DETERMINATION OF TRUCK DRAG COEFFICIENT IN WIND TUNNEL

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


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


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


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.

http://www.engineeringtoolbox.com/water-density-specific-weight-d_595.html

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


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


14

Table D.3. Difference in manometer oil height and drag coefficient without Teflon


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


Table D.4. Difference in manometer oil height and drag coefficient with Teflon


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


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DETERMINATION OF TRUCK DRAG COEFFICIENT IN WIND TUNNEL