Formula 1 Race Car Performance Improvement by Optimization of the Aerodynamic Relationship Between the Front and Rear Wings

Home , Brabham, Diffuser (automotive), F1 Race, Formula One car, Ground effect (cars), Lift coefficient

The Pennsylvania State University The Graduate School College of Engineering

FORMULA 1 RACE CAR PERFORMANCE IMPROVEMENT BY OPTIMIZATION OF THE AERODYNAMIC RELATIONSHIP BETWEEN THE FRONT AND REAR WINGS

A Thesis in Aerospace Engineering by Unmukt Rajeev Bhatnagar

Submitted in Partial Fulfillment of the Requirements for the Degree of

Master of Science December 2014

The thesis of Unmukt R. Bhatnagar was reviewed and approved* by the following:

Mark D. Maughmer Professor of Aerospace Engineering Thesis Adviser

Sven Schmitz Assistant Professor of Aerospace Engineering

George A. Lesieutre Professor of Aerospace Engineering Head of the Department of Aerospace Engineering

*Signatures are on file in the Graduate School

Abstract

The sport of Formula 1 (F1) has been a proving ground for race fanatics and engineers for more than half a century. With every driver wanting to go faster and beat the previous best time, research and innovation in engineering of the car is really essential. Although higher speeds are the main criterion for determining the Formula 1 car’s aerodynamic setup, post the San Marino Grand Prix of 1994, the engineering research and development has also targeted for driver’s safety. The governing body of

Formula 1, i.e. Fédération Internationale de l'Automobile (FIA) has made significant rule changes since this time, primarily targeting car safety and speed.

Aerodynamic performance of a F1 car is currently one of the vital aspects of performance gain, as marginal gains are obtained due to engine and mechanical changes to the car. Thus, it has become the key to success in this sport, resulting in teams spending millions of dollars on research and development in this sector each year. Although F1 car aerodynamics is at a highly advanced stage, there is always potential for further development. With the under-body aerodynamics banned by the FIA, the only significant changes that can be made to improve the aerodynamic performance of the car are by modifying the front and rear wings cross-sections, i.e. airfoils, or by developing new diffuser to modify the air flow underneath the car. Airfoil design is one of the important factors to consider while designing the car. Design of the most optimum airfoils is track-dependent, as each track has different aerodynamic requirements. The development of the F1 car is regulated by the rules sanctioned by the FIA. In recent years, the FIA has reduced the allowable operational hours for development at the wind-tunnel by a F1 team. From the 2015 season onwards, use of Computational Fluid Dynamics (CFD) software for the development of the F1 car is also being limited. This rule change will result in limited test-runs every season. This study, thus, focuses to provide a preliminary estimate of the most optimum aerodynamic loads acting on the front and rear wings for achieving the best lap times possible around a particular track.

This will effectively focus the area of development leading to targeted use of CFD simulations.

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To perform the optimization, a genetic algorithm (Covariance Matrix Adaptation Evolution

Strategy – CMA-ES) is used. In order to obtain all the telemetric information, a lap simulation tool called

AeroLap is used. For simulation, the Sepang F1 race track, which annually hosts the Malaysian Grand

Prix (GP), is selected. This track provides a perfect conundrum of whether to design the car for high downforce or low drag configuration, as it contains fast-turning corners and long straights. The optimization is performed for a given F1 car setup used for the 2010 season, with the aerodynamic loads acting on both the front and rear wings as well as the racing line being optimized. First, an optimum racing line is derived for this particular race track using CMA-ES. It is observed that the lap time is reduced by a margin between 0.542 to 1.699 seconds when compared with the best lap time for the actual race during the 2010 Malaysian GP. For this racing line, the optimum values of the aerodynamic loads in the form of lift and drag coefficients for the front and rear wings are calculated. The optimum values of lift coefficients for the front and rear wings are calculated as 1.123 and 1.651 respectively. The optimization of drag coefficients for the obtained lift coefficient values led to the conclusion that the best lap times always occurred for the least value of the drag coefficient that had been set as the lower limit for the simulation. As a result, a parametric study is performed by varying the drag and lift coefficients for the front and rear wings. The results are summarized in form of contour plots, displaying the change in lap times with variation in the aerodynamic loads for the front and rear wings. The best lap time for the minimum set of drag coefficients and the optimized lift coefficients is observed to be at least 2.02 seconds better than the lap time performed by an actual F1 car that raced in the 2010 season.

Table of Contents

List of Figures ...... vii

List of Tables ...... ix

Acknowledgment ...... x

Abbreviations Used ...... xii

Chapter 1 INTRODUCTION ...... 1

1.1 History of Development of Aerodynamics in Formula 1 ...... 2 1.2 Problem Statement ...... 7 1.3 Research Objective ...... 9 1.4 Thesis Overview...... 9

Chapter 2 LITERATURE REVIEW ...... 11

2.1 Aerodynamics of Race Cars ...... 11 2.2 Ground Effect Aerodynamics...... 22 2.3 Role of Optimization ...... 23 2.4 Motivation for this study ...... 25

Chapter 3 TOOLS USED ...... 27

3.1 Covariance Matrix Adaptation – Evolution Strategy: ...... 28 3.2 AeroLap – Lap Simulation software ...... 31 3.3 Optimization Process ...... 32

Chapter 4 OPTIMIZATION OF RACING LINE AROUND THE TRACK ...... 35

4.1 Study of a Straight Track ...... 36 4.2 Study of the Circular Track ...... 36 4.3 Study of the Oval Track ...... 41 4.4 Optimization of the Sepang F1 Race Track ...... 46

Chapter 5 OPTIMIZATION OF AERODYNAMIC LOADS ON FRONT AND REAR WINGS ...... 55

5.1 Variation in time with respect to change in aerodynamic loads ...... 56 5.2 Derivation of aerodynamic coefficients for the wings and F1 car body without wings ...... 58 5.3 Optimization of Aerodynamic Coefficients ...... 67 5.4 Results of Parametric Study ...... 72

Chapter 6 CONCLUSION AND DISCUSSION ...... 79

6.1 Summary and Results ...... 79 6.2 Assumptions ...... 81 6.3 Discussion and Conclusion ...... 81 6.4 Drawbacks ...... 83 6.5 Future Research ...... 83

References ...... 84

Appendix A Input Data Tables for Lift and Drag Coefficient and Location of Center of Pressure ...... 89

Appendix B Contour Plots for Varying Viscous Drag Coefficients of Front and Rear Wings ...... 91

List of Figures

Fig 1.1: First Formula 1 car designed in 1950. (Hanlon, 2010) ------2

Fig 1.2: (a) Lotus 49B with a front wing and a rear spoiler (MAB, 2008) (b) Lotus 49B with a front and rear wing (History of Motorsport, 2013) ------3

Fig 1.3: Lotus 72D, variant of Lotus 72. Built in 1972 (Wikipedia, 2013) ------4

Fig 1.4: McLaren M23 (World Cars List, 2013) ------4

Fig 1.5: Lotus 78 with side skirts (FormulaPassion.it, 2014) ------5

Fig 1.6: Brabham BT46B "fan car" (Wikipedia, 2014) ------5

Fig 1.7: Brabham BT55 with no hood for air intake (FormulaPassion.it, 2013) ------5

Fig 1.8 Rear end Diffuser for McLaren MP 4-5B (F1 Nutter, 2014) ------5

Fig 1.9: Active DRS in the rear wing (Wise, 2012) ------7

Fig 1.10: Front wing for a modern F1 car (Planet F1, 2014) ------7

Fig 2.1: Effect of aerodynamic downforce on tire performance (Katz, 1995) ------12

Fig 2.2: Breakdown of drag and downforce for individual components of 2009 F1 car (Toet, 2013) ------15

Fig 2.3: Front wing for a modern F1 car (Planet F1, 2014) ------16

Fig 2.4: Active DRS in the rear wing (Wise, 2012) ------18

Fig 2.5: Rear wing along with the beam wing (Michalis, 2011) ------19

Fig 2.6: Forces and moments acting on a car while making a turn (Dominy, 1992) ------20

Fig 3.1: Pictorial representation of different factors influencing evolutionary optimization. (Hansen &

Ostermeier, 2001)------30

Fig 3.2: Flowchart describing optimization process------34

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Fig. 4.1: Circular tracks with varying radii displaying the location of the racing line.------38

Fig. 4.2: Variation of Lift forces with change in radius of curvature of the track. ------40

Fig. 4.3: Optimum racing line for an oval track with 200m radius of curvature. ------43

Fig. 4.4: Optimum racing line for an oval track with 100m radius of curvature. ------43

Fig. 4.5: Comparison of speeds between the optimized path and along the centerline. ------44

Fig. 4.6: Comparison of lift forces between the optimized path and along the centerline. ------45

Fig. 4.7: Aerial view of Sepang F1 Race Track (©Google Maps) ------46

Fig. 4.8: Malaysian F1 track obtained from ArcGIS data set. ------49

Fig. 4.9: Optimal racing line obtained after running detailed set of simulations. ------52

Fig 5.1: Influence of drag and lift forces on the speed of a 2010 Formula 1 car.------57

Fig 5.2: Breakdown of drag and downforce for individual components of F1 car (Toet, 2013) ------58

Fig 5.3: Components of a rear wing assembly for a 2009 season’s F1 car (Toet, 2013) ------63

Fig 5.4: Downforce contributions in rear wing assembly of a 2009 season’s F1 car (Toet, 2013) ------64

Fig 5.5: Drag contributions in rear wing assembly of a 2009 season’s F1 car (Toet, 2013) ------65

Fig 5.6: Map of variables showing optimization of lift coefficient ------70

Fig 5.7: Variation in contours for constant value of viscous drag coefficient of the rear wing ------75

Fig 5.8: Variation in contours for constant value of viscous drag coefficient of the front wing ------76

Fig 5.9: Variation in contours with respect to L/D ratio of front wing vs. the rear wing ------77

Fig. B.1: Variation in contours for viscous drag coefficient of the rear wing = 0.04------91

Fig. B.2: Variation in contours for viscous drag coefficient of the rear wing = 0.05------92

Fig. B.3: Variation in contours for viscous drag coefficient of the rear wing = 0.06 ------93

Fig. B.4: Variation in contours for viscous drag coefficient of the rear wing = 0.07 ------94

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List of Tables

Table 2.1: Parameters affecting lap times (Toet, 2013)------13

Table 4.1: Variation of velocity, distance and lap time with change in radius across the track width ------39

Table 4.2: Initial and final positions of data points along the race track, with corresponding radius of curvature values ------41

Table 4.3: Comparison of lap times for the actual race and optimized racing line. ------54

Table 5.1: Contributions by front and rear wings and rest of the car’s components.------59

Table 5.2: Contributions by front and rear wings and rest of the car’s components. ------66

Table 5.3: Lap time comparisons for different optimized parameters. ------71

Table A.1: Lift coefficient for the whole F1 car, varying with ride height.------89

Table A.2: Lift coefficient for the body (without wings) exerting load at front wheel center ------89

Table A.3: Lift coefficient for the body (without wings) exerting load at rear wheel center ------89

Table A.4: Drag coefficient for the whole F1 car, varying with ride height ------90

Table A.5: Drag coefficient for the body (without wings), varying with ride height------90

Table A.6: Location of center of pressure for whole F1 car, varying with ride height ------90

Acknowledgment

“Nothing Happens Unless First a Dream”

- Carl Sandburg

This research has been my dream project. Ever since I was a child, Formula 1 racing and aircraft design have been the two things that fascinated me. Very few get a chance to work in the field of their dreams.

This research study has introduced me to the joy of living my dreams. The journey has just started.

Firstly, I would like to express my deepest thanks to my adviser, Dr. Mark D. Maughmer for giving me this priceless opportunity and courage to follow my heart. His guidance and invaluable advice have been the driving force towards this study. With little spare time available to him, he was always able to provide guidance to ensure that I understood the work I did before putting it into action.

His experiences and the push to reason everything one does has been priceless and highly inspirational.

Also, I would like to thank him for the financial support he has provided for the purchase of various software, required to work for this project. He has been a source of deep knowledge, motivation and encouragement.

I would also like to thank my co-advisor, Dr. Sven Schmitz, for introducing me to the field of genetic algorithms and his instrumental inputs towards my research. With great expertise in his field, he helped provide a definitive structure to this research. I would also take the opportunity to express my gratitude to the Department Head of the program, Dr. George Lesieutre, who along with Dr. Maughmer arranged for the financial support towards the software, and for providing me with excellent educational tools through the course of my Master’s program. Special thanks towards faculty and staff members of

Department of Aerospace Engineering for supporting me throughout the program and making all the necessary arrangements I requested for.

A special mention of Mr. Phil Morse, who was always present and patient enough to answer my innumerable number of queries regarding the working of AeroLap software. He was the ‘go-to’ man if I got stuck at any point while using the software. Thank you to Mr. Keith Young and Mr. Chris Savage

x for giving me their 3-D model of a Formula 1 car to help me run basic simulations. It was of great help and gave great insight into the design of different elements.

To all my friends and colleagues, I greatly appreciate your precious input and time you dedicated to listen to my issues and help me resolve them, – thank you Mr. Ethan Corle, Mr. Amandeep Premi,

Mr. Tenzin Choephel, Mr. Zhixiang Wang and Mr. Bernardo Vieira for making this journey memorable. A special mention goes out to Mr. Taylor Hoover, for letting me use your desk space. Also,

I would like to thank my friends Mr. Debanjan Das, Mr. Vaibhav Rajput, Mr. Balaji Raman, Mr.

Vignesh Ramchandran and Mr. Aditya Choudhary for acting as stress busters and making my last semesters a lot of fun. A special mention goes out to Mr. Udit Narayanan and Mr. Nitish Vasudevan, who have helped peer review this thesis.

The best has to be always reserved for the end. I would have been nowhere without their guidance. They stood as strong as a pier of the bridge, supporting me throughout my life. They bore all kinds of pressure and stress in their life, just to fulfill my dreams. They made the flight of my dreams possible, by being there day and night, 24/7. They made their comforts and happiness secondary to mine. I can’t even quantify my gratitude towards them, as words will never be sufficient to justify their love and affection for me. They are my family. This is for my Mom, Dad and my little sister. The success of this dream is dedicated to you.

Thank you all !!

Abbreviations Used

DRS : Drag Reduction System

F1 : Formula One

FIA : Fédération Internationale de l'Automobile kph : Kilometers per hour mph : Miles per Hour

KERS : Kinetic Energy Recovery System

CFD : Computational Fluid Dynamics

CMA-ES: Covariance Matrix Adaptation Evolution Strategy

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Chapter 1 INTRODUCTION

The sport of Formula 1 (F1) has been a proving ground for race fanatics and engineers for more than half a century. With every driver wanting to go faster and beat the previous best time, research and innovation in the engineering of the car are really essential. Due to the marginal gains that can be obtained by engine and mechanical changes to the car, aerodynamics of a F1 car has become one of the major aspects of performance gain. Thus, it is a key element to success in the sport, resulting in teams spending millions of dollars on research and development in this area every year.

A Formula 1 car is an open-wheeled rear-wheel drive vehicle, the design of which is regulated and governed by strict set of rules. According to Willem Toet (Toet, 2013), the key attributes of a current F1 car are:

 A top speed of up to 350kph (217.5mph), depending on the circuit.

 Acceleration of 2.6 secs for speed range of 0 – 100kph (0 – 62.14mph).

 Deceleration from 200 – 0kph (124.27 – 0mph) in about 2.0 secs.

 Loads of ≥ 5g in braking (longitudinal acceleration) and ≥ 4g in cornering (lateral

acceleration).

 No movable aerodynamic components of bodywork, except F1 body regulated DRS system.

The cars are optimized for each and every circuit, starting from a base design at the start of the season. The base design changes every year in order to comply with the new sets of rules and regulations imposed. The rate of evolution of the car over a season is so high, that the end of the season car is normally a full second faster than the base design at the start of the season (Toet, 2013).

This evolution, over a cycle, is generally brought about by the optimization of aerodynamics, tires, and suspension of the car.

1.1 History of Development of Aerodynamics in Formula 1

[Source: (Wikipedia, 2014), (F1 Nutter, 2014), (Formula 1 - Dictionary, 2014)]

Fig 1.1: First Formula 1 car designed in 1950. (Hanlon, 2010)

The first Formula 1 championship season started in the year 1950. The era from 1950-1957

mainly comprised of front-engine, front-wheel drive cars, with absolutely no aerodynamics

involved. As there were no strict rules about designing the car (mainly the dimensioning of the car),

there was a lot of freedom for designers to come up with innovative designs. Through the seasons,

the cars’ height started becoming lower (to seek advantages of low center of gravity), radiators and

air intakes became bigger to increase engine power and efficiency, wheels became smaller and

wider to provide more stability and grip, engines started moving towards the rear, and so on, all in

order to go faster.

Engineers started realizing in the 1960’s that tire adhesion and weight of the car were one of the

primary factors essential to reduce lap times and increase speed of the cars over the corners. In

order to increase tire adhesion without increasing the weight of the car, ‘inverted wings’ were

introduced, that produced negative lift or ‘downforce’. Thus, 1968 saw the birth of aerodynamics in

the world of Formula 1, with Team Lotus introducing front wings and raised spoiler in the Monaco

grand prix of 1968 (Fig 1.2(a)). This led to frantic efforts of trying to acquire aerodynamic

2 supremacy, leading to new designs in almost every other race. In response to Lotus’s introduction of front wings, competitors like Brabham and Ferrari mounted full width wings over the struts high above the driver. Lotus replied by mounting and connecting a full width wing directly over the rear suspension (Fig 1.2(b)). The wing was mounted really high, to avoid turbulent flow generating from the wake of the front of the car. To get the maximum out of the wings, teams used highly mounted wings, directly connected to both the front and rear unsprung wheel assemblies. In conjunction with this, variable incidence mechanisms were used to maximize the downforce.

However, as the designs were unregulated, the whole assembly resulted in a highly vulnerable structure, susceptible to accidental damage. There were many cases of strut, suspension, and wing failures, which led to severe crashes. This brought about many regulatory changes and governance of design by the FIA; the major rulings being ban on high-mounted wings and no movement of the chassis bodywork components.

(a) (b) Fig 1.2: (a) Lotus 49B with a front wing and a rear spoiler (MAB, 2008) (b) Lotus 49B with a front and

rear wing (History of Motorsport, 2013)

In the 1970s and 1980s, the role of vehicle aerodynamics gained significant prominence. Wind tunnels were constructed and gained prominence during this period in order to facilitate the understanding of the flow around different elements of the race cars. Engineers realized the importance of wing-in-ground-effect that immensely increased the downforce of the car, with a slight increase in drag. For the cars equipped with higher levels of grip around the corners, lap times started to tumble quickly. It started with Lotus 72 (Fig 1.3), with base model designed in

1970, which moved the radiator in the wake of the front tires for the rear-engine drive to

3 accommodate a wedge-shaped nose. This design reduced the frontal area and improved air penetration, which in turn, reduced drag. It also featured an overhanging rear wing far behind the rear axle to increase downforce. McLaren (Fig 1.4) and Ferrari designed the future cars on this philosophy, building championship winning cars for the seasons from 1974-77.

Fig 1.3: Lotus 72D, variant of Lotus 72. Fig 1.4: McLaren M23 (World Cars List, 2013) Built in 1972 (Wikipedia, 2013)

1977 was the year that led to the birth of ground-effect aerodynamics. Developed by accident while trying to study the complexities of airflow underneath the car, Lotus incorporated the ground- effect in their Lotus 78 (Fig 1.5) and 79 car models by using wing-shaped sidepods that were sealed to the ground by a sliding skirt. The design was perfected for the 1978 season by Lotus, while the other teams played catch-up to them. Brabham introduced a ‘fan car’ (Fig 1.6), where a fan was placed at the rear of the car, underneath the rear wing. This created a favorable pressure gradient at the rear of the car, thus increasing the airflow velocity underneath the car, which in turn provided more downforce. However, this car participated in only one race, as it was withdrawn by the constructors before a ban could be sanctioned.

The fight for domination on race track led to extensive research in ground-effect aerodynamics, with teams designing new underbody ‘tunnels’ to improve airflow, thus increasing downforce.

Lotus overplayed the use of ground-effect, intending to run a car without the drag-inducing wings; the downforce coming purely due to underbody aerodynamics. However, this concept failed and they were not competitive throughout the season of 1980. In 1981 and 82, the ground-sliding skirts were abolished and a minimum of 6 cm (2.36 in.) ground clearance was required. In 1983, the

4 underbody tunnels were banned as some racers lost their lives, with ground effects indirectly blamed. Thus, it was required that every race car have a flat bottom in area between the trailing edge of the front wheel and leading edge of the rear wheel of the car. This shifted the emphasis on optimizing the design of the rear wing to obtain maximum downforce. In mid to late -80’s various developments occurred to improve the flow in front of the rear wing. In 1983, as the height for the rear wing was limited, the Toleman team placed an airfoil ahead of the centerline of the rear wing to generate more rear downforce. In 1986, Brabham BT55 (Fig 1.7) canted their engine by 720 to improve airflow over the rear wing. However, due to increase in drag and air intake issues, the power was significantly reduced, resulting in poor performance. Drivers also tried to sit as low as possible in order to reduce the frontal area of the car to reduce drag. After 1986, due to allowance in

F1 rules, the rear wing comprised of multi-element airfoils (in most cases it was comprised of two elements – the main wing and a flap). The second element, i.e. the flap, cambered the overall profile of the wing to produce more downforce, but at the same time drag was increased.

Fig 1.5: Lotus 78 with side skirts Fig 1.6: Brabham BT46B "fan car" (Wikipedia, (FormulaPassion.it, 2014) 2014)

Fig 1.7: Brabham BT55 with no hood for air Fig 1.8 Rear end Diffuser for McLaren MP 4-5B (F1 Nutter, 2014) intake (FormulaPassion.it, 2013) 5

In 1990, McLaren used a rear diffuser in its car MP4-5B (Fig 1.8) to increase the pressure of the airflow from the underbody. This was done to reduce drag. The front nose was raised and splitters were added underneath so that a high-energy feed of airflow can be provided and made to rush under the flat bottom. With the advancement of electronics and computational technology, the early part of 90’s saw a huge inflow of driver aids and active control systems. Using them, the ride heights at the front and rear axles could be optimized as per the required downforce.

Perceiving a huge rise in the running costs as well as loss of public interest due to minimization of driver errors, the active suspensions were banned at the beginning of 1994 season. This on the contrary made it really difficult for the drivers, as they were unable to manage with the continuous change in weight distribution over the course of a lap around a race track. Also the aerodynamic performance of the car became really sensitive to ride heights, which caused oversteer and understeer in quick succession while steering through a turn. The balance issues, caused due to both mechanical and aerodynamic forces, resulted in far more serious accidents over the initial half of the season than any other. Two drivers lost their lives; one being a three-time driver’s championship winner, Ayrton Senna. After these incidents, rule changes governed by FIA were focused on increasing driver’s safety. As a result, there were strict regulations on all the aerodynamic devices.

Since 1994, as the focus of FIA completely changed towards driver safety, there were no significant changes. Aerodynamicist realized that in order to get the best out of the car over the season, the car has to be designed specifically for each track, within the confinements of the strict rules imposed. As all the teams were brought to a level playing field due to restrictions, advantages were obtained by giving attention to the minutest of the details. Any improvement in the car, with a reduction in lap time of the order of one-tenths of a second, was considered worthwhile. From 2004 onwards, FIA started the initiative of making Formula 1 more eco- friendly and proposed various cost-cutting methods. In order to keep the keep the thrills and

6 excitement levels the same, options like KERS and DRS were introduced in 2011. DRS allowed the rear-wing flap to be raised along certain parts of the track when the following driver was separated by a time of 1.0 sec to the driver ahead. Fig 1.9 displays configuration of an open DRS rear wing. As a result the drag was reduced by at least 20%, and overtaking possibilities increased. To date, this is the only movable component allowed on the car.

Fig 1.9: Active DRS in the rear wing (Wise, 2012)

Fig 1.10: Front wing for a modern F1 car (Planet F1, 2014)

1.2 Problem Statement

Although the F1 car aerodynamics is at a highly mature stage, there is always the possibility for further development. As the under-body aerodynamics are banned by FIA, the only significant

7 aerodynamic change allowed is that of changing the front and rear wing cross-sections, i.e. airfoils, or by developing new diffuser to improve the airflow underneath the car. Development of the most optimum airfoils becomes a track property, as each track has a different aerodynamic requirement. The design of the car and airfoils is also dependent on the rule changes introduced by the FIA at the start of every season. Aerodynamicists always have to find the correct balance to get the maximum out of the car; the design being based on whether the lap times are influenced more by drag reduction or by increments in downforce levels. As the FIA is on the road for cost- reductions, limited time is allowed for wind-tunnel studies. Due to availability of computational power and strong CFD tools, the preliminary designs are tested and optimized using CFD, in conjunction with different optimization algorithms. However, starting 2015, there has been added limitation to the amount of CFD technologies as well (Formula 1, 2014). Hence, in order to make the most efficient use of the limited hours and power available, this present study is undertaken.

The problem statement of this thesis, can thus, be summarized as follows:

“The purpose of this study is to provide optimized parametric study tables for a particular track, defining the sensitivity of the lap time across the track to different ranges of lift and drag coefficient of the front and rear wings. This study, hence, provides the designer a preliminary estimate of a lap time possible for a given track, for a given set of Cl and Cd values for the front and rear wings.”

Using the methodology provided by this study, a designer can establish a range of Cl and Cd for the front and rear wings. Based on these values, an optimized airfoil shape can be obtained using the techniques presented in various studies conducted and commercial CFD software available in market. Details for the optimization studies on airfoil design conducted are provided in Chapter 2. This study helps in localizing the problem and thus, will save both time and computational power required to conduct a CFD analysis. A lap simulation tool called AeroLap is

8 used, which is described in detail in Chapter 3 (Section 3.3). It simulates the car performance over a lap and provides the necessary information required for this parametric study.

1.3 Research Objective

The main objectives of the research are stated as following:

a) To successfully create an interaction between the optimization code written in Matlab,

and the lap-simulation software AeroLap.

b) To obtain the most efficient path across a circular track, an oval track and Malaysian F1

race track using Covariance Matrix Adaptation Evolution Strategy.

c) To understand the variation of the aerodynamic forces and the center of pressure with

change in radius of curvatures for the tracks considered.

d) To obtain the optimum lift and drag coefficient values for the front and rear wings,

independent of the ride height of the car.

e) To study the sensitivity of lap times, for the Malaysian F1 track with change in the lift

and drag coefficient values for the front and rear wings, as well as change in center of

pressure.

1.4 Thesis Overview

The thesis has been organized into the following chapters in order to realize all the research tasks:

(1) Chapter 2 provides an insight into the research conducted in the field of aerodynamics

of Formula 1 cars. The importance of the front and rear wings, their contribution to

aerodynamic loads acting on the car, and the studies performed for obtaining the most

optimum airfoil design for them are also discussed in brief. A brief discussion on the

ground effect aerodynamics and their contribution to downforce acting on a F1 car is

also presented in the end.

(2) Chapter 3 presents the theory for the optimization technique using Covariance Matrix

Adaptation Evolution Strategy. It presents the advantages and disadvantages over other

techniques used. This chapter also highlights the salient features of the lap simulation

software, AeroLap, used for this study. It concludes by defining how the optimizer and

the lap simulator work with each other to determine the most optimum lap times.

(3) Chapter 4 discusses the technique used to obtain the most effective racing line for a

particular race track. A detail study is conducted initially, discussing the variation of the

racing line, with respect to change in radius of curvatures and aerodynamic forces for

simple elemental tracks, for e.g. circular track. This is followed by highlighting the

process of optimizing the racing line for Malaysian F1 grand prix track.

(4) Chapter 5 showcases the calculation of most optimum values for aerodynamic forces,

with the racing line, obtained from the previous chapter, serving as an input. These

values are calculated independent of change in ride height of the car along the track.

The chapter concludes by providing a sensitivity of the lap time to change in the

aerodynamic forces.

(5) Chapter 6 summarizes the research and lists the conclusions drawn from the obtained

results. It also suggests future research work that can be undertaken following the work

done as a part of this research.

Chapter 2 LITERATURE REVIEW

This chapter discusses in brief, various studies conducted by researchers with regards to aerodynamics of Formula One race cars, and the various techniques used to optimize the bodywork and its components in order to reduce lap times. This discussion has been categorized in the following order:

i. Aerodynamics of race cars

ii. Ground effect aerodynamics

iii. Optimization of airfoils and aerodynamic forces

2.1 Aerodynamics of Race Cars

The aerodynamics of racing cars is similar to the aerodynamics of an airplane. While the

main purpose of the components of an aircraft is to generate lift, most of the components on a

racing car are designed to create downforce, i.e. lift in negative direction. Aerodynamics allow

tons of downforce to be created on the front and rear axles, without increasing the weight of the

body; thus increasing tire adhesion, which, in turn, allows the race car to negotiate a turn at a

higher speed.

Tire adhesion is defined as the ratio of the longitudinal or lateral force acting along the

horizontal plane of the car, at the axle, to the normal force exerted at that axle. Assuming, the x-

direction is aligned with the vehicle’s longitudinal axis, while the y-direction is aligned with the

lateral direction (refer Fig. 2.1); mathematically, tire adhesion can be represented by the symbol µ

as follows:

퐹푥 (표푟 퐹푦) 휇 = 퐹푧 … Eq. (2.1)

where, Fx = Longitudinal force, or braking force.

Fy = Lateral force, or cornering force 11

Fz = Normal force (weight of the car, downforce etc.)

Based on what force is being considered in the numerator, µ is called either braking coefficient or cornering coefficient. The maximum value of this tire adhesion coefficient, µmax, gives the classical friction coefficient. Hence, if the tire adhesion coefficient exceeds the friction coefficient value, the tire will start to slide or slip. From Eq. (2.1), it can be seen that increasing the normal force helps in reducing the tire adhesion coefficient. This signifies that for the same value of µ, if the normal force is increased, it will result in increasing the brake force as well as the cornering force capacity of the tire. Therefore, adding downforce at the tire improves its capacity to brake and/or negotiate a turn at a faster speed. Katz (Katz, 1995) summarized this phenomenon by developing a polar diagram, as shown in Fig. 2.1, to deduce tire’s performance. The radial units, highlighted as circles made from broken lines, are defined in terms of gravitational acceleration.

The vehicle is assumed to be moving to the left.

Fig 2.1: Effect of aerodynamic downforce on tire performance (Katz, 1995)

From the above figure, it can be seen that on addition of downforce equal to the weight of the vehicle (1g), there is tremendous increase the braking and cornering forces. Downforce aids traction as well; however, the improvement is not as large when compared to braking or cornering. The acceleration is limited by the engine power available. When the car turns and 12 brakes simultaneously, a significant point to be noted is that even though the resultant force

(represented by vector D) available to the tires is equal to pure cornering or braking force, the individual capacities of the components of the resultant force (cornering and braking force) are actually lesser. It can be concluded that braking while negotiating a turn will reduce the cornering force, leading to an unprecedented slide. Thus, levels of downforce have to be increased sufficiently to prevent the slide from occurring. Theoretically, higher the downforce acting on the car, faster it can make the turn without sliding.

Nothing comes free of charge though. The penalty for excessive downforce comes in the form of increased drag (Katz, 1995). Hence, the main task for aerodynamicists in designing the bodywork is to optimize the balance between downforce and drag forces created in order to obtain fastest possible time around a race track. Aerodynamics of race car not only involves creation of downforce, but also includes areas such as airflow to engine box and radiators, cooling of brakes, limiting fuel sloshing, release of exhaust gases, etc. However, this work primarily focuses on the issue of obtaining optimized downforce and drag values for F1 cars. For the remaining part of the thesis, the term lift, if used, will be considered the same as downforce. For e.g. CL being referred to as lift coefficient will actually mean dimensionless value of downforce.

Willem Toet conducted lap simulation studies and roughly calculated how much a percentage change in different parameters would affect the lap time (Toet, 2013). This is shown in

Table 2.1:

Table 2.1: Parameters affecting lap times (Toet, 2013)

Parameter % change in Effect on Lap Time Parameter Grip from tires, suspensions etc. 10% 3.2% Vehicle mass 10% 1.9% Center of gravity height 10% >0.4% Engine and transmission of power 10% 1.5% KERS Application for 6 secs 0.5% Aerodynamics 10% 1.0%

At first sight, it is seen that aerodynamics isn’t the key performance changer. However, most of the other parameters are controlled by the rules suggested by FIA, and every team is provided with tires from the same manufacturer. Hence, aerodynamics becomes a key differentiator for a team’s car total performance.

Fig 2.2 shows the contribution of different components of a 2009 F1 car to the total downforce and drag (Toet, 2013). It is to be observed that the break-down is for a general configuration. These values will differ for every car; the percentage of change depending on the downforce requirement of the track. From the figure, it is observed that some components produce lift. However, if this component is redesigned to neutralize this effect, either the total downforce levels might decrease, or it might affect the stability of the car. From these results, an important realization can be deduced that all parts of the car are interacting with each other and work as a whole. The values shown in the figure are due to this interaction. For example if a component is lost during a collision, such as the front wing, the loss of downforce will not be 30%, it will be higher. The front wing is designed to divert the flow away from the tires and provide a channelized flow to pass underneath the floor. The drag and downforce values of these components will change accordingly. For preliminary design purposes, it is possible to discretize these contributions; however, the observed effects might be different than expected. Also, if the wings are to be designed for a particular track having a high aerodynamic requirement, it will be difficult to isolate and optimize the lift forces obtained from a base setup that is configured for low aerodynamic requirement. Therefore, it is necessary to base the preliminary design on car’s configurations developed for similar track profiles.

Similar observations to that of Toet were made by Hilhosrt (Hilhorst, 2013) and Huminic et.al. (Huminic & Huminic, 2008). It is very clear from the chart that the key downforce providing components are the wings on the car and the floor-diffuser combination, with each providing a

55% contribution. As F1 rules authorize the use of a flat bottom, no design variations can be

14 studied for the floor. Hence, this effort primarily discusses the optimization of the lift and drag- coefficients of the front and rear wings. Insight to these components is provided in the following subsections.

Drag and Downforce of 2009 F1 car

Floor and Diffuser

Rear Wing Assembly

Rear Wheel Suspension, Brake Duct Drag Chassis, Bodywork Downforce

Front Wheel, Suspension, Brake Duct

Front Wing Assembly

-20 -10 0 10 20 30 40 50 60 % contribution to the total value

Fig 2.2: Breakdown of drag and downforce for individual components of 2009 F1 car (Toet, 2013)

2.1.1 Front Wings

The front wing is considered to be, aerodynamically, the most efficient and the most

important device on the F1 car, as it is the first component of the bodywork to come in contact

with the undisturbed air. It is primarily responsible for generating downforce to aid high

cornering speeds. The wings are inverted (in comparison to aircraft), with the suction side of

the airfoil towards the ground, in close proximity to it. This is done to take advantage of

ground effect, which are discussed in detail in Section 2.2. The design of the front wing is

mainly dictated by the flow field required by the rest of the car. It might be required to

sacrifice some downforce at the front wing to improve the overall aerodynamic balance and

performance of the car.

The length and, to some extent, the design of the front wing is governed by the rules stated by the FIA. Currently, the front wing span is fixed at 1650 mm (65 in.) (Fédération

Internationale de l’Automobile, 2011). A typical front wing for the current F1 car is displayed in Fig 2.3.

Fig 2.3: Front wing for a modern F1 car (Planet F1, 2014)

The center portion of the front wing is a neutral section as it just regulates the airflow and doesn’t contribute to the downforce (Toet, 2013). The modified splitter plates that structurally connect the body with the wing are used to channelize the flow underneath the floor. As seen in the figure, there are multiple flaps, guide vanes and specially crafted endplates on the front wing. The main purpose of the flaps is to increase the downforce levels. The slots and guide vanes are used to divert the airflow, either towards the radiators and underneath the bodywork or around the tires.

The performance of the front wing is also dependent on the presence of the front wheels.

Agathangelou & Gascoyne (Agathangelou & Gascoyne, 1998) stated that as the wheel rotates at high velocities, a high-pressure zone forms in region upstream of the tire contact patch, due to the jetting effect. It was a result of interaction between the contact patch of the tire and the airstream incident on it. The airflow squeezes out from either sides of the wheel, thus creating

jet vortices that reduce the effectiveness of the front wing. Jasinski & Selig (1998) made a

similar observation. Hence, the endplates and guide vanes on modern F1 cars are designed in

such a way that they divert the flow away from the tires. Endplates also help control the wing-

tip effects associated with finite spans. This also helps in increasing the downforce at wing tips

(Dominy, 1992).

The wake generated from the front wings influences the airflow over rest of the body, in

particular towards the radiators, thus influencing the cooling performance of the car. The

greater the incidence of the airfoil, the greater is the influence of its wake on the cooling flow

(Dominy, 1992). Thus, the design of the front wing is based on downforce requirements at the

front and airflow to different components of the car. Jasinski & Selig (1998) identified crucial

factors affecting the design of the front wing. It was noted that a flap deflection of 100

improved the CL by 0.5, with no significant change in drag coefficient. The added area due to

the use of flap also aided to the increment of the lift coefficient of the wing, due to increase in

aerodynamic loading. The authors also stated that increase in endplate planform improves

overall lift coefficient of the wing and also aids in reduction of drag coefficient.

2.1.2 Rear Wings

While the front wing is considered to be the most efficient component of the F1 car, the

rear wing, on the contrary, is highly inefficient. The airflow over the rear wing is very

complex, as it is exposed to unsteady, turbulent flows arising from the front wing, wheels,

cockpit, radiators, air intake, exhaust and other numerous other upstream influences. Due to

these turbulent flows, the performance of the rear wing is significantly reduced. Aerodynamic

stability of the car is important for peak performance of the car as well as driver’s safety. To

maintain the aerodynamic balance, the rear wing is required to produce almost similar, if not

more, levels of downforce. Huge oversteer occurs for the car while negotiating a turn, if the

location of center of gravity is aft of that of the aerodynamic center of pressure along the car’s

length. Therefore, to generate large amount of downforce from the rear wing exposed to 17 turbulent air the flaps are deflected to very high angles. Such high values of camber, however, results in significant increase in drag. As seen from Fig 2.1, the rear wing contributes 28% of the total drag. Thus, the design of rear wing becomes a tradeoff study between management of lift and drag values. Optimum balance being dependent on the lift coefficients of the wings, therefore, becomes a function of the track.

Even with such high drag penalty, rear wing is a very important component of the F1 car. Generally, it is structurally stronger as compared to front wing. No changes are made to the rear wing once the setup is complete. The aerodynamic balance is changed by changing the incidence of the front wing. Loss of the rear wing during a collision can create loss up to 40 % of the total downforce (Toet, 2013), and render the car un-drivable. Loss of front wing can be fixed by changing the nose section, which can be replaced in less than 10 secs. A typical rear wing for a present day F1 car is shown in Fig 2.4.

Fig 2.4: Active DRS in the rear wing (Wise, 2012)

As seen in Fig 2.4, the flap in the rear wing is raised. This is the DRS system, governed by the rules prescribed by the FIA. The flap is allowed to be raised or “opened” by a certain amount defined by the FIA in order to reduce the drag and downforce. This action can be performed only during certain sectors of race track, normally straights that require less levels of downforce. The system is allowed to function only if the car is following a car, separated by less than a 1.0 sec margin. The drag and downforce levels decrease by 20-25% (Toet, 2013)

18 when the flap is raised, thus making the car go faster that allows more overtaking opportunities. Up until the 2013 season, a supplementary wing, known as ‘beam wing’, was added below the rear wing. The main function of the beam wing was to seek advantage of the gases being released from exhaust to increase downforce. As a result of close proximity of the suction side of the rear wing to the pressure side of the beam wing, there was decrease in downforce on the rear wing. However, if looked upon as a single unit, there was an increment in total downforce obtained. Fig 2.5 shows the beam wing (horizontal wing in black, underneath the rear wing). This element was eliminated for the 2014 season and onwards.

Fig 2.5: Rear wing along with the beam wing (Michalis, 2011)

A performance study was conducted by Parra & Kontis (Parra & Kontis, 2005), where it was observed that the flow of exhaust gases over the rear wing improve the overall downforce of the car. The authors determined that the exhaust gases allowed to increase the downforce generated without an increase in drag. They also calculated the optimum diameter ratio for the exhausts.

Most of the other research was performed in obtaining the optimal design for the airfoils to be used in the rear wing. These are discussed in Section 2.3.

2.1.3 Aerodynamic Balance and Stability

As stated earlier, aerodynamic balance is of utmost importance for stability and handling

of the F1 car. Many severe accidents, some of which have tragically ended in the death of the

driver, have occurred due to ignorance of this property. As most of the driver aids are not

allowed by the FIA to be used in F1, it is of prime importance for reason of safety to find the

right balance of the car. Aerodynamic stability also improves transient phases, such as braking,

acceleration and cornering.

Dominy (Dominy, 1992) stated that the relationship between aerodynamic performance

and mechanical properties of the car was crucial to understand the handling properties of the

car. The author performed a cornering analysis of the car, where all forces were equated for the

condition of a car negotaiting a corner. He formulated equations to calculate weight transfer.

The author was able to deduce that the downforce of the car played a major influence on the

cornering speed of the car, with similar results being observed for center of pressure on balance

of the car. Fig. 2.6 displays the forces acting on each axle while performing a steady-state

cornering maneouvour.

Fig 2.6: Forces and Moments acting on a car while making a turn (Dominy, 1992) 20

Here Wf and Wr are wheel loadings, and Lf and Lr are the weight transfers due to overturning moment, at the front and rear tires respectively. The wheel loadings are basically the sum of the weights and downforce components acting at that axle.

Katz (Katz, 1995) stated that by changing the levels of downforce, the stability of the vehicle could be improved or degraded. The locations of center of gravity and center of pressure are of extreme importance, as they governed the stability of the car. Higher stability in a vehicle causes the vehicle to understeer, i.e. the vehicle will turn less than the input provided by the driver. On the other hand, oversteer is a condition of an unstable vehicle that causes the car to turn more than the desired level. While turning at high speeds, oversteer would require the driver to make corrections by turning the steering in the opposite direction. Understeer would result in reducing the speed of the car, in order to negotiate the turn successfully. Hence for optimum driveability, the car should corner with a near-neutral attitude. The location of center of pressure is governed by the downforce acting on the front and rear axles. Thus, from stability point of view, the most optimum levels of downforce might not be the best levels of downforce that can be obtained from the car.

As stated earlier in the section, downforce helps in increasing braking force. Due to some variable road conditions or suspension motion, if the vehicle pitches forward, the front wing downforce increases tremendously, making the car less stable. This might result in the front wing bottoming out, leading to structural damage as well as loss of downforce. This phenomenon is called pitch sensitivity of the car. Suspension details of the car are configured to prevent bottoming out of any of its portion.

Lateral stability of the car also highly depends on the relative location of the center of pressure and center of gravity, especially in conditions when wind forces act at an angle to the car. Side-winds cause side forces to act at the center of pressure. It is desirable to have the center of gravity (c.g.) forward of center of pressure (c.p.), as the side forces cause the car to

turn into the wind. To correct the course, the driver should steer in a direction away from the

wind to counter the effects of side forces. For the reverse condition, where the c.g. is aft of the

c.p., the driver is required to steer into the wind, as the car drifts away in the direction of the

wind. Also, the oscillations for the first case have less amplitude and dampen out faster, when

compared to the second case.

Overall, it can be inferred that drivability and stability of the car are highly significant

reasons for determining the optimum downforce levels acting at the front and the rear axles.

2.2 Ground Effect Aerodynamics

Ground effect aerodynamics is basically generation of downforce by maintaining low clearances between the underside of the car and the track. The air flowing beneath the car accelerates as it channels under the nose of the F1 car towards the floor of the car. At the same time, as the freestream velocity over the car is slower when compared the bottom, it creates a pressure differential. This results in generation of downforce. Dominy (Dominy, 1992) states that the undertray of the F1 car is situated little behind the nose, as the designers need to push the low pressure zone to as rearward position as possible. A separation bubble is formed at the leading edge of the undertray, where the flow seperates due to it’s sharp edge and forms underneath it.

With the area of entry reduced even further, the airflow gains more acceleration. Addition of diffuser behind the undertray creates a further low pressure zone around the transition point between the undertray and the diffuser. The generation of a low pressure zone helps in drawing out the accelerating airflow beneath the car. As shown in Fig 2.1, the floor and diffuser combination provides for 55% of the total contribution.

Ground effect also help increase the effective downforce generated by the wings in close proximity to the ground. Investigations conducted by Zerihan and Zhang (Zerihan & Zhang,

2000) and Ranzenbach and Barlow (Ranzenbach & Barlow, 1996) found out that under the influence of ground effects the downforce on a single element increased up to a certain maximum.

Beyond a critical ride height, which was calculated as 0.1c by Zerihan and Zhang (2000), the wing stalled and the loss of downforce was observed. This loss was attributed to the occurrence of adverse pressure gradients underneath the wing as a result of ride height reduced below the critical level. Zerihan and Zhang (Zhang & Zerihan, 2003) extended this study and performed a

3D experimental analysis on a single element airfoil. The authors evaluated the effect of incident angle of the front wing and varying ride height on the wake structure behind the wing. The results showed that as the ride height was reduced, vortex strength increased that lead to increase in wake size as it moved downstream. Due to boundary layer separation at the suction side of the wing stationed below the critical ride height, the edge vortex broke down that resulted in loss of downforce. Soso & Wilson (Soso & Wilson, 2006) studied the aerodynamics of the front wing of a F1 car when following in the wake of a leading car. It was observed that when in the wake flow of the leading car, the downforce levels dropped more at higher ride heights as compared to lower ride height. This was mainly due to upwash created due to the upstream bluff body. The wing was loaded more near the mid-span as compared to the wing tips at lower ride heights; trends reversing for greater ride heights.

2.3 Role of Optimization

As stated earlier, the objectives of aerodynamic development are: (Hilhorst, 2013)

i. Reduction in aerodynamic drag – leads to better acceleration.

ii. Increase in downforce – gives more tire grip, and thus, better traction and higher

cornering speed.

iii. Aerodynamic stability – improves transient phases such as braking, acceleration and

cornering.

Based on the track requirements, each of the above three parameters need to be optimized to get the best possible lap time; however, the optimization is inter-dependent on each of the aerodynamic variables mentioned above. This section discusses various strategies used in the past by various researchers to optimize certain parameters. 23

Gerhardt et al. (Gerhardt, Kramer, Zakowski, & Barth, 1986) summarized the issues surrounding the Formula 1 team of Zakspeed and provided optimum locations of the wings as well as shape of the body of the car to maximize the performance. They suggested that the front wing should have the lowest possible clearance and must be placed further upfront the front axles.

This was due to the reduction of downforce values with a corresponding increase in drag values being observed as a result of the vortices generated by the wing tips interacting with the tires. For the central body section, they suggested to go ahead with rectangular planform instead of using the wedge shape. This idea was proposed in order to increase the downforce from underbody ground effects. For the rear wing, using a three component airfoil positioned higher and further aft of the location of rear axles was suggested.

Muzzupappa & Pagnotta (Muzzupappa & Pagnotta, 2002) were one of the few to use genetic algorithms to optimally design the rear wing; however, the design of the rear wing was optimized for structural integrity rather than for aerodynamic loads. The main objective was to design the thinnest possible wing with the smallest vertical deflection between the tip and the wing. Wloch & Bentley (Wloch & Bentley, 2004) used genetic algorithms in association with a lap simulation software to obtain the best lap times for a given circuit. The lap simulation software used was ‘Formula One Challenge ’99-’02’ by Electronic Arts. Being a computer game, it didn’t take the values of drag and lift co-efficeints as input. Also, the software didn’t specify the actual values of the setup variables. The optimization changed the variables by entering a factor for the parameters that were allowed by the software to be altered.

Melvin & Martinelli (Melvin & Martinelli, 2008) examined optimizing the shape of multi- element airfoil by developing an adjoint based optimization system. They used the pressure distribution, lift coefficient and L/D ratio as input. Based on the best values of CL and L/D obtained by the software, the optimizer modified the pressure distribution of the given airfoil in order to generate a new airfoil shape. A special case of a F1 car’s multi-element front wing was

24 considered. The boundary conditions were defined as the the range of ride heights possible for the car. The optimizer determined the best possible CL pressure distribution, and generated the coordinates of a new airfoil. From the data, it was observed that the CL and L/D values increased with decrease in ride height until it reached the critical value of 0.1c. This was in accordance with the study conducted by Zerihan & Zhang (Zerihan & Zhang, 2000). This optimizer was, however, dependent on the initial airfoil coordinates and pressure distribution. Gopalarathnam & Selig

(Gopalarathnam & Selig, 1997) developed a methodology for designing high-lift airfoils for use on low aspect ratio wings. Goto & Sakurai (Goto & Sakurai, 2006) performed a parameteric study to determine the optimal flap chord length of a given two-element airfoil. The authors observed the variation in lift coefficient and L/D ratio with changes in flap chord length, flap angle and distance between the main airfoil and the flap. From these studies, it was inferred that a flap with chord length within the range of 40-70% of the main airfoil yielded the highest downforce values.

However, it was desired to keep the flap’s chord length to a minimum value within the specified range in order to reduce the drag. Zhigang et al. (Zhigang, Wenjun, & Qiliang, 2011) performed a parametric optimization study where multiple cases of parameters affecting the aerodynamic charactereistics of the rear wing were considered (for e.g. camber, angle of attack, thickness of airfoil etc.). For each of those parameters, a range was specified as an input to FLUENT and a

CFD analysis was conducted. The best results were selected once all the simulations were completed.

2.4 Motivation for this study

From the research studies listed above, it was deduced that many tried to optimize the

airfoil for either the front or the rear wing, without considering the following issues:

i. There was a particular set of input values for which the iterations were performed. The

true optimum might not exist within the specified range.

ii. The main objective of the optimization was to get the maximum lift-to-drag ratio for a

given airfoil without accounting the aerodynamic requirements of the F1 car. 25

iii. The simulations did not account for race track properties.

With the stated issues, it is very hard to state whether an airfoil is optimized or not.

Toet (Toet, 2013) suggested the following steps to test and design new airfoil:

1) Lap simulations

2) CFD simulations, design and testing

3) Wind – tunnel testing

4) On-track testing.

Currently, the rules stated by FIA (Fédération Internationale de l’Automobile, 2011) limit the on- track testing and also have capped the usage of wind – tunnel testing in order to make the sport cost-efficient. For the 2015 season this cap will be reduced even further, and an additional cap on

CFD usage will also be imposed. Hence, running an optimizer in association with a CFD solver will exhaust precious resources available to the team. This study, thus, focuses on the lap simulations aspect.

Using lap simulations to get an optimized lap time based on different parameters will provide a preliminary estimate of the optimal range of aerodynamic coefficients to be used for designing the wings. Also, optimizing the aerodynamic values for minimum lap time will address two of the three stated issues observed for the conducted research. To resolve the first issue, use of an evolutionary genetic optimizer will be made. The algorithm will provide the optimized range of CL and L/D values for a particular track. Using the parametric charts in form of contour plots, a designer can select a suitable range of coefficients based on the lap time for which the car is required to be designed. This range can be fed to any one of the previously mentioned methodologies in order to obtain the most optimum airfoil shape. The design can be tested along with the complete car model in any of the available CFD software. Thus, a great load of computational time and testing time can be saved.

Chapter 3 TOOLS USED

This chapter discusses the tools used to optimize various parameters of the F1 car to get the best possible lap time for a particular track. The present research study is an optimized parametric study analyzing the effects of change in lift, drag and aerodynamic stability on lap times. The term

‘optimized parametric study’ is used to imply that the parameters will be optimized for a given setup of a F1 car over a track, one at a time. Once completed, the trends for variation of lap times will be analyzed and reported. Performing such a comprehensive study assures that the best lap times for a given setup are listed after inspecting various possible scenarios. Thus, the designer can make an informed decision of selecting the required range of aerodynamic coefficients for which an optimum airfoil can be designed. Using this set of information, the aerodynamic stability of the car can also be predicted.

To perform the optimization we make use of a genetic algorithm based optimization strategy called Covariance Matrix Adaptation – Evolution Strategy (CMA-ES). Details of this algorithm are provided in Section 3.1.

Why to use genetic algorithms? It is an optimization model of machine learning, whose behavior is derivative of the natural process of evolution and natural selection (survival of the fittest). Genetic algorithms typically require a form of representation or distribution of the domain used for learning trends and guidance, and an objective function to validate the fitness of each of its population members. The evolution usually starts with populating the domain with randomly generated individuals. The fitness of each of these individual solutions is evaluated. The fitter individuals are then selected for reproduction to create a new generation of individuals. These individuals possess hereditary data from all the selected individuals of the previous generation. As the algorithm is iterative, each generation of new population is an iteration step. The process is

27 repeated until the optimization criteria or the boundary condition(s) are met. The process can be summarized as follows:

1. Initialization of the Genetic algorithm

2. Selection of fitter solutions

3. Mutation or recombination of the selected solutions Iterative Process

4. Reproduction of new generation

5. Termination: when boundary conditions are met.

The advantage of genetic algorithms is that they can be used to solve multi-dimensional non- linear problems. As there is no requirement to provide an initial set of state variables to start the optimization, it serves as an excellent platform to solve such kind of problems. A major limitation of using these algorithms is that they can be computationally intensive and might not converge to a solution at a faster rate.

3.1 Covariance Matrix Adaptation – Evolution Strategy:

Covariance Matrix Adaptation – Evolution Strategy (CMA-ES) is one example of an

adaptive genetic algorithm. The term adaptive is self-explanatory, i.e. the solution that is

generated at every iteration tries to adapt to the best value possible. This is brought about by

adapting the covariance matrix of the distribution. For the genetic representation, the solution is

sampled according to multivariate normal distribution. Users are asked to enter the standard

distribution at which the data need to be sampled and processed, along with the optimization

bounds and the objective function that are required to completely define the optimization problem.

A default value of 30% of the difference between the limits is used for standard deviation, which

is suggested for the use of this optimization tool.

The basis for adaptation is derived from two main principles (Wikipedia, 2014):

i. Maximum likelihood principle: On the basis of this principle, CMA-ES tries to increase

the likelihood, i.e. the probability of event occurring (in our case it is minimization of the

lap time) for a given set of successful candidates.

ii. Multiple path approach: CMA-ES works on two evolution paths derived for the

distribution mean of the given sample. One path is used to update the adaptation

covariance matrix, while the other is used to conduct step size control. These paths

contain information regarding the correlation between consecutive steps. Adaptation

determines the direction of convergence towards the true solution by correlating data

between different generations. On the other hand, the step size control helps in preventing

premature convergence and, at the same time, leads to the true solution at a faster rate.

Thus, it ensures that the solution obtained has a true minimum fitness value, and it is not

stuck in the local minima.

A brief description of the methodology used by the CMA-ES is provided as follows. Covariance of a distribution is basically the square of the standard deviation (σ) of the distribution. Thus, the covariance matrix calculated for the initialization step includes only σ2 terms in its diagonal matrix. This step basically defines the area of the distribution along which the members are to be populated as shown in the leftmost segment of Fig.3.1. The circle is defined for a radius of length of σ2. For the first iteration, the covariance of the distribution is updated using the equation stated by Hansen & Ostermeier (Hansen & Ostermeier, Completely Derandomized Self-Adaptation in

Evolution Strategies, 2001). A diagonal matrix D is calculated by assuming the square roots of the eigenvalues of the covariance matrix along its diagonals. The D matrix is responsible for expanding or shrinking the search (populated) area. This is shown in the central section of Fig.

3.1. The roots of the eigenvalues align along the major and minor axis, as denoted by the ellipses drawn with solid and broken lines respectively. The evolution path variables are used to update the covariance matrix based on the fitness of the population selected for mutation. The covariance

29 matrix orients the search area towards the expected convergent, optimized solution with information of correlation and evolution paths stored in it. A visual representation of the orientation is depicted by the rightmost image in Fig. 3.1. The step-control evolution path variable allows the search area to move towards the optimized solution. It is basically used to update the D matrix and is therefore responsible of the size of the search area.

Fig 3.1: Pictorial representation of different factors influencing evolutionary optimization. (Hansen & Ostermeier, Completely Derandomized Self-Adaptation in Evolution Strategies, 2001)

The algorithm for explaining the process is provided as follows:

i. Set the number of members to be populated / child solutions to be calculated (λ).

ii. Initialize all parameters, where mean and standard distribution of the distribution are

provided by the user. Covariance matrix is initialized as σ2I, where I is an identity

matrix.

iii. Populate the search area with λ members, by random selection.

iv. Determine the fitness of all the members using the objective function. Sort the fitness

values.

v. Select population members for generating new solution. The higher the fitness value,

the greater is the percent contribution to the next generation.

vi. Update the mean, standard distribution, covariance matrix of the distribution and

calculate the evolution path variables.

vii. Iterate until a specified accuracy is met.

viii. Terminate by providing the values for the best solution and its fitness value.

A comparison study was performed by Hansen et al. (2010) to check the performance of

CMA-ES against other available evolutionary algorithms. A total of 31 Genetic Algorithms were benchmarked on 24 noiseless functions. The runtime for each algorithm was measured in terms of number of function evaluations along with its convergance rate to evaluate each algorithm’s performance. For benchmarking, the best algortihm would be able to solve 75% of the functions up to a precision of 10-6, while the median algorithm would be able to complete about 30%. For multimodal functions, the rate dropped to 50% for the best and 20% for the median. It was determined that CMA-ES performed the best for difficult multi-dimensional, multimodal fucntions; however, they were considered median for low dimensions.

Thus, for high-dimensional non-linear functions, CMA-ES is benchmarked to perform well.

Hence, this algorithm was used in this study.

3.2 AeroLap – Lap Simulation software

AeroLap is a commercial product of Ansible Design Limited, based in UK. It is a lap simulation tool used for analyzing and maximizing performance of race cars over a given race track. The configurations and setup for the car can be defined though various components and extensions offered by the software. The following components of the car can be defined using

AeroLap:

i. Aerodynamic Component: AeroLap allows the information regarding downforce, drag

and center of pressure location of the whole car, for either a fixed or variable ride height

configuration, to be set. For variable configurations, tabulated data is required as an

input, where aerodynamic values varying with respect to change in front and rear ride

height. For force calculations, the frontal projection area of the car is considered.

ii. Chassis Component: The dimensions and mass of the car serve as an input to this

component. The car is assumed to be a rigid-frame spring-mass model, with all the

vertical loads assumed to be acting on the wheelbase.

iii. Engine Component: The specifications of engine to be used such as the throttle, RPM,

fuel consumption etc. can be defined.

iv. Gear Ratios: The total number of gears along with the number of teeth on each of them

and the RPM limit for each gear can be provided through this component.

v. Suspension Component: All suspension details can be entered in this section. This

allows for the design of the suspension for each wheel separately.

vi. Tire Component: Data representing change in grip with temperature and effective rate

of degradation is to be provided in this section.

vii. Environment Component: AeroLap allows for the simulation of weather by allowing

to define the specification of wind gust speed, density of air etc., for a realistic

simulation of a race.

viii. Track Component: The x- and y co-ordinate of the track are set as an input in this

section. Details, such as bank of the curve, elevation, radius of curvature, grip and

various other offsets, can be entered.

3.3 Optimization Process

For running the optimization to get the best possible lap time, CMA-ES requires an objective function. AeroLap serves that purpose. Thus, there is always an exchange of data between AeroLap and the optimizer for every evaluation step.

The process is activated by providing input to the optimizer. This file includes the starting value of the parameter being optimized (xmean), the upper and lower limits for the parameter, the standard deviation (σ) of the distribution, and control flags, or boundary conditions, that govern the terminal condition for the optimizer. The parameter being optimized can be a single-valued variable or a multiple-valued variable in the form of a column vector. The standard deviation and the boundary limits need to be of the same order as the input variable. Another important variable that needs to be defined is the population size (λ). This defines the number of child solutions to be

32 created in each generation by the CMA-ES to populate the search area. Once the input variables have been defined, the optimizer is called by the input file. The optimizer initiates all the variables it requires to carry out the process. Once all the values are set, it randomly generates λ child solutions within the search space, such that the standard deviation of the distribution is maintained as specified. For each of these created λ solutions, optimizer calls AeroLap to perform a lap simulation with the specified F1 car attributes. AeroLap returns lap times associated with each of the child solutions, which defines their fitness. These child solutions are then sorted based on increasing levels of fitness. Based on intrinsic methods of the optimizer, the best solutions are selected for mutation. These mutations lead to generation of next set of child solutions. All the fitness values and those of the child solutions are stored in history. The covariance (C) matrix, along with the diagonal matrix (D) and evolution paths are all updated based on the newly generated solutions. The fitness values of the previous set of child solutions are compared with those in the history of the optimizer. If better, they are stored in a ‘best-ever’ matrix, which stores the values of the fittest solution and its associated variables. This process is repeated until one of the specified boundary conditions is met. The flowchart in Fig 3.2 provides pictorial details of the functioning of the optimizer and demonstrates how CMA-ES, with the aid of AeroLap, achieves the optimum value.

Start

Initiate Input file Enter xmean, σ, λ, boundary conditions for the optimizer

CMA-ES (Optimizer) Initiates all variables (only first run) Generates random λ solutions

Yes Objective Function Is Prepare variables Count<= λ? (xmean) in format

fitness compatible with

xmean No AeroLap

Optimizer

fitness Sorts the fitness values Selects parents for mutation AeroLap Stores information in history Simulates a flying lap Updates C matrix, evolution Stores output paths, σ

Is fitness Yes Record Data in value better than bestever history before?

No Any Boundary Conditions Met?

Yes

Stop

Fig 3.2: Flowchart describing optimization process.

Chapter 4 OPTIMIZATION OF RACING LINE AROUND THE TRACK

With the theoretical background and purpose of the study defined, the methodology of performing the optimization on various parameters is discussed in this chapter. To obtain true optimum values of the lift and drag coefficients corresponding to the minimum lap time, all the components of the race car are required to be optimized. However, many factors are either limited by governing rules (like engine power), or due to the configuration dataset provided, for example suspension setting, chassis, tires, etc. The given data may or may not be optimized, but true optima for the aerodynamic coefficients of the wings can be obtained for the given setup. This concept forms the underlying basis of this study.

The Formula 1 car setup used for this study was an example setup file provided as a complementary package with the AeroLap software. The chassis setup corresponded to the rules and regulations of the 2010 season. The aerodynamic setup, however, resembled to that of the 2001 season. This was not a concern as the optimal values would be calculated for the dimensions specific to the 2010 season. An assumption was made that the total contributions from the body of the car without wings did not change significantly.

In addition to optimizing the aerodynamic properties of the wings of the F1 car, the racing line around the track, specified as a set of X and Y coordinates, must also be optimized. This results in the establishment of an optimal, or the fastest racing path around the circuit for which the best possible values of the lift and drag coefficients would be calculated. Thus, the first step of the research was to optimize the racing line of the track. An assumption was made that the effect of change in aerodynamic forces on the racing line could not be significant. The changes might be of the order of few centimeters at some locations, which would be considered negligible for a racing

35 driver. Before the results of the optimization of the track are presented, the variation of racing lines around tracks of simple geometries are presented to get an understanding of how the dynamics of the car changes. This study forms an excellent basis for validating the optimizer’s performance, to check whether the results obtained are as expected in real-life situations.

4.1 Study of a Straight Track

As the name suggests, a straight track consists of a track with absolutely no turns (like a

runaway for an aircraft). The position of the car at the start within the boundaries of the track is

immaterial. The fastest path is the shortest linear path between the start and the finish line that is

perfectly parallel to the edges of the track. Therefore, optimization of racing line is not required.

Usually the race car traverses along the centerline of the track to provide enough room on both

sides of the car. The main issue to be tackled is to increase the velocity of the car. As discussed in

Section 2.1, this objective can be achieved either by increasing downforce or by reducing drag for

a given car setup. Generally, drag reduction is the solution for achieving maximum pace on a

straight track for a given car. With the car operating at maximum engine power, the addition of

downforce can help only to a certain extent by improving the traction of the car (as shown in Fig.

2.1). Increasing the downforce also results in increasing the induced drag on the car. Therefore,

the lift forces are optimized in order to provide sufficient traction force without excessive levels

of induced drag.

4.2 Study of the Circular Track

A circular track is the most basic form of circuit track. With a uniform turning radius, the

performance parameters of different components are expected to remain constant throughout the

lap. Therefore, a study of the vehicle dynamics and aerodynamic properties around such a track

can highlight some basic trends that will help in understanding the results obtained further along

in this research.

In order to observe the trends, three different circular tracks were considered with radii of the centerline being 50 m, 100 m and 200 m. The width of each of these tracks was considered to be

15 m, which is approximately the average width of most of the Formula 1 tracks. There was no elevation or banking considered for any of the tracks. The optimum racing line for each of the tracks was obtained using CMA-ES. Four data points were selected on the centerline for each of the three circular tracks as displayed in Fig. 4.1. They served as input variables to CMA-ES.

These data points are indicated by blue square markers on the centerline. The change in the position of these data points was limited in the direction normal to the center line at their starting locations. This path is signified by the brown lines connecting the inner and outer boundaries of the track, as shown in Fig. 4.1. This restriction in the free movement of the data points within the boundaries of the track helps to remove redundant results. Considering the coordinates of the data points in polar co-ordinates, the restrictions help to omit data points that would have the same pole length, but different angular coordinates. With the performance parameters not changing along the circular path, restrictions to these points reduces the number of optimization iterations and the racing path is sufficiently defined by the four data points considered.

The optimization results are pictorially represented in Fig. 4.1 with the optimized racing line displayed as a broken black line. The locations of the data points along the optimized path are depicted by blue circular-shaped markers. It is observed in Fig 4.1 that the most optimum location of the racing line is along the inside boundary of the track, irrespective of the radius of the centerline of the track. This concluded that the levels of downforce generated by the F1 car were sufficient to overcome the centrifugal forces acting on the car. Another deduction that could be made looking at the results is that the velocity gradient from the inner to outer edge of the track does not change as significantly when compared to the distance traversed.

Fig. 4.1: Circular tracks with varying radii displaying the location of the racing line.

To verify this claim, a mathematical check was performed as follows:

When the car negotiates a turn, the centrifugal forces are balanced by the side force acting on the tires (as given by Eq. (2.1)). Thus, at a given instant of time:

Tire Side Force = Centrifugal force

푚푣2 휇 × 퐹 = 푧 푟 … Eq. (4.1)

푚푣2 휇 × (푚푔 + 퐿) = 푟 … Eq. (4.2) 1 퐿 = 휌푣2푆푐 2 퐿 … Eq. (4.3)

휇푚푔푟 푣 = √ (푚 − 1⁄ 휌푆푐 푟휇) 2 퐿 … Eq. (4.4) where,

휇 = Cornering coefficient of the tire

Fz = Side Force acting on the tire.

mg = Normal weight component of the car acting on the axle

r = Radius of curvature of the turn

v = Velocity of the car

L = Lift force (downforce) acting on the axle

S = Planform area of the wing being considered

ρ = Density of air at that location

푐퐿 = Lift coefficient of the wing.

As at a given point in time on a circular track all the variables in Eq. (4.4) remain constant, the expected lift forces are considered to remain constant throughout a racing path. When moving from the outer edge to the inner edge of the track, it is observed that all parameters on the right- hand side of Eq. 4.4 except ‘r’ remain constant. Thus, the resultant change in velocity is dependent on the change of radius only, with it being directly proportional to that change. On running the simulation with the selected F1 car model, results that included variation of velocity, distance, and lap time with change in radius are obtained for the track with 100 m radius at centerline. Table 4.1 summarizes these results. This analysis confirms the results obtained from the optimization.

Table 4.1: Variation of velocity, distance and lap time with change in radius across the track width

Radius (m) 92.5 100 107.5 Velocity (kph) 191.50 202.17 212.66 Distance (m) 581.20 628.32 675.44 Lap time (sec) 10.92 11.19 11.43

14000 Downforce - Front Wing Downforce - Rear Wing 12000 Total Downforce

10000

8000

6000

Downforce (N) Downforce

4000

2000

0 50 100 150 200 Radius of Curvature (m)

Fig. 4.2: Variation of lift forces with change in radius of curvature of the track.

Fig. 4.2 displays the variation in lift forces acting on the car. The data from the table and the plot imply that the lift forces increase with increase in radius. This can be attributed to the increase in the velocity with increase in radius of curvature of the track. Also, the drop in downforce is higher as the radius of curvature is reduced from 100 m to 50 m, when compared to the drop occurring when radius is reduced from 200 m to 100 m. This directly implies that the drop in velocity would be higher with reducing radius. Based on this observation and the results obtained, it can be inferred that the car will try to move towards the outer edge of the track as the radius of curvature reduces. It implies that the gain in maximum speed is comparatively higher than the gain in distance around the track. Analysis conducted for the current set of circular tracks doesn’t show this observation, as the downforce created is sufficient enough for the side forces to overcome the centrifugal forces acting along the inner edge of the track.

This phenomenon is true only for the case of a turn with constant radius of curvature. In the following sections, trends for turns with varying radii of curvature will be established.

4.3 Study of the Oval Track

An oval track is another basic track, where a turn with constant radius of curvature is followed by a straight. In this section, observations in changes to the racing line and the lift forces acting on the car around the lap will be examined and summarized. For this study, two oval tracks are considered. The first oval track consists of 200 m long straights and turns with 200 m radius of curvature, while the other oval track had the same straight length, but turns with 100 m radius of curvature. Data points on both tracks are considered along the centerline as shown in Fig. 4.3 and

Fig. 4.4 for initiating the optimization. The data point number in the figure is associated with that given in Table 4.2. The direction of car’s motion is from data point 1 to data point 6. The optimized racing line is obtained by feeding the initial positions of data points to CMA-ES.

Figs. 4.3 and 4.4 also display the optimized racing line for both oval tracks. The initial and final location of the data points for both the tracks are tabulated in Table 4.2, along with radius of curvature values at corresponding points. The initial location of these data points in Fig 4.3 and

Fig. 4.4 are signified by brown markers, while the corresponding final locations are marked by blue markers. The center of curvature for points 1, 5 and 6 was located at (100, 0), while that for points 2, 3 and 4 was located at (-100, 0). As stated earlier, the movement of the data point is restricted along a path normal to the direction of the center line to remove redundancies. This normal path is signified by the brown lines connecting the inner and outer boundaries of the track.

Looking at Fig. 4.3, it can be inferred that for the track with 200 m radius of curvature the dynamics of the F1 car are very similar to that of a circular track, i.e. for track with no banking the most optimum path would be the inside edge of the track. The increase in velocity due to decrease in centrifugal force is not proportional to increase in distance, as the radius of curvature of the turn increases. Hence, the most optimum path around the track is the smallest distance around the track.

Table 4.2: Initial and final positions of data points along the race track, with corresponding radius of curvature values

Track 1 (200 m Straight and 200 m Radius of Curvature)

Data Initial Final Point # X Y Radius X Y Radius (m) (m) (m) (m) (m) (m) 1 100.00 200.00 200.00 100.00 192.50 192.50 2 -100.00 200.00 200.00 -100.00 192.50 192.50 3 -300.00 0.00 200.00 -292.50 0.00 192.50 4 -100.00 -200.00 200.00 -100.00 -192.50 192.50 5 100.00 -200.00 200.00 100.00 -192.00 192.50 6 300.00 0.00 200.00 292.50 0.00 192.50 Track 2 (200 m Straight and 100 m Radius of Curvature)

Data Initial Final Point # X Y Radius X Y Radius (m) (m) (m) (m) (m) (m) 1 100.00 100.00 100.00 100.00 102.89 102.89 2 -100.00 100.00 100.00 -100.00 92.50 92.50 3 -200.00 0.00 100.00 -193.40 0.00 93.40 4 -100.00 -100.00 100.00 -100.00 -105.01 105.01 5 100.00 -100.00 100.00 100.00 -92.50 92.50 6 200.00 0.00 100.00 193.00 0.00 93.00

Fig. 4.3: Optimum racing line for an oval track with 200m radius of curvature.

Fig. 4.4: Optimum racing line for an oval track with 100m radius of curvature.

However, observing Fig. 4.4, the optimum path for track with 100 m radius of curvature does not follow the same trend as that for the track with 200 m radius. It appears that the path followed the inside edge of the track when entering the turn. The car then abandons the inside edge when it reaches the middle of the turn and moves towards the outside edge of the track until the turn ends.

It then takes a diagonal path across the straight instead of taking the shortest distance. Fig. 4.5 shows the variation of the speed of the car along the optimized path as well as along the centerline of the track. It can be observed that when entering a turn (at data points 2 and 5), the car decelerates under braking for both cases. The significant change arises at data points 3 and 6. The car is under uniform motion when driven along the centerline, but starts to accelerate when driven along the optimized path. This point is called the apex of the curve. Data from Fig 4.5 shows that although the distance covered is slightly less than that along the centerline, the variation in speed is tremendous. This results in significant improvement in the lap time.

Fig. 4.5: Comparison of speeds between the optimized path and along the centerline.

As generally observed, racing cars try to slow down until the apex point of the curve and accelerate when past it. The apex is generally the point that splits the curve in perfect symmetry, when a normal line to the curve is drawn at that point. Track geometry and radius of curvature of

44 the curve govern whether a car negotiates a curve by passing through the apex or not. Fig. 4.6 displays the variation of downforce along the track. Assuming the car follows the same optimal path as before, the drop in speed and lift forces would be higher with reduction in the radius of the curve. It is due to the increase in centrifugal forces as given by Eq. 4.4. This might result in the car braking at a further distance before the turn, thus reducing the peaks between data points 1 and

2, and 4 and 5 in Figs. 4.5 and 4.6, resulting in massive loss of time. In order to avoid this loss in time, the car tries to negotiate the curve by increasing the radius of the turn by moving towards the outer edge of the track instead of following the inner edge. Thus, referring to Fig. 4.4, data points 2 and 5 would move towards the outer edge with reducing radius of curvature to carry more speed into the curve. Therefore, to negotiate a tight corner, the suggested racing line would be to place the car near the extreme outside of the track before the turn, hit the apex of the curve while braking, and accelerate out of the curve moving towards the outside edge of the track again. A car with good aerodynamics can generate higher levels of downforce that will allow it to go even faster around the curve.

Fig. 4.6: Comparison of lift forces between the optimized path and along the centerline.

4.4 Optimization of the Sepang F1 Race Track

For the present study, the Sepang F1 race track is selected, which annually holds the national

Formula 1 race of Malaysia. The selection was made as this track is a perfect combination of long straights and fast corners, along with couple of slow corners. This is a perfect situation, where an aerodynamicist will face the dilemma of whether to base the design of the car on high downforce configuration that allows it to go fast around the curves but slower on the straights, or have a low drag configuration that makes the car faster on the straights but slower around the corners. Both strategies are equally capable of producing the fastest time. Hence, the genetic algorithm is a perfect tool that can be used for making this decision. The optimizer would select the best path around the track based on the car setup provided. Fig 4.7 shows an aerial of the Sepang F1 race track.

Fig. 4.7: Aerial view of Sepang F1 race track (©Google Maps)

A Geographic Information System (GIS) software is used to capture the data points of the track from the above image to be used in the AeroLap software. The software used is ArcGIS.

4.4.1 Use of ArcGIS:

To perform the optimization, X and Y coordinates of the track are required to be fed into

AeroLap. ArcGIS software is used to extract information in Cartesian coordinate form from an aerial image of the track. ArcGIS is a geo-mapping software, which contains geo-referenced data. It allows creation and usage of maps that contain geographical data measured with respect to an inertial reference point. Therefore, it is used to determine the track geometry. Thus, all distances calculated are highly accurate and correspond to the actual track. In order to obtain the coordinates of the track, the following steps were undertaken:

i. A layer containing a map with geo-referenced data was used as a base layer to extract

coordinates from a particular image. The geo-referenced map image was called the raster

image.

ii. An image of the actual track in JPEG format was then superimposed on the raster image.

The top image layer was made translucent that allowed views of both the images.

iii. Control points were added to make the overlying image superimpose perfectly with the

raster image. Distinctive features on both the images were selected as control points, as

they helped to define the accuracy of the superimposition. For best case scenarios, it was

advised to use at least 4 control points with one in every direction. 6 control points are

generally recommended by the software.

iv. After defining the control points on the top layer, the image was stretched and dragged

using these points until the features, and therefore, the images aligned with one another.

v. Once the image was superimposed as accurately as possible, the data points were

collected along the centerline. A reference centerline of the track was drawn as a spline

using AutoCad on the overlying image using the data points thus obtained.

The point at the start-finish line was selected as the origin. All the lengths and track coordinates were calculated using this point as a datum.

4.4.2 Determination of Sepang F1 track’s optimal racing line:

A reference centerline was recreated in AutoCAD as a spline. An offset of the spline was created at a distance of 7.5 m on both sides to determine the corresponding inner and outer boundary elements’ coordinates in normal directions. The minimum width of the track was considered to be 16 m, as reported officially on the Formula 1 website. The width of the actual track is slightly more at the start – finish straight and at the final curve (the one connecting the two long straights), due to pit entry and pit exit lanes. As the difference in width, when compared to rest of the track, was not significantly large, the width of the track was considered to be 16 m everywhere. The FIA regulations limit the maximum width of a car to be 1.8 m between the wheelbases, which is also known as the ‘track’ of the car. Thus, the inner and outer edges were required to be set at 7.1 m from the centerline to keep the car within the track boundaries. This distance was calculated as the difference between half of the track and half of the car width. An additional width was provided for the car to maneuver, as curbs are generally used along every corner of the track. Hence, an effective width of 7.5 m was calculated for the inner and outer edges of the track on both sides of the centerline. The extracted set of data point coordinates for the inner and outer edge served as boundary conditions for the optimization problem. The resultant geometry of the track obtained is displayed in Fig. 4.8.

To initiate the optimization, the necessary inputs required are:

 An input variable or column vector.

 Upper and lower boundary limits for the input variable or the column.

 Standard deviation for the distribution.

 An objective function that determined the fitness value of the distribution.

The X coordinates of all the data points along the centerline formed a column matrix that was used as the input to the optimizer. As the coordinates for the inner and outer boundaries were calculated with reference to the datum, they did not necessarily correspond to the lower

48 and upper limits for the corresponding X coordinate of a particular data point. Therefore, the lower limit was selected as the lesser of the values of the X coordinate of the inner and outer boundary for a corresponding data point. Similarly, the upper limit was selected as the larger one of the two values. These limits were also represented in forms of column vectors. The standard deviation was assumed to be 30 percent of the distance between the inner and outer boundary data points. AeroLap software was considered to be the objective function that determined the fitness value of the distribution, which was equal to the lap time for the new path created. The data points were limited to displacements only along the normal direction to the centerline, as was considered for previous test cases. This methodology was used to avoid the overlapping and cross-over of data points that would lead to erroneous results.

Fig. 4.8: Malaysian F1 track obtained from ArcGIS data set.

The track component required three key variables to define a race track, namely:

i. The X-coordinate of a point on the track

ii. The Y-coordinate of a point on the track

iii. The radius of curvature for each of the corresponding data points.

The values of some other variables values are also required to define the track geometry, for example elevation (Z coordinate of the point on the track), banking angle, tire grip, etc. Due to minimal changes in elevation for the Sepang race track and the negligible bank values of the turns, these variables were not considered for this study (zero value). Also, as the aim of this study was to find the fastest possible way around the track, the tire grip was maintained at 100 percent at all times.

To define the radius of curvature for all data points, a concept developed by Gabriel

Taubin (Taubin, 1991) was used. It was primarily developed to represent and estimate complex planar curves, surfaces, and non-planar space curves using implicit equations. These non-planar curves could be a product of the intersection between two 3-D surfaces, boundaries of patterns on a particular object, or due to small smooth surface patches of objects. Although this technique was developed as a generalization for various forms of both planar and non-planar curves, N. Chernov developed a code in Matlab to use Taubin’s method to fit a circle for the given distribution. The code performs a circle-fit to a given set of data points and determines the center and radius of the circle for the given data set. Moments of the data points are obtained with respect to the centroid of the distribution to calculate the parameters mentioned, which are later substituted into the characteristic equation obtained from Taubin’s work. The results are iterated using a modified Newton’s method to determine the correct roots of the characteristic equation, while minimizing the error. The center and the radius are calculated using the roots obtained.

Al-Sharadqah & Chernov (2009) conducted an error-analysis study, comparing the mean- square error obtained for various geometric and algebraic circle fitting algorithms. Geometric fitting algorithms generally provided solutions with higher accuracy, but were computationally intensive. The Taubin’s method was judged to be the best amongst the commonly used algebraic fitting algorithms; however, they were adjudged to have slightly lesser accuracy when compared to the geometric fit algorithm. Taubin’s method was selected to estimate the radius of curvature for the curves, as it was comparatively faster in converging to a solution than the geometric fit algorithms, and had negligible differences in error values for lesser number of data points.

A total of seven data points were considered to calculate the radius for a particular data point, three each in preceding and following positions of a point along the track. AeroLap associates a negative value with a left handed turn, a positive value with a right-handed turn, and zero value for straights. The radius values associated with each data point were changed accordingly. The radii of curvature values were equated to 0 m for any value of radius greater than 2500 m, as these high values equated to the data points lying on the straights.

Once the track file was updated with all the information, the simulation was initiated. The

Matlab code of CMA-ES optimizer interacted with the AeroLap software, exchanging information to calculate the best possible path. The simulation followed the process as described in the flowchart depicted in Fig. 3.2. With 303 data points, the simulation became computationally intensive. Hence, once the standard deviation of the data set reduced to a significantly small value (up to the order of 10-2), a new simulation was initiated with a lesser number of data points and new boundary limits to the corresponding data points. The limits were reduced proportionately to the associated standard deviation value. The data points were reduced from 303 to 183, with the deleted data points selected from parts of the path which were well defined by the previous simulation. The final simulation was launched to determine

51 the optimum racing line for the reduced data set. The optimal path obtained by the simulation is showcased in Fig 4.9.

Optimal racing Optimal line obtainedafter arunning detailedset of simulations.

: :

9 Fig. 4.

From Fig. 4.9, it can be observed that the optimized path follows the trends discussed for the optimized racing line for the oval track with a smaller radius of curvature. As turns 1 and 2 are tight curves with longer arc lengths, the path enters the curve from the extreme outer edge carrying as much velocity as possible into the curve, while simultaneously moving towards the inner edge. This path followed is very similar to the one observed for the 100 m oval track (Fig.

4.4). Turn 3 on the other hand is made of almost an arc of a circle with a large radius. This allows the car to accelerate and negotiate the long turn while moving towards the outer edge of the straight leading to the sharp turn 4. As turn 4 is highly acute with very small radius (<20 m) of curvature and arc length, the path moves from extreme outer edge, touches the apex of curve and then deviates to the outer edge again. These three basic trends are repeated around the lap of the track.

Table 4.3 shows a comparison of lap times; the one acquired for the centerline as well as that obtained for the optimum racing line. These times were compared with the fastest lap time ever recorded and the fastest lap of the 2010 season at the Sepang race track (Formula 1, 2014).

It was observed that the distance covered along the optimal racing path was slightly less than the actual distance traversed as per the official track data (Formula 1, 2014). The difference between the track lengths was 69 m, which was 1.2% of the total track length. This error might have been generated due to minute mapping error caused while superposition of the track image on the raster image in ArcGIS. The optimization might or might not have contributed to the reduction in length. However, a gain or loss of a fraction of a second is of significant value to the development of these F1 cars. For making a fair comparison, the lap times are normalized to the officially reported track length. To normalize the lap time, the actual track distance is divided by the average speed of the car around the optimized path. This value is also highlighted in the table.

Results show that by optimization of racing line itself, the lap time for a F1 car with the configuration setup of the 2010 season is reduced by 1.699 secs. Assuming that the reduction in 53 the race distance per lap was entirely due to mapping error, the lap time across the corrected lap length with the same average speed was still faster by 0.542 sec. Therefore, in a race consisting of 56 laps around this race track, just by tweaking the racing line, a total time of at least 30.352 secs can be gained, which in terms of track distance, is almost one-third of a lap.

Table 4.3: Comparison of lap times for the actual race and optimized racing line.

Race Distance Average Speed Lap time per lap m kph m/s min : sec Best Ever time (2005) 5543.0 215.54 59.871 1:32.582 Best Lap in 2010 season 5543.0 213.32 59.257 1:33.542 Optimized Racing Line 5474.0 214.57 59.602 1:31.843 Normalized Lap Time 5543.0 214.57 59.602 1:33.000

This analysis provides an excellent validation for the experimental data set used as an alternative for actual telemetric dataset of a recent F1 car, which is never published in any available public literature. The expectation is that optimization of the aerodynamic coefficients of the front and rear wings will reduce the lap time further, which is discussed in the following chapter.

Chapter 5 OPTIMIZATION OF AERODYNAMIC LOADS ON FRONT AND REAR WINGS

The purpose of obtaining an optimum racing line was to determine the fastest way around the track after eliminating driver error and utilizing the maximum potential of the car. As per the analysis performed in the previous chapter, optimization of the track itself improves the best lap time set by an actual F1 car of similar configuration. With all the other parameters defining the car’s configuration being set and established, the performance of the car can now be enhanced only by optimizing the aerodynamic loads acting on the car. With the prime objective of this research being focused on attaining the preliminary optimum lift and drag values for the front and rear wings, the downforce generated by the underbody is not optimized.

In his study, Toet (Toet, 2013) provided the analysis data for a F1 car used in the 2009 season.

The 2009 season had significant changes introduced with respect to aerodynamics of the car. The front wing was made wider, along with a longer neutral section, i.e. section not providing any downforce in the center portion of the wing beneath the nose section of the car. The rear wing was made smaller. The main aim of applying changes to the rear wing was to reduce the wake of the car, resulting in less turbulent air, or ‘dirty air’ in the car’s wake. Increasing the width of front wing allowed generation of higher downforce at the front of the car. The resulting reduction of drag of the leading car and increase in downforce at the front of the trailing car allowed better handling of the trailing car; thus, allowing more overtaking opportunities. These regulations have barely changed until the present season. As a result, the analysis data for the 2009 car provided by Toet is used and applied for the current F1 car setup, tuned in accordance with the 2010 season. The only change related to aerodynamics of the car brought about for the 2010 season was increase in body length by

20-22 cm. This would have a slight impact on the contribution of the major components towards the

55 total aerodynamic load, but due to this being the only available reference, the changes that might be introduced are neglected.

5.1 Variation in time with respect to change in aerodynamic loads

As discussed in Section 2.1, aerodynamics allows the cars to go faster around the track. The

lift forces increase with the square of the speed of the car, resulting in better traction and

cornering. However, the drag forces are also directly proportional to the square of the speed,

thereby reducing the overall performance. The puzzle is to find the correct balance suitable for

any given track.

A small sensitivity study was performed to understand how the aerodynamic loads should

vary in order to get the best performance out of the car. Fig 5.1 shows the variation of speeds

across the Malaysian race track that follows the optimal path obtained in section 4.4. The current

aerodynamic configuration of the car was set as the baseline against which different scenarios

were compared. The different scenarios highlight percentage increase or decrease in drag and lift

forces. Various combinations were tried and the variations were summarized.

From Fig. 5.1, it can be inferred that the reduction in drag has a more profound effect in

reducing the lap times rather than increase in lift forces. A reduction of 20% in drag forces from

the baseline values results in lap time being reduced by 1.311 seconds, while an increase in the lift

forces by 20% from the base values reduces the lap time marginally by 0.511 seconds. However,

if both cases are considered simultaneously, the lap time improved by 2.326 seconds. It can,

therefore, be inferred that the track contains features where both drag reduction and lift increase

are to be given equal priority. Via observation, it is found that for the cases of only drag reduction

(highlighted by turquoise solid line), as well as that of both drag reduction and lift increment

(dark blue solid line), the trends for speed variation are very similar until 1800 m into the lap from

the start-finish line. This is approximately past turn 4 on the track. After that turn, the increased

download case starts dominating over the reduced drag case, allowing much higher speeds to be

taken into the turns. This trend is followed until 4000m, which is approximately the location 56 between turns 12 and 13. For the following segment until the finish, similar trends are observed again for both the cases. Increment in lift with drag reduction allowed the car to go faster by 1.015 seconds when compared to the lap times of the case of drag reduction only, which is a tremendous

gain.

Influence Influence of draglift forcesand on the ofspeed 2010 a 1Formula car.

: :

1 Fig 5.Fig

Based on this observation, it can be deduced that the track is a combination of both

strategies of aerodynamic design. The segment of the track between 1800 m – 4000 m requires

high downforce that can improve lap times by approximately a half to a full second for the same

drag. The rest of the track suggests that the car can be designed for low drag designs, as this

allows the car to reach higher speeds and provides gains of approximately a second. The split-up

of the track is nearly 50-50. Some designers might have an edge towards using the drag reduction

strategy, as it allows the car to be faster in the overtaking zones; thus, sacrificing middle sector

and lap times. It is highly difficult to attain both designs simultaneously, as an increase in lift

causes a significant increase in the induced drag. Therefore, this track is a perfect for the

optimization problem.

5.2 Derivation of aerodynamic coefficients for the wings and F1 car body

without wings

As shown in Fig. 5.2, Toet has explained the contribution of major individual components

of a Formula 1 car. For the present study, the components have been clubbed together as front

wing assembly, rear wing assembly, and the car body. The car body includes contributions from

the wheels, suspension, ducts, chassis, bodywork, floor and diffusers. For our reference, the

contributions have been summarized in Table 5.1.

Drag and Downforce of 2009 F1 car

Floor and Diffuser

Rear Wing Assembly

Rear Wheel Suspension, Brake Duct Drag Chassis, Bodywork

Front Wheel, Suspension, Brake Duct

Front Wing Assembly

-20 -10 0 10 20 30 40 50 60 % contribution to the total value Fig 5.2: Breakdown of drag and downforce for individual components of F1 car (Toet, 2013) 58

Table 5.1: Contributions by Front and rear wings and rest of the car’s components.

Contribution to Contribution Components Lift Forces to Drag Forces

Front Wing Assembly 30 % 20 %

Rear Wing Assembly 25 % 30 %

Body 45% 50%

According to Toet, these contributions are only significant when working in cohesion as a

single system. If these components are selected individually, their contribution will be different.

Hence, the values obtained for the effective aerodynamic loads acting on the wing will

incorporate all the inter-component interactions. The freestream velocity is considered to be the

same at all points. The lift and drag forces are calculated using the formulae stated below:

1 퐿 = 휌푉2푆퐶 … (Eq. 5.1) 2 퐿

1 퐷 = 휌푉2푆퐶 … (Eq. 5.2) 2 퐷

1 푞 = 휌푉2 … (Eq. 5.3) 2 where,

L, D = Lift and Drag forces respectively. (N)

ρ = Density of air at 40 m above sea level (Altitude of Sepang F1 track) (kg/m3)

V = Freestream velocity of the F1 car. (Assuming no wind conditions) (m/s)

S = Planform area of the wing or frontal area of the car’s body (m2)

CL = Lift coefficient of the wing

CD = Drag coefficient of the wing

q = Dynamic Pressure acting on the system (Pascals or N/m2)

In accordance with Toet’s work, it can be stated that the equations to calculate the contributions for individual components as listed in Table 5.1 are:

퐿푐푎푟 = 퐿퐹푤 + 퐿푅푤 + 퐿퐵표푑푦 … (Eq. 5.4)

퐷푐푎푟 = 퐷퐹푤 + 퐷푅푤 + 퐷퐵표푑푦 … (Eq. 5.5)

These equations simply say that lift and drag forces acting on the car are equal to the sum of the lift and drag forces generated by front and rear wings, and the body (without wings). Substituting Eq.

5.1, 5.2 and 5.3 into the above equations, yields:

푞. (푆푐퐿)푐푎푟 = 푞. (푆푐퐿)퐹푤 + 푞. (푆푐퐿)푅푤 + 푞. (푆푐퐿)퐵표푑푦 … (Eq. 5.6)

푆 푐 = 푆 푐 + 푆 푐 + 푆 푐 … (Eq. 5.7) 푐푎푟 퐿푐푎푟 퐹푤 퐿퐹푤 푅푤 퐿푅푤 퐵표푑푦 퐿퐵표푑푦

Similarly,

푞. (푆푐퐷)푐푎푟 = 푞. (푆푐퐷)퐹푤 + 푞. (푆푐퐷)푅푤 + 푞. (푆푐퐷)퐵표푑푦 … (Eq. 5.8)

푆 푐 = 푆 푐 + 푆 푐 + 푆 푐 … (Eq. 5.9) 푐푎푟 퐷푐푎푟 퐹푤 퐷퐹푤 푅푤 퐷푅푤 퐵표푑푦 퐷퐵표푑푦

From Eq. 5.7 and Eq. 5.9, it can be inferred that in order to optimize the aerodynamic loads for a particular race track and a given F1 car setup, the aerodynamic coefficients for the front and rear need to be optimized as the planform area of the wings (and the frontal area of the body) remain constant during a lap around the track. The contribution of the body is assumed to be constant.

AeroLap works on a similar principle. It considers two-dimensional tables showing variation of the values of the aerodynamic coefficients with change in front and rear ride heights. As stated earlier, due to a lack of information, the aerodynamic data is being considered for a F1 car is from the 2002 season, which was provided as a complimentary package with AeroLap license purchase.

These data might be inaccurate, but the validation performed in Section 4.4 shows that the data is usable. An additional table is listed as an input, which shows the variation of the center of pressure with change in ride heights at front and rear ends. The location of the center of pressure is represented as percentage of the wheelbase, i.e. distance between the wheel centers of the front and 60 rear tires, from the rear wheel center. For example, a value of 40% means the center of pressure is located at a distance of 40% of total wheelbase from the rear wheel center. By definition, the center of pressure is the location along the body, where the moment due to all aerodynamic forces is zero.

Therefore, assuming that F is the total aerodynamic force acting at the center of pressure and L being the total wheelbase, the forces acting at the front and rear wheel will be balanced by the equations stated as follows:

… (Eq. 5.10) ∑ 푀 퐶푃 = 0

−푥 × (퐿 − 푦) + (퐹 − 푥) × 푦 = 0 … (Eq. 5.11)

푦 푥 = … (Eq. 5.12) (퐿 − 푦) (퐹 − 푥)

where,

x = Downforce acting on the front wheel center

y = Length to the rear wheel center from the location of center of pressure

From Eq. 5.12, it is observed that the ratio of the downforce acting at the front wheel to that acting at the rear wheel is equal to the ratio of distance of the center of pressure from the rear wheel center to that measured from the front wheel. Therefore, if the rear wheel is located at a distance of 40% from the center of pressure, then the downforce acting on the front wheel will be 40% of the total downforce acting at the center of pressure. As drag forces act through the center of pressure, they do not cause any moment, and hence are defined for the whole car only. The center of pressure will be denoted by %F for future references.

Using this methodology and the input tables provided for lift and drag coefficients, and the center of pressure location, AeroLap provides the downforces acting at all wheel centers and drag force on the whole body. However, for this research purpose, the downforce provided by the wing components is set to be optimized. Hence, to change the input tables in accordance to the change in the values of the aerodynamic coefficients of the front and rear wings, the contribution of the car’s

61 body without wings is isolated. As percentage contribution for the front wing and the rear wing towards the total value was known (shown in Fig 5.1), the cL values for them were calculated as:

푆푐푎푟푐퐿푐푎푟 … (Eq. 5.13) 푐퐿퐹푤 = × 30% 푆퐹푤

푆푐푎푟푐퐿푐푎푟 … (Eq. 5.14) 푐퐿푅푤 = × 25% 푆푅푤

Similarly, values for cD were calculated using appropriate percentage contribution values.

There were certain correction factors applied to the cL and cD values, which are highlighted in the following points:

i. Ground Effect Correction: The front wing is in close proximity to the ground, resulting in

ground effects adding additional downforce produced by the wing profile. Katz (Katz,

1995) in his textbook has stated that, generally, the ground effects increase the downforce

of the front wing approximately by 30% - 50%. This is mathematically represented by Eq.

5.15. As an exercise to analyze this statement, a lap simulation was performed for the given

setup over the optimum racing path. Telemetric data containing the aerodynamic loads,

coefficients, ride heights and %F for the whole car, and their components acting at the

wheel centers were recorded as outputs. The values of cL for the front wing were calculated

using the equations stated above. Based on the front ride height values, approximate

corrections were applied in accordance with Eq. 5.15, with higher percentages (50%) used

for lower ride heights and lower values (30%) used for higher ride heights. The corrected

values of cL for the front wing at different track positions lied between the range of 1.01 –

1.04, which has an error range of less than 3%. Therefore, a correction of 30% was applied

to the values of the lift coefficient considered for the front wing, as the least amount of gain

due to ground effects was considered. Although approximated, the effect of variation in

ride heights is countered by using the correction. Incorporating ride height variation for

calculation of the wing’s aerodynamic coefficients would lead to severe complications. The

time saved for overcoming an inaccuracy of 3% would have been tremendous.

푐퐿퐹푤 = 푐푙푓푤 + (푐푙푓푤) × 퐶. 퐹

푐퐿 … (Eq. 5.15) 푐 = (1 + 퐶. 퐹) 푐 표푟 푐 = 퐹푤 퐿퐹푤 푙푓푤 푙푓푤 (1 + 퐶. 퐹)

Where,

푐푙푓푤 = Lift coefficient of front wing only, without ground effects C.F = Correction Factor applied (30% for present study)

There is no correction factor considered for the drag coefficient of the front wing. ii. Contribution of rear wing towards total lift and drag for rear-wing assembly: In

present day Formula 1, the rear wing is a complex system comprising various components

other than the main rear wing, all of those contribute to the total downforce and drag

produced by the system. Fig. 5.3 displays the components of rear wing assembly used for

2009 season F1 car.

End Plates Rear Main Wing Rear Wing Flap

Rear Zing

Pylons Lower Rear Wing (Beam Wing)

Fig 5.3: Components of a rear-wing assembly for a 2009 season’s F1 car (Toet, 2013)

Toet (Toet, 2013) in his study gave a breakdown of the contributions of the various

components of the rear wing assembly to the lift and drag forces generated by the entire

assembly. This breakdown is described in form of the pie charts presented in Figs 5.4 and

5.5. As the main focus is to determine the optimum lift and drag coefficient values of the

rear wing, it is considered as a system comprising of the rear main wing, rear wing flap,

and end plates. The end plates increase the effective aspect ratio of the wings, which helps

in reducing the induced drag and increasing the lift generated. Based on the stated

contributions, correction factors of 83% and 82% are applied to the downforce and drag

forces generated by the rear wing assembly. This can be represented by Eq. 5.16.

푐푙푟푤 = (0.83) 푐퐿푅푤 푎푛푑 푐푑푟푤 = (0.82) 푐퐷푅푤 … (Eq. 5.16)

Rear Wing Rear Zing Pylons Endplate 2% 1% 2% Rear Wing Flap Lower Rear Wing 10% 14%

Rear Main Wing 71%

Fig 5.4: Downforce contributions in rear-wing assembly of a 2009 season’s F1 car (Toet, 2013)

Again, due to minimal changes in design of rear wings, these contributions, as obtained for a

2009 season car, are assumed to be the same for the 2010 season car.

Pylons Rear Zing Rear Wing Lower Rear Wing 1% 1% Endplate 16% 4% Rear Wing Flap 44%

Rear Main Wing 34%

Fig 5.5: Drag contributions in rear-wing assembly of a 2009 season’s F1 car (Toet, 2013)

With knowledge of the lift and drag coefficients for the front and rear wings, the coefficients for the body without wings can be calculated using Eqs. 5.7 and 5.9. It is to be noticed that formula requires uncorrected values of the coefficients of the wings to determine the ones for the body.

Separation of the tables for body without wings that contains the variation of lift and drag coefficients with change in ride heights of the car is the key. After accounting for all the effects, the newly generated cl and cd values of the wings can be added to these tables in order to obtain the input tables as required by AeroLap. The base tables used for initiating the simulation, showing the variation of lift and drag coefficients for the whole car and body without wings are listed in

Appendix A.

To populate the third input table, displaying variation of %F with change in ride heights, the lift forces acting at the front and rear wheel centers due to body without wings were required to be calculated. Using the base tables for %F and the lift coefficients for the whole car, the values for lift coefficients for the body without wings at front and rear wheel centers were calculated as follows:

Lift for body w/o wings @ front = (Total lift acting at front wheel) –

(Lift for the front wing)

(푆 푐 × %퐹⁄ ) − (푆 푐 ) … (Eq. 5.17) 푐푎푟 퐿푐푎푟 100 퐹푤 퐿퐹푤 푐퐿퐵표푑푦_푓 = 푆퐵표푑푦

Similarly, @ rear center

(푆 푐 × (1 − %퐹⁄ ) − (푆 푐 ) … (Eq. 5.18) 푐푎푟 퐿푐푎푟 100 푅푤 퐿푅푤 푐퐿퐵표푑푦_푟 = 푆퐵표푑푦

These values are tabulated in Appendix A as well. An interesting point to be noted is that the area considered for both the front and rear sections are same. This is due to frontal area being considered for the car and the body without wings. The %F just distributes the load proportionately by changing the effective lift coefficient.

On the basis of the equations stated above, the average values of the lift coefficients for the front and rear wings were calculated as 1.02 and 2.60 respectively, after incorporating the corrections. The corresponding values of the drag coefficients were 0.30 and 1.12. The planform and frontal areas for the different components of the F1 car considered are listed in Table 5.2.

Table 5.2: Contributions by Front and rear wings and rest of the car’s components.

Planform Area Frontal Area Components (m2) (m2) Front Wing Assembly 0.766 0.250 Rear Wing Assembly (Flap, main wing, Endplates) 0.276 0.092 Body without wings - 0.870 Whole car - 1.212

5.3 Optimization of Aerodynamic Coefficients

In the previous section, the base tables for the aerodynamic coefficients for the car’s body

without wings were established. Using the tables in Appendix A and the values of the lift and drag

coefficients simulated by the CMA-ES optimizer, the tables for the lift and drag coefficients for the

whole car, including the wings, could be determined using Eq. 5.7 and 5.9. These tables define the

input aerodynamic component of AeroLap and show the variation with respect to ride heights; thus,

accounting for ground effects.

The drag forces acting on a body are generally of two types: viscous and induced. Induced

drag is caused due to the generation of lift, while viscous drag is due to the boundary layer

interaction between the body and the fluid (in this case, air). The total drag coefficient for a finite

wing is thus calculated as in Eq. 5.19. Induced drag is calculated as shown in Eq. 5.20.

… (Eq. 5.19) 푐퐷 = 푐퐷0 + 푐퐷푖

푐2 … (Eq. 5.20) 푐 = 퐿 퐷푖 휋푒퐴푅 Where,

푐퐷0 = Viscous drag coefficient

푐퐷푖 = Induced drag coefficient

푐퐿 = Lift coefficient 푒 = Span efficiency factor 퐴푅 = Aspect ratio = 푏2⁄푆 b = Span of the wing (m) S = Planform area of the wing (m2)

The span efficiency factor was selected as 0.65 for the front wing and 0.7 for the rear wing. These

values were selected as the lift distribution over the rear wing is the same as a rectangular wing,

while the front wing has a distribution that is somewhat between that of a triangular and rectangular

wing. On the basis of Eqs 5.19 and 5.20, the viscous component of drag coefficient can be

computed for the front and rear wings and can be tabulated in the form of input tables required for

AeroLap. These equations are of importance as the contribution of lift towards total drag for a F1

67 car can be as high as 90% (Toet, 2013). Therefore, to select an optimum value of the lift coefficient for the front and rear wings, induced drag plays a key role. Applying the above set of equations, the average value of the viscous drag coefficient for the front and rear wings were calculated as 0.08 and 0.03, respectively. This difference in the values of viscous drag coefficients of front and rear wings arises due complex geometry of the front wing, with various flaps and guide vanes attached to it. They are assumed to be of constant value when determining the optimum lift coefficient, and made to vary while calculation of optimum drag coefficient.

As every iteration involves generation of new cL and cD tables for the whole car, the value of

% F needs to be calculated for each step as well. To generate the elements of new %F table:

푐퐿푓푟표푛푡 … (Eq. 5.21) %퐹 = × 100 푐퐿푐푎푟

where, (푆 푐 ) + (푆 푐 ) 퐹푤 퐿퐹푤 퐵표푑푦 퐿퐵표푑푦_푓 푐퐿푓푟표푛푡 = 푆푐푎푟

With the input tables of 푐퐿, 푐퐷 and % F for the whole car calculated at every step, the optimization study can be initiated. As the simultaneous optimization for lift and drag coefficients for both the front and rear wings will become highly cumbersome and computationally intensive, the optimization is performed in steps. Initially, for the given values of drag coefficients of the wings, the optimum lift coefficient is calculated. Once the optimum lift coefficients of the wings have been established, the drag coefficients are optimized. The details of the optimization are listed in the following subsections.

5.3.1. Optimization of Lift Coefficient:

The lift coefficient is optimized by using the CMA-ES algorithm in the same way as it was

used to optimize the track path. The following steps define the methodology adopted and values

of variables considered during the process:

i. The input variable to the optimizer is a 2 x 1 column vector containing the two values of

the lift coefficients of the front and rear wings of the car. These values are assumed to be

mean value of the upper and lower limits set for the optimizer.

ii. The upper limit for the lift coefficients is set at 4.5, while the lower limit is set at 0.5 for

both the front and rear wings. The standard deviation used for the distribution is 30% of

the difference between the limits, i.e. 1.2.

iii. AeroLap is used as the objective function again, with fitness of the function being

calculated as the lap time. Using information from tables in Appendix A, the input tables

are calculated.

iv. To obtain the lift coefficient input table for the whole car, the values generated by the

optimizer are used, and the corrections were applied in accordance with Eqs 5.15 and

5.16 to get 푐퐿퐹푤 and 푐퐿푅푤. The values of 푐퐿푏표푑푦_푓 and 푐퐿푏표푑푦_푟 are used from Table A.2

and Table A.3 to calculate 푐퐿푏표푑푦. These values were then substituted into Eq. 5.7 to

get 푐퐿푐푎푟.

v. As stated earlier, to calculate the drag coefficient table of the whole car, constant drag

values of viscous drag is to be used (0.08 for front wing, 0.03 for the rear). The induced

drag is calculated using the lift coefficients generated by the optimizer using Eq. 5.19 and

5.20.

vi. The values of 푐퐷푏표푑푦 are obtained from Table C.5. These values are then used to

determine 푐퐷푐푎푟 with aid of Eq. 5.9. vii. The new %F table is calculated in accordance with Eq. 5.21. viii. After storing the newly generated tables in the working directory of AeroLap, a

simulation is performed. The lap time generated is sent back to the optimizer, which

stored it as the fitness value for the function and the selected input parameters.

ix. The iterations are continued until the difference between the lap times for 15 consecutive

iterations is less than 0.001 sec.

The various data points selected by the optimizer during different iterations are highlighted

in Fig. 5.6. The most optimum value of lift coefficients obtained by the optimizer were 1.123 and

1.651 for the front and rear wings, respectively. The associated lap time for these lift coefficients

and drag coefficients of 0.08 and 0.03 was 90.675 seconds, providing a gain of 1.168 seconds,

when compared to the time obtained along the same path for the car with the given aerodynamic

setup.

Fig 5.6: Map of variables showing optimization of lift coefficient

5.3.2. Optimization of drag coefficient:

The process of optimizing the drag coefficients is performed in a very similar manner to that

of the lift coefficients. The only difference is that the lift coefficient is kept constant and, for

those, the viscous drag coefficient values are changed. The lift coefficient values obtained from

optimization performed in the previous section were considered. The limits for the optimization

were 0.004 to 0.2 for drag coefficients for both the front and rear wings. The starting point was 70

the mean value of the limits, which was equal to 0.098. The standard deviation of the distribution

was considered to be 30% of the distribution, i.e. 0.06. Also, as none of the values of lift

coefficients changed during the due course of the simulation, the %F table also remained

constant.

As only viscous drag was made to vary within the limits, with the contribution of induced

drag remaining constant, the optimizer moved the solution towards the lowest limit as the lap

time improved with reduction in drag. Therefore, the most optimized values for viscous drag

coefficients were generated as 0.004 for both the front and rear wings. The total drag coefficients

for the front and rear wing were calculated as 0.141 and 0.457, respectively. The associated lap

time was calculated as 90.383 seconds, which improved the lap time by a further 0.292 seconds

when compared to the lap time attained after optimization of lift coefficients.

Table 5.3: Lap time comparisons for different optimized parameters.

Average 풄풍 풄풍 풄풅 풄풅 Lap time Distance/lap Speed 풇풘 풓풘 ퟎ_풇풘 ퟎ_풓풘 m kph min : sec Best Ever time (2005) 5543.0 215.54 - - - - 1:32.582 Best Lap in 2010 season 5543.0 213.32 - - - - 1:33.542 Optimized Racing Line 5474.0 214.57 1.04 2.60 0.08 0.03 1:31.843 Normalized Lap Time 5543.0 214.57 1.04 2.60 0.08 0.03 1:33.000 Optimized Lift Coefficient 5474.0 217.33 1.123 1.651 0.08 0.03 1:30.675 Normalized Lap Time 5543.0 217.33 1.123 1.651 0.08 0.03 1:31.818 Optimized Drag Coefficient 5474.0 218.03 1.123 1.651 0.004 0.004 1:30.383 Normalized Lap Time 5543.0 218.03 1.123 1.651 0.004 0.004 1:31.522

Table 5.3 displays the change in lap times with optimization of every parameter considered in

this study. The lap times have been calculated for both the optimized track length and the actual

track length to observe the improvement on level terms. On the basis of the optimization conducted

for all parameters, it is observed that the total lap time is reduced by 3.159 seconds for the optimal

racing length and by 2.02 seconds for the actual racing length for the simulated car relative to the

best lap performed by an actual F1 car. The times correspondingly improve by a margin of 2.199 71

and 1.06 seconds when the simulated car’s lap times are compared with the best-ever recorded time,

which occurred during the 2005 season. Thus, the optimization helps in improving the performance

of the car enormously.

However, it should be noted is that the optimized value of the viscous drag coefficient for

both the front and rear wings is very low and might not be attainable with their complex designs.

There are various components added to the front and rear wings to channelize the flow to different

regions of the car; thus, proving it difficult to manage with the values obtained through

optimization. Therefore, a parametric study is performed, to develop contour plots depicting

variation of lap times for different sets of lift and drag coefficient values. This study is explained in

the following section.

5.4 Results of Parametric Study

In order to study the changes in lap times for various combinations of lift and drag

coefficients for the front and rear wing, a parametric study is performed. Based on the results

obtained from optimization, the limits for each parameter were decided. The limits for the lift

coefficients for both the front and rear wings were set from the range of 0.750 to 2.250 with an

interval of 0.125. Thus, if only lift coefficients are made to vary, a matrix of 13x13 would be

created that would contain lap times for each combination. The viscous drag coefficients for the

front wing varied within a range of 0.06 to 0.10, while those for the rear wings varied from 0.03 to

0.07. The range of drag coefficients for the front wing is higher, as due to its much more complex

geometry it is expected to have more viscous drag compared to the rear wing. The values for total

lift and drag coefficients as well as location of center of pressure are calculated in the same manner

as done in the previous section. A total of 25 contour plots are generated for each combination of

viscous drag coefficient for the front and rear wing, using a data plotting software. Each of these

contour plots show variation of time with respect to change in lift coefficients for the front and rear

wings. Fig 5.7 describe a series of plots, for which the drag coefficient of the rear wing is kept

constant at 0.03 and that of the front wing is made to vary. Fig 5.8 shows plots for constant value of

72 the drag coefficient of front wing at 0.08, with the value for the rear wing being varied. The observations in the change of trends are listed below.

From Fig. 5.7, for a constant value of the viscous drag coefficient of rear wing (푐푑0_푟푤) at

0.03, the optimized lap times are calculated for varying degrees of the front wing viscous drag

coefficient (푐푑0_푓푤). The optimal region is defined by the dark brown region. It is observed that, as

the value of 푐푑0_푓푤 increases, the lap times increase as well.

For 푐푑0_푓푤 = 0.06, the optimal zone has a minimum value of 90.9 seconds. As the value of

푐푑0_푓푤 increases, this region becomes smaller and smaller and gets superseded by the next contour level of 91.0 seconds when 푐푑0_푓푤 = 0.07. The optimal lap time lies within the range of 1.0 to 1.2 for the front wing lift coefficient. This range becomes smaller as the value of 푐푑0_푓푤 increases. The optimal range of the rear wing lift coefficient reduces as well, as the front wing drag coefficient increases. This reduction is significantly faster when compared to that of the front wing. For 푐푑0_푓푤

= 0.06, the optimal time lies between 1.35 and 1.75. The plot for 푐푑0_푓푤 = 0.09 gives a clearer picture, as to what might the smallest optimal zone for the lift coefficient for both the wings might be. It can, therefore, be concluded that, for most cases, the optimal lap time will lie within a range of 1.05 to 1.18 for the front wing lift coefficient, and within 1.55 to 1.7 for the rear wing lift coefficient.

Fig. 5.8 shows similar trends as shown in the previous figure. This time the viscous drag coefficient for the front wing is kept constant, while that for the rear wing is made to vary. The optimal value range for both the front and rear wing reduces, but not as drastically, as was the case when rear wing drag coefficient was made to change. The dark brown region shrinks at a slower rate, compared to the previous scenario, with the least optimal lap time contour level having the value of 91.0 seconds. The optimal values for the front wing lift coefficient starts with a range of

1.1 – 1.2, which reduces to a range of 1.11 – 1.15, as 푐푑0_푟푤 increases from 0.03 to 0.07. Similarly,

73 within the same level of increment, the optimal value range for the rear wing lift coefficient starts with 1.42 – 1.9, and reduces down to 1.55 – 1.7.

From these figures it can be concluded, that the change in drag coefficient of the front wing has a more profound effect on lap times than the rear wing drag coefficient. Also, the optimal values of the lift coefficient for the front wing lie within a range of 1.11 to 1.15, while the same for the rear wing lie within a range of 1.55-1.7; irrespective of the value of the drag coefficients for both the front and rear wing. Thus, the most optimum lap time will always lie within the above specified range, for any given set of drag coefficients for the front and rear wing.

(b) 풄 = 0.07, 풄 = 0.03 풅ퟎ_풇풘 풅ퟎ_풓풘 (d) 풄풅ퟎ_풇풘 = 0.09, 풄풅ퟎ_풓풘 = 0.03

(e) 풄풅ퟎ_풇풘 = 0.10, 풄풅ퟎ_풓풘 = 0.03

Fig 5.7: Variation in contours for constant value of viscous drag coefficient of the rear wing 75

(b) 풄 = 0.08, 풄 = 0.04 풅ퟎ_풇풘 풅ퟎ_풓풘 (d) 풄풅ퟎ_풇풘 = 0.08, 풄풅ퟎ_풓풘 = 0.06

(e) 풄풅ퟎ_풇풘 = 0.08, 풄풅ퟎ_풓풘 = 0.07

Fig 5.8: Variation in contours for constant value of viscous drag coefficient of the front wing

The contour plots depicting variations with respect to change in the values of 푐푑0_푓푤 for the values of 푐푑0_푟푤 = 0.04, 0.05, 0.06 and 0.07 are listed in Appendix B. In order to compress all the information from all the 25 different plots, a contour plot was generated for lift-to-drag ratio of the front wing vs. lift-to-drag ratio of the rear wing. This is shown in Fig. 5.9.

Fig 5.9: Variation in contours with respect to L/D ratio of front wing vs. the rear wing

From the figure, the florescent green zones, lying with the pink zones, depict the most optimal region. It can be observed that many different values of L/D ratios provide the most optimal zone.

However, the higher the L/D ratio, the better it might be. All these plots were generated for the optimal racing distance of 5474 m. The user can select any given time, and normalize the track length, using the speed-distance-time relationship.

With the results obtained from this chapter, any user can determine the optimal time lap that can be obtained for a given set of coefficients. Combining the information generated from the two sets of contour plots, most optimum lap time can be figured. Thus, with the knowledge of what best possible lap time can be attained for a given track and a given car setup, the area of development can be focused on attaining the aerodynamic coefficients for the front and rear wing while determining the airfoil shape of the wings. This will save significant amount of time during the car’s development cycle. Also, results of various strategies adopted during the race can be visualized. As many assumptions are made to simplify the process, the information is best used for preliminary estimates.

The higher the accuracy of the preliminary estimates, the better and more accurate will be the solution.

Chapter 6 CONCLUSION AND DISCUSSION

6.1 Summary and Results

i. The objective of this study was to develop a methodology to create plots depicting the

sensitivity of lap times performed by a F1 car around a particular track to the aerodynamic

coefficients of the front and rear wings.

ii. After providing a historical background of the sport of Formula 1 and developments in the

field of race aerodynamics in this sport, a literature review was conducted to develop a

focus area for this research study.

iii. An introduction was provided for the genetic algorithm (CMA-ES) used for optimization of

key variables. Also, AeroLap, which is a lap simulation tool used for simulating an actual

F1 race, was presented and the key components being used for the study were briefly

described.

iv. Malaysian F1 GP track was selected for the study, as this track is a perfect mix of long

straights that require drag reduction as main design philosophy, and fast corners that

require high levels of downforce.

v. The track boundaries and centerline were established using a geo-mapping tool called

ArcGIS. With X and Y coordinates for the data points collected being established, the

radius of curvature was calculated for the turns, using Taubin’s method.

vi. An optimization using CMA-ES was performed to determine the optimal racing line used

along the track. The resulted best lap time was 1.699 seconds faster than the lap performed

by an actual F1 car with similar setup. However, the distance covered was 69 m lesser.

Assuming that the total error was due to mapping issues only, the normalized lap time with

same distance as an official lap length was found out to be 0.542 seconds quicker.

vii. It was established that the total value of aerodynamic forces can be calculated as

superposition of the aerodynamic forces generated by individual components, namely, front

wing assembly, rear wing assembly, and body without wings. viii. The base values of the aerodynamic coefficients of body without wings were established, as

the front wing and rear wing coefficients were required to be optimized with no change in

the aerodynamic setup of the body.

ix. Once the base values of all the required components were established, and the car setup’s

variables being locked, the optimum values of lift and drag coefficients for the front and

rear wings were calculated using CMA-ES and AeroLap.

x. With values of 1.123 and 1.651 for the front and rear wings’ lift coefficients, for a viscous

drag coefficient of 0.08 and 0.03, respectively, the lap time improved by a total margin of

2.867 seconds, when compared with the best time by an actual F1 car of 2010 season.

When normalized, the time was still found to be better by 1.724 seconds.

xi. On optimizing the viscous drag coefficients for the optimum values of lift coefficients, it

was found that the optimizer always selects the lowest possible limit defined. Hence, a

parametric study was undertaken, where all the four coefficients were changed and their

impact on the lap times was analyzed using contour plots. For an optimum viscous drag

coefficient of 0.004 for both the front and rear wing, the lap time improved overall by 3.159

seconds, and by 2.02 seconds after the track being normalized.

xii. On conducting the parametric study, it was established that the lap times were more

sensitive to the change in viscous drag coefficient of the front wing, rather than that of the

rear wing. The optimal zone was found out to be always within the range of 1.11 to 1.17 for

the front wing lift coefficient and within the range of 1.55-1.7 for the rear wing,

irrespective of the value of the drag coefficients for both the front and rear wing. The

reduction in drag coefficients brought about the reduction in lap times within this range for

the lift coefficients.

xiii. A lift-to-drag ratio contour plot was prepared that summarized all the plots for different

coefficients considered for this study. It was found that multiple L/D ratios lied in the

optimal zone. Best results were expected to be achieved by combining the information with

the lift and drag coefficient plots.

6.2 Assumptions

i. The optimized racing path doesn’t change significantly with change in aerodynamic

coefficients with changes expected to be of the order of a few centimeters.

ii. The contributions of individual components of the F1 car towards the total aerodynamic

loads acting on the car were assumed on basis of Toet’s work. This allowed for application

of the superposition principle, as the contributions were calculated for the car as a whole,

and not discretely for each component.

iii. As the contributions of individual components were listed for a 2009 F1 car model, they

were assumed to work in the same way for a 2010 F1 car, as there were no major changes

related to the aerodynamics of the car.

iv. The ground effect increased the lift coefficient of the car by a minimum of 30%. This

allowed for negligence of the effect that ride heights have on the front wing.

v. The span efficiency factor was considered to be 0.65 for the front wing and 0.7 for the rear

wing, as the distribution over the rear wing was very similar to that of a rectangular wing,

while that over the front wing was somewhat similar to the rectangular and triangular wing.

6.3 Discussion and Conclusion

It was found that on a track, which is equally divided into long straights and fast corners,

either strategies of drag reduction or downforce increment can work equally well. For half the track

length that lied in the middle portion, the downforce increment strategies were found to be very

useful. However, the other half consisted of sections where the possibility of hitting the maximum

81 speeds was really high. This would lead to adoption of drag reduction strategies, as lower the drag, the higher will be the maximum speed. With increase in lift causing increase in lift-induced drag, which is a major component of the total drag (up to 90%), adopting the downforce increment strategy would definitely hurt the pace of the car in other sectors. The results of this study also show a similar trend. The values of the lift coefficients for the front and rear wings were considerably low, compared to what downforce levels being generally expected of F1 car wings.

Thus, the drag reduction strategy was required to be adopted to get the best lap time out of the car.

The main purpose of this study was to present information in the form of contour plots to a designer, signifying the effect of change of the aerodynamic coefficients on the lap times. Although it is well established that aerodynamics plays a major role in the setup of the car’s performance over the track, no research study seemed to focus on the importance of the dependency of the aerodynamic coefficients on the track parameters. Most of the studies concentrated on developing high downforce generating airfoils without establishing whether these airfoils might be helpful to the performance of the car. This study was an attempt to bridge this gap, where the designer can determine the downforce and drag requirements on the basis of the results obtained from this study and then determine which of the newly developed airfoils presented in various studies can be useful. The contour plots developed as a part of the study provide a good estimate of the lap times and also of the aerodynamic coefficients associated with those lap times. This methodology can determine what strategy is required to be adopted for a particular race. It is expected to accelerate the decision-making system and the development of the F1 car for a particular race. With more and more restrictions being applied by the FIA on the Formula 1 teams to make the sport cost efficient, this particular study helps in reducing the operations’ cost and focus the area of development. This methodology can, thus, be extremely useful for teams working with smaller budget.

6.4 Drawbacks

i. The data considered is an example set of data. With actual wind-tunnel study’s data on an

actual F1 car, the accuracy of the information provided will improve significantly.

ii. Interaction between different components of the car were assumed to be incorporated into

the contributions suggested. Based on the location and design of the wings, the

contributions might change.

iii. As no wind tunnel study was performed, the impact of ground effects on the wings is

calculated based on the theoretical data only. Effect of change in ride heights of the car can

be incorporated to make the data more accurate.

6.5 Future Research

i. This study can be expanded to develop the airfoils for the front and rear wings for the given

car setup and track by using the present data.

ii. A sensitivity study can be conducted that can determine the impact of significant factors,

which are being assumed in the present study. With knowledge of the impact of each

variable, a more accurate lap time can be predicted.

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Appendix A Input Data Tables for Lift and Drag Coefficient and Location of Center of Pressure

Table A.1: Lift Coefficient for the whole F1 car, varying with ride height.

Rear Ride Height in mm 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 0 2.760 2.825 2.879 2.924 2.961 2.992 3.017 3.038 3.055 3.069 3.080 3.088 3.093 3.094 3.091 3.082 3.067 3.045 5 2.715 2.781 2.837 2.883 2.921 2.952 2.979 3.001 3.019 3.034 3.047 3.056 3.062 3.064 3.062 3.054 3.040 3.019 10 2.667 2.734 2.791 2.838 2.877 2.910 2.937 2.961 2.980 2.996 3.010 3.020 3.027 3.030 3.029 3.023 3.010 2.990 Front 15 2.615 2.684 2.742 2.790 2.830 2.864 2.893 2.917 2.938 2.955 2.969 2.981 2.989 2.993 2.993 2.988 2.977 2.957 Ride 20 2.561 2.630 2.689 2.738 2.780 2.815 2.845 2.870 2.892 2.910 2.926 2.938 2.948 2.953 2.954 2.950 2.940 2.921 Height 25 2.503 2.573 2.633 2.684 2.726 2.762 2.793 2.820 2.843 2.862 2.879 2.893 2.903 2.910 2.912 2.909 2.900 2.882 in mm 30 2.441 2.513 2.574 2.626 2.669 2.707 2.739 2.766 2.790 2.811 2.829 2.844 2.855 2.863 2.866 2.865 2.856 2.840 35 2.377 2.450 2.512 2.564 2.609 2.648 2.681 2.710 2.735 2.757 2.775 2.791 2.804 2.813 2.817 2.817 2.810 2.795 40 2.309 2.383 2.446 2.500 2.546 2.586 2.620 2.650 2.676 2.699 2.719 2.736 2.750 2.760 2.765 2.766 2.760 2.746 45 2.238 2.313 2.377 2.432 2.479 2.520 2.555 2.586 2.614 2.638 2.659 2.677 2.692 2.703 2.710 2.711 2.707 2.694 50 2.163 2.240 2.305 2.361 2.409 2.451 2.488 2.520 2.548 2.574 2.596 2.615 2.631 2.643 2.651 2.654 2.650 2.639

Table A.2: Lift Coefficient for the part of the body (without wings) exerting load at front wheel center

Rear Ride Height in mm 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 0 0.258 0.295 0.325 0.350 0.375 0.400 0.420 0.440 0.460 0.479 0.498 0.521 0.539 0.560 0.581 0.601 0.620 0.641 5 0.204 0.240 0.273 0.297 0.321 0.345 0.365 0.385 0.404 0.423 0.441 0.464 0.482 0.504 0.525 0.540 0.559 0.580 10 0.156 0.190 0.222 0.249 0.273 0.292 0.311 0.330 0.349 0.371 0.390 0.408 0.426 0.447 0.468 0.484 0.503 0.521 Front 15 0.113 0.146 0.176 0.202 0.225 0.243 0.266 0.284 0.303 0.321 0.339 0.357 0.375 0.396 0.413 0.433 0.452 0.465 Ride 20 0.068 0.103 0.131 0.156 0.178 0.200 0.218 0.236 0.254 0.272 0.289 0.311 0.329 0.346 0.366 0.382 0.401 0.415 Height 25 0.031 0.065 0.092 0.116 0.137 0.158 0.175 0.192 0.214 0.231 0.249 0.266 0.283 0.300 0.320 0.336 0.356 0.369 in mm 30 -0.007 0.028 0.054 0.077 0.100 0.121 0.137 0.154 0.171 0.188 0.205 0.222 0.243 0.259 0.275 0.295 0.310 0.324 35 -0.040 -0.007 0.021 0.043 0.065 0.085 0.101 0.121 0.137 0.154 0.170 0.187 0.203 0.219 0.235 0.255 0.270 0.284 40 -0.068 -0.037 -0.010 0.014 0.035 0.054 0.069 0.085 0.101 0.117 0.133 0.152 0.169 0.185 0.200 0.216 0.235 0.245 45 -0.094 -0.064 -0.036 -0.014 0.007 0.025 0.039 0.058 0.073 0.088 0.104 0.119 0.135 0.151 0.170 0.185 0.200 0.214 50 -0.118 -0.087 -0.061 -0.039 -0.020 0.000 0.014 0.032 0.046 0.061 0.076 0.091 0.106 0.122 0.137 0.155 0.170 0.180

Table A.3: Lift Coefficient for the part of the body (without wings) exerting load at rear wheel center

Rear Ride Height in mm 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 0 1.473 1.476 1.480 1.483 1.481 1.476 1.471 1.464 1.456 1.445 1.433 1.415 1.400 1.379 1.356 1.331 1.303 1.268 5 1.498 1.503 1.506 1.510 1.510 1.505 1.502 1.497 1.489 1.479 1.469 1.452 1.438 1.417 1.395 1.374 1.347 1.312 10 1.516 1.523 1.528 1.530 1.531 1.532 1.530 1.526 1.519 1.507 1.497 1.485 1.472 1.452 1.430 1.411 1.384 1.354 Front 15 1.526 1.537 1.543 1.547 1.549 1.552 1.548 1.544 1.539 1.531 1.522 1.512 1.499 1.480 1.464 1.440 1.414 1.388 Ride 20 1.538 1.546 1.555 1.560 1.565 1.565 1.566 1.563 1.559 1.553 1.545 1.531 1.520 1.506 1.486 1.467 1.442 1.416 Height 25 1.538 1.548 1.559 1.567 1.572 1.574 1.576 1.575 1.568 1.563 1.556 1.548 1.537 1.524 1.505 1.487 1.462 1.437 in mm 30 1.537 1.547 1.560 1.569 1.573 1.576 1.580 1.580 1.578 1.574 1.569 1.561 1.547 1.536 1.521 1.501 1.480 1.456 35 1.530 1.543 1.554 1.564 1.570 1.575 1.580 1.578 1.577 1.575 1.570 1.563 1.555 1.544 1.531 1.511 1.491 1.468 40 1.515 1.530 1.544 1.553 1.561 1.567 1.573 1.576 1.577 1.575 1.572 1.563 1.555 1.546 1.533 1.518 1.496 1.477 45 1.497 1.514 1.527 1.538 1.547 1.555 1.563 1.564 1.566 1.566 1.563 1.559 1.553 1.544 1.529 1.514 1.497 1.475 50 1.473 1.492 1.506 1.520 1.530 1.537 1.546 1.548 1.551 1.553 1.551 1.548 1.543 1.535 1.525 1.508 1.491 1.474

Table A.4: Drag Coefficient for the whole F1 car, varying with ride height

Rear Ride Height in mm 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 0 0.978 0.978 0.980 0.981 0.983 0.984 0.986 0.986 0.987 0.986 0.986 0.984 0.982 0.980 0.978 0.977 0.975 0.975 5 0.983 0.984 0.985 0.987 0.989 0.990 0.991 0.992 0.993 0.992 0.992 0.990 0.989 0.987 0.985 0.983 0.982 0.981 10 0.987 0.988 0.990 0.991 0.993 0.995 0.996 0.997 0.998 0.998 0.997 0.996 0.994 0.993 0.991 0.989 0.988 0.987 Front 15 0.991 0.992 0.993 0.995 0.997 0.999 1.001 1.002 1.002 1.002 1.002 1.001 0.999 0.997 0.996 0.994 0.993 0.993 Ride 20 0.994 0.995 0.996 0.998 1.000 1.002 1.004 1.005 1.006 1.006 1.005 1.004 1.003 1.001 1.000 0.998 0.997 0.997 Height 25 0.995 0.997 0.999 1.001 1.003 1.005 1.007 1.008 1.009 1.009 1.008 1.007 1.006 1.005 1.003 1.002 1.001 1.001 in mm 30 0.997 0.998 1.000 1.002 1.004 1.007 1.008 1.010 1.011 1.011 1.010 1.010 1.009 1.007 1.006 1.004 1.003 1.003 35 0.997 0.998 1.000 1.003 1.005 1.007 1.009 1.011 1.012 1.012 1.012 1.011 1.010 1.009 1.007 1.006 1.005 1.005 40 0.997 0.998 1.000 1.003 1.005 1.007 1.009 1.011 1.012 1.012 1.012 1.012 1.011 1.010 1.008 1.007 1.007 1.007 45 0.995 0.997 0.999 1.002 1.004 1.007 1.009 1.010 1.012 1.012 1.012 1.012 1.011 1.010 1.008 1.007 1.007 1.007 50 0.993 0.995 0.997 1.000 1.003 1.005 1.007 1.009 1.010 1.011 1.011 1.011 1.010 1.009 1.008 1.007 1.006 1.007

Table A.5: Drag Coefficient for the body (without wings), varying with ride height

Rear Ride Height in mm 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 0 0.681 0.681 0.683 0.683 0.685 0.685 0.687 0.687 0.687 0.687 0.687 0.685 0.684 0.683 0.681 0.681 0.679 0.679 5 0.685 0.685 0.686 0.687 0.689 0.690 0.690 0.691 0.692 0.691 0.691 0.690 0.689 0.687 0.686 0.685 0.684 0.683 10 0.687 0.688 0.690 0.690 0.692 0.693 0.694 0.694 0.695 0.695 0.694 0.694 0.692 0.692 0.690 0.689 0.688 0.687 Front 15 0.690 0.691 0.692 0.693 0.694 0.696 0.697 0.698 0.698 0.698 0.698 0.697 0.696 0.694 0.694 0.692 0.692 0.692 Ride 20 0.692 0.693 0.694 0.695 0.697 0.698 0.699 0.700 0.701 0.701 0.700 0.699 0.699 0.697 0.697 0.695 0.694 0.694 Height 25 0.693 0.694 0.696 0.697 0.699 0.700 0.701 0.702 0.703 0.703 0.702 0.701 0.701 0.700 0.699 0.698 0.697 0.697 in mm 30 0.694 0.695 0.697 0.698 0.699 0.701 0.702 0.704 0.704 0.704 0.704 0.704 0.703 0.701 0.701 0.699 0.699 0.699 35 0.694 0.695 0.697 0.699 0.700 0.701 0.703 0.704 0.705 0.705 0.705 0.704 0.704 0.703 0.701 0.701 0.700 0.700 40 0.694 0.695 0.697 0.699 0.700 0.701 0.703 0.704 0.705 0.705 0.705 0.705 0.704 0.704 0.702 0.701 0.701 0.701 45 0.693 0.694 0.696 0.698 0.699 0.701 0.703 0.704 0.705 0.705 0.705 0.705 0.704 0.704 0.702 0.701 0.701 0.701 50 0.692 0.693 0.694 0.697 0.699 0.700 0.701 0.703 0.704 0.704 0.704 0.704 0.704 0.703 0.702 0.701 0.701 0.701

Table A.6: Location of Center of Pressure for whole F1 car, varying with ride height

Rear Ride Height in mm 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 0 36.7 37.5 38.1 38.6 39.1 39.6 40.0 40.4 40.8 41.2 41.6 42.1 42.5 43.0 43.5 44.0 44.5 45.1 5 35.4 36.2 36.9 37.4 37.9 38.4 38.8 39.2 39.6 40.0 40.4 40.9 41.3 41.8 42.3 42.7 43.2 43.8 10 34.2 35.0 35.7 36.3 36.8 37.2 37.6 38.0 38.4 38.9 39.3 39.7 40.1 40.6 41.1 41.5 42.0 42.5 Front 15 33.1 33.9 34.6 35.2 35.7 36.1 36.6 37.0 37.4 37.8 38.2 38.6 39.0 39.5 39.9 40.4 40.9 41.3 Ride 20 31.9 32.8 33.5 34.1 34.6 35.1 35.5 35.9 36.3 36.7 37.1 37.6 38.0 38.4 38.9 39.3 39.8 40.2 Height 25 30.9 31.8 32.5 33.1 33.6 34.1 34.5 34.9 35.4 35.8 36.2 36.6 37.0 37.4 37.9 38.3 38.8 39.2 in mm 30 29.8 30.8 31.5 32.1 32.7 33.2 33.6 34.0 34.4 34.8 35.2 35.6 36.1 36.5 36.9 37.4 37.8 38.2 35 28.8 29.8 30.6 31.2 31.8 32.3 32.7 33.2 33.6 34.0 34.4 34.8 35.2 35.6 36.0 36.5 36.9 37.3 40 27.9 28.9 29.7 30.4 31.0 31.5 31.9 32.3 32.7 33.1 33.5 34.0 34.4 34.8 35.2 35.6 36.1 36.4 45 27.0 28.0 28.9 29.6 30.2 30.7 31.1 31.6 32.0 32.4 32.8 33.2 33.6 34.0 34.5 34.9 35.3 35.7 50 26.1 27.2 28.1 28.8 29.4 30.0 30.4 30.9 31.3 31.7 32.1 32.5 32.9 33.3 33.7 34.2 34.6 34.9

Appendix B Contour Plots for Varying Viscous Drag Coefficients of Front and Rear Wings

(d) 풄풅ퟎ_풇풘 = 0.09, 풄풅ퟎ_풓풘 = 0.04 (b) 풄풅ퟎ_풇풘 = 0.07, 풄풅ퟎ_풓풘 = 0.04

(e) 풄풅ퟎ_풇풘 = 0.10, 풄풅ퟎ_풓풘 = 0.04 Fig. B.1: Variation in contours for viscous drag coefficient of the rear wing = 0.04

(a) 풄풅ퟎ_풇풘 = 0.06, 풄풅ퟎ_풓풘 = 0.05 (c) 풄풅ퟎ_풇풘 = 0.08, 풄풅ퟎ_풓풘 = 0.05

(d) 풄 = 0.09, 풄 = 0.05 (b) 풄풅ퟎ_풇풘 = 0.07, 풄풅ퟎ_풓풘 = 0.05 풅ퟎ_풇풘 풅ퟎ_풓풘

(e) 풄풅ퟎ_풇풘 = 0.10, 풄풅ퟎ_풓풘 = 0.05 Fig. B.2: Variation in contours for viscous drag coefficient of the rear wing = 0.05

(d) 풄풅ퟎ_풇풘 = 0.09, 풄풅ퟎ_풓풘 = 0.06 (b) 풄풅ퟎ_풇풘 = 0.07, 풄풅ퟎ_풓풘 = 0.06

(e) 풄풅ퟎ_풇풘 = 0.10, 풄풅ퟎ_풓풘 = 0.06 Fig. B.3: Variation in contours for viscous drag coefficient of the rear wing = 0.06 93

(d) 풄풅ퟎ_풇풘 = 0.09, 풄풅ퟎ_풓풘 = 0.07 (b) 풄풅ퟎ_풇풘 = 0.07, 풄풅ퟎ_풓풘 = 0.07

(e) 풄풅ퟎ_풇풘 = 0.10, 풄풅ퟎ_풓풘 = 0.07 Fig. B.4: Variation in contours for viscous drag coefficient of the rear wing = 0.07 94