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Aerodynamic Vehicle Design and Analysis
Thesis · November 2016
DOI: 10.13140/RG.2.2.31696.10244
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School of Engineering
Blackpool and the Fylde College
Foundation Degree in Motorsport Engineering
Introduction to Aerodynamics
Aerodynamic Vehicle Design and Analysis
22/04/2016
Jason Moffat
30112030
Wayne Gater
School of Engineering
Foundation Degree Motorsport Engineering
Jason Moffat
30112030
Table of Contents
Introduction
Page 3
Aerodynamic Basics
Page 3
BMW Z4
Page 4
Concept Modelling
Page 5
Initial Testing
Page 6
Addition of Front Splitter, Side Skirts & Rear Diffuser
Page 8
Addition of Rear Wing
Page 11
Final Concept
Page 13
Vehicle Dynamics
Page 14
Mathematical Validation
Page 15
Conclusion
Page 16
Future Work Recommendations
Page 17
References
Page 18
Appendices
Page 19
Page 2 of 25
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Introduction
In this project it necessary to build a 3D surface model of a BMW Z4 concept car using industry
standard aerodynamic CAE (Computer Aided Engineering) software then import the model CFD
(Computational fluid dynamics) software. Using the CFD software an appropriate simulation will be
created for accurate post processing. Using the results gathered from the CFD simulation a critical
analysis will be made on the vehicles drag, lift and flow characteristics under, over and around the
vehicle. After analysing the data from the CFD simulations, improvements will be made to the model
to improve downforce while minimise the increase in drag.
Basics of Aerodynamics
Aerodynamics is the study of the interaction between moving bodies and the fluids around, under and
though them. Aerodynamics first came from aeronautical engineers studying aircraft wing design for
aircrafts that fly within the earth’s atmosphere. Aerodynamics are used in designing many different
things including building design, bridge design and motorsports/automotive vehicle design and many
more.
(Jim Lucas, 2014)
Drag
Aerodynamic drag is the force opposing the vehicles direction of movement. (See figure 1) The main
contributor to vehicle drag is the high pressure acting on the front of the vehicle, surface friction and
the relatively negative pressure left behind the vehicle.
(Jim Lucas, 2014)
The formula to mathematically calculate aerodynamic drag is as followed.
𝐷 = 𝑐𝐷 1⁄2 𝜌𝑣 2 𝐴
D = Drag Force N
CD = Drag Coefficient
½ = Mathematical Constants
ρ = Air Density Kg/m3
V2 = Speed m/s
A = Frontal Area m2
Figure 1 Lift & Drag Direction (NASA, 2014)
(Joseph Katz, 2006)
Page 3 of 25
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Lift
Lift is force acting perpendicular to the motion of the vehicle. It is essential for aircrafts to create
positive lift to fly, this is unwanted in motorsports. In motorsports negative lift it sought after to force
the vehicle into the ground, this force acting on the vehicle helps increase vehicle grip which leads to
faster cornering speed. The formula to mathematically calculate aerodynamic lift is similar to the drag
formula although the drag coefficient is replaced with a lift coefficient and is as followed.
𝐿 = 𝑐𝐿 1⁄2 𝜌𝑣 2 𝐴
L = Lift N
CL = Lift Coefficient
A = Frontal Area (Vehicle) Plan Area (Aerofoil) m2
(NASA, 2016) (Joseph Katz, 2006)
Downforce
Downforce is a motorsports/automotive term that applies to negative lift. This is the force that pushes
a vehicle into the ground this is a sought after affect in the motorsport industry to increase cornering
speeds.
Lift and drag coefficients
Lift and drag coefficients are a number given to a model and is affected by from vehicle shape,
surface friction (drag) and angle of attack (lift).
𝐷𝑟𝑎𝑔 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 =
𝐿𝑖𝑓𝑡 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 =
𝐷𝑟𝑎𝑔
1⁄ 𝜌𝑣 2 𝐴
2
𝐿𝑖𝑓𝑡
1⁄ 𝜌𝑣 2 𝐴
2
(Joseph Katz, 2006)
BMW Z4
The BMW Z4 is a rear wheel drive sports car, with a 3 litre straight six engine mounted at the front. It
has a wheel base of 2497mm and an overall length of 4091mm, width of 1781mm and a height of
1268mm. The Z4 weighs 1395Kg and has a 50:50 weight distribution. The quoted drag coefficient of
the Z4 is 0.34 and has a frontal area of 1.91m 2 given it a drag coefficient/area ratio of 0.65.
(Car Foilo, 2006)
Page 4 of 25
Figure 2 BMW Z4 (BMW, 2016)
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Concept Modelling
A concept model was designed using the basic dimension and canvas images of the BMW Z4. The
software chosen for this was Autodesk Alias SpeedForm 2016. Autodesk Alias is an industry standard
surface modelling package, Alias is a great visualization tool for automotive design and has powerful
rendering features.
(Autodesk, 2016)
See figure 3 below for an image of the Z4 concept during the modelling stage. During this design
process a symmetry plane was used (Green line though the vehicle in figure 3), this was used to help
keep the vehicle symmetrical, any modifications made to one side would automatically update on the
other side. See appendix 1 for more images of the Z4 concept during the design process.
Figure 3 Z4 Concept Design Stage
See figure 4 below for a rendered image of the BMW Z4 concept design rendered in Alias SpeedForm
with added wheels designed in Autodesk Inventor. See appendix 2 for more rendered images of the
conecept.
Figure 4 Z4 Concept Render image
Page 5 of 25
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Initial Testing
Initial testing was carried out on the Z4 concept vehicle using CD-Adapco’s Star CCM+ CFD
(Computational Fluid Dynamics) Software. Star CCM+ is a powerful CFD solver that can handle
multiple physics and complex geometries, as well as being a CFD solver Star CCM+ solve for heat
transfer and stress.
(CD-Adapco, 2016)
All tests were carried out at 120kph, and in a wind tunnel measuring 65m in length, 10m wide and
10m high. The air density used for testing is 1.2255kg/m 3 which is the air density at sea level at 10°C.
(Richard Shelquist, 2016)
Analysing Results
The result from initial testing are as followed,
𝑊𝑖𝑒𝑔ℎ𝑡 𝐹𝑜𝑟𝑐𝑒 (𝐾𝑔) =
𝐹𝑜𝑟𝑐𝑒
9.81 (𝐺𝑟𝑎𝑣𝑖𝑡𝑦)
Drag
506.33 N (51.61Kg)
Lift
314.48 N (32.06Kg)
Frontal Area
1.74 m
Drag Coefficient
0.43
Lift Coefficient
0.27
Drag/Lift Ratio
1.61
Lift/Drag Ratio
0.62
After analysing the results from initial testing its shown that the Z4 concept is actually creating lift, this
is unwanted in a sports type vehicle, this could lead to undesirable handling characteristics e.g.
under/over steer. When analysing the vector scene (see figure 5) it is clearly visable what the main
contributers to the vehices lift are. The first being the large amount slow velocity air traveling under
the vehicle, this low velocity will be creating a high pressure area which will try to force the vehicle
upwards. The second biggest area creating lift is the high velocity air passing over the roof of the
vehicle, this high velocity creates a low pressure passing over the vehicle, the low pressure area pulls
the car upwards creating lift.
Flow Separation
Figure 5 Initial Testing Vector Scene
Page 6 of 25
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Figure 6 Initial Testing Resampled Volume Scene
After further post processing its clear what the main areas that are creating drag are. The biggest
contributor to the vehicles drag is the wake, a large, slow speed turbulent area of air behind the
vehicle trying to pull the vehicle back this can be seen in the vector scene and the resampled volume
scene. (See figure 5 & 6) Another area creating vehicle drag is the high pressure zone on the front of
the vehicle this can be seen in the scalar scene (see figure 7). The pressure acting on the front of the
vehicle tries to push the vehicle in the opposing direction to the vehicles movement.
Figure 7 Initial Testing Scalar Scene
Another part creating drag is the flow separation areas at the bottom of the windscreen and over the
rear window these are highlighted in the vector scene. (See figure 5) The flow separation areas create
turbulence on the surface of the vehicle, also within these separation areas the velocity of the air
almost becomes zero. The final main area to cause drag are the vehicles wheel arches, the wheel
arches reduce the velocity of the air creating high pressure zone. These high pressure zones try
pushing the vehicle in the direction opposing its motion in the same way it occurs on the front of the
vehicle. This can be seen in the resampled volume scene. (See figure 6) For more images from initial
testing see appendix 3.
Page 7 of 25
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Addition of Front Splitter, Side Skirts & Rear Diffuser
To create more downforce from the underbody design of the vehicle the amout of air allowed to travel
under the vehicle must be reduced and the velocity of the air that is allowed to travel under the vehicle
needs to be increased which will create a low pressure area which will pull the car into the ground. In
an attempt to increase vehicle downforce a front splitter, side skirts and rear diffuser where designed
and added to the Z4 concept.
Front Splitter
A front splitter consists of a parallel extension attached at the bottom of the front bumper. The way the
front splitter creates downforce is by creating a high pressure area above the splitter and low pressure
below, this high pressure is drawn to the low pressure forcing the front end of the vehicle into the
ground.
(Formula 1, 2016)
Side Skirts
Side skirts are used to minimise the clearance between the vehicle body and the ground at the side of
the vehicle. The reason for side skirts is to help aerodynamic downforce this is done by reducing the
amount of high pressure from around the vehicle being drawn to the low pressure area under the
vehicle. The effectiveness of the side skirts depend on the clearance from the ground, 2cm or less is
best practice anything above this any bigger ground clearance diminishes quickly.
(Formula 1, 2016)
Diffusers
Diffusers are designed to increase the volume at the rear of the under body of the vehicle, this
increase in volume creates a void which needs to be filled this increases the velocity of the air
traveling at the rear of the vehicle. This high velocity air will create a low pressure area which
increases the downforce.
(Racecar Engineering, 2016)
There are different types of diffusers that very for the normal design (see figure 8). One of the designs
is a blown diffuser, this design uses the high velocity gases from the exhaust to help draw the air from
under the vehicle. The downside to this design is that it can increase the size of the wake behind the
vehicle.
Figure 8 Diffuser Design
Page 8 of 25
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Modelling of the Front Splitter, Side Skirts and Rear Diffuser
To model the front splitter, side skirts and rear diffuser additions Autodesk alias this can be seen in
figure 9 the red wire frame shows the additions added to the initial design. The chosen diffuser design
was a blown diffuser the reason this was chosen was because this design can have the biggest
increase in vehicle downforce is correctly designed.
Figure 9 Wire frame model of Splitter, Side Skirts & Blown Diffuser
Results from the addition of front splitter, side skirts & rear diffuser
The results from the second test are in the table below,
Drag
Lift
Frontal Area
Drag Coefficient
Lift Coefficient
Drag/Lift Ratio
Lift/Drag Ratio
778.15 N (97.32Kg)
-410.82 N (-41.87Kg)
1.88 m
0.61
-0.32
-1.89
-0.53
After anaylising the results with the additional componets added, there is a noticable increase in
vehicle downforce but also an increase in vehicle drag. One area where downforce was created is at
the front splitter as you can see in figure 10 that there much higher pressure acting on the top of the
splitter compared to the low pressure under the splitter therefore creating downforce.
Figure 10 Scalar Scene Showing Pressure Between -2500 & 600 Pa on a the XZ Plane
Page 9 of 25
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In figure 10 it is visible the reduced ground clearance compared to the initial test this reduced ground
clearance which consequently reduces the air allowed to flow under the vehicle, the less amount of air
allowed under the vehicle the more down force can be created. In figure 11 you can see the air
velocity tremendously increased around the blow diffuser where the high velocity exhaust gases draw
the air around. The outer diffuser channels are not working affectively as the centre channels, this is
because there is no exhaust gases to help increase the flow. These outer channels could be
improved by reducing the steepness of the angle the diffuser was designed at. Another area of
interest is the effect the vehicles wheels and tyres have on the velocity behind them creating very low
velocity areas.
Slow Velocity
Figure 11 Velocity Scalar Applied to the Vehicles Surface
As you can see in figure 12 the side skirt design has
effectively stopped the slow turbulent air flow from the
side of the vehicle bleeding under the vehicle and
disturbing the underbody flow. After further post
processing it is apparent there has been an increase in
the vehicle drag and drag coefficient the main
contributors to this are the increase in frontal area and
also the increase in the vehicles wake. For further
images from post processing for the added front splitter,
side skirts and rear diffuser see appendices 4.
Figure 12 streamlines down the Side of the Z4 Concept
Figure 13 Vector Scene Showing Velocity on Z4 with Additional Components
Page 10 of 25
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Addition of Rear Wing
To further increase the vehicle downforce a rear wing was designed and attached to the rear of the
vehicle. The rear wing design uses an inverted aerofoil shape taken from aircraft creating lift when the
aerofoil is inverted this will create downforce. The design will utilize a Gurney flap attached to the top
side of the trailing edge of the aerofoil. The gurney flap helps to slow down the air traveling over the
top of the aerofoil therefore increasing the pressure needed to create higher downforce.
The AOA (angle of attack) relates to the angle of the chord line of an aerofoil in relation to the
direction of motion the vehicle takes. The AOA has a great effect of the amount of drag and lift
created by an aerofoil, greater the AOA the greater the drag and downforce. After a certain AOA the
aerofoils lift will start to decrease this is called aerodynamic stall. To work out the optimal AOA for the
designed aerofoil simulations where carried out on every AOA until stall occurred the result are below.
(See figure 14)
After analyising the AOA data it can be seen
that the stall point ouccurs at 21°. The choosen
angle for the aerofoil on the Z4 concept was 5°
this was choosen because it gives the best
drag to lift ratio at -0.099.
Now that the optimal angle of attack has been
chosen the aerofoil now has to be positioned in
an effective area, when analysing figure 13 it is
noticeable that the aerofoil position has to
extend higher than the slow velocity air
traveling down and over the rear of the car.
After the AOA and position has been decided
the end plates and aerofoil legs could be
designed see figure 15 for the finished rear
wing design.
Figure 14 Angle of Attack Data
Figure 15 Aerofoil Design
Page 11 of 25
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Results from the addition of the rear wing
The results from the addition of the rear wing are as followed,
Drag
Lift
Frontal Area
Drag Coefficient
Lift Coefficient
Drag/Lift Ratio
Lift/Drag
840.65 N (82.02Kg)
-1035.86 N (-105.59Kg)
1.92 m
0.64
-0.79
-0.81
-1.23
Figure 17 Vector Scene Showing the Velocity with the Rear Wing Addition
After analysing the results, it is clear there is a dramatic increase in downforce created from the rear
wing. In figure 16 you can see the rear wing working well creating a high velocity area under the wing
and a much slower velocity over the wing. The high velocity under the rear wing creates low pressure
which draws the high pressure created from the low velocity speed above the wing creating
downforce. After further post processing it is visible there is a 62.5 N of drag increase this is due to
two factors first is the increase in frontal area but the main factor is the increase in the vehicles wake
this can be seen when comparing figure 13 and 16, it’s visible that the slow turbulent area is much
larger. After analysing figure 17 it is noticeable that the rear wing support arms are having an effect on
the drag creating low velocity air behind them. For more post processing images from the rear wing
additions see appendices 5.
Figure 16 Resampled Volume with Rear Wing Addition
Page 12 of 25
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Final Concept
See figure 18 and 19 for a rendered image of the final Z4 concept design, the software used to render
the image is Alias SpeedForm. For more images of the final render see appendices 6.
Figure 18 Side Render of Z4 Concept
Figure 19 Rear Diffuser Rendered Images
Page 13 of 25
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Vehicle Dynamics
Due to the aerodynamic modification created for Z4 concept the vehicle dynamics have changed
these could affect the handling characteristics and the weight distribution. The vehicles as standard
had a 50/50 weight distribution while the vehicle is static. To work out the weight force of the vehicle
at the simulation speed of 120 kph the lift force figure must be subtracted from the vehicles static
weight the formula for this is below.
𝑊𝑖𝑒𝑔ℎ𝑡 𝐹𝑜𝑟𝑐𝑒 (𝐾𝑔) = 𝑆𝑡𝑎𝑡𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 (𝐾𝑔) − 𝐿𝑖𝑓𝑡𝐹𝑜𝑟𝑐𝑒 (𝐾𝑔)
Initial Test
1362.94𝐾𝑔 = 1395𝐾𝑔 − 32.06𝐾𝑔
Addition of Front Splitter, Side Skirts & Rear Diffuser
1436.87𝐾𝑔 = 1395𝐾𝑔 − −41.87Kg
Addition of Rear Wing
1500.59𝐾𝑔 = 1395𝐾𝑔 − −105.59Kg
On the final simulation the new vehicle weight distribution can be calculated with the addition of the
rear wing, the addition of the rear wing will increase the weight over the vehicles rear end. To
calculate the new weight distribution, it’s first needed to calculate the weight under each axle before
the addition of the rear wing.
𝑊𝑖𝑒𝑔ℎ𝑡 𝐹𝑜𝑟𝑐𝑒 𝑂𝑣𝑒𝑟 𝐴𝑥𝑙𝑒 (𝐾𝑔) = 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑊𝑖𝑒𝑔ℎ𝑡 𝐹𝑜𝑟𝑐𝑒 (𝐾𝑔) × 𝑊𝑒𝑖𝑔ℎ𝑡 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑂𝑣𝑒𝑟 𝐶ℎ𝑜𝑜𝑠𝑒𝑛 𝐴𝑥𝑙𝑒
Front Weight = 718.435Kg
Rear Weight = 718.435Kg
To calculate the new weight distribution must add additional weight force from the rear wing to the
rear weight. The force weight created from the rear wing is 63.72Kg.
𝐹𝑟𝑜𝑛𝑡 𝑊𝑒𝑖𝑔ℎ𝑡 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 = 718.435 ÷ 1500.59 = 0.48
𝑅𝑒𝑎𝑟 𝑤𝑒𝑖𝑔ℎ𝑡 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 = (718.435 + 63.72) ÷ 1500.59 = 0.52
The Z4 Concept with all additions has a new weight distribution 48/52 bias to the front of the vehicle.
This new weight distribution could affect the vehicles handling, depending on other vehicle set up
could make the vehicle under steer.
Page 14 of 25
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Mathematical Validation
In this section is mathematical validation to back up the CFD simulation software.
Using the simulation from the additional rear wing the lift and drag will be calculated.
Lift
𝐿 = 𝑐𝐿 1⁄2 𝜌𝑣 2 𝐴
L = Lift (N)
CL = Lift Coefficient
½ = Mathematical Constant
𝜌 = Density (Kg/m3)
V2 = Velocity (m/s)
A = Area (m)
𝐿 = −0.79 × 0.5 × 1.2255 × 33.332 × 1.92 = −1032.48 𝑁
Drag
𝐷 = 𝑐𝐷 1⁄2 𝜌𝑣 2 𝐴
D = Drag (N)
CD = Drag Coefficient
½ = Mathematical Constant
𝜌 = Density (Kg/m3)
V2 = Velocity (m/s)
A = Area (m)
𝐷 = 0.64 × 0.5 × 1.2255 × 33.332 × 1.92 = 836.44 𝑁
The mathematical validation is very close to the simulation figures, the reason for slight discrepancies
is down to rounding of the figures.
Page 15 of 25
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Conclusion
To conclude this project basic aerodynamics principals have been researched thoroughly and applied
to CFD simulations data. A concept vehicle was created using industry standard surface modelling
software, and then successfully modified to increase vehicle downforce. A detailed analysis of the
CFD simulations where done and a critical analysis of the model where made.
After analysing all 3 simulations there had been an increase in downforce of 1350.34 N and an
increase in drag of 334.32 N. This increase in downforce and drag will increase fuel consumption by
increasing the drag coefficient and the rolling resistance. For motorsports application this increase in
fuel consumption is irrelative when downforce is the most important.
From the three simulations it’s clear that the final model created the highest downforce figure but also
the highest drag. Between all three simulations, simulation 2 (addition of front splitter, side skirts and
rear diffuser) has the best drag lift ratio.
Due to previous experience with the software packages used there was no potential problems that
occurred and with further development, there is possibilities of potential manufacture problems.
Simulation
Drag
Lift
Cd
Cl
314.48
Frontal
Area
1.74
1
506.33
2
3
0.27
Drag/Lift
Ratio
1.610054693
Lift/Drag
Ratio
0.621096913
0.43
778.15
-410.82
1.88
0.61
-0.32
-1.894138552
-0.527944484
840.65
-1035.86
1.92
0.64
-0.79
-0.811547893
-1.232213168
Lift & Drag
1000
FOCE (N)
500
0
-500
-1000
-1500
1
2
3
Drag
506.33
778.15
840.65
Lift
314.48
-410.82
-1035.86
SIMULATION
Drag
Lift
Figure 20 Lift & Drag Graph
Page 16 of 25
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Future Work & Recommendations
Future work would include trying to optimise the current additional components this would include
lower the angle of the outer diffuser channel to reduce the flow separation with these channels.
Another area that would be included in future work would be modification the wheel arches to allow
air to flow more freely out of them which would reduce the pressure within them, the way this could be
done is by designing vents to release and redirect the air flow smoothly around the vehicle.
One more recommendation would to try and eliminate the flow separation happening around the
bottom of the wind screen and down the rear window this could be done by decreasing the angle of
the windscreen, and for the rear window it needs to have a less of a recess and to be made more
flush with the bodywork.
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References
Autodesk. (2016). Alias. Available: http://www.autodesk.com/products/alias-products/overview. Last
accessed 2nd May 2016.
BMW. (2016). Suspension and damping on the BMW Z4 Coupé. Available:
http://www.bmw.com/com/en/newvehicles/z4/coupe/2006/allfacts/engine_suspension.html. Last
accessed 3rd May 2016.
CD-Adapco. (2016). Star CCM+. Available: http://www.cd-adapco.com/products/star-ccm%C2%AE.
Last accessed 2nd May 2016.
Car Foilo. (2006). 2006 BMW Z4 Coupé 3.0si. Available:
http://www.carfolio.com/specifications/models/car/?car=137995. Last accessed 3rd May 2016.
Formula 1. (2016). Side Skirts. Available: http://www.formula1-dictionary.net/side_skirts.html. Last
accessed 4th may 2016.
Formula 1. (2016). Splitter and air dam. Available: http://www.formula1-dictionary.net/splitter.html.
Last accessed 4th May 2016.
Joseph Katz (2006). Race Car Aerodynamics. Cambridge: Bentley.
NASA. (2014). What Is Drag. Available: https://www.grc.nasa.gov/www/k-12/airplane/drag1.html. Last
accessed 1st May 2016.
NASA. (2016).What is Lift. Available: https://www.grc.nasa.gov/www/k-12/airplane/lift1.html. Last
accessed 3rd May 2016.
Jim Lucas. (2014). What Is Aerodynamics. Available: http://www.livescience.com/47930-what-isaerodynamics.html. Last accessed 1st May 2016.
Racecar Engineering. (2016). Diffusers. Available: http://www.racecar-engineering.com/technologyexplained/diffusers-engineering-basics-aerodynamics/#. Last accessed 4th may 2016.
Richard Shelquist. (2016). An Introduction to Air Density. Available:
http://wahiduddin.net/calc/density_altitude.htm. Last accessed 2nd May 2016.
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Appendices
Appendix 1 – Z4 Concept Design Stage
Appendix 2 – Z4 Concept Rendered Images
Appendix 3 – Initial Testing
Appendix 4 – Addition of Front Splitter, Side Skirts & Rear Diffuser
Appendix 5 – Addition of Rear Wing
Appendix 6 – Final Renders
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Appendix 1 – Z4 Concept Design Stage
Below are some extra images of the different views of modelling stages of the BMW Z4 concept.
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Appendix 2 – Z4 Concept Rendered Images
Below are more images of the initial design rendered in Alias SpeedForm at different views.
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Appendix 3 – Initial Testing
Below are more images form post processing from initial testing.
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Appendix 4 – Addition of Front Splitter, Side Skirts & Rear Diffuser
Below Are images from post processing from simulation 2.
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Appendix 5 – Addition of Rear Wing
Below Are images from post processing from simulation 3.
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School of Engineering
Foundation Degree Motorsport Engineering
Jason Moffat
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Appendix 6 – Final Renders
Final Rendered images with all additional modifications.
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