Aspect Ratio Effects on the
Drag Coefficient of a Cylinder
Group 8 Final Report
Aaron Shakin
Syed Habib
April 27, 2011
ME 241-Professor Gans
Abstract:
We measured the drag coefficient of a finite length cylinder in the University of
L
Rochester wind tunnel within a range of aspect ratios ( D ) from 2.03 to 6.5 for Reynolds
numbers on the order of 104 - 105. The greatest obstacle in collecting data was eliminating the
vibration of the cylinder. A mechanical filter helped to resolve this, and allowed us to compare
values of drag coefficients over our Reynolds number range with the curve for an infinite
cylinder. We found that the curves for larger aspect ratios (such as 6.5 and 4.33) seem to deviate
more from the infinite case than the curves for smaller aspect ratios (such as 2.03 and 2.71).
1
Introduction:
Drag on objects in a flow is usually described by a drag coefficient, which supposes that
the force is proportional to the dynamic pressure of the flow and the area perpendicular to the
flow:
1
𝐹𝐹 = 2 𝐶𝐶𝐷𝐷 ρV 2 A
(1)
where CD denotes the drag coefficient, ρ the density of the fluid, V the speed of the fluid and A
the area. The flow itself can be characterized by the Reynolds number:
𝑉𝑉𝑉𝑉
𝑅𝑅𝑅𝑅 = ν
(2)
where D denotes a characteristic length and ν the kinematic viscosity. Typical fluids texts and
handbooks give relations between Re and CD for different shapes. (For examples see [1] and
[2].)
The relations between CD and Re for cylinders are based on experiments that attempt to
model infinitely long cylinders. Figure 1 shows such a curve, taken from [2]. We intended to
measure the drag force on finite cylinders in a wind tunnel to see how the drag coefficient
depends on some aspect ratio:
L
where L denotes the length of the cylinder.
λ=D
(3)
We plotted CD vs. Re for various values of λ for comparison with the data shown in
Figure 1. We expected that the drag coefficient relation will approach that for the infinite
cylinder in the limit λ —> ∞.
Procedure:
The objective of our project was to change the exposed length of three different diameter
cylinders and to find a relationship between the aspect ratio, Reynolds number and drag
coefficient. For our cylinder material we used hollow PVC pipes of 50.8, 76.2 and 101.6 mm (2,
3 and 4 inch) diameters. Most of the purchases for the experiment including the pipes were made
at Lowe’s, totaling $61.67 for all purchased materials (Table 1).
2
Figure 2: A picture of the set up below the wind tunnel.
Next, we developed a method which would allow different lengths of the cylinders to be
exposed in the wind tunnel testing chamber. We designed and built an adjustable shelf that we
could move up and down in increments of 127 mm (5 inches) (Figure 2). We did this by placing
corner braces at two different levels. At each level, we screwed the shelf and corner braces
together to avoid instability. The frame of the shelf was attached to the mounting board with two
additional corner braces and the cross-bar could be moved to different heights in order to expose
different lengths of the cylinder as shown in Figure 2. The shelf cross-bar was made from an
acrylic material with a pre-cut groove to hold the load cell in place. The load cell had a thin
metal cylindrical piece screwed on top of it, and together they were placed on top of the shelf
(Figure 3). A threaded rod of 7.94 mm (5/16 inch) and 18 threads per inch was screwed into the
thin metal cylindrical piece and connected to the bottom of the PVC pipe. With this design, the
rod was able to transmit the force experienced by the PVC pipe into the smaller cylindrical piece
connected to the load cell. All three pipes purchased were approximately 609.6 mm (2 ft) long.
The available space under the wind tunnel testing chamber made it necessary to cut each pipe
down to 457.2 mm (18 inches) with a band saw.
3
Figure 3: Picture of the attachment piece to the load cell.
Figure 4a: Design idea of the rectangular block inside the bottom of the cylinder which created attempted
to create a rigid connection between the pipe and the load cell. On the left is the cross section of the pipe
and the right shows a top-down view.
4
Figure 4b: Picture of the PVC pipe with the wooden plug secured into place.
In order to secure the upper end of the rod into the PVC pipe, our design idea was to
place a rectangular wooden block inside the pipe to join the pipe to the load cell as one rigid
body. We cut out three 203.3 mm (8 inch) long rectangular blocks using a band saw and drilled
holes through the center of the block’s thickness. We attached to this, a square wooden piece just
large enough to cover the pipe opening, perpendicular to the longer rectangular piece (that would
go inside the pipe). Next, we drilled a hole into each of the pipe covers that aligned with the hole
through the internal wooden piece (Figure 4a). Then, we fit each wooden plug into its
corresponding PVC pipe. We drilled two wooden screws into either side of the pipe to secure
the internal wooden piece to the pipe wall. At this point, we screwed the threaded rod through
the wooden end cap, and the internal rectangular piece that went inside of the cylinder (Figure
4b). We screwed the bottom end of the threaded rod into the attachment piece that connected to
the load cell. Once the threaded rod was tightly secured into place, it was able to transmit the
force experienced by the cylinder under increasing amounts of wind speed. The strain gages in
the load cell converted the deformation into electrical signals which was converted to forces that
the cylinder experienced from the wind. We also purchased flat caps that were glued with epoxy
to seal the top opening that would be exposed to the wind. After the pipe and pipe attachment
were assembled, we cut a hole in the mounting board, about 127 mm (5 inches) in diameter,
which was large enough for the largest pipe to fit through with extra room for clearance. The
hole-saws we had access to were not able to make this sized hole in the mounting board, so
several smaller holes were drilled into the board, and a hand saw was used to cut out the
remaining wood.
5
Prior to testing our cylinders, we needed to calibrate the wind tunnel to obtain a
relationship between percent power and wind speed, as well as the load cell being used to obtain
a relationship between voltage and force applied in the x direction. We had done the wind tunnel
calibration using a pitot static tube in a prior lab and our data had a strong linear correlation
which we used for this lab (Figure 5). The data from this calibration allowed us to construct a
table of power increments (by 5%) and their corresponding wind speeds (Table 2). We
constructed a VI on LabView 2009 to calibrate the load cell labeled Mod 5. Masses in
increments of 0.5 kg all the way up to 8 kg were hung off the load cell in the x direction. We
obtained a linear trend between the applied force and the corresponding voltage (Figure 6) that
would allow us to begin mounting the cylinders to collect data. The relationship we obtained
between force and voltage is shown in Equation (4):
𝐹𝐹 = −81500 × 𝑉𝑉 − 3.67
(4)
where V denotes the voltage in mV and F is the force in the x-direction. After calibration, we
constructed a VI on LabView for testing the PVC pipes that closely resembled our load cell
calibration VI.
We tested three different diameter cylinders of 50.8, 76.2 and 101.6 mm (2, 3 and 4
inches) exposed at an increasing wind velocity, initially from 4.74 to 41.7 m/s. We also chose to
vary the exposed length for each cylinder, with two increments of approximately 203.2 mm and
330.2 mm (8 and 13 inches), to obtain a total of 6 different aspect ratios ranging from 2.03 to 6.5.
We mounted the cylinder upright in the tunnel and positioned it in the center of the chamber, in
order to minimize wall effects. At each exposed length, we changed the wind velocity and
recorded the drag force that was measured by the Mod 5 load cell.
During our first set of testing, we noticed significant wobbling of the cylinders. We
suspected that this was because our adjustable shelf was hanging free under the tunnel, and was
not secured to any large stationary objects. To remedy this problem, we trimmed down about
35.6 mm (1.4 inches) off of the side pieces of our shelf. With this extra room, we were able to fit
a flat wooden board about 12.7 mm (.5 inches) in thickness which we attached to the shelf with
wooden screws. We then clamped the board underneath the shelf to the large metal frame that
we had previously removed from underneath the testing chamber for our initial testing. We
started the testing process again with the 101.6 mm (4 inch) diameter cylinder at the 203.2 mm
(8 inch) exposure length, turning the wind speed up from 0, up to 35% power (20 m/s) in
increments of 5%. Even at the lower speeds, the cylinder continued to vibrate. This seemed to
be because the attachment of the threaded rod to the load cell was not robust enough. We added
a low pass filter to add to our LabView VI to help filter out the “noise” that was coming from the
pipe’s unwanted movement. The correlations from the filtered data were identifiable, but not
very strong.
6
With one week’s time remaining for lab work, we devised a type of damper, which would
mechanically filter out the vibrations of the cylinder. To do this we obtained slow-recovery
super cushioning polyurethane foam, type 3 blue, to fill the empty space between the pipe edges
and the wooden mounting board (Figure 6). This memory foam was a soft material which
absorbed vibration and took several seconds to return to its original shape after compression, but
did not wear out after repeated use [3]. It was a very soft, non-flammable material, with a
thickness of approximately 25.4 mm (1 inch). More foam was needed to fill the gap for the
smallest pipe, and less foam was used in the assembly for the largest pipe. The foam was duct
taped securely into place, but was temporary enough that it could be removed for the various setups for each size cylinder. The memory foam appeared to work in practice, dampening most of
the vibrations experienced. The low pass filter was still incorporated into LabView to eliminate
some of the remaining “noise” in the data. With this final set-up we were able to collect data for
all three pipes, each at 2 different exposure lengths, from 10 to 70 % power (4.74 to 41.7 m/s).
Figure 6: Blue Memory foam used to fill the space between the pipe and the mounting board.
The pipe fits through the hole in the center.
Results:
Once we had secured our testing apparatus and applied the memory foam padding to the
hole in the mounting board, we were able to obtain raw data that mirrored the results that were
previously filtered on LabView. Using Equation (4) we were able to convert the voltages from
LabView into corresponding drag forces on the PVC pipe at each aspect ratio tested. Every
aspect ratio had 100 voltage readings taken and recorded onto excel at each power setting,
starting at 10 % (4.74 m/s), going up to 70% (41.7 m/s) in 5% increments.
With 100 drag forces at each power setting, we were able to take the mean force value
and convert it to a drag coefficient using Equation (1) to solve for CD. The values of the drag
coefficients spanned from 0.087 to 2.05 over a Reynolds number range of 6×104 -2.8×105. Using
7
Equation (2), we were able to determine the Reynolds number that corresponded to each drag
coefficient, knowing the velocities at each power setting from Table 2 and using the kinematic
viscosity of air at room temperature (20 °C) of 1.82 ×10-5 m2/s [4]. Tables 3a-3f show the data of
Reynolds number and drag coefficient for each aspect ratio. Figure 7 shows a plot of these
results on a log-log scale, where each colored line represents one of the six aspect ratios (ranging
from 2.03 to 6.5) calculated with Equation (3). The dark line on the graph with the shallowest
downward slope shows the plot of an infinite aspect ratio for a smooth cylinder [2]. We found
the standard deviation of the drag coefficients from the list of force values at each power setting
and plotted error bars onto Figure 7. The error bars show two standard deviations added to and
subtracted from each data point. Assuming a Gaussian distribution, these means that 95% of the
values recorded for each Reynolds number will fall between the upper and lower limits of the
error bars on the plot.
Figure 7 shows that the slopes for all six aspect ratios have negatively sloping lines of
comparable magnitudes. However, the aspect ratios of 4.33, 6.5 and 3.25 have lines that begin to
have a steeper slope at higher Reynolds numbers. From the error bars, it is clear that the
standard deviation becomes much larger at these points. If the data points at the largest
Reynolds number for these three aspect ratios are positioned towards the upper extreme of the
error bar, these lines closely resemble the lines for the smaller aspect ratios. While the slopes of
the lines appear to be similar for the majority of the points in all six aspect ratios, there does
seem to be a trend showing a decrease in the apparent y-intercept based on the range of Reynolds
numbers plotted.
Another way that we analyzed our drag coefficients was by creating a log-log plot of the
data on a linear scale. To have plots that are easier to read, the data were divided into the three
aspect ratios from the three pipes at a 203.2 mm (8 inch) exposure length (Figure 8) and the
aspect ratios for all three pipes at the larger 330.2 mm (13 inch) exposure length (Figure 9).
Figure 8 shows a stronger linear correlation for the aspect ratios at the smaller exposure length,
with slopes that are relatively close to one another, ranging from -1.65 to -2.38. The aspect ratios
for the larger exposure length plotted on Figure 9 shows both a weaker linear trend as well as
steeper slopes that are farther apart, ranging from -2.01 to -3.98.
Discussion/Conclusion:
Our results show that there is a general downward slope for all aspect ratios in Figure 7,
plotting the drag coefficient vs. Reynolds number on a logarithmic scale. From this decrease in
drag coefficient, we hypothesize that the Reynolds number range tested is where the flow over
the pipe was transitioning from laminar to turbulent. As the Reynolds number approaches the
order of 105 for a smooth pipe, the value of the drag coefficient begins to decrease due to
boundary layer separation [1]. The wake of the air after it flowed past the cylinder began to
8
spread apart and the drag coefficient was smaller. However, the drag force was likely to continue
increasing at higher Reynolds numbers, because of its dependence on the velocity squared
(Equation (1)). The greater the Reynolds number became, the steeper the slope was. This would
continue to occur until the Reynolds number approached 106, when the flow would be
completely turbulent. The plot of the assumed infinite aspect ratio [2], suggests that that the drag
coefficient would begin to increase again.
It is clear from Figure 7 that the drag coefficient began to decrease more rapidly at lower
Reynolds numbers for the finite cases we tested than for the infinite case, where the curve has a
very shallow slope until the Reynolds number approaches 3×105. For the case of a finite
cylinder, the top end of the pipe is exposed freely to the oncoming wind stream. There was an
additional wake generated for flow over the top of the pipe. For a infinite model, the wake only
has a presence along the lateral surface of the cylinder. Prior experiments that have measured
drag forces on a finite length cylinder have found that the wake caused by flow over the free end
of the cylinder creates disturbances in the lateral wake, potentially causing the boundary layer to
separate pre-maturely [5]. The experiment in [5] found that values of drag force in a finite length
cylinder are lower than the values predicted from existing data for an infinite cylinder. Since
drag force is directly proportional to drag coefficient, a finite cylinder in this experiment would
also have a drag coefficient that is lower than an infinite case. Our results agreed with these
findings. The flow over the top of the cylinder is one possible explanation for the lower values
of drag coefficients.
In looking at Figure 1 [2], it is clear that the curve for drag coefficients on a rough
cylinder deviates from the curve for a smooth cylinder in the Reynolds number region that were
analyzing. The drag coefficient for a rough cylinder decreases rapidly, at lower Reynolds
numbers, just as our data appears to do on Figure 7. Therefore, we decided to superimpose our
curves from Figure 7, onto the curve from Figure 1, focusing in on the boxed region (Figure 10).
This shows us that our data more closely resembles the trend for a rough cylinder than a smooth
cylinder. At lower velocities in particular, our data become more and more like the drag
coefficients for a rough cylinder as the aspect ratios get larger. If we consider our pipes to be
rough cylinders, it makes sense that the larger aspect ratios behaved more like the infinite case
than the smaller ratios did. It is possible that the smaller aspect ratios acts like smoother pipes
than the larger aspect ratios. The apparent similarity between our pipes and rough cylinders
might be because of flaws in the pipes that make them less smooth such as screw heads
protruding out of the lateral area. The roughness could also be coming from the caps on the pipe
tops, which dip inward slightly as opposed to remaining at the same level as the cylinder
opening.
Another analysis was done with the log of the drag coefficients and Reynolds numbers on
the plots in Figures 8 and 9. All three slopes on Figure 8 are all within a factor of 0.73. While
these are relatively close trend line equations, the values are still not consistent enough to
generate a universal relationship between the Reynolds numbers and drag coefficients among the
9
aspect ratios on this plot (2.03, 2.6, and 4.06). For Figure 9, (aspect ratios of 3.25, 4.33 and 6.5)
which represents the three larger exposure lengths, the slopes are not as consistent, with a range
of 1.97 between the three lines. This means that exposing larger portions of the pipe made it
more vulnerable to various experimental errors that compromised some of the integrity of our
data.
Several sources of error may have negatively impacted the data we obtained throughout
the lab, especially for the testing done at larger exposure lengths. Even though we attempted to
create a rigid connection between the load cell and the pipe in the tunnel, the wooden plug we
were using only went 203.2 mm (8 inches) deep inside the pipe and was not long enough to run
all the way up to the top. A tighter attachment to the load cell would have helped confirm that
the forces measured by the load cell were equal to the exact drag force on the cylinder.
As mentioned in the procedure, significant vibrations of the pipe occurred before the
memory foam was put into place. The data taken without the memory foam were filtered on
LabView for the power range tested (0-35 % power). After putting the memory foam in place,
the data seemed to indicate that the drag forces found with the foam were similar to the values
found without the foam. However, further investigation showed that there was about a
15 to 25 % error between the drag forces. This meant that the foam was absorbing about 15 to
25 % of the force (equivalent to approximately one Newton) that was intended to be exerted
entirely on the cylinder. As a result, the values for drag coefficients were proportionately
different than they were without the foam in place. Figure 7 shows the drag coefficients with the
memory foam. The Reynolds number range without the foam was fairly small since the
wobbling made it unadvisable to run the wind tunnel at high speeds. Therefore, there was not
enough data taken without the foam to create a plot for comparison. Furthermore, even with the
foam in place, as the wind tunnel speed reached faster velocities the pipe no longer remained
perfectly upright, perpendicular to the mounting board below it. The pipe was titled at an acute
angle, due to the force of the high wind speed. As a result, some of the data collected at the
higher velocities generated drag coefficients that were inconsistent with the trends that seemed to
be developing. Therefore, some of the data at higher velocities were neglected, and are not
shown in the Table 3 or the corresponding plots for some of the aspect ratios.
With more time and resources we would be able to conduct further testing to compile
more data. A larger range of aspect ratios would have been helpful for generating a correlation
that more clearly shows how the drag coefficient depends on this parameter. Our largest ratio is
only 6.5, which is far from being a candidate for an infinite ratio compared to the other ratios we
tested which were all above 2. The size of the wind tunnel testing chamber limited us by not
being large enough to test very large lengths or diameters. We could attempt to compensate for
this by testing much smaller ratios relative to our largest ratio of 6.5, making the 6.5 ratio appear
as a relatively infinite case. However, we still may have to confront the wall effects of the
chamber ceiling at this large an exposure length.
10
References:
[1] Young, Donald F., Bruce R. Munson, Theodore H. Okiishi, and Wade W. Huebsch. "Chapter
9." A Brief Introduction to Fluid Mechanics. 4th ed. Hoboken, NJ: Wiley, 2007. Print.
[2] Janna, William S. Introduction to Fluid Mechanics, 25. 2nd ed. PWS-KENT, 1987. Print.
[3]"Slow-Recovery Super-Cushioning Polyurethane Foam." McMaster-Carr. Web. 27 Apr. 2011.
<http://www.mcmaster.com/#memory-foam/=c25x1o>.
[4] " Air Properties ." Engineering ToolBox . N.p., n.d. Web. 17 Mar. 2011.
<http://www.engineeringtoolbox.com/air-properties-d_156.html>.
[4] Luo, S. C., T. L. Gan, and Y. T. Chew. "Uniform Flow Past One (or Two in Tandem) Finite
Length Circular Cylinder(s)." Science Direct 5.1 (1996): 69-93. Print.
11
Figure 1: Sample plot of drag coefficient vs. Reynolds number. The range we are looking at for a smooth
cylinder is boxed in red [2].
Velocity vs. Power Percent
70
60
y = 0.6207x - 1.8778
R² = 0.9989
Velocity (m/s)
50
40
30
20
10
0
-10
0
20
40
60
80
100
Power Percent
Figure 5: Calibration graph of the velocity vs. power percent of the wind tunnel.
12
120
Force vs. Voltage X-Calibration
90
80
y = -7.6648E+01x
70
Force (N)
60
50
40
30
20
10
0
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
Voltage (mV)
Figure 6: Calibration graph for the load cell, of the force vs. voltage in the x-direction.
10
Drag Coefficient vs. Reynolds Number
(Error Bars)
Drag Coefficient
2.03125
1
3.00E+04
3.00E+05
3.25
2.7083
4.333
6.5
0.1
Infinite
4.0625
0.01
Reynolds Number
Figure 7: Graph of the drag coefficient vs. Reynolds number for each aspect ratio.
13
log(Cd) vs. log(Re) (8 1/8" Exposure)
0.4
y = -2.3829x + 12.345
0.3
2.03125
0.2
2.7083
log(Cd)
0.1
0
-0.1 4
-0.2
4.5
5
y = -1.6803x + 8.4728
5.5
6
4.0625
Linear
(2.03125)
-0.3
-0.4
y = -1.6506x + 8.5094
-0.5
-0.6
log(Re)
Figure 8: Graph of log(Cd) vs. log(Re) for the 8 1/8 inch exposure cylinders.
log(Cd) vs. log(Re) (13" Exposure)
0.6
0.4
y = -3.9779x + 19.626
0.2
3.25
log(Cd)
0
-0.2 4
4.5
5
5.5
6
6.5
-0.4
Linear (3.25)
-0.6
-0.8
-1
-1.2
4.3333
Linear (4.3333)
y = -2.0116x + 10.044
y = -3.3467x + 16.31
log(Re)
Figure 9: Graph of log(Cd) vs. log(Re) for the 13 inch exposure cylinders.
14
Linear (6.5)
Figure 10: Graph of the drag coefficient vs. Reynolds number for each aspect ratio compared to an
infinite rough cylinder (dashed line).
Table 1: Experimental Materials
Items Purchased
2” PVC pipe (2 ft. long)
3” PVC pipe (2 ft. long)
4” PVC pipe (2 ft. long)
Two 2” PVC flat caps
Two 3” PVC flat caps
Two 4” PVC flat caps
1” x 6” x 6’ wooden board
5/16”-18 Threaded rod
16 corner braces and screws
Roll-On Black Ink (Office Depot)
Total Price: $61.71
Additional Materials
Epoxy
Scrap wood
Scrap metal
Wooden screws
Slow-Recovery Super-Cushioning
Polyurethane Foam
15
Table 2: Data of height difference and velocity at a corresponding power percent
Power Percent
0
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
Height Difference (m)
0
0.0014
0.0033
0.0069
0.0095
0.0161
0.0243
0.03
0.0424
0.0523
0.0643
0.0783
0.0939
0.1085
0.1248
0.1429
0.163
0.1839
0.2047
0.2277
Velocity (m/s)
0
4.735
7.270
10.512
12.334
16.057
19.726
21.918
26.057
28.940
32.089
35.410
38.777
41.683
44.705
47.837
51.090
54.267
57.254
60.385
Table 3a: Data for the 2.03 Aspect Ratio
Re
108012.4
132693.1
147438.3
175280.6
194674
215856.7
238196.5
260845.6
280393.8
CD
1.650218
1.133379
0.941051
0.688206
0.590511
0.491107
0.429252
0.377807
0.342534
Table 3b: Data for the 3.25 Aspect Ratio
Re
70712.26
82968.51
108012.4
132693.1
147438.3
175280.6
CD
1.736824
1.400757
0.921735
0.635075
0.472982
0.254505
Table 3c: Data for the 2.708 Aspect Ratio
Re
110578.7
131460.4
146005.5
161892.5
178647.3
195634.2
210295.3
CD
2.046662
1.417152
1.111862
0.889766
0.678753
0.543661
0.44143
Table 3d: Data for the 4.33 Aspect Ratio
Re
62226.39
81009.33
99519.84
110578.7
131460.4
CD
1.593092
0.822386
0.472516
0.291116
0.121088
Table 3e: Data for the 4.063 Aspect Ratio
Re
62226.39
81009.33
99519.84
110578.7
131460.4
CD
1.593092
0.822386
0.472516
0.291116
0.121088
Table 3f: Data for the 6.5 Aspect Ratio
Re
66346.56
73719.15
87640.29
97336.99
107928.4
119098.2
130422.8
140196.9
CD
2.121165
1.680276
1.081307
0.776853
0.510928
0.337327
0.188801
0.087026
16
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