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DETERMINATION OF TRUCK DRAG COEFFICIENT IN WIND TUNNEL
Patrick Atkinson
Christine Cammenga
Mary Arwen La Dine
Matt Schulstad
2
Introduction
The Indiana Department of Transportation has assembled an engineering team to find the drag
coefficient of a truck cab. This can be done through the use of a wind tunnel. Engineers use wind
tunnels in many industries when developing products such as airplanes, cars, and structures.
Wind tunnel testing is important in engineering because it can predict the response of real world
engineering problems by testing a model on a much smaller scale. Since finding the drag
coefficient of a full scale truck is difficult, the goal of this experiment was to determine the drag
coefficient of a small scale yellow truck cab model. We successfully completed our goal through
the use of a wind tunnel.
System & Model
Table 1. Nomenclature
Symbol Term Units
𝑓𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 Friction force Newtons
𝑔 Acceleration of gravity 𝑚𝑠2⁄
𝑚 Mass of truck kg
N Normal force Newtons
𝜃 Angle of slip degrees
𝐹𝑑𝑟𝑎𝑔 Drag force Newtons
𝐶𝑑𝑟𝑎𝑔 Drag coefficient —
𝑏 Truck width m
𝑎 Truck height m
𝜌𝑜𝑖𝑙 Density of manometer oil 𝑘𝑔𝑚3⁄
V Velocity of wind 𝑚𝑠⁄
𝑃𝑎𝑡𝑚 Atmospheric pressure atm
𝜌𝑎𝑖𝑟 Density of air 𝑘𝑔𝑚3⁄
𝑃𝑠𝑡𝑎𝑔 Stagnation pressure atm
𝑉𝑠𝑡𝑎𝑔 Stagnation velocity 𝑚𝑠⁄
ℎstag Manometer fluid height for stagnation pressure in
ℎ𝑎𝑡𝑚 Manometer fluid height for atmospheric pressure in
SGoil Specific gravity of water —
ρwater [1] Density of water 𝑘𝑔𝑚3⁄
𝐶𝑑𝑤𝑖𝑡ℎ Drag Coefficient with Teflon —
𝐶𝑑𝑤𝑖𝑡ℎ𝑜𝑢𝑡 Drag Coefficient without Teflon —
𝐶𝑑 Coefficient of drag —
3
The coefficient of friction between the truck and the wind tunnel platform is necessary to find the
truck’s drag coefficient. In order to find the coefficient of friction we performed an auxiliary
experiment outside of the wind tunnel. This was accomplished by tilting the truck and platform
until the truck slipped on the platform. The angle of slip was recorded. In this setup the truck’s
weight and the normal force between the platform and truck are the only forces acting on the
truck. When this surface is tilted, the friction between the truck’s wheels and surface keep the
truck from slipping. The truck will slip once the weight of the truck is equal to the opposing
force of static friction. Slipping occurs when the truck and platform are tilted through a large
enough angle. The forces acting on the truck are shown in Figure 1.
Figure 1: Forces acting on a truck cab on an inclined plane
The maximum friction force is found using the sum of the forces parallel to the inclined slope
and the measured angle. This calculation is shown below.
𝑓𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 = 𝑚 𝑔 sin(𝜃) (1)
In this equation 𝑓𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 is the friction force between the truck and the platform, 𝑚 is the mass of
the truck, 𝑔 is the acceleration due to gravity, and 𝜃 is the angle of slip.
After the force of friction is obtained, a separate truck model is used which accounts for the force
due to the wind in the tunnel. This is used to find the drag force on the truck. Figure 2 shows the
forces present on the truck when tested in the wind tunnel, where N is the normal force between
the platform and the truck and 𝐹𝑑𝑟𝑎𝑔 is the force of drag from the wind.
4
Figure 2: Forces acting on the truck cab in the wind tunnel
Figure 2 shows the drag force can be found by summing the forces parallel to the surface. This
is shown in equation 2.
𝐹𝑑𝑟𝑎𝑔 = 𝑚 𝑔 sin ( 𝜃) (2)
The force of drag is also modeled as
𝐹𝑑𝑟𝑎𝑔 = 𝐶𝑑𝑟𝑎𝑔 𝑏 𝑎 ( 𝜌𝑜𝑖𝑙 𝑉2
2) (3)
where 𝐶𝑑𝑟𝑎𝑔 is the drag coefficient, 𝑏 is the width of the truck, 𝑎 is the height of the truck, 𝜌𝑜𝑖𝑙 is
the density of the manometer oil (see Appendix B), and 𝑉 is the velocity of the wind.
Solving equations 2 and 3 yields the equation for 𝐶𝑑𝑟𝑎𝑔 shown below.
𝐶𝑑𝑟𝑎𝑔 =
𝑚 𝑔 sin ( 𝜃)
𝑏 𝑎 ( 𝜌𝑜𝑖𝑙 𝑉2
2 )
(4)
Apparatus and Method
To relate the velocity, 𝑉, to the difference between the stagnation and atmospheric pressures we
used Bernoulli’s equation
𝑃𝑎𝑡𝑚 + 𝜌𝑎𝑖𝑟 (
𝑉2
2) = 𝑃𝑠𝑡𝑎𝑔 + 𝜌𝑎𝑖𝑟 (
𝑉𝑠𝑡𝑎𝑔2
2)
(5)
𝑃𝑎𝑡𝑚 is the atmospheric pressure, 𝜌𝑎𝑖𝑟 is the density of air, 𝑃𝑠𝑡𝑎𝑔 is the stagnation pressure, and
𝑉𝑠𝑡𝑎𝑔 is the stagnation velocity. By definition the stagnation velocity is equal to zero. We then
solved for𝑉2
2.
𝑉2
2=
(𝑃𝑠𝑡𝑎𝑔 − 𝑃𝑎𝑡𝑚)
𝜌𝑎𝑖𝑟
(6)
5
Knowing the general relation
where 𝑃 is pressure, 𝜌 is density, and ℎ is fluid height, we were able to derive an equation for the
difference between stagnation and atmospheric pressures.
Substituting equation (8) into equation (6) we get
𝑉2
2=
𝜌𝑎𝑖𝑟 𝑔 (hstag − ℎ𝑎𝑡𝑚)
𝜌𝑎𝑖𝑟
(9)
Substituting equation (9) into equation (4) yields the Data Reduction Equation.
𝐶𝑑𝑟𝑎𝑔 =𝑚 sin ( 𝜃)
𝑎 𝑏 𝜌𝑜𝑖𝑙 (hstag − ℎ𝑎𝑡𝑚) (10)
The mass of the truck is found using a digital scale. The area of the truck is found using calipers
to measure the width and height of the truck’s frontal area. The angle of slip is found by using a
digital protractor. This is done using two different surfaces, Teflon and Plexiglas, and a
randomization test sequence generated using MatLab to ensure test procedure accuracy. To
determine the truck’s drag coefficient we used an Aerolab wind tunnel apparatus (see Appendix
A).
The accuracy of the scale, digital protractor, as well as the readability of the scale, digital
protractor, manometer, and calipers are taken into account when finding the total uncertainty of
the drag coefficient (see Appendix C).
Once the angle measurements are taken, the truck is attached to a platform using fishing line.
The platform is screwed into the wind tunnel so that it will remain fixed while running the
experiment. The truck and platform assembly is depicted in Figure 3.
Figure 3: Top view of truck tied to platform within wind tunnel.
𝑃 = 𝜌𝑔ℎ (7)
𝑃𝑠𝑡𝑎𝑔 − 𝑃𝑎𝑡𝑚 = 𝜌𝑎𝑖𝑟 𝑔 (hstag − ℎ𝑎𝑡𝑚) (8)
6
The wind tunnel is turned on and the wind speed is slowly increased until the truck begins to
slip. At this point, the heights of the pitot-static and atmospheric manometer tubes are recorded.
This is repeated multiple times using both surfaces in a randomized test sequence.
A tube is connected from a pitot-static probe in the wind tunnel to a manometer bank containing
Dwyer gage oil. The pitot-static probe measures stagnation pressure in the wind tunnel. This is
done because the wind velocity cannot be directly measured in the wind tunnel. A different
column of the manometer bank is attached a static probe which measures atmospheric pressure
(Figure 4). The heights of the oil are used to find wind speed.
Figure 4: Model of a pitot-static tube inside a wind tunnel
Data and Analysis
Several measurements gathered were completed with tools that were already calibrated for their
measurements (see appendices A and C). Table 2 displays the measured and constant values for
necessary to find the drag coefficient of the truck.
Table 2. Constant experimental values
Parameter Representative Value
m .239 kg
SGoil 0.826
ρwater 977.8 𝑘𝑔
𝑚3
b .059 𝑚
a .065 𝑚
7
A total of 34 data points were acquired for the angle measurements because the angle at which
the truck beings to slip has the largest source of uncertainty. The mean angle was found and used
for the calculations. A sample of the mean angle measurements can be found in Tables 3 and 4
(see Appendix D for complete data sets).
Table 3. Angle of slip with Teflon surface attached to the platform
Trial number Angle (degrees)*
1 21.5
2 26.5
3 24.6
4 27.8
5 28.2
6 24.7
*Reported values were converted to radians for calculations
Table 4. Angle of slip without the Teflon surface
Trial number Angle (degrees)*
1 44
2 44.5
3 45.7
4 43.9
5 45.2
6 43.1
*Reported values were converted to radians for calculations
Measurements for the difference in oil height were gathered from the manometer attached to the
wind tunnel. The drag coefficient found from the data reduction equation was paired with the
change in manometer oil height. A small data sample of these values can be found in Tables 5
and 6 (see Appendix D).
Table 5. Difference in manometer oil height and drag coefficient without Teflon
Trial Number Height Difference (inches)* Drag Coefficient
1 3.5 0.62
2 3.3 0.64
3 3.5 0.62
4 3.3 0.66
5 3.8 0.57
6 3.4 0.64
* Reported units were converted to SI units for calculations
8
Table 6. Difference in manometer oil height and drag coefficient with Teflon
Trial Number Height Difference (inches)* Drag Coefficient
1 2.1 0.54
2 1.5 0.75
3 1.5 0.75
4 1.5 0.75
5 1.6 0.71
6 1.6 0.73
* Reported units were converted to SI units for calculations
The full data set for the drag coefficients are displayed graphically in figures 5 and 6.
Figure 5. Time series plot of drag coefficient with Teflon Figure 6. Time series plot of drag coefficient without
Teflon
The final drag coefficient for each sample run can be observed in the plots above. Both data sets
appeared to have two outliers each. This was confirmed using Thompson’s Tau technique [4].
Averaging these drag coefficients leaves a final drag coefficient for the truck for each test
surface.
𝐶𝑑𝑤𝑖𝑡ℎ = 0.73 ± 0.08
𝐶𝑑𝑤𝑖𝑡ℎ𝑜𝑢𝑡 = 0.59 ± 0.02
The drag coefficient of the truck is independent of the surface it is resting on. In order to obtain a
more accurate representation of the drag coefficient, two surfaces were used during testing. From
these two surfaces, we averaged the two drag coefficients from both surfaces together and took
the root sum square of the associated errors to get a final drag coefficient of
𝐶𝑑 = 0.66 ± 0.09
9
Conclusion
The drag coefficient of the truck was found with the use of a wind tunnel. Wind tunnel testing is
important in engineering because it can predict the response of real world engineering problems
by testing a model on a much smaller scale. In this case, a small scale truck is used to model a
full scale truck. Two different surfaces were used to get a more accurate representation of the
truck’s drag coefficient. The results for the drag coefficient of the truck on both surfaces differed
more than expected. If repeated, it would be better to try a few more surfaces. Additionally, if a
force gauge was positioned behind the truck, the slipping point could be more easily determined.
The force gauge could also be used to better find the truck’s angle of slip. These changes could
eliminate discrepancies by the human eye needing to distinguish when slipping occurs.
References
1. Water - Density and Specific Weight [Online]. Available:
http://www.engineeringtoolbox.com/water-density-specific-weight-d_595.html
2. Mayhew, J. (2016, January 6). Private conversation.
3. Mech, A. (2016, January 5). Private conversation.
4. “Thomson tau technique.” “Workshop 3: Establishing technical credibility. “Handout.
Measurement Systems. (Professor Ryder Winck.) Rose-Hulman Institute of Technology.
Dec. 2016. Print.
10
Appendices
Appendix A: Auxiliary Angle Experiment
The error for the angle of slip, 𝑤𝑎𝑛𝑔𝑙𝑒,𝑡𝑜𝑡𝑎𝑙, is equal to the root sum square of the random and
systematic errors.
𝑤𝑎𝑛𝑔𝑙𝑒,𝑡𝑜𝑡𝑎𝑙 = √𝑤𝑎𝑛𝑔𝑙𝑒,𝑟𝑎𝑛𝑑
2 + 𝑤𝑎𝑛𝑔𝑙𝑒,𝑠𝑦𝑠𝑡2
(A.1)
Where 𝑤𝑎𝑛𝑔𝑙𝑒,𝑟𝑎𝑛𝑑 is the random error in the angle of slip and 𝑤𝑎𝑛𝑔𝑙𝑒,𝑠𝑦𝑠𝑡 is the systematic
uncertainty in the angle of slip.
Total random error in the angle is equal to the Student’s t times the standard deviation in the data
divided by the square root of the number of data points taken. The Students’s t value (𝑡𝑛) is
dependent on the number of data points.
𝑤𝑎𝑛𝑔𝑙𝑒,𝑟𝑎𝑛𝑑 =𝑡𝑛𝑆𝑎𝑛𝑔𝑙𝑒
√𝑛 (A.2)
Where n is the number of data points and 𝑆𝑎𝑛𝑔𝑙𝑒 is the standard deviation in the angle of slip.
Total systematic error is equal to root sum squares of all the individual systematic errors for each
measurand.
𝑤𝑎𝑛𝑔𝑙𝑒,𝑠𝑦𝑠𝑡 = √𝑤𝑎𝑛𝑔𝑙𝑒,𝑎𝑐𝑐
2 + 𝑤𝑎𝑛𝑔𝑙𝑒,𝑟𝑒𝑠2
(A.3)
Where 𝑤𝑎𝑛𝑔𝑙𝑒,𝑎𝑐𝑐 is the accuracy in the angle measurement and 𝑤𝑎𝑛𝑔𝑙𝑒,𝑟𝑒𝑠 is the resolution of the
digital protractor.
The partial derivative is hidden in this equation because it is equal to one.
Table A.1. Systematic uncertainties in measurement equipment
Tool Name Symbol value
Digital Protractor angle accuracy 𝑤𝑎𝑛𝑔𝑙𝑒,𝑎𝑐𝑐 .0017 rad
Digital Protractor angle resolution 𝑤𝑎𝑛𝑔𝑙𝑒,𝑟𝑒𝑠 .0017 rad
11
Appendix B: Density of oil calculation
Information about the oil in the manometer was reported by the specific gravity of the oil at
seventy degrees Fahrenheit. This value was multiplied by the density of water at 70 degrees
Fahrenheit to find the density of the oil.
𝑆𝐺𝑜𝑖𝑙 = 0.826 (B.1)
𝜌𝑤𝑎𝑡𝑒𝑟 = 977.8 𝑘𝑔
𝑚3 (B.2)
𝜌𝑜𝑖𝑙 = 𝜌𝑤𝑎𝑡𝑒𝑟 𝑆𝐺𝑜𝑖𝑙 = 808 𝑘𝑔
𝑚3 (B.3)
Appendix C: Drag coefficient Error
The error for the drag coefficient,𝑤𝐶𝑑,𝑡𝑜𝑡𝑎𝑙, is equal to the square root of the sum of the squares
of random and systematic error.
𝑤𝐶𝑑,𝑡𝑜𝑡𝑎𝑙 = √𝑤𝐶𝑑,𝑟𝑎𝑛𝑑2 + 𝑤𝐶𝑑,𝑠𝑦𝑠𝑡
2 (C.1)
Where 𝑤𝐶𝑑,𝑟𝑎𝑛𝑑 is the random error in the drag coefficient and 𝑤𝐶𝑑,𝑠𝑦𝑠𝑡 is the systematic
uncertainty in the drag coefficient
Total random error is equal to the student’s t times the standard deviation in the data divided by
the square root of the number of data points taken. The Students’s t value (𝑡𝑛) is dependent on
the number of data points.
𝑤𝐶𝑑,𝑟𝑎𝑛𝑑 =𝑡𝑛𝑆𝐶𝑑
√𝑛 (C.2)
Where 𝑆𝐶𝑑 is the standard deviation in the drag coefficient and n is the number of data points.
Total systematic error, 𝑤𝐶𝑑,𝑠𝑦𝑠𝑡, is equal to root sum squares of all the individual systematic
errors for each measurand multiplied by the individual partial derivatives.
𝑤𝐶𝑑,𝑠𝑦𝑠𝑡 =
[ (
𝜕𝐶𝑑
𝜕𝑚)2
𝑤𝑚,𝑟𝑒𝑎𝑑2 + (
𝜕𝐶𝑑
𝜕𝜃)2
𝑤𝜃,𝑡𝑜𝑡𝑎𝑙2 + (
𝜕𝐶𝑑
𝜕ℎ𝑠𝑡𝑎𝑔)
2
𝑤ℎ𝑠𝑡𝑎𝑔,𝑟𝑒𝑠2
+ (𝜕𝐶𝑑
𝜕ℎ𝑎𝑡𝑚)2
𝑤ℎ𝑎𝑡𝑚,𝑟𝑒𝑠2 + (
𝜕𝐶𝑑
𝜕𝑏)2
𝑤𝑏,𝑟𝑒𝑠2 + (
𝜕𝐶𝑑
𝜕𝑎)2
𝑤ℎ,𝑟𝑒𝑠2
] 12
(C.3)
The partial of the drag coefficient with respect to mass is:
12
𝜕𝐶𝑑
𝜕𝑚=
sin (𝜃)
𝜌𝑜𝑖𝑙 (ℎ𝑠𝑡𝑎𝑔 − ℎ𝑎𝑡𝑚) 𝑏 𝑎
(C.4)
The partial of the drag coefficient with respect to theta is:
𝜕𝐶𝑑
𝜕𝜃=
𝑚 cos(𝜃)
𝜌𝑜𝑖𝑙 (ℎ𝑠𝑡𝑎𝑔 − ℎ𝑎𝑡𝑚) 𝑏 𝑎
(C.5)
The partial of the drag coefficient with respect to stagnation manometer height is:
𝜕𝐶𝑑
𝜕ℎ𝑠𝑡𝑎𝑔=
−𝑚 sin (𝜃)
𝜌𝑜𝑖𝑙 (ℎ𝑠𝑡𝑎𝑔 − ℎ𝑎𝑡𝑚)2 𝑏 𝑎
(C.6)
The partial of the drag coefficient with respect to atmospheric manometer height is:
𝜕𝐶𝑑
𝜕ℎ𝑎𝑡𝑚=
𝑚 sin (𝜃)
𝜌𝑜𝑖𝑙 (ℎ𝑠𝑡𝑎𝑔 − ℎ𝑎𝑡𝑚)2 𝑏 𝑎
(C.7)
The partial of the drag coefficient with respect to the width of the cross-sectional area of the
truck cab is:
𝜕𝐶𝑑
𝜕𝑏=
−𝑚 sin (𝜃)
𝜌𝑜𝑖𝑙 (ℎ𝑠𝑡𝑎𝑔 − ℎ𝑎𝑡𝑚) 𝑏2 𝑎
(C.8)
The partial of the drag coefficient with respect to the height of the cross-sectional area of the
truck cab is:
𝜕𝐶𝑑
𝜕ℎ=
−𝑚 sin (𝜃)
𝜌𝑜𝑖𝑙 (ℎ𝑠𝑡𝑎𝑔 − ℎ𝑎𝑡𝑚) 𝑏 𝑎2
(C.9)
Where all uncertainties are defined in Table C.1.
Table C.1. Systematic uncertainties in measurement equipment
Tool Name Symbol value
Scale Mass readability 𝑤𝑚,𝑟𝑒𝑎𝑑 0.0001 kg
Manometer Stagnation height resolution 𝑤ℎ𝑠𝑡𝑎𝑔,𝑟𝑒𝑠 0.00127 m
Manometer Atmospheric height resolution 𝑤ℎ𝑎𝑡𝑚,𝑟𝑒𝑠 0.00127 m
Digital Caliper Caliper resolution 𝑤𝑏,𝑟𝑒𝑠 0.00000254 m
Digital Caliper Caliper resolution 𝑤ℎ,𝑟𝑒𝑠 0.00000254 m
13
Appendix D: Data
Table D.1. Angle of slip with Teflon surface attached to the platform
Trial Number Angle (degree)*
1 21.5
2 26.5
3 24.6
4 27.8
5 28.2
6 24.7
7 21.5
8 26.5
9 24.6
10 27.8
11 28.2
12 24.7
13 21.1
14 19.1
15 17.5
16 18.5
17 20.7
18 17.2
19 19.8
*Reported values were converted to radians for calculations
Table D.2. Angle of slip without Teflon surface attached to the platform
Trial Number Angle (degree)*
1 44
2 44.5
3 45.7
4 43.9
5 45.2
6 43.1
7 48.4
8 49.2
9 46.8
10 54.1
11 46.9
12 45.4
13 46.9
14 49.4
15 48.8
*Reported values were converted to radians for calculations
14
Table D.3. Difference in manometer oil height and drag coefficient without Teflon
Trial Number Height Difference (inches)* Drag Coefficient
1 3.5 0.62
2 3.3 0.64
3 3.5 0.62
4 3.3 0.66
5 3.8 0.57
6 3.4 0.64
7 3.4 0.64
8 3.5 0.62
9 3.8 0.57
10 3.8 0.57
11 3.9 0.56
12 3.8 0.57
13 3.7 0.59
14 3.7 0.59
15 4.0 0.54
16 3.6 0.60
17 3.8 0.57
* Reported units were converted to SI units for calculations
Table D.4. Difference in manometer oil height and drag coefficient with Teflon
Trial Number Height Difference (inches)* Drag Coefficient
1 2.1 0.54
2 1.5 0.75
3 1.5 0.75
4 1.5 0.75
5 1.6 0.71
6 1.6 0.71
7 1.5 0.75
8 1.5 0.75
9 1.5 0.75
10 1.6 0.71
11 1.5 0.75
12 1.7 0.67
13 1.7 0.67
14 1.3 0.87
15 1.4 0.81
* Reported units were converted to SI units for calculations
15
