To: Professor Anderson
From: RJ Hojnacki, Wes Wall, Sam Caruso
Date: 3/8/12
Subject: 1/12th Scale NASCAR Lift and Drag Findings
Purpose: We were asked to look into the Lift and Drag forces acting on the 1/12th scale stock car model.
This memo was made to report on our wind tunnel findings of the NASCAR lift and drag testing.
Findings: The wind tunnel test results showed that the lift coefficient achieved Reynold’s Number
independence after a Reynold’s Number of approximately 5X105. However, the drag coefficient never
seemed to achieve Reynold’s Number independence, although it seemed to be approaching Reynold’s
Number independence near the upper speed constraints of the wind tunnel.
Figure 1: The lift and drag coefficients versus Reynold’s Number. The force coefficients at a Reynold’s Number of 82817 were
removed from the plot (see Attachment 4 for more detail).
Since lift reached Reynold’s Number independence, beyond a Reynold’s Number of 5.0X105 the forces
exerted on the model can be scaled to a full size car. I would recommend re-running the test in a wind
tunnel capable of achieving higher speeds. This would provide more opportunity to confirm Reynold’s
Number independence was reached for the lift coefficient and to potentially achieve Reynold’s Number
independence for the drag coefficient. Additionally, I would recommend using a dynamometer with
higher precision. This would significantly reduce the uncertainty in our lift and drag measurements,
especially at low wind tunnel speeds.
Experiment Setup: The dynamometer was connected to a data acquisition system which was connected
to the computer. Voltage readings relating to lift, drag, and air speed within the tunnel were recorded at
wind tunnel speeds varying from 4.84m/s to 41.78m/s. At each speed 100 readings from each
dynamometer channel were recorded and input into Excel for further analysis. (See Attachment 5)
Analysis: The lift and drag coefficients displayed in Figure 1 offer valuable insight into the aerodynamics
of the stock car. The force coefficients represent the relative magnitude of the lift/drag forces on the
vehicle. The lift and drag coefficients are directly related to the change in momentum of particles
flowing past that object, as well as how abruptly those changes in momentum occur. Stock cars
designers use a technique called streamlining to achieve minimal lift and drag forces in order to
maximize vehicle performance.
Using Figure 1, we can see that the maximum lift coefficient (CL = 0.624) occurred at a Reynold’s
Number of 95890 (see Lift Coefficient 1). After LC 1, the lift coefficient decreased until reaching the
Reynold’s Number 308062 (see LC 4). LC 4 was the minimum lift coefficient throughout the experiment
(CL = 0.116). After this minimum value, the lift coefficient gradually increases until reaching a Reynold’s
Number of approximately 5X105(see CL 7). From Reynold’s Number 5X105 to 7.15X105 (CL 7 to CL 11) the
lift coefficient remained at approximately CL = 0.161. Since the lift coefficient remained stable from CL 7
to CL 11, we can conclude that Reynold’s Number independency was most likely achieved. Therefore,
beyond a Reynold’s Number of 5X105, dynamic similarity exists between our model and a full size
prototype. As a result, our lift forces for Reynold’s Numbers greater than 5X105 can be scaled to the lift
forces a full size car would experience. Note, due to the limitations of the wind tunnel it was assumed
that the lift coefficient will remain constant beyond a Reynold’s Number of 7.15X105(final data point).
The maximum drag coefficient occurs at a Reynold’s Number of 95890 (Drag Coefficient 1).
From DC 1, the drag coefficient decreases and reaches its lowest point at a Reynold’s Number of 244449
(see DC 3). After this point the drag coefficient creeps up to CD = 0.26 at a Reynold’s Number of 715607
(see DC 11). Observe DC 7 to DC 11 and notice that the rate of change between points is decreasing. This
lead us to believe that if the wind tunnel could produce higher wind speeds we would see convergence
of the drag coefficient at a higher Reynold’s Number. Since the drag coefficient doesn’t reach Reynold’s
Number independence, dynamic similarity does not exist prior to a Reynold’s Number of 7.15X105.
Generally, both the lift and drag coefficients follow a similar trend. They begin high, decrease
rapidly, and then increase gradually. To offer an explanation for this we can make an analogy to flow
around a sphere because we understand that behavior and it relates to our results. In laminar (low
Reynold’s Number) flow around a sphere, separation occurs at the midpoint of the sphere. This leaves a
low pressure area on the back of the sphere which increases the drag. As the flow becomes turbulent
and gains more momentum, the separation occurs past the mid-point and the low pressure area on the
back of the sphere gets small. As a result, the drag is decreased. Given that, it is possible that flow
around our NASCAR model may behave similarly.
To find the uncertainty in the force coefficients, we considered the uncertainty in velocity, the
uncertainty in force measurements, and the uncertainty in density (See Attachment 3). The average
uncertainty in the lift/drag coefficients were CL ± 0.024 and CD ± 0.067. The error bars on Figure 1 show
the percent uncertainty for each force coefficient. For both the lift and drag the uncertainty decreased
as Reynold’s Number increased (see Attachment 4).This is what we would expect because at low wind
tunnel speeds the lift and drag forces are very small, especially with regards to the dynamometers
precision. As the forces became larger (i.e. at higher wind tunnel speeds) the percent uncertainty
reduced significantly.
Conclusion: From the provided force coefficient data we reached a few conclusions. The lift coefficient
converges at approximately CL = 0.161 and therefore reaches Reynold’s Number independence. This
statement is limited by the speeds achievable with our wind tunnel. The drag coefficient does not reach
Reynold’s Number independence, but it does seem to be converging prior to reaching the top speed of
the wind tunnel. If our wind tunnel had the ability to produce higher speeds I predict that the drag
coefficient would reach Reynold’s Number independence. Next time this test is done a dynamometer
with higher precision should be used because of the relative magnitude of forces on this streamlined
stock car design. If you need any further assistance in any way feel free to contact us at
NASCAR_labteam@gmail.com
Attachments:
1,2 - All Data for Lift, Drag, and Wind Tunnel Velocity
3 - Uncertainty Analysis for Lift and Drag Coefficients
4 - Explanation for the Removal of Data for Reynold’s Number 82817
5 - Experimental Setup and Schematic
Attachment 1
Table 1: All Calculated Lift Results
Data Point Label Reynold's Number Velocity Uncertainty Lift Force Lift Coefficient Uncertainty
CL #
Re
V
δV
FL
CL
δCL
82817
4.84
0.771
0.19
0.462
0.219
1
95890
5.60
0.474
0.34
0.624
0.074
2
176372
10.30
0.382
0.52
0.285
0.040
3
244449
14.27
0.356
0.59
0.168
0.030
4
308062
17.99
0.364
0.64
0.116
0.021
5
369669
21.58
0.390
1.00
0.125
0.019
6
428999
25.05
0.425
1.54
0.143
0.016
7
487720
28.48
0.466
2.12
0.152
0.014
8
545954
31.88
0.510
2.74
0.157
0.014
9
603775
35.25
0.556
3.47
0.163
0.014
10
660036
38.54
0.602
4.19
0.165
0.013
11
715607
41.78
0.648
4.81
0.161
0.012
Table 1 includes all the relevant results computed from the dynamometer readings at various wind
tunnel speeds.
Table 2: Raw Experimental Lift Data
Raw Lift Data
Avg. Volt STDEV. Volt % STDEV. Volt STDEV. F(N) Avg. F(N)
0.008
0.001
7.73%
0.007
0.185
0.020
0.001
3.97%
0.010
0.335
0.035
0.001
3.43%
0.015
0.519
0.041
0.003
8.43%
0.042
0.586
0.046
0.004
7.75%
0.043
0.645
0.075
0.005
6.83%
0.062
0.998
0.119
0.006
4.80%
0.070
1.539
0.167
0.007
3.92%
0.080
2.120
0.218
0.008
3.68%
0.098
2.738
0.278
0.011
3.78%
0.128
3.468
0.338
0.011
3.16%
0.130
4.195
0.389
0.011
2.93%
0.138
4.813
δF(N)
0.101
0.102
0.104
0.130
0.132
0.159
0.171
0.188
0.219
0.274
0.278
0.294
Table 2 contains additional raw data that was necessary for the calculations made to get the values in
Table 1.
Attachment 2
Table 3: All Calculated Drag Results
Data Point Label Reynold's Number Velocity Uncertainty Drag Force Drag Coefficient Uncertainty
CD #
Re
V
δV
FD
CD
δCD
82817
4.84
0.771
0.03
0.30
0.719
1
95890
5.60
0.474
0.05
0.32
0.257
2
176372
10.30
0.382
0.06
0.11
0.137
3
244449
14.27
0.356
0.04
0.04
0.086
4
308062
17.99
0.364
0.09
0.06
0.059
5
369669
21.58
0.390
0.15
0.07
0.047
6
428999
25.05
0.425
0.39
0.13
0.034
7
487720
28.48
0.466
0.61
0.16
0.029
8
545954
31.88
0.510
0.95
0.20
0.024
9
603775
35.25
0.556
1.31
0.23
0.022
10
660036
38.54
0.602
1.70
0.25
0.021
11
715607
41.78
0.648
2.14
0.26
0.022
Table 3 includes all the relevant results computed from the dynamometer readings at various wind
tunnel speeds.
Table 4: Raw Experimental Drag Data
Raw Drag Data
Avg. Volt STDEV. Volt % STDEV. Volt STDEV. F(N) Avg. F(N)
0.016
0.003
17.0%
0.003
0.032
0.028
0.005
17.8%
0.006
0.046
0.036
0.007
18.6%
0.008
0.056
0.023
0.009
38.4%
0.010
0.040
0.062
0.009
14.1%
0.010
0.086
0.118
0.022
18.5%
0.026
0.152
0.318
0.015
4.7%
0.017
0.386
0.508
0.023
4.5%
0.027
0.610
0.800
0.019
2.3%
0.022
0.952
1.107
0.030
2.7%
0.035
1.313
1.438
0.035
2.4%
0.041
1.702
1.808
0.052
2.9%
0.061
2.137
δF(N)
0.100
0.101
0.101
0.102
0.102
0.112
0.106
0.114
0.109
0.122
0.129
0.158
Table 4 contains additional raw data that was necessary for the calculations made to get the values in
Table 3.
Attachment 3
Calculation of uncertainty in the lift coefficient
First the uncertainty in the density had to be calculated. We concluded that the uncertainty of the
density was about 3% of the density, so δρ = 1.186*(0.03) = 0.03558.
Next, the uncertainty in the velocity was derived from the equation for the velocity
2𝛥𝑃
𝜌
𝑉=√
(1)
Next, the percent uncertainty of the velocity had to be calculated using the equation
𝑈𝑉 =
𝛿𝑉
𝑉
= √((0.5) ×
2
𝛿𝛥𝑃
( 𝛥𝑃 ))
2
+ ((0.5) × (
𝛿𝜌
( 𝜌 ))
(2)
This value was then calculated at each wind speed. The percent uncertainty in the lift coefficient was
derived from the equation
𝐶𝐿 = 1
𝐹𝐿
(3)
𝜌𝑉 2 𝐴𝑇
2
Next, the equation for the percent uncertainty of the lift coefficient had to be derived, as
shown
𝑈𝐶𝐿 =
𝛿𝐶𝐿
𝐶𝐿
𝛿𝐹𝐿 2
)
𝐹𝐿
= √(
𝛿𝜌
𝜌
2𝛿𝑉 2
)
𝑉
+ ( )2 + (
(4)
In order to find the uncertainty of the lift force, the percent uncertainty in the lift force needed to be
multiplied by the lift force at each wind tunnel speed.
𝛿𝐹
2
𝛿𝜌 2
2𝛿𝑉 2
)
𝑉
𝛿𝐶𝐶𝐿 = 𝐶𝐶𝐿 ∗ √( 𝐹 𝐿 ) + ( 𝜌 ) + (
𝐿
(5)
The calculation in equation 5 was carried out for each wind speed.
Equations 3-5 can be applied for the drag coefficient in the same way. Note that the area used for drag
is the frontal area of the model rather than the area of the top of the model used for finding lift force.
Attachment 4
Figure 2: Percent Uncertainty in Lift Coefficient versus the Lift Force
Figure 2 shows the percent uncertainty in the lift force with respect to the size of the lift force. As you
can observe, as the force gets bigger the percent uncertainty decreases. This is likely because the
dynamometer’s precision. The highest precision the dynamometer can measure is a thousandth.
Relative to a thousandth, our lift forces are quite small, therefore any fluctuation or inaccuracy in the
data results in a high uncertainty.
Figure 3: Percent Uncertainty in Drag Coefficient versus the Drag Force
Figure 3 shows a similar trend. However, note the magnitude of the drag forces in comparison to the lift
forces in Figure 2. Since these force measurements are even smaller on the scale of tenths (for the first
few forces shown), any fluctuation in testing causes a huge uncertainty relative to the force measured.
We chose not to include the points represented by the first data bars, on Figure 1, because of the large
percent of uncertainty for both lift and drag.
Attachment 5
Experimental setup
The NASCAR model was pre-mounted to the dynamometer in the wind tunnel so that the wheels were
just touching the floor of the wind tunnel and the front of model was facing the direction of the air flow.
The dynamometer was attached to a data acquisition box which was plugged into a computer. A Pitot
probe was positioned in the wind tunnel and attached to a pressure transducer using Tygon tubing
which was also then connected to the data acquisition box.
Then dynamometer was zeroed using the attached knobs. Results pertaining to the Pitot probe, the lift,
and the drag were represented by voltages. Data was recorded while the motor was at 10 Hz (4.8 m/s)
and repeated in intervals of 4 Hz up to and including 54 Hz (41.8 m/s). Care was taken to ensure that the
flow in the wind tunnel was steady before taking any readings. One student watched the behavior of the
car to record any suspicious movements (i.e. the front of the model tipping upwards).
Figure 4: Schematic of Experimental Setup
“We completed this assignment to our full academic honest - [RJ, Wes, Sam]”