect: In fixed-wing aircraft, ground effect is the increased lift (force) and decreased aerodynamic drag that an aircraft's wings generate when they are close to a fixed surface. When landing, ground effect can give the pilot the feeling that the aircraft is "floating".

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1.3 AIM & OBJECTIVE AIM: Development of a desktop application to create and simulate single and multielement airfoils for formula vehicle aerodynamics and high lift applications. OBJECTIVE: 1. DESIGN OBJECTIVE: 1.1. Airfoil plotting by user data (in form of .txt or .dat file) 1.2. Airfoil plotting of NACA 4 and 5 series 2. SIMULATION OBJECTIVE: 2.1. Calculation of Coefficient of lift, drag, pressure, friction etc. 2.2. Post Processing Visual aid. 2.3. Ground Effect calculation and variation of airfoil properties according to it. In the first chapter, an introduction was given so as what to this project was aimed at. The aim and objective as given above will be tackled in the chapters to come one by one. The programming language Python is used for the coding purposes. The second chapter will give an introduction to 2D vortex panel method, which is used in the code to calculate the flow velocity around a 2D dimensional airfoil, and eventually calculate the coefficient of lift. The third chapter is an extension of the lift calculation of flow around 2D airfoil into a simple three dimensional wing. For the sake of simplicity, lifting line theory was used to calculate the reduction in coefficient of lift due to induced drag generated by a finite wing. The subsequent chapter talks about what ground effect is and how it affect the performance of an infinite wing.

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CHAPTER 2 2D VORTEX PANEL METHOD 2.1 INTRODUCTION: The aerodynamic properties of most 2D airfoils can be predicted quite accurately using 2D inviscid panel method or vortex panel method. This approach is applicable for any arbitrary 2D body and is simpler in approach as compared to other more complex methods such as computational fluid dynamics (CFD). The vortex panel method not only increases the overall simplicity of the calculations but also provides great flexibility in testing by considerably reducing the overall time required to get the results. 2.2 PANEL DISTRIBUTION: The first step in the solution of the potential function is to divide the airfoil into small panels. Many different method are available to give the distribution of panels over an airfoil. In our case, to generate the panels, the airfoil is first plotted by importing its points in x, y format from a .DAT extension file. Then, a circle is plotted with its diameter equal to the chord length of the airfoil such that the leading and trailing edge of the airfoil touch the circle at two ends. The circle is then superimposed onto the airfoil by generating the y coordinates using interpolation. These coordinates are then used to generate the panels by defining a function. These steps can be clearly seen in the code in the Appendix .

Fig 2.1

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2.3 BOUNDARY CONDITIONS: There are two boundary conditions to be followed to satisfy the flow over the airfoil. The boundary conditions are as follows: 1.

The flow through the surface i.e. Vn = 0

2.

Kutta condition at the trailing edge.

For N number of panels, there would be N+1 number of variables to be solved, the boundary condition of no flow through the surface is applied at the centre of each panel thereby giving N equations in N+1 unknowns. The (N+1)th unknown is found by applying kutta condition at the trailing edge of the airfoil.

Fig 2.2

The diagram (Fig 2.2) shows an airfoil divided into N panels with N+1 points where the solution of the flow is to be determined.

6

2.4 LINEAR EQUATION: The above two boundary conditions give rise to a system of linear equations which allow the solution for the required distribution of strengths to be found. The system of linear equations are given in matrix form as follows:

Fig 2.3

Once the values of strengths are determined, these values are used to find the tangential velocity at each point. The coefficient of pressure can be found using the formula CP =

1

The coefficient of lift is then calculated by integrating the coefficient of pressure acting in the y direction. The above process has been converted into a Python code which asks the user to input the location of the airfoil data file. The code then automatically creates the panels with optimum number of panels. The angle of attack is asked next, after which the code gives the Cl value as an output. The output also contains a Cp V/S x graph and pressure contour over the airfoil. The code for 2D vortex panel over an airfoil is provided in Appendix . Aero-python course by Professor Lorena A. Barba has been the primary source in the preparation for the above code.

7

CHAPTER 3 LIFTING LINE THEORY FOR FINITE WING 3.1 FINITE AND INFINITE WINGS The major difference between an infinite (2D airfoil) and finite wing is the induced drag generated in a finite wing at its tips due to the trailing vortices. This induced drag thereby reduces the overall lift coefficient of the wing.

Fig 3.1 As can be seen from the figure 3.1, trailing vortices are created at the wing tip of a finite wing. These vortices induce what is known as vortex drag or induced drag. 3.2 LIFTING LINE THEORY: The Prandtl Lifting line theory is a method to predict lift distribution over a 3D wing based on its shape. It is also known as the Lanchester–Prandtl wing theory. This method basically uses the concept of circulation and kutta Joukowski theorm. L(y) = ρVГ(y)

This method is basically applicable for following configuration of wings 1. Unswept or slightly swept wings. 2. Wings with Aspect ratio more than or equal to 4. 3.2.1 IMPORTANT POINTS IN LIFTING LINE THEORY 1. Place the bound vortex system at the quarter chord line.

8

2. The strength of the bound vortex system at any spanwise location Г(y) is proportional to the local lift acting at that location l(y). 3. At any spanwise location y, the sum of the strengths of all of the vortex filaments in the bundle at that station is Г(y). 4. The strengths of the trailing vortex at any location y is equal to the change in the strength of the bound-vortex system. 3.2.2 TRAILING VORTICES & DOWNWASH The strength of the trailing vortex is given by ∆Г =

Г

∆

The resultant induce velocity at y1 due to all the trailing vortices is known as the downwash is given by Г

Wy = +

This resultant velocity or downwash has a downward direction and has the effect of titling the undisturbed airflow. The drag force is a consequence of the lift developed by a finite wing and is termed vortex drag or the induced drag or the drag due to the lift. This lift at any point y is given by, l(y) = ρ∞V∞ Г(y) and the vortex drag is given by, dv(y) = -ρ∞ Wy Г(y) integrating over the entire span of the wing, the total lift is given by, L=

ρ∞V∞ Г y

and the total vortex drag is given by, Dv = -

ρ∞W y Г y dy

3.3 INPUT DATA 1. Angle of attack alpha, span b, planform area s. 2. Root chord, tip chord and the taper ratio ʎ.

9

3. The lifting curve slope at both root and tip along with their zero lift angle of attack. 3.4 SUMMARY The lifting line theory is a simple method to predict lift characteristics if the user is equipped with the above mentioned data. To this date, although being a simplified method, the lifting line theory is used for preliminary aircraft design considerations. For this project too, the theory proved sufficient to account for the induced drag. The code generated for lifting line theory takes input in the form of the above mentioned data. As far as the lifting curve slope is concerned, that value solely depends on the user, whether he wants to take the value provided by the vortex panel method from the previous chapter or from experimental data.

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CHAPTER 4 GROUND EFFECT 4.1 DEFINITION: Ground effect is the increased lift (force) and decreased aerodynamic drag that an aircraft's/formula car’s wings generate when they are close to a fixed surface. When landing, ground effect can give the pilot the feeling that the aircraft is "floating". In cars, Ground effect is a term applied to a series of aerodynamic effects used in car design, which has been exploited to create downforce, particularly in racing cars. In our case, we are going to look into what changes do ground effect have in creating the required downforce for the formula car. But before we look into calculations involving ground effect, there are a few assumptions which are obvious with the above mentioned definition of ground effect, these are 1. The coefficient of lift for the airfoil will increase as the distance between the airfoil and ground decreases. 2. The drag (induce) is decreased due to the ground effect. 3. The optimum height above the ground can be found by varying the height of as a function of the chord length (basically non dimensional variable).

4.2.1 GROUND EFFECT CALCULATIONS USING PYTHON Ground effect can be seen as the flow around a 2D airfoil in the vicinity of a line at which the normal velocity is zero. This line is treated as the ground in our case. To actually incorporate this condition into real calculations, we do something known as mirror imaging of the body with respect to a line which in effect is our ground.

11

Fig 4.1 Consider the above diagram with an airfoil at 5 degree angle of attack. In case of ground effect, we create a mirror image of the airfoil along the x axis. The calculations are done for two airfoils instead of one in this case using the same python code. Their effect on each other give rise to the ground effect. The results for ground effect have been discussed in the coming sections.

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CHAPTER 5 RESULTS – VORTEX PANEL METHOD OVER AN AIRFOIL 5.1 OPTIMUM NUMBER OF PANELS Deciding upon the number of panels which would give values as close to the experimental results as possible has been the first puzzle to be solved. For a preliminary investigation, the angle of attack was fixed along with the airfoil to be used (NACA2412). Afterwards, the number of panels were varied for the given airfoil and angle of attack. The results have been tabulated in the graph given below (Figure 5.1).

Cl

Cl vs number of panels at AOA of 3 degrees 0.6 0.58 0.56 0.54 0.52 0.5 0.48 0.46 0.44 0.42 0.4

Cl Calculated Cl at AOA 3(Experimental)

0

20

40

60 80 number of panels N

100

120

Fig 5.1 An interesting thing can be noted here, the coefficient of lift does not wary linearly with the increase in number of panels. In fact, the value of cl reached a peak value before decreasing to further down continually. Finally it is established that the value of coefficient of lift is varying with the change in the number of panels. Looking into the experimental data, we realise that the coefficient of lift for NACA2412 at 3 degree angle of attack is 0.53 (approx.). Looking into the above graph, it can be seen that the optimum number of panels which are required to get a value close to the experimental data is 50.

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So we figured out the number of panels required for a particular angle of attack. But what about a higher angle of attack, say 9 degrees. The graph below shows the same graph as fig 5.1 but with angle of attack to be 9 degrees.

Cl Vs Number of panels at AOA 9 Degrees 1.34 1.32

Cl at AOA 9 Calculated

1.3

Cl at AOA 9 (Theortical)

1.28

Cl

1.26 1.24 1.22 1.2 1.18 1.16 1.14 0

20

40

60

80

100

120

140

160

N

Fig 5.2 The number of panels to satisfy the calculated cl with the experimental cl have drastically increased with an increase in the angle of attack.

The correct number of panels to be put hence solely depends on the intuition of the user of the code. In order to facilitate this process. A lift curve is shown below (fig 5.3) with different number of panels. It still depends on the user to as to how many panels he chooses to use for the program depending on the time required to process the results and the need of accuracy.

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CL vs Alpha at different number of panels 1.6 1.4 1.2

CL

1 0.8 0.6 0.4 0.2 0 0

2

4

6

8

10

12

Number of panels N

N = 40

N = 60

N = 80

N = 100

N = 120

Experimental data

Fig 5.3

Alpha Cl @

Cl @

Cl @

Cl @

Cl @

Cl @

Cl @

Cl @ Experimental

N

N

N

N

N

N

N

N

=40

=50

=60

=70

=80

=90

=100

=120

values of Cl

0

0.165 0.165 0.165 0.165 0.165 0.165 0.165 0.071

0.25

1

0.292 0.275 0.259 0.243

0.23 0.218 0.209 0.194

0.35

2

0.419 0.401 0.384 0.368 0.355 0.343 0.333 0.318

0.44

3

0.545 0.527

0.53

4

0.671 0.653 0.636 0.619 0.604 0.591

0.58 0.564

0.63

5

0.797 0.779 0.761 0.743 0.728 0.715 0.704 0.687

0.75

6

0.923 0.905 0.886 0.868 0.852 0.838 0.827 0.809

0.86

7

1.049

0.95 0.932

0.95

8

1.174 1.155 1.135 1.116

1.1 1.085 1.073 1.054

1.07

9

1.299

1.24 1.223 1.207 1.195 1.175

1.165

10

1.423 1.404 1.383 1.363 1.346 1.329 1.317 1.296

1.29

1.03

0.51 0.494 0.479 0.467 0.456 0.441

1.01 0.992 0.976 0.962

1.28 1.259

Table 5.1 variation of lift curve with number of panels

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5.2 GROUND EFFECT The lift curve for an airfoil in ground effect gave the following results at different values of h/c are given below in figure 5.4 and 5.5

CL vs H/C at AOA 3

0.8

Cl Ground Calculated

0.75

Cl unbound air calculated

0.7

Cl unbound air theortical

0.65

Cl Ground Theortical

CL

0.6

0.55 0.5 0.45 0.4 0

0.1

0.2

0.3

H/C

0.4

0.5

0.6

0.7

Fig 5.4

The change in coefficient of lift with respect to its height from the ground is given in figure 5.4. The dotted yellow curve in figure 5.4 represents the change in Cl as calculated using the following expression Cl (ground) = Cl x (H/C) -0.11 The solid blue line represents the coefficient of lift using the python code. As can be seen, there is a stark difference between the values of Cl between the two curves. The maximum error calculate is around 17%. This error can be further reduced using higher order panel method.

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CHAPTER 6 RESULTS – FINITE WINGS 6.1 INTRODUCTION The major difference between a finite and an infinite wing (2D airfoil) were previously discussed in the chapter 3. This chapter is concerned with the results obtained by using lifting line theory for a finite wing of a user defined span and area. A plot showing the difference between the changes in the lift curve for the finite and infinite wings has also been shown. Another major variation in coefficient of lift is also observed when the aspect ratio is changed for the wing. This variation is plotted along the cl vs alpha curve for different aspect ratios. 6.2 LIFT CURVE FOR FINITE WING 6.2.1 INPUT PARAMETERS 1. Angle of attack = 0 to 10 degrees 2. Span of the wing = 4 m 3. Area of the wing = 4 square meters 4. Root chord = 1m 5. Tip chord = 1m 6. Lift curve slope for root = 6.28 7. Lift curve slope for tip = 6.28 8. Zero lift AOA for root = -1 degree 9. Zero lift AOA for tip = -1 degree 10. Number of control points = 10 11. Airfoil = NACA2412

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6.2.2 LIFT CURVE The graph shown below (Fig 6.1) clearly shows a decrease in the coefficient of lift generated by a finite wing as compared to an infinite wing. This decrease in lift gives rise to induce drag for finite wings. Figure 6.2 shows the change in coefficient of induced drag with respect to the angle of attack.

Cl Vs Alpha 1 0.9 0.8 0.7 0.6

Cl

0.5 0.4 0.3 0.2 0.1

-2

0 -0.1 0

2

4

6

8

10

12

AOA

Fig 6.1

Cd (inducecd) Vs Alpha 0.045 0.04 0.035

Cd

0.03 0.025 0.02 0.015 0.01 0.005 0 -2

0

2

4

6

8

10

12

AOA

Fig 6.2 Since coefficient of drag is directly proportional to the square of the coefficient of lift, the drag curve is not linear. 6.3 LIFT CURVE WITH RESPECT TO ASPECT RATIO

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The lift of a 3D wing is also dependent on the aspect ratio. The following graph shows the lift curve at different aspect ratio. NACA 0012 was taken for this case as its wind tunnel data was available in the literature. 6.3.1 INPUT PARAMETERS 1. Angle of attack = 10 degrees 2. Span of the wing = 4 to 6 m 3. Area of the wing = 2 square meters 4. Root chord = 1m 5. Tip chord = 1m 6. Lift curve slope for root = 5.73 7. Lift curve slope for tip = 5.73 8. Zero lift AOA for root = 0 degree 9. Zero lift AOA for tip = 0 degree 10. Number of control points = 10 11. Airfoil = NACA0012

Cl at AOA 10 Degree 0.8 0.7 0.6

Cl

0.5 0.4

Experimental Data

0.3

Calculated Data

0.2 0.1 0 3

3.5

4

4.5

5

5.5

6

6.5

Aspect Ratio

Fig 6.3 6.3.2 COEFFICIENT OF LIFT AND DRAG AT DIFFERENT ASPECT RATIO The coefficient of lift is directly proportional to the aspect ratio of the wing. To illustrate this, the chart in figure 6.3 shows the lift curve at AR = 4, 6.25, 9. These

19

aspect ratios were achieved by keeping the area constant while changing the span of the wing. The second chart in figure 6.4 shows the variation of coefficient of drag along with the angle of attack at different AR = 4, 6.25, 9.

Cl Vs Alpha 1 0.9 0.8 0.7

Cl

0.6 0.5

AR=4

0.4

AR=5

0.3

AR=6

0.2 0.1 -2

0 -0.1 0

2

4

6

8

10

12

AOA

Fig 6.4

Cd (induced) Vs Alpha 0.06 0.05

Cl

0.04 AR=4

0.03

AR=5

0.02

AR=6 0.01 0 -2

0

2

4

6

AOA

Fig 6.5

8

10

12

20

The above two charts clearly illustrate the effect aspect ratio has on the lift and drag. The code for lifting line theory given in appendix is pretty straight forward as compared to other programs. Though simple, it provides a powerful insight into the difference between a finite and an infinite wing.

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CHAPTER 7 RESULTS: MULTI ELEMENT AIRFOIL 7.1 INTRODUCTION A multi-element airfoil uses two or more airfoils to meet the lift requirement. In this project, and in the case of formula sports in general, it is important to calculate the lift produced by a combination of multiple airfoils together so as to better understand their behaviour whilst in each other’s neighbourhood. A python code (Appendix 6) was developed using the vortex panel method to calculate the overall lift generated by a pair of airfoil at their individual angle of attacks. The position of the two airfoil with respect to each other has been decided as follows: 1. The two airfoil coordinates are imported by the user. 2. The user is asked to give the distance between the leading edge of the two airfoil with respect to the primary airfoil. The distance between the two leading edges parallel to the chord is defined as x, while the distance perpendicular to the chord between the two leading edges is y. 3. The airfoils are then plotted Figure 7.1. 7.2 PANEL METHOD The vortex panels is used to calculate the lift just like in the case single element airfoil. A modified version of the code for ground effect was used in this case. Since the code for multi element airfoil can adjust up to any sensible number of airfoils, the number of airfoils in the mentioned code (Appendix 6) can thus be modified by the user to account for any number.

Fig 7.1

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7.3 LIFT CURVE FOR MULTI ELEMENT AIRFOIL The chart below figure 7.1, the blue curve showcases the overall Cl of the multi element airfoil at different angle of attacks for x = 1 and y= 0.2 distance from the leading edge of the primary airfoil. The orange curve shows Cl at different angle of attacks for x = 50m and y = 20m.

Cl(Overall) Vs Alpha 2.823

3

2.577 2.329

2.5

2.081 1.832

Cl

2

1.582 1.332

1.5

1.082 0.981 0.831 0.78 1 0.58 0.578 0.376 0.329 0.5 0.174

1.182

1.383

1.582

1.783

1.983

2.182

X= 1, Y = 0.2 X = 50, Y = 20

0 0

2

4

6

8

10

12

AOA

Fig 7.2 The coefficient of lift for a single NACA 2412 airfoil at any angle of attack (at 3 degree Cl = 0.527) is half of the value shown in the orange curve in figure 7.2. This is because of the fact that at far away distance, the two airfoils have very little effect on each other, thus the overall lift coefficient is simply the sum of the Cl of the two individual airfoils.

23

7.4 VARIATION OF Cl WITH DISTANCE FROM THE TRAILING EDGE The variation of coefficient of lift by changing the distance between the primary and secondary airfoil is plotted below in figure 7.3 and figure 7.4.

Cl vs Y distance 0.82

0.811

0.81

Cl

0.8

0.792

0.79

0.78

0.78 0.77

0.783 0.777 0.772 0.771 0.768 0.766 0.7630.7620.7630.765

X = 1m

0.76 0.75 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Y distance from leading edge in m

Fig 7.3

Cl vs X distance 0.89

0.881 0.874

0.88

0.866

0.87 0.86

Cl

0.85 0.837

0.84

Y= 0.1 m

0.83 0.82

0.811

0.81 0.8 0

0.2

0.4

0.6

0.8

1

1.2

X distance from the trailing edge

Fig 7.4

1.4

1.6

1.8

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CHAPTER 8 DISCUSSION OF RESULTS 8.1 2D PANEL METHOD In the process of finding the coefficient of lift of an airfoil section, it was noted that these properties cannot be entirely predicted at any single number of panel, in fact the number of panels required to give a near experimental value depended on the angle of attack. For an airfoil section (NACA2412) of chord length 1m, at an angle of attack of say 3 degree, the coefficient of lift calculated using panel method was 0.527 at number of panels N = 50. The experimental value for the same was 0.530. The problem arise when we increased the angle of attack to 9 degree. Then the value of Cl was 1.28 at N=50, while the experimental value was 1.165. To account for this change in value of Cl, the number of panels were increased to 120 which gave the value of Cl to be 1.175. 8.2 GROUND EFFECT The ground effect has been the prime focus of this project due to its application in formula one. The basic goal while trying to account for the ground effect was to increase the lift and see its variation as the height from the ground was increased or decreased. The results obtained clearly showed an increase in lift at the same angle of attack when the airfoil was subject to ground proximity. This increase was more apparent when the airfoil was brought closer to the ground. The results were then matched with CFD results from the available literature and a 10% variation in results was seen. 8.3 LIFTING LINE THEORY The lifting line theory was used to extend the calculations to simple 3D wings. To account for the same, results were plotted to show a comparison between 2D section result and 3D wing result at a particular aspect ratio. There was a significant decrease in the lift for the 3D wing as compared to the 2D wing section. A lift curve was then plotted for three wings of different aspect ratio (AR = 4, 6.25, 9). The results showed that the Cl value increased as the AR was increased. This

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also showed that the induced drag coefficient also increased with an increase in aspect ratio. 8.4 PANEL METHOD ON TWO ELEMENT AIRFOIL The panel method was then used to figure out the lift coefficient for a two element airfoil. Although the code is prepared for two elements, it is easier for the user to modify it a bit to account for any number of elements. The lift curve generated was similar to the one generated for a single element airfoil. Two other curves were also generated to see the effect on coefficient of lift with the change in position of secondary airfoil w.r.t the primary airfoil in x and y direction. The angle of attack was taken to be 3 degree. In case of increasing the x distance from the leading edge of the primary airfoil, the coefficient of lift was found to increase at a steep rate. In the second case, increasing the y distance from the leading edge first saw an increase in the coefficient of lift and then decreases.

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CHAPTER 9 CONCLUSION AND FUTURE WORK 9.1 REMARKS The vortex panel method was used to get primary data for any given airfoil section. The following conclusions were drawn from this project  The results from vortex panel method give a pretty close approximation (under 8%) to the experimental data, depending on the number of panels chosen by the user.  The lift coefficient varies with number of panels. The right number of panels required also depend on the angle of attack. For low angle of attack, the panel requirement is also low and for high number angle of attack the panel requirement is also high.  Ground effect is expected to increase the coefficient of lift. The same is seen when we applied vortex panel method on NACA2412.  The second thing in ground effect is the change in coefficient of lift with respect to the change in height from the ground. It was seen that the coefficient of lift decreased with an increase in height from the ground.  In case of multi element airfoils, we see that the overall lift coefficient increases.  The variation in coefficient of lift was also seen with the distance between the primary and secondary airfoils. The lift coefficient can be seen to increase when the distance increases along the x axis. Although, along the y axis, the lift coefficient decreases and then increases. 9.2 FUTURE WORK  A higher order panel method can be used for getting the more precise result. Also, it can reduce the variation in lift with number of panels.  For finite wings, lifting line theory is used in this project. But, lifting line theory is limited to simple rectangular wings. Three dimensional

27

panel method can be excellent tool for finding out the lift coefficient of a wing of any arbitrary size and shape.  Ground effect calculations can be extended to finite wings.

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REFERENCE(S): 1. http://www.engapplets.vt.edu/fluids/vpm/vpminfo.html 2. Smith, A.M.O.: The panel method: Its Original Development, AIAA 1989. 3. John D. Anderson, Fundamentals of Aerodynamics, 3rd edition, McGraw-Hill 2001 4. Prof. Lorena A. Barba, Aero-python series, 2012 5. Hess, J. L., and Smith, A.M.O.: Calculation of Non-lifting Potential Flow about Arbitrary Three-Dimensional Bodies. Douglas Aircraft Company Report No. ES 40622 (March 1962). 6. Hess, J. L., and Smith, A.M.O.: Calculation of Potential Flow about Arbitrary Bodies. Progress in Aeronautical Sciences, Vol. 8, Pergammon Press, New York (1966).

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Appendix 1 PYTHON CODE FOR SINGLE ELEMENT AIRFOIL PLOTTING

""" Created on Wed Jan 25 09:25:06 2017

author: Dipak Kumar Sisodiya (UR13AE021)

""" ################ Import and plot single element airfoil ################

import numpy from matplotlib import pyplot import math

##asking the user to select a file in Dat format def input_file(): file1 = input("Enter here: ") return file1

file1 = input_file()

x1, y1 = numpy.loadtxt(file1, dtype=float, unpack=True)

xc = numpy.linspace(x1.min(), x1.max(), num = 65,dtype = float) yc = numpy.linspace(0.0, 0.0, num = 65,dtype = float)

# using th angle of attack###

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AOB = float(input("Enter the Angle of attack:")) con1 = (math.pi / 180) AOA = AOB * con1 xaoa = (x1-0.25) * math.cos(AOA) + (y1-0.0) * math.sin(AOA)+0.25 # =L4*(COS(2 * 3.14/180))-M4*(SIN(2*3.14/180)) yaoa = (y1-0.0) * math.cos(AOA) - (x1-0.25) * math.sin(AOA)+0.00 yaoa1 = numpy.empty_like(yaoa) for i in range(130): yaoa1[i] = -yaoa[i] - 0.1200344 -.1

# plot geometry val_x, val_y = 0.1, 0.2 xp_min, xp_max = xaoa.min(), (x1).max() yp_min, yp_max = yaoa1.min(), (yaoa).max()

chord_length = math.sqrt(((x1.max() - x1.min())*(x1.max() - x1.min())+ ((y1.max() - y1.min())*(y1.max() - y1.min()))))

# plot limits xp_start, xp_end = xp_min-val_x*(xp_max-xp_min), xp_max+val_x*(xp_maxxp_min) yp_start, yp_end = yp_min-val_y*(yp_max-yp_min), yp_max+val_y*(yp_maxyp_min)

# priliminary plotting the airfoil size = 10 pyplot.figure(figsize=(size, (yp_end-yp_start)/(xp_end-xp_start)*size)) pyplot.grid(True)

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pyplot.xlabel('x', fontsize=16) pyplot.ylabel('y', fontsize=16) pyplot.xlim(xp_start, xp_end) pyplot.ylim(yp_start, yp_end) pyplot.plot(xaoa,yaoa1,color='k', linestyle='-', linewidth=2); pyplot.plot(xaoa,yaoa,color='k', linestyle='-', linewidth=2);

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Appendix 2 PYTHON CODE FOR MULTI ELEMENT AIRFOIL PLOTTING

""" Created on Wed Jan 18 15:49:55 2017

author: Dipak Kumar Sisodiya (UR13AE021)

################ Import and plot two element airfoil ####################

""" import numpy from matplotlib import pyplot import math from scipy import integrate, linalg

""" asking the user to select a file in Dat format

""" print("input primary airfoil file location")

file1 = input("Enter here: ")

print("input secondary airfoil file location")

file2 = input("Enter here: ")

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x1, y1 = numpy.loadtxt(file1, dtype=float, unpack=True) xn, yn = numpy.loadtxt(file2, dtype=float, unpack=True) a = numpy.size(xn) b =float(input("Enter the x distace from the leading edge of the primary airfoil: ")) c = float(input("Enter the y distace from the leading edge of the primary airfoil: ")) xp = numpy.linspace(b,b,num =a ,dtype= float) yp = numpy.linspace(c,c,num =a ,dtype= float) x2,y2 = (xn+xp),(yn+yp)

xc = numpy.linspace(x1.min(), x1.max(), num = 65,dtype = float) yc = numpy.linspace(0.0, 0.0, num = 65,dtype = float)

# using th angle of attack### AOB1 = float(input("Enter the Angle of attack for primary airfoil:")) con1 = (math.pi / 180) AOA1 = AOB1 * con1 xaoa1 = (x1-0.25) * math.cos(AOA1) + (y1-0.0) * math.sin(AOA1) # =L4*(COS(2 * 3.14/180))-M4*(SIN(2*3.14/180)) yaoa1 = (y1-0.0) * math.cos(AOA1) - (x1-0.25) * math.sin(AOA1)

# using th angle of attack### AOB2 = float(input("Enter the Angle of attack for secondary airfoil: ")) con2 = (math.pi / 180) AOA2 = AOB2 * con2 xaoa2 = (x2-0.25) * math.cos(AOA2) + (y2-0.0) * math.sin(AOA2) # =L4*(COS(2 * 3.14/180))-M4*(SIN(2*3.14/180)) yaoa2 = (y2-0.0) * math.cos(AOA2) - (x2-0.25) * math.sin(AOA2)

# plot geometry

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val_x, val_y = 0.1, 0.2 xp_min, xp_max = xaoa1.min(), (x2).max() yp_min, yp_max = y1.min(), (y2).max()

# plot limits xp_start, xp_end = xp_min-val_x*(xp_max-xp_min), xp_max+val_x*(xp_maxxp_min) yp_start, yp_end = yp_min-val_y*(yp_max-yp_min), yp_max+val_y*(yp_maxyp_min)

# priliminary plotting the airfoil size = 10 pyplot.figure(figsize=(size, (yp_end-yp_start)/(xp_end-xp_start)*size)) pyplot.grid(True) pyplot.xlabel('x', fontsize=16) pyplot.ylabel('y', fontsize=16) pyplot.xlim(xp_start, xp_end) pyplot.ylim(yp_start, yp_end) pyplot.plot(xaoa1,yaoa1,color='k', linestyle='-', linewidth=2); pyplot.plot(xaoa2,yaoa2,color='k', linestyle='-', linewidth=2);

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Appendix 3 VORTEX PANEL METHOD ON 2D AIRFOIL """ Created on Fri Jan 13 10:29:59 2017

author: Dipak Kumar Sisodiya (UR13AE021)

## Panel Method over an airfoil for calculation of lift ## """

# importing libraries and modules import numpy from scipy import integrate, linalg from matplotlib import pyplot import math

# load geometry from data file using function Single_airfoil() def Single_airfoil(): print("input primary airfoil file location")

file1 = input("Enter here: ")

x1, y1 = numpy.loadtxt(file1, dtype=float, unpack=True)

# using th angle of attack### xaoa = x1 yaoa = y1 return xaoa, yaoa

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x,y = Single_airfoil() AOB = float(input("Enter the Angle of attack:")) con1 = (math.pi / 180) AOA = AOB * con1

# plot geometry val_x, val_y = 0.1, 0.2 xp_min, xp_max = x.min(), x.max() yp_min, yp_max = y.min(), y.max()

# plot limits xp_start, xp_end = xp_min-val_x*(xp_max-xp_min), xp_max+val_x*(xp_maxxp_min) yp_start, yp_end = yp_min-val_y*(yp_max-yp_min), yp_max+val_y*(yp_maxyp_min)

# priliminary plotting the airfoil size = 10 pyplot.figure(figsize=(size, (yp_end-yp_start)/(xp_end-xp_start)*size)) pyplot.grid(True) pyplot.xlabel('x', fontsize=16) pyplot.ylabel('y', fontsize=16) pyplot.xlim(xp_start, xp_end) pyplot.ylim(yp_start, yp_end) pyplot.plot(x, y, color='k', linestyle='-', linewidth=2);

#### CLass named panle to compute panel number and size###################

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class Panel: """Contains information related to a panel.""" def __init__(self, xa, ya, xb, yb): """function to create a panel.

Parameters ---------_ xa, ya: float Coordinates of the starting-point. xb, yb: float Coordinates of the ending-point. """ self.xa, self.ya = xa, ya

# panel starting-point

self.xb, self.yb = xb, yb

# panel ending-point

self.xc, self.yc = (xa+xb)/2, (ya+yb)/2

# panel center

self.length = numpy.sqrt((xb-xa)**2+(yb-ya)**2) # panel length

# orientation of panel (angle between x-axis and panel's normal) if xb-xa 0.0: self.beta = numpy.pi + numpy.arccos(-(yb-ya)/self.length)

# panel location if self.beta ') pyplot.fill([panel.xc for panel in panels1], [panel.yc for panel in panels1], color='k', linestyle='solid', linewidth=2, zorder=2) pyplot.xlim(x_start, x_end) pyplot.ylim(y_start, y_end) # computes the pressure field cp = 1.0 - (u**2+v**2)/freestream.u_inf**2

# plots the pressure field size=12 pyplot.figure(figsize=(1.1*size, (y_end-y_start)/(x_end-x_start)*size)) pyplot.xlabel('x', fontsize=16) pyplot.ylabel('y', fontsize=16) contf = pyplot.contourf(X, Y, cp, levels=numpy.linspace(-2.0, 1.0, 100), extend='both') cbar = pyplot.colorbar(contf) cbar.set_label('$C_p$', fontsize=16) cbar.set_ticks([-2.0, -1.0, 0.0, 1.0]) pyplot.fill([panel.xc for panel in panels1], [panel.yc for panel in panels1], color='k', linestyle='solid', linewidth=2, zorder=2) pyplot.xlim(x_start, x_end) pyplot.ylim(y_start, y_end) pyplot.title('Contour of pressure field');

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Appendix 6 PYTHON CODE FOR VORTEX PANEL METHOD ON MULTI ELEMENT AIRFOIL

""" Created on Wed Jan 18 15:49:55 2017

author: Dipak Kumar Sisodiya (UR13AE021)

################ Import and plot two element airfoil ####################

""" import numpy from matplotlib import pyplot import math from scipy import integrate, linalg

""" asking the user to select a file in Dat format

""" print("input primary airfoil file location")

file1 = input("Enter here: ")

print("input secondary airfoil file location")

file2 = input("Enter here: ")

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x1, y1 = numpy.loadtxt(file1, dtype=float, unpack=True) xn, yn = numpy.loadtxt(file2, dtype=float, unpack=True) a = numpy.size(xn) b =float(input("Enter the x distace from the leading edge of the primary airfoil: ")) c = float(input("Enter the y distace from the leading edge of the primary airfoil: ")) xp = numpy.linspace(b,b,num =a ,dtype= float) yp = numpy.linspace(c,c,num =a ,dtype= float) x2,y2 = (xn+xp),(yn+yp)

xc = numpy.linspace(x1.min(), x1.max(), num = 65,dtype = float) yc = numpy.linspace(0.0, 0.0, num = 65,dtype = float)

# using th angle of attack### AOB1 = float(input("Enter the Angle of attack for primary airfoil:")) con1 = (math.pi / 180) AOA1 = AOB1 * con1 xaoa1 = (x1-0.25) * math.cos(AOA1) + (y1-0.0) * math.sin(AOA1) # =L4*(COS(2 * 3.14/180))-M4*(SIN(2*3.14/180)) yaoa1 = (y1-0.0) * math.cos(AOA1) - (x1-0.25) * math.sin(AOA1)

# using th angle of attack### AOB2 = float(input("Enter the Angle of attack for secondary airfoil: ")) con2 = (math.pi / 180) AOA2 = AOB2 * con2 xaoa2 = (x2-0.25) * math.cos(AOA2) + (y2-0.0) * math.sin(AOA2) # =L4*(COS(2 * 3.14/180))-M4*(SIN(2*3.14/180)) yaoa2 = (y2-0.0) * math.cos(AOA2) - (x2-0.25) * math.sin(AOA2)

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# plot geometry val_x, val_y = 0.1, 0.2 xp_min, xp_max = xaoa1.min(), (x2).max() yp_min, yp_max = y1.min(), (y2).max()

# plot limits xp_start, xp_end = xp_min-val_x*(xp_max-xp_min), xp_max+val_x*(xp_maxxp_min) yp_start, yp_end = yp_min-val_y*(yp_max-yp_min), yp_max+val_y*(yp_maxyp_min)

# priliminary plotting the airfoil size = 10 pyplot.figure(figsize=(size, (yp_end-yp_start)/(xp_end-xp_start)*size)) pyplot.grid(True) pyplot.xlabel('x', fontsize=16) pyplot.ylabel('y', fontsize=16) pyplot.xlim(xp_start, xp_end) pyplot.ylim(yp_start, yp_end) pyplot.plot(xaoa1,yaoa1,color='k', linestyle='-', linewidth=2); pyplot.plot(xaoa2,yaoa2,color='k', linestyle='-', linewidth=2);

#enter angle of attack

class Panel: def __init__(self, xa, ya, xb, yb): self.xa, self.ya = xa, ya

# panel starting-point

self.xb, self.yb = xb, yb

# panel ending-point

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self.xc, self.yc = (xa+xb)/2, (ya+yb)/2

# panel center

self.length = numpy.sqrt((xb-xa)**2+(yb-ya)**2) # panel length

# orientation of panel (angle between x-axis and panel's normal) if xb-xa 0.0: self.beta = numpy.pi + numpy.arccos(-(yb-ya)/self.length)

# panel location if self.beta