Development and performance evaluation of undertray diffusers during racing manuevers Bachelor’s thesis in Mechanics and Maritime Sciences WILLEM DE WILDE JACOB GUNNARSSON LENA IVARSSON LINNÉUS KARLSSON OSKAR KOLTE DANIEL OLANDER DEPARTMENT OF MECHANICS AND MARITIME SCIENCES CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2021 www.chalmers.se www.chalmers.se Bachelor’s thesis 2021:12 Development and performance evaluation of undertray diffusers during racing manuevers WILLEM DE WILDE JACOB GUNNARSSON LENA IVARSSON LINNÉUS KARLSSON OSKAR KOLTE DANIEL OLANDER Department of Mechanics and Maritime Sciences Division of Vehicle Engineering and Autonomous Systems Chalmers University of Technology Gothenburg, Sweden 2021 Development and performance evaluation of undertray diffusers during racing manuev- ers WILLEM DE WILDE JACOB GUNNARSSON LENA IVARSSON LINNÉUS KARLSSON OSKAR KOLTE DANIEL OLANDER © WILLEM DE WILDE, JACOB GUNNARSSON, LENA IVARSSON, LINNÉUS KARLSSON, OSKAR KOLTE, DANIEL OLANDER, 2021. Supervisor: Erik Josefsson, Division of Vehicle Engineering and Autonomous Sys- tems Examiner: Simone Sebben, Division of Vehicle Engineering and Autonomous Sys- tems Bachelor’s Thesis 2021 Department of Mechanics and Maritime Sciences Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Cover: Velocity streamlines around the Chalmers Formula Student car and Cp visualized on the surface while driving straight. Typeset in LATEX, template by Magnus Gustaver Printing /Department of Mechanics and Maritime Sciences Gothenburg, Sweden 2021 iv Abstract A potential to develop the monocoque and diffuser on the Chalmers Formula Stu- dent (CFS) car, as to increase its downforce, was identified by CFS. Downforce is the downward aerodynamic lifting force that is obtained when a pressure difference is created between the top and bottom of the car. This effect is crucial for the grip of the car in driving scenarios like cornering, accelerating and braking. With more downforce, accelerating and braking can be done faster, and higher speeds can be maintained in cornering. The objective of this project is to develop a methodology for modeling of aerody- namic forces during different racing maneuvers of a CFS car. These new methods are purely computational. Further, the methods are used in the development of a new diffuser concept. This is done with the aim of providing CFS with knowledge and proof-of-concept of the implementation of a new diffuser, which could increase aerodynamic performance of the car. It was shown that straight ahead driving, braking, and cornering were the most critical driving scenarios. These were used to perform three types of simulations. Different diffuser designs were simulated based on these scenarios. It was further shown that the following parameters had an effect on aerodynamic performance: the expansion angle of the diffuser, the starting point of the diffuser, the radius of the diffuser throat, and the implementation of strakes and side floors. Differences in the performance robustness of the different designs were observed. Two diffusers provided the greatest downforce: one with a 13° expansion angle and the other with a 19° expansion angle. The diffusers were in other regards iden- tical. Lastly, the 13° diffuser was chosen as the best contending design, due to its robust performance in each of the simulated driving scenarios. Keywords: CFD, aerodynamics, Formula Student, downforce, lift coefficient, drag coefficient, diffuser, racing manuevers, cornering, braking. v Sammandrag En potential att vidareutveckla monocoque och diffusor på Chalmers Formula Student- bilen (CFS-bilen), med avsikt att öka bilens downforce, identifierades av CFS. Down- force kallas den nedåtriktade aerodynamiska lyftkraft, som uppstår vid en tryckskill- nad mellan bilens ovan- och undersida. Denna effekt är avgörande för bilens grepp i körscenarier såsom kurvtagning, acceleration och inbromsning. Större downforce tillåter kraftigare acceleration och inbromsning och att högre hastighet kan hållas under kurvtagning. Detta projekts syfte är att utveckla en metodik för att modellera aerodynamiska krafter under de olika manövrar som en CFS-bil gör under en tävling. Dessa metoder är baserade på simuleringar. Vidare ska metoderna användas för att assistera utveck- lingen av ett nytt diffusor-koncept. Detta görs med målet att tillhandahålla CFS kunskap om, och konceptvalidering för implementeringen av en ny diffusor, vilket kan förbättra bilens aerodynamiska prestanda. Det visades att rak körning, inbromsning och kurvtagning var de mest kritiska körscenarierna. Tre typer av simulering baserades på dessa scenarier. Olika typer av diffusorer undersöktes med dessa simuleringstyper. Vidare visades att följande parametrar hade en inverkan på bilens prestanda: diffusorns expansionsvinkel, dif- fusorns övergångsradie, samt implemetering av skiljeväggar och sidogolv. Skillnader i de olika diffusorernas prestandas robusthet observerades. Två diffusorer genererade störst downforce: en med expansionsvinkeln 13°, och den andra med expansionsvinkeln 19°. I andra hänseenden var diffusorerna identiska. Slutligen valdes diffusorn med 13° expansionsvinkel som den bäst presterande av de undersökta. Detta baserades på dess robusta prestanda i alla undersökta körscenar- ier. Nyckelord: CFD, aerodynamik, Formula Student, downforce, lyftkraftskoefficient, motståndskoefficient, diffusor, manövrar, kurvtagning, inbromsning. vi Acknowledgements We would like to express our greatest gratitude to our supervisor Erik Josefsson, Ph.D student in the Road Vehicle Aerodynamics research group, for all the help and guidance throughout this bachelor thesis. We would also like to thank Simone Sebben, professor in Aerodynamics and manager of the division Vehicle Engineering and Autonomous Systems, for the opportunity to carry out this thesis. Simone Sebben has given us lectures that have been very useful in the project. Finally we would like to thank the Chalmers Formula Student 2021 team for their cooperation. We are particularly grateful for Christian Svensson’s invaluable source of knowledge and willingness to help in this project. The authors, Gothenburg, May 2021 vii viii Contents Nomenclature and Abbreviations xiii List of Figures xv List of Tables xix 1 Introduction 1 1.1 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Working Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Delimitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Theory 5 2.1 Fundamental Fluid Dynamics Theory . . . . . . . . . . . . . . . . . . 5 2.1.1 Bernoulli’s Principle and the Continuity Equation . . . . . . . 5 2.1.2 Dimensionless parameters . . . . . . . . . . . . . . . . . . . . 5 2.1.2.1 Reynolds Number . . . . . . . . . . . . . . . . . . . 6 2.1.2.2 Drag Coefficient, CD . . . . . . . . . . . . . . . . . . 6 2.1.2.3 Lift Coefficient, CL . . . . . . . . . . . . . . . . . . . 6 2.1.2.4 Pressure Coefficient, Cp . . . . . . . . . . . . . . . . 6 2.1.2.5 Skin Friction Coefficient, Cf . . . . . . . . . . . . . . 6 2.1.3 Vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.4 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . 7 2.1.5 RANS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.6 Flow Past Boundary . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.6.1 Flow Separation . . . . . . . . . . . . . . . . . . . . 7 2.1.6.2 The Logarithmic Overlap Law . . . . . . . . . . . . . 8 2.2 CFD Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.1 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . 9 2.2.2 The k − ε Turbulence Model . . . . . . . . . . . . . . . . . . . 9 2.2.3 The k − ω Turbulence Model . . . . . . . . . . . . . . . . . . 9 2.2.4 The SST k − ω Model . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Vehicle Dynamics Theory . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.1 Steering Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.2 Body Slip Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.3 Pitch Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.4 Roll Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 ix Contents 2.3.5 Ride Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.6 Aerodynamic Influence on Vehicle Dynamics . . . . . . . . . . 12 2.4 Working Principles of a Race Car Diffuser . . . . . . . . . . . . . . . 13 2.4.1 Effect of Different Design Parameters of the Diffuser . . . . . 14 2.4.1.1 Differences in the Angles of the Diffuser . . . . . . . 14 2.4.1.2 Differences in Area Ratio . . . . . . . . . . . . . . . 14 2.4.1.3 Implementation of Strakes . . . . . . . . . . . . . . . 15 2.4.1.4 Different Throats of the Diffuser . . . . . . . . . . . 15 2.4.2 Venturi Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Methods 17 3.1 CAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.1.1 Current diffuser . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.1.2 Design Methodology . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 CFD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.1 Whole-car Versus Half-car Symmetry Simulation . . . . . . . . 20 3.2.2 Physics Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2.3 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2.3.1 Surface Wrapper . . . . . . . . . . . . . . . . . . . . 21 3.2.3.2 Volume Mesh . . . . . . . . . . . . . . . . . . . . . . 21 3.2.4 Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2.6 Post Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Choice of Simulated Driving Scenarios . . . . . . . . . . . . . . . . . 24 3.3.1 Assertion of Reynolds-independence . . . . . . . . . . . . . . . 26 3.3.2 Driving Straight Ahead . . . . . . . . . . . . . . . . . . . . . . 26 3.3.3 Cornering Driving Scenario . . . . . . . . . . . . . . . . . . . 27 3.3.4 Braking Driving Scenario . . . . . . . . . . . . . . . . . . . . . 28 4 Results and Analysis 29 4.1 General Differences for Driving Scenarios . . . . . . . . . . . . . . . . 29 4.1.1 Differences in Velocity Magnitude . . . . . . . . . . . . . . . . 32 4.1.2 Differences in Pressure Coefficient . . . . . . . . . . . . . . . . 35 4.1.3 Differences in Skin Friction Coefficient . . . . . . . . . . . . . 38 4.1.4 Differences in Vorticity Magnitude . . . . . . . . . . . . . . . 40 4.2 Starting Point of the Diffuser . . . . . . . . . . . . . . . . . . . . . . 41 4.3 Expansion Angle of the Diffuser . . . . . . . . . . . . . . . . . . . . . 43 4.4 Implementation of Strakes . . . . . . . . . . . . . . . . . . . . . . . . 47 4.5 Implementation of Side Floors . . . . . . . . . . . . . . . . . . . . . . 51 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5 Discussion 57 5.1 Impact on Vehicle Dynamics . . . . . . . . . . . . . . . . . . . . . . . 57 5.1.1 Vehicle Dynamics While Cornering . . . . . . . . . . . . . . . 57 5.1.2 Vehicle Dynamics While Braking . . . . . . . . . . . . . . . . 58 5.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 Implications of a New Diffuser on the Car’s Subsystems . . . . . . . . 59 x Contents 5.2.1 Implications on Non-aero Vehicle Subsystems . . . . . . . . . 59 5.2.2 The Diffuser’s Synergy with the Aero Package and Possible Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Comments on Methodology . . . . . . . . . . . . . . . . . . . . . . . 60 5.3.1 Whole-car Versus Half-car Simulations . . . . . . . . . . . . . 60 5.3.2 Cornering Left and Right . . . . . . . . . . . . . . . . . . . . 60 5.3.3 Mesh Inaccuracies . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3.4 Approximation of Center of Rotation for Roll and Pitch . . . 61 5.3.5 Mean of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.3.6 No Wind Tunnel Validation of the Data . . . . . . . . . . . . 62 6 Conclusion 63 6.1 Description of The Final Design Proposal . . . . . . . . . . . . . . . . 63 6.2 Final choice of Simulated Driving Scenarios . . . . . . . . . . . . . . 63 6.3 Recommendations for Further Research . . . . . . . . . . . . . . . . . 64 Bibliography 65 A Appendix 1 I A.1 Lift Coefficient for Side Wings . . . . . . . . . . . . . . . . . . . . . . I A.2 Pressure Coefficient for Diffusers with and without Side Floors . . . . II xi Contents xii Nomenclature and Abbreviations Fluid Mechanics µ Dynamic viscosity [kg ·m · s−1] ν Kinematic viscosity [m2 · s−1] ρ Density [kg ·m−3] τ Shear stress [N ·m−2] τw Wall shear stress [N ·m−2] ~ζ Flow vorticity vector [s−1] ~u Flow velocity vector[m · s−1] CD Coefficient of drag [-] Cf Coefficient of skin friction [-] CL Coefficient of lift in the direction of -ẑ [-] Cp Coefficient of pressure [-] FD Drag force [N] FL Lift force [N] p Static pressure [Pa] p∞ Freestream flow pressure [Pa] u Flow velocity in x-direction or |~u| [m · s−1] u∞ Freestream flow velocity [m · s−1] V Volume [m3] v Flow velocity in y-direction [m · s−1] w Flow velocity in z-direction [m · s−1] y+ Dimensionless distance from wall [-] Re Reynolds number [-] Vehicle Dynamics x̂ Coordinate axis pointing towards the rear of the car ŷ Coordinate axis pointing towards the right side of the car ẑ Coordinate axis pointing upwards Other Symbols g Gravitational acceleration [m · s−2] Abbreviations CAD Computer-aided Design CFD Computational Fluid Dynamics CFS Chalmers Formula Student COG Center Of Gravity COP Center Of Pressure FS Formula Student FVM Finite Volume Method xiii Nomenclature and Abbreviations M&D Monocoque and Diffuser SNIC Swedish National Infrastructure for Computing Software CATIA V5 Software package for CAD MATLAB Mathematical software used for calculations and data analysis Simcenter Star-CCM+ Software used for simulations in CFD xiv List of Figures 1.1 3D representation of the CFS21 car. The diffuser is marked in red. . . 1 2.1 Flow close to a wall. The flow enters and acts laminar and then changes and becomes turbulent at Rex > 106. . . . . . . . . . . . . . 8 2.2 Steering angles, δfl for the left wheel and δfr for the right wheel. These angles can differ from each other if the car has Ackermann steering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 A body slip angle β. The car’s longitudinal direction is pointing in the x direction while it is moving in a different direction ~v thus creating a slip angle β. The centripetal acceleration ac is acting on the car’s center of mass towards the corner’s center. . . . . . . . . . . . . . . . 11 2.4 A pitch angle θ around the car’s lateral axis. . . . . . . . . . . . . . . 11 2.5 A roll angle φ around the car’s longitudinal axis. . . . . . . . . . . . . 12 2.6 A ride height Hr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.7 3D-schematic showing the typical vortex pair that form as high- pressure ambient air flows into the low-pressure cavity of the under- tray diffuser (red arrows). Note the direction of the vortex rotation. The blue streamlines are grossly simplified. . . . . . . . . . . . . . . . 15 3.1 Workflow for the development of a diffuser. . . . . . . . . . . . . . . . 17 3.2 CAD model with the aerodynamic parts marked in different colors. Front wing (orange), deflector (red), monocoque (grey), side wing (green), rear wing (blue) and diffuser (yellow). . . . . . . . . . . . . . 18 3.3 Outlines of the three different diffuser designs when iterating the starting point of the diffuser. The 500 mm design is displayed as purple, the 300 mm design as red and the 0 mm design as blue. . . . 19 3.4 Outlines of the four different diffuser designs when iterating the angle between the flat bottom of the monocoque and the top of the diffuser. The 13° design i displayed as blue, the 15° design as purple, the 17° design as red and the 19° design as green. . . . . . . . . . . . . . . . 19 3.5 Diffuser with both strakes (orange) and side floor (blue). . . . . . . . 20 3.6 Volume mesh for the simulations. . . . . . . . . . . . . . . . . . . . . 22 3.7 The Wall y+ for the inner prism layer around the car. . . . . . . . . . 23 3.8 GPS data from a lap of the 2016 Formula Student Germany En- durance circuit from the CFS16 car. A histogram of longitudinal acceleration in the forward direction is illustrated. . . . . . . . . . . . 26 xv List of Figures 3.9 GPS data from a lap of the 2016 Formula Student Germany En- durance circuit from the CFS16 car. A typical 12.5 m radius corner where the average recorded speed is approximately 40 km/h is marked in the figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.1 Accumulated CL and CD while driving straight for the whole car with the small diffuser attached to the original CFS20 and CFS21 monocoque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Accumulated CL for the braking scenario compared to the straight scenario for a diffuser starting after the monocoque. . . . . . . . . . 31 4.3 Velocity magnitude plots for driving straight, braking and cornering with the small 0 mm starting point baseline diffuser. The visualised xy-plane is located at z = 88 mm above ground. . . . . . . . . . . . . 32 4.4 Velocity magnitude plots for the air surrounding the car. The visu- alized xz-plane is located in the middle of the car at y = 0. Only the straight and braking scenarios are included. . . . . . . . . . . . . . . 33 4.5 Pressure coefficient on the underside surface of the car during the three driving scenarios straight, braking and cornering. . . . . . . . . 35 4.6 Pressure coefficient of the air surrounding the car. The visualized xz- plane is located in the middle of the car at y = 0. Only the straight and braking scenarios are included. . . . . . . . . . . . . . . . . . . . 36 4.7 Skin friction coefficient on the car surface seen from below. . . . . . . 38 4.8 Vorticity magnitude for driving straight, braking and cornering. The visualized yz-plane is cutting through the diffuser 230 mm behind the monocoque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.9 Skin friction for different diffuser starting points while driving straight and during cornering. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.10 Skin friction while driving straight for diffusers with different angles. . 44 4.11 Skin friction while cornering for diffusers with different expansion angles. Note the differences in Venturi vortex generation between the different cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.12 Difference in accumulated lift coefficient CL, relative to the 19◦ dif- fuser for the straight and curved driving scenarios. A higher CL im- plies a greater downforce. . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.13 Vorticity magnitude while driving straight for the 13◦ and 19◦ dif- fusers with and without strakes. The visualized yz-plane cuts through the diffuser 27 mm behind the monocoque. . . . . . . . . . . . . . . . 48 4.14 Pressure coefficient on the underside surface of the car while driving straight. The four different diffuser designs with expansion angles 13◦ and 19◦, with and without strakes, are presented. . . . . . . . . . . . 49 4.15 Vorticity magnitude while cornering for the four diffusers with 13◦ and 19◦ angles with and without strakes. The visualized yz-plane cuts through the diffuser 27 mm behind the monocoque. . . . . . . . 50 4.16 Skin friction on the underside of the car. The four different diffuser designs with expansion angles 13◦ and 19◦, with and without strakes, are presented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 xvi List of Figures 4.17 Pressure coefficient for the 13° and 19° diffusers with and without side floors while driving straight are illustrated. The visualized yz-plane cuts through the diffuser 27 mm into the back of the monocoque. . . 52 4.18 Vorticity magnitude with and without side floors while driving straight. The visualized yz-plane cuts through the diffuser 27 mm behind the monocoque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.19 Velocity magnitude during cornering. The visualized xy-plane is lo- cated at z = 78 mm above ground. . . . . . . . . . . . . . . . . . . . 54 4.20 Skin friction under the car with and without side floors while cornering. 55 A.1 Pressure coefficient under the car with and without side floors while driving straight. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II xvii List of Figures xviii List of Tables 3.1 The settings used in Star-CCM+. . . . . . . . . . . . . . . . . . . . . 21 3.2 The boundaries for a straight simulation. . . . . . . . . . . . . . . . . 23 3.3 The boundaries for a cornering simulation. For an accurate repre- sentation the system has to be viewed in a rotating reference frame. The reference frame contains the fluid volume and outer boundaries which rotates while the car is fixed. . . . . . . . . . . . . . . . . . . . 24 3.4 Angles for simulating cornering at a corner radius of 12.5 m. For details regarding the different parameters, see Section 2.3. . . . . . . 28 4.1 Values for CD, CL for different components and aero balance (rear- wards) for different driving scenarios using the small diffuser starting at the end of the monocoque. M&D stands for Monocoque & Diffuser, FW for Front Wing, RW for Rear Wing and SW for Side Wing. . . . 29 4.2 Values for CD, CL, CL M&D and aero balance (rearwards) for the different diffuser starting points at the driving scenarios straight, cor- nering and braking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.3 Values for CD, CL, CL M&D and aero balance (rearwards) for different diffuser expansion angles at the driving scenarios straight, cornering and braking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Values for CD, CL, CL M&D and Aero balance (rearwards) for 13◦ and 19◦ diffusers with and without strakes. . . . . . . . . . . . . . . . 47 4.5 Values for CD, CL, CL M&D and aero balance (rearwards) for 13° and 19° diffusers with and without side floors. . . . . . . . . . . . . . 52 4.6 Values for CD, CL, CL M&D and aero balance (rearwards) for different driving scenarios comparing the small 0 mm diffuser and the best performing 13° diffuser with strakes and side floors. . . . . . . . . . . 56 A.1 The side wings’ lift coefficient for all the straight simulations . . . . . I A.2 The side wings’ lift coefficient for all the cornering simulations . . . . I xix List of Tables xx 1 Introduction Formula Student (FS) is a competition where universities and engineering students design and build formula cars. Chalmers team is called Chalmers Formula Student (CFS) and has been building electric formula cars since 2015. The average speed in competition for a Formula Student car is about 50 km/h, while the top speed is about 110 km/h [1]. The current CFS car has a diffuser, but it is believed that there is great potential in improving the flow around the floor and the diffuser to create more downforce. Downforce is a force that refers to the downward aerodynamic lifting force that is obtained on a car when a pressure difference is created between the top and bot- tom of the car. As downforce increases in a racing car, the tires generate more friction, which results in increased grip against the ground. This means that the car can accelerate more without the wheels spinning and that higher speeds can be maintained in corners. A diffuser is an aerodynamic part belonging to the bodywork at the rear of the car, see Figure 1.1. By applying a diffuser, the pressure under the car can decrease in relation to the pressure above the car thus generating more downforce. A diffuser also plays an important role in returning the air flow under the car to the ambient air in a smooth way to minimize energy losses, reducing the drag. Figure 1.1: 3D representation of the CFS21 car. The diffuser is marked in red. Different dynamic driving scenarios such as cornering, braking and acceleration af- fect how well all the car’s aerodynamic components (aero package) work. The airflow when the car is driving straight ahead at a constant speed does not interact with the car in the same way as when the car is cornering. Thus, the pressure distribution 1 1. Introduction and consequently the downforce in the two cases differ. For a formula car, downforce is most critical in dynamic driving scenarios where the car’s grip is most important. Implementation of different dynamic driving scenarios in aerodynamic simulations has rarely been done by the CFS team. However, this is something the team believe will bring benefits for future aerodynamic development. A structured method for modeling of such dynamic scenarios would be favorable for the 2021/2022 CFS team (CFS22) and future CFS teams. 1.1 Historical Background Motorsport began developing in the end of the 19th century following the invention of the petrol fueled internal-combustion engine. Until 1960, the development of rac- ing cars consisted mainly of strengthening the engines and generating more grip by developing chassis and wheel suspension. This required costly investments, which led to only the largest manufacturers being able to afford continued development and thus the opportunity to fight for a top position during competition. For the smaller manufacturers, new cheaper development areas were therefore needed, and the development of aerodynamics came into focus. Previously, the development of car aerodynamics had been intended to reduce air resistance, but in the late 1960s, motorsport engineers observed that downforce was also important for faster corner- ing. This led to innovative designs of cars that used downforce, which resulted in drastically improved lap times. Since then, aerodynamics has played a central role in motorsports [2]. In Formula 1, in particular, aerodynamic improvements have been of great im- portance. With increased knowledge about aerodynamics, higher speeds can be maintained, but also lower fuel consumption, lower wind noise and higher comfort are achieved. In the late 60’s, the first F1 car with a front wing and a rear spoiler, the Lotus 49B, was presented. During this period experiments of having wings high above the body of the car were carried out on, for example, the Matra MS10. Since the 60’s, lots of technical innovations in aerodynamics have been tested. Among other things, skirts have been added to the body work, in order to get the maximum ground effect. Lots of different wing and underbody designs have also been tested [3]. One of the most radical F1 cars of all time was the Brabham BT46B, also known as the ”fan car”. The car was designed by Gordon Murray and was equipped with a large fan that generated an immense amount of downforce by extracting air from the underside of the car. The car only raced once at the Swedish Grand Prix in Anderstorp in 1978, where the car also won before the concept was withdrawn [4]. It is still an ongoing process for motorsport engineers to look for areas for aerodynamic improvement in order to find that little extra performance that is crucial in a race. 1.2 Objective The objective of this project is to develop a methodology for modeling of aerody- namic forces during different racing maneuvers of a CFS car. These new methods 2 1. Introduction will then be used in the development of a new diffuser concept. This is done with the aim of providing CFS with knowledge and proof-of-concept of the implementation of a new diffuser, which could increase aerodynamic performance of the car. 1.3 Working Methods The modeling of aerodynamic forces on the CFS car has been done using Computa- tional Fluid Dynamics (CFD). Setups for relevant driving scenarios such as driving straight ahead, cornering and braking were simulated using a range of new diffuser design concepts. This was done in order to draw conclusions of how well a design concept works in dynamic driving scenarios, i.e. how aerodynamically robust it is. Design concepts were developed using 3D Computer Aided Design (CAD). A base model of the CFS21 car was obtained from CFS and used as a baseline. Iterations of a new diffuser concept were systematically developed and simulated using CFD. Furthermore, CFS shared three post-processing scripts. These were edited and used to produce necessary data and plots to analyze the simulations. 1.4 Delimitations The project is delimited in several regards. For instance the diffuser concepts de- veloped in the project will not be manufactured. It will only be visualized in three dimensional CAD, and simulated using CFD. When developing a new diffuser, the design is based on the rules for Formula Student in 2022 [5]. However, the regulations tend to look different from year to year when it comes to powered aero devices such as fans, which is the reason for the project to develop a diffuser design that performs good even without fans. Should it be of interest to add fans, then it should be for the purpose of tuning an already good performance. Hence the project is based on a diffuser without fans. In an aerodynamic package of a formula car, the aerodynamic components inter- acts in complex and often unpredictable ways with each other. Because of this complexity and the project’s limited time and computational resources, it is neces- sary to have practical limitation of how in-depth the work can be. Therefore, the design development will only be focused on the diffuser. Design of other parts of the car’s aero package will thus not be covered in this project. The project also does not include any mesh study. Mesh studies aim, among other things, to ensure convergence and independence of mesh for the solution in the sim- ulation. Instead of doing a mesh study, CFS current mesh was used with some refinements since CFS has already performed one [6]. More about the refinements are mentioned in Section 3.2. Only CFD simulations will be performed in order to analyze the aerodynamics of 3 1. Introduction the formula car, an thus no wind tunnel experiments. Due to the fact that no wind tunnel experiments will be carried out, no validation will be made in the form of correlation between simulations in CFD and wind tunnel testing. This will not be possible due to, among other things, that no diffuser will be manufactured. Wind tunnel experiments are also resource and time consuming. Assumptions of stationary flow is made for all simulations in the project. This means that the car’s position and quantities, such as velocity and pressure, don’t change with time. This assumption is done due to the fact that non-stationary flow would be too time consuming to base the simulations on. Thermodynamic effects on the flow are not taken into account in the project either. This is because its impact is considered to have a relatively insignificant impact on this project. The project will primarily focus on generating as high downforce as possible. Drag force is of secondary priority. Downforce always comes with the penalty of increased dragforce. However, in Formula Student it has been shown that increased downforce results in faster lap times almost regardless of any realistic drag force penalty [7]. This is true up to a factor of ∆CD/∆CL = 3, where ∆CD is a unit of gained drag coefficient (see Section 2.1.2.2) and ∆CL a unit of gained lift coefficient (see Section 2.1.2.3), as shown by previous CFS team members. No major consideration is given to the packaging of electronic components when designing the diffuser. When testing how early a diffuser concept can begin, parts of the electronics and battery storage are cut off. However, a plausibility assessment is made in order to decide how much of the electronics can be moved in order to let the diffuser start earlier. 4 2 Theory In this chapter, the fundamental physics phenomena which are central to the project are described. Further, the working principles of an undertray diffuser are investi- gated. 2.1 Fundamental Fluid Dynamics Theory The relevant portion of fundamental fluid dynamics theory is presented in this sec- tion. These principles are relevant to describe aerodynamics as a subject. In addi- tion, most of them are utilized by the CFD program Star-CCM+ which is further described in Section 3.2. The presented variables are described in Nomenclature. 2.1.1 Bernoulli’s Principle and the Continuity Equation In incompressible isentropic flow, i.e. a flow where a negligible amount of energy is lost to non-reversible processes such as heat in turbulent flows, one can derive Bernoulli’s principle from the conservation of energy u2 2 + p ρ + z · g = constant, (2.1) where z is the vertical position of a fluid element. This law states that an increase in velocity implies a decrease in pressure and vice versa. This is the working principle for wing profiles and diffusers [8]. Further, the continuity equation for mass flow provides a relationship between flow area in a closed channel and flow velocity ∂ρ ∂t + ρ∇ · ~u = 0 =⇒ Aflow · u = ṁ ρ = constant, (2.2) where Aflow represents the swept area of the fluid flow, t time and ṁ the mass flow rate through Aflow. 2.1.2 Dimensionless parameters Dimensionless parameters are used to make a parameter comparable with parame- ters of other types of flow. Below the relevant parameters are presented. 5 2. Theory 2.1.2.1 Reynolds Number The Reynolds number, Re, is a dimensionless number which tells the relation be- tween the inertia and viscosity in a newtonian fluid and is defined as Re = ρV L µ = { Inertia Viscosity } (2.3) V and L are the characteristic velocity and length of the flow. A high Re is associated with a fast, large-scale turbulent flow, whereas a low Re corresponds to a slow, viscous flow. Fast flows of gas imply a relatively high Re [8]. 2.1.2.2 Drag Coefficient, CD The drag coefficient of a body, CD, is a dimensionless number indicative of drag force. CD is defined as CD = FD 1 2ρu 2 ∞A , (2.4) where A is the projected area from the front of the body and the denominator (1 2ρu 2 ∞) is the dynamic pressure of the free stream [8]. 2.1.2.3 Lift Coefficient, CL The lift coefficient of a body, CL is defined as CL = FL 1 2ρu 2 ∞A , (2.5) where A is projected area from the front of the body. CL is a dimensionless coefficient that increases with lift force [8]. In this study, CL will indicate the lift in negative ẑ-direction, i.e. a high downforce will correspond to a large CL. 2.1.2.4 Pressure Coefficient, Cp The pressure coefficient , Cp, is a dimensionless parameter that describes the relative pressure. Cp is defined as Cp = p− p∞ 1 2ρu 2 ∞ , (2.6) where p is the static pressure at the point where Cp is calculated, while p∞ is the static pressure in the free stream [8]. 2.1.2.5 Skin Friction Coefficient, Cf The skin friction coefficient, Cf , is a dimensionless parameter defined as [8] Cf = τw 1 2ρu 2 ∞ . (2.7) 6 2. Theory 2.1.3 Vorticity The vorticity, ~ζ, is defined as the curl of the velocity field, ∇ × ~u, which relates to the rotation of a velocity field. If the vorticity is zero the flow has no rotation and is called irrotational [8]. The voriticity in x direction is given by ζx = ∂w ∂y − ∂v ∂z . (2.8) 2.1.4 Navier-Stokes Equations A fluid can be described with the Navier-Stokes equations if it is incompressible and newtonian [8] as ρ ( ∂~u ∂t + (~u · ∇)~u ) = ρ~g −∇p+ µ∆~u. (2.9) 2.1.5 RANS In turbulent flow there are fluctuations that causes rapid changes in the velocity and pressure in the Navier-stokes equations (Eq. 2.9). These rapid changes can be easier to account for if the velocity and pressure are split into a mean variable ū and a fluctuation variable u′. The velocity in the x-direction is therefore u = ū+u′. If those splits is put into the Navier-stokes equations we get the Reynold’s Averaged Navier Stokes (RANS) equations after some derivations [8]. The RANS in x-direction is ρ du dt = −∂p ∂x + ρgx + ∂ ∂x ( µ ∂u ∂x − ρu′2 ) + ∂ ∂y ( µ ∂u ∂y − ρu′v′ ) + ∂ ∂z ( µ ∂u ∂z − ρu′w′ ) . (2.10) 2.1.6 Flow Past Boundary Bodies, such as plates, immersed in a fluid stream create a boundary layer flow. The boundary layer is defined as the region where the flow velocity is less than 99 percent of the velocity of the external flow (u∞). The boundary layer flow is made up of two parts, the laminar part and the turbulent part: δ x ≈  5 Re1/2 x 103 < Rex < 106 laminar flow 0.16 Re1/7 x 106 < Rex turbulent flow . (2.11) δ is the thickness of the boundary layer and Rex the Reynolds number at x [8]. An illustration of he boundary level flow can be seen in Figure 2.1. 2.1.6.1 Flow Separation Backflow, i.e. flow going in the opposite direction of the freestream, at the wall indicates that the flow is separated. The point of separation is when the wall shear is 0 which means the velocity gradient is 0 [8]. 7 2. Theory Laminar part δ(x) Turbulent part - - - - - u(x, y) - x 6 y - - - - - - u∞ ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ Figure 2.1: Flow close to a wall. The flow enters and acts laminar and then changes and becomes turbulent at Rex > 106. 2.1.6.2 The Logarithmic Overlap Law The turbulent flow near a wall can be broken up in three regions based on which shear stress dominates: • Wall layer: Viscous shear dominates. • Outer layer: Turbulent shear dominates. • Overlap layer: A mix of the shear types. We define the dimensionless number y+ as y+ = yu∗ ν , (2.12) where u∗ = √ τw/ρ. For y+ ∈ [0, 5] the inner layer dominates and is proportional to y+. It can be described with the dimensionless variable u+ = u u∗ = y+. (2.13) For y+ ∈ [30, 103] the overlap layers dominates and can be described as u+ = 1 κ ln y+ +B, (2.14) where κ ≈ 0.41 and B ≈ 5.0. For y+ ∈ [5, 30] the inner layer has to curve so it merges with the overlap layer. Only experimental data exists for this interval making it difficult to model [8]. 8 2. Theory 2.2 CFD Theory Numerically solving the equations that describe the airflow around the car is a complex process. It is important to determine the accuracy needed in the simulation to arrive at a balance of the resources needed for the simulation and its accuracy. Different mathematical models for the modeling of the flow can be used. These models have their respective strengths and weaknesses, and should be chosen with care. This requires a basic understanding of key concepts in CFD simulation. 2.2.1 Finite Volume Method The Finite Volume Method (FVM) is a numerical method used to solve the partial differential equations encountered in fluid mechanics. The first step in the FVM is to discretize the domain of interest into a finite amount of control volumes (cells). The variables of interest are located at the centroid of these cells. The second step is integration of the governing equations over the cell to solve for these variables. Lastly, interpolation profiles are assumed between cell centroids in order to describe the variation of the variables of interest [9]. 2.2.2 The k − ε Turbulence Model k−ε is a type of RANS turbulence model that can be used in CFD. The k−ε model solves the equations for k and ε where k is the turbulent kinetic energy and ε the rate of dissipation. k − ε is accurate around external flow problems but not very accurate in complex curves. The model is fast and has a good convergence rate [10]. 2.2.3 The k − ω Turbulence Model k − ω is likewise the k − ε model a type of RANS turbulence model . The k − ω model solves the equations for k and ω where k is the turbulent kinetic energy and ω is the specific dissipation rate. Contrary to the k − ε method, the k − ω method is especially accurate in internal flow, flow past complex curvature and in separated flow. The method is more complex than the k − ε model which leads to it being slower and more difficult to converge [10]. 2.2.4 The SST k − ω Model The SST k−ω model is a mixup of the k−ω method and the k−εmethod where SST is Shear Stress Transport. Since the k − ω method is better near walls it influences the result near the walls and because it’s not very good in the free stream there is a transition from the k − ω method to the k − ε method in the free stream [11]. 2.3 Vehicle Dynamics Theory In this section, vehicle dynamics parameters relevant to this study are described. Furthermore, aerodynamic influence on vehicle dynamics in relation to these param- 9 2. Theory eters are discussed in this section. 2.3.1 Steering Angle The steering angle is the angle between the direction of a wheel and the car’s longi- tudinal axis. The steering angle for each front wheel can be different if the car has Ackermann steering [12]. An illustration can be found in Figure 2.2. δfl δfr Figure 2.2: Steering angles, δfl for the left wheel and δfr for the right wheel. These angles can differ from each other if the car has Ackermann steering. 2.3.2 Body Slip Angle The body slip angle is an yaw angle meaning it is the angle that is created when the car rotates around its vertical axis. The rotational axis goes through the car’s center of gravity. The body slip angle is the angle between the velocity vector of the car and the car’s longitudinal direction [12]. A visual representation of the body slip angle can be seen in Figure 2.3. 10 2. Theory ~v β ac β x y Figure 2.3: A body slip angle β. The car’s longitudinal direction is pointing in the x direction while it is moving in a different direction ~v thus creating a slip angle β. The centripetal acceleration ac is acting on the car’s center of mass towards the corner’s center. 2.3.3 Pitch Angle The pitch angle is the angle that is created if the vehicle’s sprung mass (everything but the wheels) rotates around its lateral axis which is shown in Figure 2.4. The vehicle rotates around what is called its pitch center [12]. θ Figure 2.4: A pitch angle θ around the car’s lateral axis. 2.3.4 Roll Angle The roll angle is the angle that is created if the vehicle’s sprung mass rotates around its longitudinal axis, see Figure 2.5. The vehicle rotates around what is called its roll center [12]. 11 2. Theory φ Figure 2.5: A roll angle φ around the car’s longitudinal axis. 2.3.5 Ride Height The ride height is the smallest distance between the ground and the car’s floor when it is stationary, see Figure 2.6. For a Formula Student car the ride height must at least be 30 mm [5]. Hr Figure 2.6: A ride height Hr. 2.3.6 Aerodynamic Influence on Vehicle Dynamics Vehicle dynamics is a large and complex subject. Mass distribution, tires, suspen- sion setup and aerodynamics are examples of areas that greatly affect the dynamics of a car. Basic principles of aerodynamics influence, in particular on a Formula student car, is further discussed here. The main aim of aerodynamics on a Formula Student car is to increase its negative lift force (downforce). Greater downforce makes it possible for the tires to produce larger forces (grip) while accelerating/braking (longitudinal forces) and cornering (lateral forces). The basic formula describing this relationship is Flong/lat = Fzµ, where Flong/lat is the longitudinal or latitudinal force, Fz the normal force which is increased with increased downforce and µ the friction coefficient. Consequently, the speed of the car can be increased in most driving scenarios or the same grip can be achieved with less tire slip, i.e smaller slip angle, when compared to a car with less downforce. This helps reducing heat production and therefore preserve tire life. Furthermore, smaller slip angles are easier to control for the driver. The car’s stability in this case is improved compared to larger slip angles and aerodynamic parts tend to work more efficiently, improving the balance [3]. In most cases a weight and aerodynamic distribution close to 50/50 on the cars 12 2. Theory rear and front axle is desired. A car with the center of gravity (COG) closer to the front axle usually generates more understeer. Conversly the aerodynamic center of pressure (COP or aero balance), the point at which the resultant downforce is lo- cated, closer to the front usually generates more oversteer. A car with a front biased COG can therefore be counter balanced by placing the COP forwards. However, it is favourable in some cases, in CFS’s case for example, to design the aerodynamic package with a COP slightly rear biased. This can generate understeer, but under- steer is considered more stable and easier to control when compared to oversteer [3]. Vehicle dynamics parameters described in previous sections are closely related to the aerodynamic package on a given car. While the generated downforce affect body slip, pitch and roll angles, these parameters in turn affect the aerodynamic efficiency. Pitch angles for example can have a large impact on vehicle balance. When braking and pitching forwards, the front wing’s downforce is generally increased. Although, other factors such as already low static ride height or an extreme pitch angle might cause a decrease in downforce for the front wing. This also results in less air flow to the car’s floor and diffuser, decreasing the middle and rear downforce. If the front wing downforce increases, the aerodynamic balance shifts forwards, decreasing the stability of the car. While this is considered unfavourable, increased grip on the front axle also results in improved braking capabilities. On the contrary, if the downforce on the front wing decreases the balance shifts rearwards and the opposite occurs. It is more difficult to describe the influence of steering and roll angles. These affect the aerodynamics in an asymmetric manner and can cause unpredictable behaviour of the car. The same is true for body slip angles, or yaw angles, where the air flow hits the car with an angle from the longitudinal axis [3]. A diffuser expands the airflow underneath the car’s floor. This expansion forces the flow upstream to increase in speed in order to fill up the larger volume in the diffuser, decreasing the pressure upstream. In motorsport it has been observed that diffusers are particularly effective aero devices. The reason is that diffusers increase the vehicle’s downforce under the whole floor. Downforce in this region is close to the center of the car and distributed over a large aera. This results in good balance contribution and makes the vehicle less sensitive to pitch or other phenomena that affect the balance. In order for a diffuser to work efficiently, the front wing needs to supply the floor and diffuser area with sufficient airflow. Optimally the airflow’s velocity is large and laminar. Another option for increasing the airflow is by feeding the diffuser with air from the sides of the vehicle. This can however interfere with the low pressure diffuser region and increase the pressure, resulting in a decrease of downforce. A compromise between keeping the floor sealed and increasing the airflow is usually optimal. This airflow can also be used to induce vortices, as described in the next section [3]. 2.4 Working Principles of a Race Car Diffuser The main function of an undertray diffuser is to reduce the pressure upstream of the diffuser, which provides a greater amount of downforce. Additionally, the diffuser 13 2. Theory provides a smooth transition where the airflow under the car joins the wake region. Equation (2.1) and Equation (2.2), can be combined to relate pressure, flow velocity and the area swept by the flow (i.e. the space between the car and the ground): 1 2 ( ṁ ρAflow )2 + p ρ = constant, (2.15) Where the parameters are as in the previously mentioned equations. It is clear that a decrease in flow area promotes a decrease in pressure. Given the boundary con- dition of ambient pressure at the end of the diffuser, one can see that the diffuser promotes a negative pressure upstream, providing downforce. However, this simplified model describes the undertray diffuser as a closed tube, in which the flow is perfectly laminar. A more accurate model would account for the air introduced from the gap between the diffuser sidewalls and the road as well as energy lost to complex, turbulent flow. 2.4.1 Effect of Different Design Parameters of the Diffuser The geometry of a simple race car diffuser can be described by a number of key design parameters. These all have their respective effects on the airflow, and interact in complex ways. 2.4.1.1 Differences in the Angles of the Diffuser The maximum angle of the diffuser channel’s upper boundary relative to the xy- plane is referred to as the expansion angle of the diffuser. This angle determines to a great extent how well the flow attaches. A typical value is around 15°. A substantially higher angle will result in separation of the flow from the inside of the diffuser, resulting in an increase in pressure [13]. 2.4.1.2 Differences in Area Ratio The area ratio of a diffuser is related to the angle of the diffuser and the length of the diffuser volume by simple geometry. Rar = Hr + Ld · sin(θ) Hr = Ainit flow Afinal flow , (2.16) where Rar denotes the area ratio, θ the angle of the diffuser, Hr the ride height, Ainit flow the inlet area of the diffuser, Afinal flow the rear outlet area of the diffuser, and Ld the length of the diffuser volume. Combining this equation with Equation (2.15) makes it clear that lower pressures can be obtained with longer diffusers and larger diffuser angles. However, a practical limit is reached when the area ratio approaches 1:5. At this point, the high pressure ambient air starts to interfere with and spoil the airflow, decreasing downforce [13]. 14 2. Theory 2.4.1.3 Implementation of Strakes Vertical panels in the diffuser volume, with the purpose of directing the airflow are referred to as strakes. Strakes lower the characteristic length of the flow, keeping the latter laminar and lowering the pressure. Other effects can be promoted by directing the airflow in different ways [3]. 2.4.1.4 Different Throats of the Diffuser The throat or kick of the diffuser is where the underfloor transitions to the diffuser ramp surface. The lowest pressure in the airstream is located in the throat, meaning that this area provides a non-negligible amount of downforce. The radius of this throat determines to some extent how the flow behaves inside the diffuser. A small radius will make the flow prone to separation downstream of the throat, but may also increase downforce by increasing underbody area [8]. 2.4.2 Venturi Vortices The performance of the diffuser is highly dependent on that the flow remains at- tached to the internal surface of the diffuser. As discussed, this stands in contrast to the measures taken to decrease pressure in and upstream of the diffuser. To keep the flow from separating, a vortex pair can be induced. The flow in such vorticies have a short characteristic length, and stay laminar to a greater degree. Methods to induce the desired vortices include: • Direction of airflow over a sharp edge, such as over the diffuser sidewall • Implementation of turning vanes, i.e. small winglets upstream of the diffuser, which give the flow closest to the car surface sideways momentum • Implementing strakes in the diffuser channel, which contain and keep the vor- tices undisturbed. Figure 2.7: 3D-schematic showing the typical vortex pair that form as high- pressure ambient air flows into the low-pressure cavity of the undertray diffuser (red arrows). Note the direction of the vortex rotation. The blue streamlines are grossly simplified. To some extent, the vortices also keep the high-pressure outside separated from the 15 2. Theory low-pressure inside of the diffuser. This effect decreases pressure upstream of the diffuser, thus increasing aerodynamic performance [3]. 16 3 Methods The methods used in the project are presented in detail in this section. The workflow for creating and simulating a diffuser is shown in Figure 3.1. Pre Processing • CAD design • Modeling of driving scenar- ios • Surface prepa- ration and mesh Simulation • Boundary con- ditions • Physics models • Convergence conditions Post Processing • Post script • Calculation of coefficients • Figures of flow quantities Figure 3.1: Workflow for the development of a diffuser. 3.1 CAD To enable simulations of the Formula Student car a 3D-model was required. CAD was used to design all the geometries of the aerodynamics package before it was imported and pre-processed in the CFD software Star-CCM+. In order to create these CAD models, the software Catia V5 was used. The project was provided with the CAD model of the CFS21 car, i.e. the latest model, from which the design work could be based upon. The aerodynamic parts of the CFS car are shown in Figure 3.2. Since the project aims to study how a new diffuser could increase the performance of the formula car, all parts that were not connected to the diffuser could remain the same. Overall, only the diffuser and the monocoque were affected (in terms of direct contact) by the diffuser replacement. The monocoque is the self-supporting chassis of the car, which can be visualized in grey in Figure 3.2. 17 3. Methods Figure 3.2: CAD model with the aerodynamic parts marked in different colors. Front wing (orange), deflector (red), monocoque (grey), side wing (green), rear wing (blue) and diffuser (yellow). 3.1.1 Current diffuser As a starting point, a simplified version of CFS’s latest diffuser was used. However, the current diffuser features an aggressive diffuser expansion angle which makes it difficult for the flow to remain attached. CFS21 has solved this by integrating the cooling package on the diffuser, and thus creating a suction normal to the flow di- rection. This results in a higher extraction of air and thus a higher downforce. The CFS21 diffuser is shown in Figure 1.1. 3.1.2 Design Methodology To create a aerodynamically robust and well performing diffuser a number of design iterations were made. The design methodology was based on a number of basic concepts. The concepts includes the effect of how early on the monocoque the diffuser can start as well as the the expansion angle of the diffuser. After these concepts had been tested and evaluated, more detailed components were added. These components were strakes and side floors. As previously mentioned, the basic ideas of CFS’s current diffuser were used as a starting point in the design development. Therefore, the first diffuser simulated had the same monocoque as CFS21. In order to start the design iterations with a basic geometry both side floors and strakes were not included. For the remaining design iterations it was chosen to redesign the monocoque. In that way, the diffuser could start earlier on the underside of the car, thus making the expansion of air more prolonged. It is considered an interesting aspect to study in order to find out if the downforce would increase. 18 3. Methods The starting point of the diffuser, i.e how far from the back of the monocoque the diffuser starts, was iterated. The two extreme cases of starting the diffuser as early as possible (500 mm in front of the monocoque rear wall) and as late as possible (coinciding with the monocoque rear wall) were tested. An alternative concept be- tween the extreme cases was also tested, where the diffuser instead started 300 mm in to the monocoque. All three designs are displayed in Figure 3.3. The 300 mm design was similar to the 500 mm case. Their difference being that the 300 mm starting point resulted in a smaller throat radius connecting the flat underfloor and the diffuser’s final angle of 15°. It was then possible to draw conclusions regarding which design showed the most potential and attachment of flow. Figure 3.3: Outlines of the three different diffuser designs when iterating the starting point of the diffuser. The 500 mm design is displayed as purple, the 300 mm design as red and the 0 mm design as blue. The angle between the flat bottom of the monocoque and the top of the diffuser was also iterated. The angles tested were 13°, 15°, 17° and 19° which are displayed in Figure 3.4. The project group chose to test four different angles due to time lim- itations, especially in terms of simulation time. 15° was estimated to be the angle corresponding to a reasonable movement of electronics in the rear of the current CFS car. The other three angles were chosen at a linear scale to see how both slightly higher and lower angles affect the diffuser. Figure 3.4: Outlines of the four different diffuser designs when iterating the angle between the flat bottom of the monocoque and the top of the diffuser. The 13° design i displayed as blue, the 15° design as purple, the 17° design as red and the 19° design as green. 19 3. Methods After these basic design concepts had been tested, strakes and side floors were added to the designs that had performed best so far. The idea behind adding strakes was to induce more vortices and the idea behind side floors was to enlarge the low pressure surface of the diffuser, both adding downforce. Strakes were added by copying the outer walls of the diffuser and placing the copies at a fourth of the diffuser’s width symmetrically around the middle line. The side floors were designed with the same width as the current CFS car’s side floors. See Figure 3.5 for details. Figure 3.5: Diffuser with both strakes (orange) and side floor (blue). 3.2 CFD As computer performance has increased over the recent decades, computational fluid dynamics have become a more powerful tool. This section contains the setup for the simulations. 3.2.1 Whole-car Versus Half-car Symmetry Simulation Previous CFS teams have mostly used a half-car symmetry model for the simulations in the design process. The near-symmetric geometry of a car was exploited to decrease the computational resources needed. This resulted in less computing time and more simulations. However, due to the asymmetric flow during cornering, a half-car symmetry simulation is not possible for all scenarios. Previous CFS teams have also noted discrepancies between the two simulation models’ results. This might be due to small asymmetries between the car’s left and right features and geometry. Another cause might be that the symmetry plane boundary conditions of the half-car symmetry simulation are over-simplified. To ensure comparable results for the different simulated scenarios, it was decided to only do whole-car simulations. This trade-off increases simulation run time, but provides greater confidence in the simulation results. 3.2.2 Physics Model All the physics models were taken from CFS’s previous simulations, and can be seen i Table 3.1. Star-CCM+ is set to use SST k − ω turbulence modeling because it 20 3. Methods uses the advantages of both k − ω turbulence modeling near the walls, and k − ε modeling in the far-field. The properties of each turbulence model are discussed in Sections 2.2.2 to 2.2.4. Table 3.1: The settings used in Star-CCM+. Settings All y+ Wall Treatment Cell Quality Remediation Constant Density Exact Wall Distance Gas: Air Gradients k − ω Turbulence Reynolds-Averaged Navier-Stokes Segregated Flow Solution Interpolation SST (Menter) k − ω Steady Three Dimensional Turbulent Wall Distance 3.2.3 Mesh The meshing procedure in Star-CCM+ is done with three operations. First a surface wrapper is executed over the car. Subsequently, the wrapper is subtracted from the wind tunnel. Lastly an automated mesh is run on the subtract. 3.2.3.1 Surface Wrapper A surface wrapper with a base and gap closure size of 32 mm and 5 mm respec- tively were applied on the car’s surface. The target surface size was 16 mm and the minimum surface size 2 mm. Contact prevention groups were created for the geometry so parts did not ”smudge” together. Contact prevention were done on the front wing, rear wing, monocoque, roll hoop, side diffusers and rear diffuser. 3.2.3.2 Volume Mesh The base size of the volume mesh was 32 mm, the target surface size 16 mm and the minimum surface size 2 mm. Trimmed cells were used so that cell size could be increased far from the car to lower the cell count. The mesh is illustrated in Figure 3.6. Where gradients in for example pressure and velocity were large, a finer mesh was created with volumetric controls. The volumetric controls were mainly inherited from CFS. The prism layer settings were inherited from CFS and consisted of 12 layers and an increase between layers of 1.2 with the all the layers’ thickness being 21 3. Methods (a) Mesh in xy-plane located z = 5 cm above ground (b) Mesh in xz-plane through the middle of the car (c) Mesh in xy-plane located z = 25 cm above ground for cornering. Figure 3.6: Volume mesh for the simulations. 10 mm. This gave good y+-values (see Section 2.1.6.2) that were below 5 for the inner prism layer as can be seen i Figure 3.7. The total amount of cells were around 52 million for the straight case and 53 million for the curved case. 22 3. Methods Figure 3.7: The Wall y+ for the inner prism layer around the car. 3.2.4 Boundaries The boundary conditions differ slightly depending on if a straight or a cornering simulation is too be performed. The boundaries for the case when that car is driving straight can be seen in Table 3.2. Table 3.2: The boundaries for a straight simulation. Boundary Boundary Condition Value Wind tunnel inlet Velocity inlet [40, 0, 0] km/h Wind tunnel outlet Pressure outlet 0 Pa Wind tunnel roof, walls Symmetry plane - Wind tunnel ground Moving wall [40, 0, 0] km/h Wheels Rotating wall 48.6 rad/s around wheel axis Car except wheels Wall 0 km/h The boundaries for the case when the car is cornering can be seen in Table 3.3. As can be seen, the velocity of the inlet and ground is 0, that is because parts of the system is in a rotating reference frame [14]. This reference frame contains the fluid volume and outer boundaries which rotates at 0.889 rad/s. This comes from dividing the speed of the car’s center of gravity, 40 km/h, with the corner radius 12.5 m. The car on the other hand is fixed at the same point with only the wheels spinning around their own axis. 23 3. Methods Table 3.3: The boundaries for a cornering simulation. For an accurate representa- tion the system has to be viewed in a rotating reference frame. The reference frame contains the fluid volume and outer boundaries which rotates while the car is fixed. Boundary Boundary Condition Value Wind tunnel inlet Velocity inlet [0, 0, 0] km/h Wind tunnel outlet Pressure outlet 0 Pa Wind tunnel roof, walls Symmetry plane - Wind tunnel ground Moving wall [0, 0, 0] km/h Wheels Rotating wall 48.6 rad/s around wheel axis Car except wheels Wall 0 km/h 3.2.5 Simulation The initial condition for the velocity of the air was set to [40, 0, 0] km/h. Because of the turbulent nature of this simulation, the residuals will never fully converge so the simulation is run for 3000 iterations. The mean values for coefficients from the last 500 iterations are saved. This is because the simulation’s fields always has some fluctuations. 3.2.6 Post Processing The post processing consisted mainly of two types of analyses: numeric values and visualized flow fields. The numeric values were obtained by using the built in fea- tures, monitors and reports, in Star-CCM+. A report had to be set up for each variable of interest. The monitor would then store values generated by a report, one set of values for every iteration of the simulation. These values was then be averaged over the last 500 iterations to get a representative value. The visualized flow fields were obtained by visualizing the solution of one variable on either the surface of the car or on a cross section plane of the simulated volume. These flow fields were not averaged but instead only contained data from the last iteration of each simulation. For the cornering scenario a field function and sum report was used to calculate some of the force and force coefficients. This was required since the surrounding air flow was curved and the direction of the force had to be calculated in the direction of the flow. The field function was therefore defined so that at every cell the force or force coefficient was computed in the same direction as the curved flow in the specific cell [15]. These values could then be summed up to get the total value of interest. This methodology was used to get the correct values and representations for the drag coefficient, the total pressure coefficient and the skin friction coefficient. 3.3 Choice of Simulated Driving Scenarios Historically, CFS has only simulated a neutral-attitude half-car driving straight ahead during the design process. Although some simulations of different driving 24 3. Methods scenarios have been done, these have not been taken into consideration in the de- sign process as they were in this project. The most accurate data on aerodynamic sensitivity would be acquired by simu- lating as many scenarios as possible. However there is a practical limitation from the computational resources available for the simulations. Hence a small number of common and critical maneuvers need to be identified and implemented into the simulations. Each whole-car simulation takes about 4 hours to run on the computa- tional cluster provided by Swedish National Infrastructure for Computing (SNIC). This corresponds to approximately 600 core-hours. Given the available resources for both this project of 10 000 core-hours/month and for future CFS teams of 20 000 core-hours/month, the optimum number of simulated scenarios for each design was assessed to be three. The aim of each simulation is to acquire data that describe the real-world per- formance of the vehicle. There is a number of different driving scenarios where aerodynamic downforce is critical. The most demanding maneuvers (driving scenar- ios) with the highest lateral and longitudinal accelerations, i.e. where downforce is the most needed, are braking and cornering [3]. This is supported by raw data col- lected by CFS during the 2016 Formula Student Germany Endurance competition. The data also suggests that the largest portion of the lap time is spent cornering, making this maneuver the most important for improved lap times. Furthermore, Figure 3.8 illustrates that the forces during braking are larger compared to acceler- ating forwards. This implies that downforce is more important during braking. Due to the mentioned resource and time restrictions within this project, the forwards- acceleration scenario was excluded. Only cornering and braking were simulated along with straight driving with a neutral attitude, the latter for reasons mentioned below. The magnitudes of certain parameters to implement, such as speed, corner radius and steering angles are based on simulations and real world data from previous CFS teams. In addition, the current CFS21 team has been involved in discussions and has been offering guidance. 25 3. Methods Figure 3.8: GPS data from a lap of the 2016 Formula Student Germany Endurance circuit from the CFS16 car. A histogram of longitudinal acceleration in the forward direction is illustrated. 3.3.1 Assertion of Reynolds-independence Simulation data from previous evaluation of the CFS20 car was provided for analysis. The data was analyzed as to establish that the flow is largely Re-independent, i.e. that the aerodynamic forces increase with the square of u∞. The data were fitted in MATLAB to a function of the following form: Faero = a · u2 ∞, (3.1) where a is an arbitrary constant. Data for 40, 60, 80, 100 and 120 km/h were avail- able. For all forces on separate aerodynamic components, the minimum coefficient of determination (R2) of the fit was 0.996. This was regarded to be a very good fit, proving that simulations with a single value of u∞ also are representative for different values in the relevant span. 3.3.2 Driving Straight Ahead Although downforce is of the least relevance when driving straight ahead at neutral attitude, as the car doesn’t accelerate in any direction, it was decided to include it in the list of scenarios. This decision was made to provide a basic understanding of the characteristics of each diffuser design. Additionally, this made comparisons with simulation data from previous diffuser designs possible. The ride height was set to 30 mm as recommended by the current CFS21 team. Contribution from aerodynamic load on the ride height has been shown to negligible and is therefore 26 3. Methods neglected. The car’s velocity was chosen as 40 km/h to be comparable to previous CFS aerodynamic simulations and is further discussed below. 3.3.3 Cornering Driving Scenario To simulate cornering as realistically as possible, real world GPS data was used to determine the most frequent corner radius and vehicle speed. The data used was acquired by CFS16 during the endurance race of the Formula Student Germany competition in 2016 and is presented in Figure 3.9. The most frequent corner radius was determined to be 12.5 m and corresponding speed 40 km/h. Figure 3.9: GPS data from a lap of the 2016 Formula Student Germany Endurance circuit from the CFS16 car. A typical 12.5 m radius corner where the average recorded speed is approximately 40 km/h is marked in the figure. While cornering the vehicle dynamics differ from when driving straight. Steering angles for the front wheels, body slip and a roll angle need to be accounted for. These angles are discussed in Section 2.3. The quantities of these angles can be found in table 3.4, note that the angles are given for a corner with radius 12.5 m and a speed of 40 km/h. The steering and body slip angles used were the same as the ones used in CFS20 and are based on simulations. The roll angle was acquired from the suspension group of the CFS21 team as 0.7◦/g, where g is the lateral acceleration and 1g ≈ 9.81 m/s2. For a 12.5 m radius corner and vehicle speed of 40 km/h this equates to almost exactly 0.7◦. Vehicle pitch was not considered as the scenario simulated does not include longitudinal acceleration. Furthermore, only left hand corners were simulated as CFS20 cornering results showed no significant difference between left and right cornering. 27 3. Methods Table 3.4: Angles for simulating cornering at a corner radius of 12.5 m. For details regarding the different parameters, see Section 2.3. Angle Value Inner steering angle 7.18◦ Outer steering angle 7.36◦ Body slip angle 3.5◦ Roll angle 0.7◦ 3.3.4 Braking Driving Scenario While braking, the vehicle rotates with a pitch angle around an axis called the pitch center. The pitch angle for hard braking can be approximated to when the front wing almost scrapes the ground which is approximately 1.0◦. This is based on observations from the CFS team. Although such steep angles are only achieved while braking from speeds exceeding 60 km/h, a speed of 40 km/h was chosen for this scenario. The flow has been shown to be Reynolds number-independent as previously described and the chosen speed of 40 km/h will therefore not influence the normalized results. This means that the resulting fields behave the same as they would at 60 km/h. The resulting forces can be multiplied by a factor of ( 60 40 )2 to yield the same results as a simulation at 60 km/h, since the forces scale with a factor of u2. Furthermore, the braking scenario is directly comparable to the straight and cornering scenarios if they are all simulated at the same speed, in this case 40 km/h, and the scaling of data and plots is then identical. 28 4 Results and Analysis In this chapter the results of this study are presented and analyzed. The objective of the study was to identify which driving scenarios to simulate, as well as investigate the car’s aerodynamic performance during these maneuvers with several different diffuser designs. This chapter will discuss both of these issues. Much attention will be dedicated to analysis of the flow patterns and their gener- ation of downforce. Due to the small number of data points, no extrapolation nor interpolation is done from the parameter sweeps. Instead, a more discussion-based presentation of results is favored. The main focus is on downforce generation as described in Section 1.4. Firstly, a general comparison of the three driving scenarios straight, braking and cornering is made. Design iterations are later compared where, to minimize the number of figures, only the most relevant results are presented. 4.1 General Differences for Driving Scenarios In this section general observations and results for the three driving scenarios: straight, braking and cornering are compared. The smallest diffuser design, with a starting point coinciding (0 mm) with the back of original flat-bottomed monocoque used in CFS20 and CFS21 (see Figure 3.3), is used as a baseline for this comparison. Although a specific design is used, most of the observations made apply to every design iteration in this project. Table 4.1: Values for CD, CL for different components and aero balance (rearwards) for different driving scenarios using the small diffuser starting at the end of the monocoque. M&D stands for Monocoque & Diffuser, FW for Front Wing, RW for Rear Wing and SW for Side Wing. Scenario CD CL CL M&D CL FW CL RW CL SW Aero balance Straight 1.433 3.612 0.435 1.352 1.032 0.862 50.06% Cornering 1.302 3.544 0.436 1.325 0.983 0.882 49.90% Braking 1.355 2.799 0.336 0.773 1.013 0.733 67.17% The largest simulated downforce is obtained in the straight driving scenario, with cornering not far below, as can be seen in table 4.1. The losses come from the front 29 4. Results and Analysis wing, where CL decreases from 1.352 to 1.325 (2 %) and the rear wing’s CL decrease from 1.032 to 0.983 (5 %). With such small changes, the aero balance is very similar for the two cases. It is important to note that not only CL M&D is affected by the different driving scenarios, but other aerodynamic components as well. The same is true when comparing different diffuser designs as seen later in Table 4.3. In the current example, the diffuser gains downforce while the car and the other aero com- ponents are losing, resulting in a worse overall package. Therefore, total downforce is the most relevant metric to analyze. When the car pitches during braking the downforce loss and consequently aero balance change is significantly greater. The largest contributor is the front wing which loses 0.580 CL, from 1.352 to 0.773 (43 %). As a result, the aero balance is greatly shifted rearwards. Furthermore, the side wing CL decreases from 0.862 to 0.733 (15 %) and M&D CL drops from 0.435 to 0.336 (22 %). The drag force coefficient CD follows the same pattern as CL and proves that larger downforce tends to result in increased drag force. Figure 4.1: Accumulated CL and CD while driving straight for the whole car with the small diffuser attached to the original CFS20 and CFS21 monocoque. The downforce CL and drag CD distribution of the car while driving straight is pre- sented in Figure 4.1 as accumulated plots. It is clear that the downforce distribution is not uniform. Instead distinct inclinations can be seen around the front wing at x = 0 − 0.5 m, the side wings & underfloor at x = 1.3 − 2.2 m and the rear wing & diffuser at x = 2.2 − 3 m. On the contrary, the drag force distribution is more uniformly distributed with the only large increases right behind the monocoque at x = 2.6 m and the rear wing flaps at x = 2.8 m. This is due to the large wake created as seen in Figure 4.3. 30 4. Results and Analysis Figure 4.2: Accumulated CL for the braking scenario compared to the straight scenario for a diffuser starting after the monocoque. The straight and braking scenarios are compared above in Figure 4.2 where their respective accumulated lift coefficients CL are presented. The performance of the front wing is severely impaired by the limited air flow under the front wing. The fact that the lines are approximately parallel downstream of the front wing suggests that the other parts of the aerodynamic package remain largely unaffected by the difference in pitch angle. However, another distinction is seen after the step in the underfloor, located around x = 1.5. The generated downforce is further decreased after this section, again suggesting that the air flow is reduced. Small pressure differences in the diffuser are not visible here, but are noted in Figure 4.5. 31 4. Results and Analysis 4.1.1 Differences in Velocity Magnitude (a) Straight driving scenario. (b) Braking scenario. (c) Cornering scenario. Figure 4.3: Velocity magnitude plots for driving straight, braking and cornering with the small 0 mm starting point baseline diffuser. The visualised xy-plane is located at z = 88 mm above ground. The air flow velocity magnitude of the three simulated driving scenarios is presented in Figure 4.3. The air flow velocity determines how much downforce is generated. This velocity is decreased between the front wheels while braking with a pitch angle, see Figure 4.3b, compared to driving straight, see Figure 4.3a. This explains the heavily reduced downforce while braking. The lowered front wing is the cause since the area beneath it is smaller, reducing the air flow. For further visual comparison, 32 4. Results and Analysis see Figure 4.4. Lower velocity in this area affects the other aerodynamic components further downstream, reducing the available air flow to produce downforce. However, the rear wing, which is located higher than every other aerodynamic device, is com- parably unaffected and only lost 1-2 % of its downforce. It is unclear why this loss occurs since the high velocity region beneath the wing is larger during braking in Figure 4.4, implying lower pressure as described by Bernoulli’s principle in Section 2.1.1. One explanation could be that the rear wing’s angle of attack is increased past the optimum, resulting in slight detachment of the flow. The air flow while cornering differs in many ways. Although the results only point to a small decrease in downforce, the air flow behaves differently. Combined conse- quences of the curved air flow and body slip makes the air interact with the car at different yaw angles. As a result, the velocity field is asymmetric. Figure 4.3c shows that the high velocity air is pushed to the left (bottom in the figure) diffuser wall, inducing a strong vortex in this region. (a) Straight driving scenario. (b) Braking scenario. Figure 4.4: Velocity magnitude plots for the air surrounding the car. The visual- ized xz-plane is located in the middle of the car at y = 0. Only the straight and braking scenarios are included. 33 4. Results and Analysis Only the straight and braking scenarios are shown in Figure 4.4. Visualizing the curved air flow during cornering in a flat plane would not provide any useful ob- servations. The previously mentioned reduction of flow beneath the front wing is once again apparent. In addition, it seems that separation occurs as indicated by the blue low velocity area. Lower velocity and more separation tends to result in lower downforce since the pressure difference between the top and bottom of the wing is reduced. Furthermore, the air flow velocity of the side wings and diffuser is decreased. The resulting pressure differences which lead to the downforce losses are presented in the next section. 34 4. Results and Analysis 4.1.2 Differences in Pressure Coefficient (a) Straight driving scenario. (b) Braking scenario. (c) Cornering scenario. Figure 4.5: Pressure coefficient on the underside surface of the car during the three driving scenarios straight, braking and cornering. 35 4. Results and Analysis The pressure coefficient on the car’s underside can be seen in Figure 4.5. In all scenarios there is a low pressure region under the front wing, under the side wings, behind the step in the underfloor of the monocoque and where the diffuser starts. The difference is that in the cornering scenario (Figure 4.5c), the low pressure area is bigger on the left hand side (the top of the figure) than the right hand side (the bottom of the figure). This means that the downforce is larger on the inside half of the car while cornering. The braking scenario (Figure 4.5b) has an area of high pressure at the start of the front wing and a smaller low pressure area. This is a consequence of the front wing separation and reduced air flow as mentioned in the last section. Furthermore, the low pressure region behind the monocoque step in the underfloor is narrower when compared to the straight scenario. The same is true for the low pressure area at the start of the diffuser and the side wings, further indicating that the air flow and downforce is restricted. (a) Straight driving scenario. (b) Braking scenario. Figure 4.6: Pressure coefficient of the air surrounding the car. The visualized xz-plane is located in the middle of the car at y = 0. Only the straight and braking scenarios are included. In Figure 4.6 the pressure coefficient can be seen in a plane through the middle of the car (y = 0) for the straight scenario (Figure 4.6a) and the braking scenario (Figure 36 4. Results and Analysis 4.6b). As previously mentioned, cornering is excluded since a straight plane would not produce any useful information from a curved air flow. It is noticeable when comparing the two scenarios that the braking scenario shows a smaller low pressure region under the front wing. This is again due to the reduced air flow resulting from the pitch angle. In addition, the low pressure area in the throat of the diffuser is smaller. The overall higher pressure underneath the car is generating less downforce compared to the straight scenario. 37 4. Results and Analysis 4.1.3 Differences in Skin Friction Coefficient (a) Straight driving scenario. (b) Braking scenario. (c) Cornering scenario. Figure 4.7: Skin friction coefficient on the car surface seen from below. The skin friction coefficient Cf on the surface of the car for the different driving scenarios can be seen in Figure 4.7. Skin friction correlates to attachment and sepa- 38 4. Results and Analysis ration. Greater skin friction (yellow in Figure 4.7) corresponds to more attached and energetic flow, whereas smaller skin friction values (blue in the figures) corresponds to more separated flow. A separation bubble is seen in the middle of the diffuser in the straight scenario in Figure 4.7a and the braking scenario in Figure 4.7b. If this separation bubble grows too large, stalling may occur resulting in sudden loss of downforce. The bubble seems larger in the straight compared to the braking scenario. The cornering scenario in Figure 4.7c seems more similar to the straight in this regard, but the separation bubble is twisted, following the curved air flow. The yellow lines in the diffusers are in the next section shown to be vortices. Vor- tices contributes with improved flow attachment, driving the air to the ceiling of the diffuser, minimizing the separation bubble. Increased skin friction coefficient corresponds to stronger vortices in this case. Separation differences are clear on the front wing surface. As described in previous sections, considerable separation occurs during braking as seen in Figure 4.7b when compared to the straight scenario seen in Figure 4.7a. Furthermore, the high skin friction area at the step in the monocoque underfloor is reduced while braking, explained by the reduced air flow. 39 4. Results and Analysis 4.1.4 Differences in Vorticity Magnitude (a) Straight driving scenario. (b) Braking scenario. (c) Cornering scenario. Figure 4.8: Vorticity magnitude for driving straight, braking and cornering. The visualized yz-plane is cutting through the diffuser 230 mm behind the monocoque. The vorticity magnitude in the diffuser in the different scenarios can be seen in Fig- ure 4.8. What can be seen from the straight scenario (Figure 4.8a) is that there is high vorticity near the diffuser walls. There is also some vorticity in the middle of the diffuser near the ceiling that originates from the turbulent separation bubble. 40 4. Results and Analysis The vortices in the braking scenario (Figure 4.8b) are not as prevalent as in the straight scenario because of the reduced air flow to the diffuser. As described previ- ously, this corresponds to a lower skin friction coefficient. In the case of cornering, it can be seen that the vortices are more asymmetric compared to when driving straight and braking. As can be seen in Figure 4.8c, a stronger vortex is generated in the left edge of the diffuser wall. This is expected since the car is simulated turning left and is correlated to the larger skin friction coefficient on the left side of the car, seen in the previous section. The flow that passes through the diffuser from the front wing and underneath the car, as well as from the side of the car, does not occur symmetrically. This explains the asymmetric formation of vortices when cornering. 4.2 Starting Point of the Diffuser First the start of the diffuser was investigated and the data from simulations when altering the starting point can be found in Table 4.2. The different designs, visual- ized in Figure 3.3, showed similar performance in regards to drag and lift coefficients. However a difference can be seen for the M&D contribution to the lift coefficient. The 300 mm design gave better M&D performance for the straight and braking sce- narios while losing performance during cornering. The aero balance also shifts the most while braking for the 300 mm design. This suggests the design is less stable and therefore not giving the desired robustness. Table 4.2: Values for CD, CL, CL M&D and aero balance (rearwards) for the different diffuser starting points at the driving scenarios straight, cornering and braking. Design Scenario CD CL CL M&D Aero balance Straight 1.424 3.639 0.483 50.36% 500 mm Cornering 1.408 3.542 0.417 49.62% Braking 1.342 2.855 0.361 67.61% Straight 1.414 3.661 0.522 50.44% 300 mm Cornering 1.410 3.503 0.392 49.32% Braking 1.342 2.838 0.404 68.64% Straight 1.433 3.661 0.435 50.05% 0 mm Cornering 1.302 3.544 0.436 49.90% Braking 1.355 2.799 0.336 67.17% 41 4. Results and Analysis (a) 500 mm, straight (b) 500 mm, cornering (c) 300 mm, straight (d) 300 mm, cornering (e) 0 mm, straight (f) 0 mm, cornering Figure 4.9: Skin friction for different diffuser starting points while driving straight and during cornering. Furthermore, the skin friction coefficient showed a more noticeable difference be- tween all three designs. As can be seen in Figure 4.9, the biggest difference is inside the diffuser. Both the 300 mm and 0 mm designs show an increase in skin friction at the starting point of the diffuser followed by a region in the middle of the diffuser with low skin friction which can be seen in Figure 4.9c and 4.9e. The 500 mm design however created a smoother gradient in between these regions shown in Figure 4.9a which could indicate that the 500 mm design is more aerodynamically robust than the 300 mm and 0 mm case, being less prone to seperation. In the cornering scenarios the same aforementioned phenomenon is displayed but 42 4. Results and Analysis with the 0 mm design showing the smallest region of low skin friction in the diffuser. This was likely a result of having a smaller volume of expansion which means the air passing through the diffuser did not need to expand as much. The smaller expansion causes the diffuser to pull less air from the underfloor of the monocoque which can be seen in Figure 4.9. The underfloor of the monocoque shows better attachment for the 500 mm and 300 mm design compared to the 0 mm design. All things considered, the 500 mm design was chosen to carry forward. Firstly, all three designs showed similar performance but with the 500 mm and 0 mm designs showing the most robust results. Secondly the 500 mm design had a larger volume than the 0 mm design which was deemed to be useful for further development. 4.3 Expansion Angle of the Diffuser Different expansion angles of the diffuser were tested with a starting point of 500 mm from the back of the monocoque, see Figure 3.4 for visualization. The data from those simulations are found in Table 4.3. Figure 4.10 shows the skin friction for the different angles of the diffuser. As the angle increases it can be seen that the transition to the low skin friction region in the middle of the diffuser becomes more aggressive meaning that the flow detaches more drastically. The skin friction for the cornering case can be seen in Figure 4.11. Table 4.3: Values for CD, CL, CL M&D and aero balance (rearwards) for different diffuser expansion angles at the driving scenarios straight, cornering and braking. Design Scenario CD CL CL M&D Aero balance 13° Straight 1.419 3.729 0.515 49.68% Cornering 1.380 3.603 0.470 49.38% 15° Straight 1.424 3.640 0.483 50.36% Cornering 1.408 3.542 0.417 49.62% 17° Straight 1.417 3.713 0.508 49.75% Cornering 1.408 3.539 0.427 50.30% 19° Straight 1.402 3.663 0.497 49.35% Cornering 1.417 3.567 0.418 49.54% It is difficult to draw far-reaching conclusions from the few data points presented in the table above. The data correlating forces to the diffuser expansion angle are not monotonous, and the data points are too distantly spaced to see at which angles certain effects are introduced. However, it is evident from the final accumulated value of CD, that the 13◦ diffuser is best-performing of the tested. As previously discussed in Section 2.4.1.2 this is to some extent expected due to the area ratio being around 1:5 for the 13◦ diffuser. 43 4. Results and Analysis (a) Diffuser with an angle of 13◦. (b) Diffuser with an angle of 15◦. (c) Diffuser with an angle of 17◦. (d) Diffuser with an angle of 19◦. Figure 4.10: Skin friction while driving straight for diffusers with different angles. In Figure 4.10, a correlation between the area ratio and the robustness of the vortices and overall flow in the diffuser can be identified. As seen in Figure 4.10a, the flow in a low-expansion-angle diffuser is smooth and well-attached along the length of the diffuser. An uniform low-pressure area has formed. In Figure 4.10b, the steeper angle produces vortices which are asymmetrical and detach from the surface. An irregular low-pressure area has formed, however this is difficult to distinguish from the artifacts seen in the data. As the expansion angle increases further in Figure 4.10c and 4.10d, the flow becomes more turbulent. The vortices detach from the surface and a separation bubble has formed. The low pressure allows for backflow into the diffuser, disturbing the airstream. 44 4. Results and Analysis (a) Diffuser with an expansion angle of 13◦. (b) Diffuser with an expansion angle of 15◦. (c) Diffuser with an expansion angle of 17◦. (d) Diffuser with an expansion angle of 19◦. Figure 4.11: Skin friction while cornering for diffusers with different expansion angles. Note the differences in Venturi vortex generation between the different cases. It can be seen in Figure 4.11 that the flows follow the same patterns as in the straight driving scenarios. Note the robust right vortex in the 13° and 19° diffuser, and its breakdown in the 15° and 17° diffuser. However, as is evident by the sharp borders between the vortices, flow separation has occured in the 19° diffuser. Also note the lack of the left vortex. 45 4. Results and Analysis (a) Relative difference in accumulated lift coefficient for the straight driving scenario. (b) Relative difference in accumulated lift coefficient for the curved driving scenario. Figure 4.12: Difference in accumulated lift coefficient CL, relative to the 19◦ dif- fuser for the straight and curved driving scenarios. A higher CL implies a greater downforce. The accumulated CL along the length of the car clearly shows the impact of changes to the design. In Figure 4.12, the non-monotonous correlation between the diffuser expansion angle and downforce is evident. It is also clear that changes to the dif- fuser affect the performance of the whole underbody of the car. The case of the 17° diffuser shows that good performance for straight driving doesn’t imply the same for the cornering case. Since the 13° diffuser outperforms the other diffuser designs in terms of CL in both scenarios, its flow seems to be more controlled and the separation bubble is less apparent, it is chosen for further development. The 19° diffuser doesn’t differ much between the scenarios, in terms of CL, and it has the second greatest CL in cornering. It also has the largest potential in terms of area expansion and is therefore chosen 46 4. Results and Analysis for further development. 4.4 Implementation of Strakes Continuing the design process the two diffuser angles showing best performance, 13◦ and 19◦, were developed further by introducing strakes. This was also done to analyze the impact of strakes on the two extreme cases in terms of angle. The results can be found in Table 4.4. Only the straight and cornering scenarios were simulated due to resource restrictions. Table 4.4: Values for CD, CL, CL M&D and Aero balance (rearwards) for 13◦ and 19◦ diffusers with and without strakes. Design Scenario CD CL CL M&D Aero balance 13° Straight 1.419 3.729 0.515 49.68 % Cornering 1.380 3.603 0.470 49.38 % 13° strakes Straight 1.404 3.747 0.565 50.08 % Cornering 1.316 3.562 0.461 49.00 % 19° Straight 1.402 3.663 0.497 49.35 % Cornering 1.417 3.567 0.418 49.54 % 19° strakes Straight 1.403 3.687 0.528 50.14 % Cornering 1.312 3.610 0.460 49.44 % It can be observed that the 13° diffuser is generating more total downforce compared to the 19° diffuser in the straight scenario, regardless of strakes. In the cornering scenario the downforce is largest for the 19° diffuser with strakes. Adding strakes to the 13° diffuser increases the downforce in the straight scenario but decreases it while cornering. The 19° diffuser produces more downforce with strakes in both scenarios. The aero balance moves forward slightly while cornering for all designs. Additionally, CD is reduced during cornering if strakes are added. Overall, strakes seem to improve the performance in most cases. 47 4. Results and Analysis (a) 13° without strakes (b) 13° with strakes (c) 19° without strakes (d) 19° with strakes Figure 4.13: Vorticity magnitude while driving straight for the 13◦ and 19◦ diffusers with and without strakes. The visualized yz-plane cuts through the diffuser 27 mm behind the monocoque. Vorticity magnitude plots are presented in Figure 4.13. The diffusers without strakes induce two vortices aligned with the outer walls. Implementing strakes induce four vortices, adding two within the strake walls. These help to reduce the size of the separation bubble usually formed in the middle. Furthermore, the pressure drops where the vortices are located. Vortices rotate at high velocity which in theory should increase the downforce. This pressure difference is apparent in Figure 4.14. 48 4. Results and Analysis (a) 13° without strakes (b) 13° with strakes (c) 19° without strakes (d) 19° with strakes Figure 4.14: Pressure coefficient on the underside surface of the car while driving straight. The four different diffuser designs with expansion angles 13◦ and 19◦, with and without strakes, are presented. The pressure coefficient underneath the car is presented in Figure 4.14. As previously discussed, low pressure regions are apparent where the vortices are located for both diffuser angles. The pressure in the area between the strakes is lower when compared to designs without strakes, i.e generating more downforce. Table 4.4 reinforces this observation since CL for M&D is significantly increased for both angles with strakes in the straight scenario. 49 4. Results and Analysis (a) 13° without strakes (b) 13° with strakes (c) 19° without strakes (d) 19° with strakes Figure 4.15: Vorticity magnitude while cornering for the four diffusers with 13◦ and 19◦ angles with and without strakes. The visualized yz-plane cuts through the diffuser 27 mm behind the monocoque. Vorticity magnitude while cornering for the four cases is presented in Figure 4.15. Similar to the straight scenario, strakes influence the number of induced vortices in both the 13° and 19° diffusers. Two vortices are created in the left most channel, creating two low pressure stripes. Another two can be seen on the right side. These appear to break up, a phenomenon also observed without strakes. However, the separation bubble size is still reduced which is seen in Figure 4.16. 50 4. Results and Analysis (a) 13° without strakes (b) 13° with strakes (c) 19° with strakes (d) 19° without strakes Figure 4.16: Skin friction on the underside of the car. The four different diffuser designs with expansion angles 13◦ and 19◦, with and without strakes, are presented. The skin friction coefficient during cornering is presented in Figure 4.16. It is clear that the strakes influence the flow pattern considerably. The main separation bubble is reduced in size due to the new vortices, especially for the 13° diffuser. Even though this particular diffuser does not perform the best in the cornering scenario, the flow seems to be the best controlled with the most attachment. This should mean that more performance is attainable with continued design development, which is investigated in the next section. 4.5 Implementation of Side Floors Data from simulations with side floors added to the diffuser can be seen in Table 4.5. The designs that proved to perform better in the previous section, i.e. 13° and 19° with strakes, were also those who got further developed by adding side floors. By adding side floors CL and CL M&D increased in all cases except in the braking scenario for the 19° diffuser. The 13° diffuser has the overall best performance in the study in all scenarios. The only exception is in cornering where CL is on par with the 19° diffuser with side floors. Additionally, the CD performance of the 13° is the best during cornering and on par with the best diffusers in other scenarios. 51 4. Results and Analysis Table 4.5: Values for CD, CL, CL M&D and aero balance (rearwards) for 13° and 19° diffusers with and without side floors. Design Scenario CD CL CL M&D Aero balance 13° strakes Straight 1.404 3.747 0.565 50.08% Cornering 1.316 3.562 0.461 48.10% Braking 1.355 2.894 0.377 68.12% 13° strakes Straight 1.409 3.854 0.647 50.93% + side floors Cornering 1.291 3.619 0.517 49.83% Braking 1.332 3.005 0.487 66.90% 19° strakes Straight 1.403 3.687 0.528 50.14% Cornering 1.312 3.610 0.460 49.44% Braking 1.352 2.983 0.417 67.95% 19° strakes Straight 1.405 3.767 0.583 50.