determination of truck drag coefficient in wind tunnel
TRANSCRIPT
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
ππππππ‘πππ Friction force Newtons
π Acceleration of gravity ππ 2β
π Mass of truck kg
N Normal force Newtons
π Angle of slip degrees
πΉππππ Drag force Newtons
πΆππππ Drag coefficient β
π Truck width m
π Truck height m
ππππ Density of manometer oil πππ3β
V Velocity of wind ππ β
πππ‘π Atmospheric pressure atm
ππππ Density of air πππ3β
ππ π‘ππ Stagnation pressure atm
ππ π‘ππ Stagnation velocity ππ β
βstag Manometer fluid height for stagnation pressure in
βππ‘π Manometer fluid height for atmospheric pressure in
SGoil Specific gravity of water β
Οwater [1] Density of water πππ3β
πΆππ€ππ‘β Drag Coefficient with Teflon β
πΆππ€ππ‘βππ’π‘ Drag Coefficient without Teflon β
πΆπ Coefficient of drag β
3
The coefficient of friction between the truck and the wind tunnel platform is necessary to find the
truckβs 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 truckβs
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 truckβs 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.
ππππππ‘πππ = π π sin(π) (1)
In this equation ππππππ‘πππ is the friction force between the truck and the platform, π is the mass of
the truck, π is the acceleration due to gravity, and π 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 πΉππππ is the force of drag from the wind.
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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.
πΉππππ = π π sin ( π) (2)
The force of drag is also modeled as
πΉππππ = πΆππππ π π ( ππππ π2
2) (3)
where πΆππππ is the drag coefficient, π is the width of the truck, π is the height of the truck, ππππ is
the density of the manometer oil (see Appendix B), and π is the velocity of the wind.
Solving equations 2 and 3 yields the equation for πΆππππ shown below.
πΆππππ =
π π sin ( π)
π π ( ππππ π2
2 )
(4)
Apparatus and Method
To relate the velocity, π, to the difference between the stagnation and atmospheric pressures we
used Bernoulliβs equation
πππ‘π + ππππ (
π2
2) = ππ π‘ππ + ππππ (
ππ π‘ππ2
2)
(5)
πππ‘π is the atmospheric pressure, ππππ is the density of air, ππ π‘ππ is the stagnation pressure, and
ππ π‘ππ is the stagnation velocity. By definition the stagnation velocity is equal to zero. We then
solved forπ2
2.
π2
2=
(ππ π‘ππ β πππ‘π)
ππππ
(6)
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Knowing the general relation
where π is pressure, π is density, and β 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
π2
2=
ππππ π (hstag β βππ‘π)
ππππ
(9)
Substituting equation (9) into equation (4) yields the Data Reduction Equation.
πΆππππ =π sin ( π)
π π ππππ (hstag β βππ‘π) (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 truckβs 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 truckβs 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.
π = ππβ (7)
ππ π‘ππ β πππ‘π = ππππ π (hstag β βππ‘π) (8)
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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
Οwater 977.8 ππ
π3
b .059 π
a .065 π
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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
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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 Thompsonβs Tau technique [4].
Averaging these drag coefficients leaves a final drag coefficient for the truck for each test
surface.
πΆππ€ππ‘β = 0.73 Β± 0.08
πΆππ€ππ‘βππ’π‘ = 0.59 Β± 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
πΆπ = 0.66 Β± 0.09
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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
truckβs 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 truckβs 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. βThomson tau technique.β βWorkshop 3: Establishing technical credibility. βHandout.
Measurement Systems. (Professor Ryder Winck.) Rose-Hulman Institute of Technology.
Dec. 2016. Print.
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Appendices
Appendix A: Auxiliary Angle Experiment
The error for the angle of slip, π€πππππ,π‘ππ‘ππ, is equal to the root sum square of the random and
systematic errors.
π€πππππ,π‘ππ‘ππ = βπ€πππππ,ππππ
2 + π€πππππ,π π¦π π‘2
(A.1)
Where π€πππππ,ππππ is the random error in the angle of slip and π€πππππ,π π¦π π‘ is the systematic
uncertainty in the angle of slip.
Total random error in the angle is equal to the Studentβs t times the standard deviation in the data
divided by the square root of the number of data points taken. The Studentsβs t value (π‘π) is
dependent on the number of data points.
π€πππππ,ππππ =π‘πππππππ
βπ (A.2)
Where n is the number of data points and ππππππ 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.
π€πππππ,π π¦π π‘ = βπ€πππππ,πππ
2 + π€πππππ,πππ 2
(A.3)
Where π€πππππ,πππ is the accuracy in the angle measurement and π€πππππ,πππ 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 π€πππππ,πππ .0017 rad
Digital Protractor angle resolution π€πππππ,πππ .0017 rad
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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.
ππΊπππ = 0.826 (B.1)
ππ€ππ‘ππ = 977.8 ππ
π3 (B.2)
ππππ = ππ€ππ‘ππ ππΊπππ = 808 ππ
π3 (B.3)
Appendix C: Drag coefficient Error
The error for the drag coefficient,π€πΆπ,π‘ππ‘ππ, is equal to the square root of the sum of the squares
of random and systematic error.
π€πΆπ,π‘ππ‘ππ = βπ€πΆπ,ππππ2 + π€πΆπ,π π¦π π‘
2 (C.1)
Where π€πΆπ,ππππ is the random error in the drag coefficient and π€πΆπ,π π¦π π‘ is the systematic
uncertainty in the drag coefficient
Total random error is equal to the studentβs t times the standard deviation in the data divided by
the square root of the number of data points taken. The Studentsβs t value (π‘π) is dependent on
the number of data points.
π€πΆπ,ππππ =π‘πππΆπ
βπ (C.2)
Where ππΆπ is the standard deviation in the drag coefficient and n is the number of data points.
Total systematic error, π€πΆπ,π π¦π π‘, is equal to root sum squares of all the individual systematic
errors for each measurand multiplied by the individual partial derivatives.
π€πΆπ,π π¦π π‘ =
[ (
ππΆπ
ππ)2
π€π,ππππ2 + (
ππΆπ
ππ)2
π€π,π‘ππ‘ππ2 + (
ππΆπ
πβπ π‘ππ)
2
π€βπ π‘ππ,πππ 2
+ (ππΆπ
πβππ‘π)2
π€βππ‘π,πππ 2 + (
ππΆπ
ππ)2
π€π,πππ 2 + (
ππΆπ
ππ)2
π€β,πππ 2
] 12
(C.3)
The partial of the drag coefficient with respect to mass is:
12
ππΆπ
ππ=
sin (π)
ππππ (βπ π‘ππ β βππ‘π) π π
(C.4)
The partial of the drag coefficient with respect to theta is:
ππΆπ
ππ=
π cos(π)
ππππ (βπ π‘ππ β βππ‘π) π π
(C.5)
The partial of the drag coefficient with respect to stagnation manometer height is:
ππΆπ
πβπ π‘ππ=
βπ sin (π)
ππππ (βπ π‘ππ β βππ‘π)2 π π
(C.6)
The partial of the drag coefficient with respect to atmospheric manometer height is:
ππΆπ
πβππ‘π=
π sin (π)
ππππ (βπ π‘ππ β βππ‘π)2 π π
(C.7)
The partial of the drag coefficient with respect to the width of the cross-sectional area of the
truck cab is:
ππΆπ
ππ=
βπ sin (π)
ππππ (βπ π‘ππ β βππ‘π) π2 π
(C.8)
The partial of the drag coefficient with respect to the height of the cross-sectional area of the
truck cab is:
ππΆπ
πβ=
βπ sin (π)
ππππ (βπ π‘ππ β βππ‘π) π π2
(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 π€π,ππππ 0.0001 kg
Manometer Stagnation height resolution π€βπ π‘ππ,πππ 0.00127 m
Manometer Atmospheric height resolution π€βππ‘π,πππ 0.00127 m
Digital Caliper Caliper resolution π€π,πππ 0.00000254 m
Digital Caliper Caliper resolution π€β,πππ 0.00000254 m
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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
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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
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