analzying lift over a rotating cylinder
TRANSCRIPT
1
Analyzing Lift on a Rotating Cylinder
Department of
Mechanical and Aerospace Engineering
University of California: Irvine
by
Nicholas Cordero
December 9, 2014
2
Table of Contents
Nomenclature List………………………………………………….…. 2-3
Abstract…………………………………………………………….….. 3-4
Chapter 1: Introduction………………………………………….…….. 4-5
1.1 Background and Previous Work…………………………….... 5-6
1.2 Theoretical Presentation……………………………………... 6-7
Chapter 2: Experimentation…………………………………………... 7
2.1 Experimental Apparatus and Procedure…………………...….. 7-10
2.2 Results of Experimentation………………………….......….... 10-12
2.3 Interpretation of Results………………………………...…... 12-14
Chapter 3: Conclusion and Recommendations ………………...…..... 14
Acknowledgements…………………………………………...…….... 14-15
List of References………………………………………………….… 15
Appendix A……………………………………………………..……..15
A.1 Mathematical Derivation……………………………..….……15-16
Appendix B……………………………………………………..……. 17
B.1 Uncertainty Analysis …………………………………….…..17
B.2 Calibration……………………………………………..….. 18
B.3 Step-by-Step Computations……………………………….…. 18
Appendix C…………………………………………………………... 18
C.1 Computer Program(s) Used…………………………………...18
C.2 Charts, Materials, Schematics, Other Material,Etc…………........18-22
Nomenclature List
( ) Speed Ratio, non-dimensional
Coefficient of Lift per Unit Length, non-dimensional
Coefficient of Pressure per Unit Length, non-dimensional
3
Coefficient of Normal force per Unit Length, non-dimensional
R Radius of Cylinder,
Radial component of velocity,
Tangential component of velocity,
RPM Revolutions per Minute
Lift Force per Unit Length,
Density,
Free stream velocity,
Velocity,
S Platform Area,
Vortex,
Re Reynolds number, non-dimensional
θ Theta, radians or degrees
P Pressure
T Temperature
FA Axial Force
FN Normal Force
Abstract
The purpose of this experiment is to analyze the lift per unit length over a rotating
cylinder with various RPMs and different airspeeds and comparing it to the analytical potential
flow theory which is defined by a 2-D, inviscid, incompressible flow as well as an to experiment
done by Aoki and Ito (2001). A 1.5 inch diameter by 6 inch length hollow polypropylene
cylinder is tested in a low speed wind tunnel while it is rotated at RPMs of 0, 645, 850, 940, and
1725 each with incoming airflow speeds of 6.1, 11.1, 12.3, 13.8, 14.8, 16.8, 19, and 21.7 m/s. In
order to have a general relationship lift as a function of the rotation rate of the cylinder and
varying incoming air velocity, a non-dimensional speed ratio ( ) is defined as the rotation of the
cylinder over the incoming air velocity and then related to the non-dimensional coefficient of lift
per unit length. Experimental results show that as the speed ratio increases, the coefficient of lift
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per unit length increases as well but is not as high as the coefficient of lift per unit length
predicted by the potential flow theory due to the assumption of a 2-dimenionsal inviscid flow.
Experimental results are very similar to the experimental results from Aoki and Ito (2001) but
slightly off in magnitude again, which is because the cylinder, or bracket holding the cylinder,
used in our experiment does not span the width of the test section like the cylinder in Aoki and
Ito’s experiment does. Overall, it can be seen that the coefficient of lift per unit length or lift per
unit length of the cylinder increases with increasing speed ratio for a Reynold’s number range of
.
Chapter 1:
Introduction
Lift over a rotating cylinder is caused mainly by the pressure distribution on the cylinder.
As the cylinder encounters airflow, the air molecules near the surface of the cylinder tend to stick
to the surface due to viscous effects and form a small boundary layer. When the cylinder begins
rotating say in a counter-clockwise direction as shown in Figure1 below, the air molecules on the
surface tend to pull or assist the incoming air molecules from the upstream airflow near the upper
surface of the cylinder while air molecules on the bottom surface of the cylinder retard the flow.
This causes the velocity over the upper surface of the cylinder to be faster than the velocity at the
bottom surface of the cylinder, which according to Bernoulli’s equation means that the pressure
at the bottom surface of the cylinder is higher than that of the upper surface. This uneven
pressure distribution on the surface of the cylinder creates a normal component of force on the
cylinder which is commonly known as lift.
Lift over a rotating cylinder is a phenomenon that can be seen in many applications today
such as sports and can be beneficial for conserving energy. In basketball when the player is
shooting the basketball into the basket, usually the chances of the basketball going into the
basket are higher when the arc trajectory of the basketball is high. The regulation diameter of a
basket is 18 inches while the regulation diameter for a Men’s basketball is 9 inches. If the arc
trajectory of the basketball is high enough such that the basketball just before entering the basket
is coming straight down (vector coming from basketball point down and is perpendicular to
basket) into the basket, there is 9 extra inches of space for the basketball to go through the
basket. Now if the arc trajectory is low, meaning that the basketball enters the hoop at an angle,
there is less than 9 inches space for the basketball to enter through the hoop and the chances of it
going in are lower. For this reason, most basketball players have a high arc trajectory when
shooting the ball and they achieve this high arc trajectory by putting spin on the ball with the
flick of their wrist. This spin generates the lift needed to aid the ball into a higher trajectory. This
can be seen in other sports as well such as baseball, when the pitcher throws the baseball with
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spin allowing it to curve in a certain direction and confuse the batter, or in soccer when the
player kicks the ball in such a way as to apply spin to the ball and make it into the goal by
curving around the goalie.
Back in the 1920’s, German engineer Anton Flettner decided to use the phenomenon of
lift over a rotating cylinder to his advantage by designing a ship that uses rotorsails for marine
propulsion. These rotorsails large hollow cylindrical components that are powered by engines
and use the incoming airstream to help generate the lift needed to propel the ship through the
ocean. The advantage of using these rotorsails for transportation on the ocean rather than on land
is that the wind is stronger over the ocean as compared to land, meaning that the rotorsail
propulsion technology over the ocean is very efficient since it uses nature as a source of energy.
Even in contemporary marine propulsion, rotorsails are still used by ships such as the E-Ship
which was launched in 2010 by a wind turbine manufacturer company known as Enercon. With
increasing fuel prices and average ocean wind speeds continuing to increase at a rate of 0.25%
per year due to climate changes, the future of rotorsail propulsion technology seems to be a very
efficient and environmentally friendly solution for conserving energy for marine transportation.
Figure1 Flow over a rotating cylinder
1.1 Background and Previous Work
Lift over a rotating object is also known as the Magnus effect, which was first described
in 1852 by a German physicist known as Heinrich Gustav Magnus and observed by Isaac
Newton in 1672 when watching tennis players at Cambridge college. Throughout history,
various experiments have been done to achieve a better understanding of lift over a rotating
cylinder. In 2001, Aoki and Ito conducted an experiment to analyze the flow characteristics of a
rotating cylinder by numerical analysis as well as by experimentation. The cylinder used in their
experiment was a hollow cylinder made of acrylic resin and had a diameter of 42.6mm
(approximately 1.677 inches). The test section used was 1m in height, .3m in width, and 1m in
length with the cylinder spanning the length of the test section, making the analysis of the flow to
be 2-dimensional. The lift was measured by a load cell attached to a linear shaft connected to the
cylinder, while the velocity of the airflow in the wind tunnel test section was measured using a
hot-wire anemometer and the rate of rotation of the cylinder was measured by a digital
tachometer. The experiment was done for velocity increments of 5m/s starting with 5m/s and
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ending at 50m/s, while the rate of rotation of the cylinder was varied from 0-10000 revolutions
per minute (RPM). The experimental results yielded a relation between the coefficient of lift per
unit length ( ) of the cylinder and plotted as a function of the speed ratio, which is a non-
dimensional defined as the ratio of the rotation of the cylinder over the incoming air velocity.
The experimental results showed that as the speed ratio increases, the coefficient of lift per unit
length increases as well.
1.2 Theoretical Presentation
One analytical approach to solving the coefficient of lift per unit length for a rotating
cylinder can be found from the potential flow theory. The potential flow theory applies to a 2-
dimensional, inviscid, incompressible flow which yields a corresponding velocity potential (ϕ)
and stream function (ψ) for the flow that satisfies Laplace’s equation. With this velocity potential
and stream function, an elementary flow can be defined and also added or superimposed with
other elementary flows to create more complex flows, one being the flow over a rotating
cylinder. The flow over a rotating cylinder can be synthesized by superimposing the flow over a
non-lifting, or non-rotating, cylinder with a vortex of strength Γ as shown in Figure 2 below. The
non-lifting flow over a cylinder is synthesized by combing a uniform flow of free-stream
velocity V∞ with a doublet flow. From the corresponding stream and potential functions of each
elementary flow, the velocity field can be obtained by taking the derivative of the functions or if
the velocity field is already calculated for each elementary flow they can be added together to
yield velocity fields for other elementary flows. By adding the velocity field for the flow over a
non-lifting cylinder and vortex, the velocity field for the flow over a rotating cylinder will be
given. The velocity field can be broken up into tangential (Vθ) and radial (Vr) velocity
components and the velocity (V) at the surface (r=R) of the cylinder can be determined by taking
the magnitude of these components. Since the radial velocity at the surface of the cylinder is
zero, the velocity at the surface is simply the tangential velocity (V= Vθ) and with this velocity at
the surface of the cylinder, the pressure coefficient per unit length (cp) can be obtained. The
pressure coefficient can then be integrated around the cylinder to yield the coefficient of lift per
unit length of the cylinder. It can be seen that the coefficient is directly proportional to the vortex
strength and inversely proportional to the radius (R) of the cylinder and incoming free-stream
velocity. With this coefficient of lift per unit length, the lift per unit length of the cylinder can be
determined which further shows that this lift per unit length is directly proportional to the free-
stream velocity, free-stream density, and vortex strength and is known as the Kutta-Joukoski
theorem (See Appendix A for detailed mathematic derivation of coefficient of lift per unit
length). In this experiment, the circular cylinder will be modeled as a vortex with strength Γ,
which is essentially the rate of rotation of the cylinder.
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Figure 2 Flow over a non-lifting, or rotating, cylinder plus a vortex of strength Γ yielding flow
over a rotating cylinder
Chapter 2:
Experimentation
2.1 Experimental Apparatus
1. The Low Speed Wind Tunnel, provided by University of California, Irvine, used for
this experiment uses air drawn from the open room and that same air is exhausted back out.
Inside the wind tunnel is a closed test section for steady airflow in the downstream direction. In
order to keep the downstream airflow in steady state, there is a honeycomb section, which
reduces swirl in the flow, and a screen to dampen velocity fluctuations at the entrance of the
wind tunnel. A nozzle is placed directly after the entrance to accelerate airflow and to thin out
the boundary layer thickness for the test section area. A diffuser is placed after the test section to
minimize the pressure drop and a variable-speed fan is used to control a steady, user-specified
speed in the test section.
2. The Setra Differential Pressure Transducer used in this experiment will be used to
measure the air velocity in the test section. This is done by using Bernoulli’s Equation in which
if the stagnation pressure, static pressure, and density of fluid are known, the velocity can be
calculated. There are two configurations for the Setra Differential Pressure Transducer, one
measuring the velocity with a Pitot-tube and the other measuring the velocity using the static
taps. Measuring the velocity using the Pitot-tube is more accurate than using the static taps
because the Pitot-tube uses stagnation pressure measured within the test section while the static
taps uses the ambient pressure as the stagnation pressure. There is a drop in stagnation pressure
when the flow goes from the inlet of the wind tunnel to the test section because of the frictional
losses that occur through the honey comb and screen. This drop means that the stagnation
8
pressure in the test section is less than the ambient pressure and if the static taps are used, the
velocity reading will be higher than what the velocity really is in the test section. Although the
Pitot-tube is more accurate, using it may interf with the incoming flow and hence lead to error in
the lift measurements which is why the will be used instead. However, the relationship between
the measured velocities from the static taps and Pitot-tube were available in lab which were used
in order to set the appropriate velocities needed for the experiment.
3. The 6-degree of freedom Force and Moment Transducer, also known as the sting
balance, used in this experiment measures the axial force (Fz), two normal forces (Fx and Fy), and
three corresponding moments, which are usually known as the roll, yaw, and pitch. These
measurements come from the change in tension or compression from the strain gauges in the
sting balance, which in turn outputs a voltage that the ATI software from the computer processes
and turns into forces and torques by the use of vector math. The only force that will be of interest
for this experiment is the normal force (Fx), which is essentially the lift force. The schematic for
the sting balance is shown in Figure 7 in Appendix C.
4. The motor used for this experiment to rotate the cylinder was a 400-3500RPM, 3-12V,
High Torque Cylinder Electric Mini DC Motor and was bought on amazon. The motor was 1.3
inches in diameter and .7 inches in length, with a shaft diameter of .08 inches and net weight of
40 grams.
5. A Neiko 20713A Professional Digital Laser Photo Non-Contact Tachometer was used
to measure the revolutions per minute (RPM) of the cylinder which has a range of 2.5-99,999
RPM with an accuracy of .05% and detecting distance of 50-500mm (or 1.9685-19.685 inches).
White reflective strips as well as batteries were provided in the Laser Tachometer kit.
6. A standard breadboard was used to wire the electronic components used to control the
motor. The electronic components include a 5V voltage regulator, 10K Ohm 15 turn
potentiometer, 2.5K Ohm resistor, diode as well as 12V power adapter to supply power to the
motor (9V Battery was also used). The RPM of the motor shaft was able to be controlled by
turning the potentiometer. The breadboard and electronic components can be seen in Figure 11 in
Appendix C.
6. A 1.5 inch diameter plastic hollow cylinder was used which is made of polypropylene.
The final length of the cylinder was 6 inches when the capped ends (which will be explained in
more detail below) were attached to the ends.
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7. The bracket for the cylinder (shown in Figures 8,9, and 10 of Appendix C) was
designed to be as light weight as possible in order to make sure it did not break the sting balance
since the maximum Ty for the sting balance is 500N*mm. It was designed on Solidworks and
then 3-D printed at UCI FabWorks with a density infill percentage of 25% so that it was light
weight but at the same time sturdy. The bracket holds the motor in place at one end by press
fitting it into the larger diameter hole (1.3 inches) while the smaller diameter hole (.30 inches)
holds the carbon fiber rod which is press fitted as well. Two 1.5 inch diameter by .25 inch height
capped ends were 3D printed as well with one capped end having a center hole of diameter .08
inches to fit the motor shaft and the other capped end having a center hole of diameter .3 inches
to fit the carbon fiber rod. The capped ends were super glued to the ends of the cylinder
concentrically, making the overall length of cylinder 6 inches. The cylinder was placed in
between the bracket, where the motor shaft and carbon fiber rod held it in place. Since the hole
on the capped end for the motor shaft came out to be a little big, the motor shaft was inserted into
the hole and then super glued to ensure that it was tightly secured. The 3-D printed model of the
bracket can be seen in Figure 12 in Appendix C.
Experimental Setup
Attach two reflective strips, one on top of the other, to the middle section of cylinder.
When using the Laser Tachometer to measure the RPM of the cylinder, the RPM displayed by
the Laser Tachometer must be divided by two since two reflective strips are on the cylinder.
Dividing by two will give the actual RPM of the cylinder. Starting with the sting balance in the
horizontal position (0˚), secure the bracket-cylinder assembly onto sting balance with set screw
and washer (thin square plate), ensuring that cylinder axis is parallel to test floor section. Have
the breadboard on a table near the test section and when closing the test section door, be aware of
the wires that connect the motor to the breadboard so that they do not get cut by door. Connect
power adapter to an outlet and have the leads of the power adapter ready to connect to
breadboard to power motor when experiment starts. Since the static taps are being used to
measure velocity, make sure one of the tubes connected to the Setra Differential Pressure
Transducer is open to the atmosphere while the other should be connected to static taps. Obtain
density from excel sheet by entering the temperature, pressure, and humidity values read by
sensor in lab and input density into Labview. To compensate for the tare weight of assembly,
buck out all forces on Labview, except for Ty (Ty cannot exceed 500N*mm), while the wind
10
tunnel is off (0m/s) and be sure to buck out the speed as well. Make sure test section is closed
and no one walks in front of wind tunnel when experiment is running. Once setup is complete,
the procedures can be followed. Pictures of experimental setup can be seen in Figures 13 and 14.
Experimental Procedure
1. Starting with the cylinder at 0 RPM (non-rotating) and sampling rate set to 1000Hz, turn
on the wind tunnel and adjust speed so that a speed of approximately 6.1 m/s (Conversion
to actual test section speed is made on excel sheet) is read on Labview.
2. Once speed is obtained, change sampling rate to 5000Hz and then record the mean value
of Fx.
3. Repeat steps 1-2 for wind tunnel speeds of approximately 11.1, 12.3, 13.8, 14.8, 16.8,
19, and 21.7 m/s
4. Turn off wind tunnel and open the test section.
5. Have one person connect the power adapter leads to breadboard so that motor is powered
and turn the potentiometer to adjust speed of motor while another person measures the
RPM with the Laser Tachometer pointed at the white reflective strips on cylinder.
6. Adjust potentiometer accordingly until the Laser Tachometer displays a value of
approximately 1290 RPM, which after dividing by two yields the actual RPM of cylinder
(645RPM).
7. Once RPM is set and the cylinder is spinning, close test section and turn on the wind
tunnel.
8. Adjust speed to 6.1m/s, and record the mean value of Fx.
9. Repeat step 8 for wind tunnel speeds of approximately 11.1, 12.3, 13.8, 14.8, 16.8, 19,
and 21.7 m/s. If motor burns out and cylinder stops spinning during the experiment,
disconnect power adapter leads and wait approximately 3-5min for motor to cool down
then reconnect leads to power motor.
10. Repeat steps 4-9 for RPMs of 1700, 1880, and 3450 displayed by Laser Tachometer,
which after dividing by two yields actual RPMs of 645, 850, 940, and 1725.
2.2 Results of Experimentation
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Figure 3 Plot of Theoretical Lift per Unit Length as a function of Air Speed at various RPM’s
Figure 4 Plot of Experimental Lift per Unit Length as a function of Air Speed at various RPM’s
0
2
4
6
8
10
12
0 5 10 15 20
Th
eore
tica
l L
ift
per
Un
it L
ength
(N/m
)
Air Speed (m/s)
850 RPM
940 RPM
645 RPM
0
0.5
1
1.5
2
2.5
3
3.5
4
0 5 10 15 20
Exp
erim
enta
l L
ift
per
Un
it L
ength
(N/m
)
Air Speed (m/s)
850 RPM
940 RPM
645 RPM
0 RPM
12
Figure 5 Plot of Experimental and Theoretical Coefficient of Lift per unit length as a function of
Speed Ratio
Figure 6 Digitized Plot of Experimental Coefficient of Lift per unit length as a function of Speed
Ratio at a Reynold’s number of from Aoki and Ito (2001) (See Figure 15 in Appendix C
for the non-digitized actual plot)
2.3 Interpretation of Results
It can be seen from the experimental results in Figure 3 that the lift per unit length of the
cylinder increases with increasing velocity and increasing RPM as well. The results from Figure
3 almost seem to vary linearly as a function of increasing velocity for each RPM. Further
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.2 0.4 0.6 0.8
Co
efff
icie
nt
of
Lif
t p
er U
nit
Len
gth
(cl)
Speed Ratio (α)
TheoreticalCl850 RPM
940 RPM
645 RPM
0
0.2
0.4
0.6
0.8
1
1.2
0 0.2 0.4 0.6 0.8 1 1.2
Exp
erim
enta
l C
oef
fici
ent
of
Lif
t
per
Un
it L
ength
fro
m A
ok
i an
d
Ito (
2001)
(cl)
Speed Ratio (α)
13
generalizing the relation between lift as a function of air speed and RPM, Figure 5 shows a non-
dimensional plot of coefficient of lift per unit length versus speed ratio (α). By doing this, it can
be seen that as the speed ratio increases, the coefficient of lift per unit length increases as well
and that the plot seems to vary linearly as a function of increasing spin ratio. Comparing Figure 3
to the theoretical potential flow theory results in Figure 4, it can be seen that there is some error.
The potential flow theory predicts a higher lift per unit span with increasing air flow velocity and
RPM than that produced by experimental results. The same error applies to Figure 5 where the
theoretical plot shows a higher coefficient of lift per unit length with increasing speed ratio than
that of experimental results. This error is due to assumptions made for the potential flow theory
that do not quite apply to experimental solutions. One of the assumptions was that the flow is
inviscid, but in reality viscous effect do indeed play a role on the lift for the rotating cylinder.
Since the experimental results are for a range of Reynold’s number of
, it can be assumed that a boundary layer forms near the surface of the cylinder due to
viscosity. When the cylinder is modeled as a point vortex for the potential flow theory, it says
that the velocity at each point of the surface of the cylinder is non-zero but in reality, the velocity
within the boundary layer section on the surface of the cylinder is zero due to the no slip
condition from viscosity. This means that shear stresses get introduced within the boundary layer
and contribute to a downward force on the cylinder meaning a decrease in lift or coefficient of
lift per unit length of the cylinder which can clearly be seen in Figures 3 and 4 when compared to
potential flow theory results. Another assumption made from the potential flow theory was that
the flow is 2-dimensional. Since the cylinder spanned the length of the holder, 3-dimensional
flow effects from the cylinder can be neglected but since the holder itself did not span the width
of the test section, 3-dimensional flow effects cannot be neglected for the holder which brings in
another source of error. Although there is error caused by making these assumptions, the same
trend between both theoretical and experimental results can be seen: the coefficient of lift per
unit length increases with increasing spin ratio.
Figure 6 shows experimental results from Aoki and Ito (2001) that can be compared to
our experimental results in Figure 5. By comparing the plot of Figure 6 , which is for a slightly
higher Reynold’s number than our range, it can be seen that the error when comparing to
experimental results in Figure 5 is far less than the error associated when comparing it with the
theoretical plots. The coefficient of lift per unit length from Aoki and Ito’s experimental results
14
are still slightly higher than our experimental results, due to the fact that Aoki and Ito’s test
cylinder spanned the width of the test section meaning 3-dimensional flow effects can be
ignored. This means that the holder or cylinder not spanning the width of the test section leads to
a smaller coefficient of lift per unit length than that of a cylinder spanning the width of a test
section. Also Aoki and Ito used an acrylic resin cylinder which is different than the material of
cylinder we used for experimentation and introduces another variable, surface roughness, to
consider which is not quantified in Aoki and Ito experiment as well as our experiment.
Chapter 3:
Conclusion and Recommendations
Overall, whether assuming flow is inviscid or visous and assuming a 3-dimensional flow
or 2-dimensional flow, the coefficient of lift increases with increasing spin rate regardless. In
order to eliminate 3-dimensional effects for the contribution of lift on the rotating cylinder, it
would be best for the holder to span the width of the test section so that only 2-dimensional
effects contribute to the lift. There also were some vibrations caused by the motor attached to the
holder which lead to fluctuations in the Fx or lift force read by the sting balance. These vibrations
could have been reduced by having some sort of dampening material between the motor and the
holder. The potential flow theory is a somewhat unrealistic model to follow since in reality
viscous effects do play a role on lift from the rotating cylinder, but is a helpful model in
understanding the trend that can be found on lift over a rotating cylinder. A better theoretical
solution for lift over a rotating cylinder can be obtained by using the Navier-Stokes equations
and including the viscosity term however, this process requires numerical analysis and can be
quite complex to solve.
Acknowledgements
I would like to take the opportunity to thank instructor Alejandro Puga for teaching me
the concepts of aerodynamics and wind tunnel testing. He provided me with invaluable
knowledge which helped me become a better experimenter throughout the couse.
A special thanks to teaching assistant Baolong Nguyen for his time and effort in helping
me perform this experiment as well as to my group members, Enrique Gurrola, Jennifer Song,
Khai Dao, Kevin Ren, Justin Williams, and Max Sun, for contributing to the successful
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completion of the experiment. Another special thanks to Sarah Hovsepian, director of Fabworks
at UC Irvine, for allowing us to use the Airwolf 3D printers to print our bracket.
I greatly appreciate the University of California, Irvine allowing us me use their wind
tunnel facility for this experiment.
List of References
Abbot, Ira H., “Report No. 824”, NACA
Anderson, John D. Jr. Fundamentals of Aerodynamics. New York: McGraw-Hill, 1991. Print.
"CiNii Article - Flow Characteristics around a Rotating Cylinder." CiNii Articles. Aoki and Ito.
Web. 9 Dec. 2014. <http://ci.nii.ac.jp/naid/110000031224/en>.
"Flettner Marine Propulsion." Monorotor Wind Propulsion for Cargo Ships. Web. 9 Dec. 2014.
<http://www.monorotor.com/history/>.
"Lift of a Rotating Cylinder." Lift of a Rotating Cylinder. Web. 9 Dec. 2014.
<http://www.grc.nasa.gov/WWW/k-12/airplane/cyl.html>.
"Magnus Effect." Wikipedia. Wikimedia Foundation, 29 Nov. 2014. Web. 9 Dec. 2014.
<http://en.wikipedia.org/wiki/Magnus_effect>.
"NBA.com - The Game Court." NBA.com - The Game Court. Web. 9 Dec. 2014.
<http://www.nba.com/canada/Basketball_U_Game_Court-Canada_Generic_Article-
18039.html>.
"Rotor Ship." Wikipedia. Wikimedia Foundation, 12 Mar. 2014. Web. 9 Dec. 2014.
<http://en.wikipedia.org/wiki/Rotor_ship>.
Appendix A:
A.1 Mathematical Derivation
Derivation of theoretical coefficient of lift and lift per unit length for a cylinder
Radial and tangential velocities for the elementary flow over a rotating cylinder,
(
) ( )
(
) ( )
Velocity at surface of cylinder (r=R),
16
( )
Substituting V into coefficient of pressure per unit length equation,
(
)
( ( )
)
( ( ) ( )
(
)
)
Coefficient of lift per unit length equation with skin friction equal to zero is,
∫
∫
Converting to polar coordinates by substituting,
With,
and interval
Then,
∫
∫
Since cylinder is symmetrical, and are the same and equal to
∫
Substitute into ,
∫ ( ( ( )
( )
(
)
))
∫ (( ( )
( )
(
)
))
Since,
∫
∫
∫
The then yields,
Where the Lift per unit span is,
17
With S=2R,
Appendix B:
B.1 Uncertainty Analysis:
Error associated with Sting Balance,
, , , , ( )
,
Measured Values from Barometer
, , ,
( )
( ) ( )
(
)
Where
, and
, ((
)
)
( )
(
) ,
(
) ,
(
),
(
)
((
)
(
)
(
)
(
)
)
=
( ) ( )
( )
( ),
( ),
( ) ( )
((
)
(
)
(
)
)
(( ( )) ( ) ( ( )) ( ) ( ( ) ( )) ( ) )
18
By using , for the first Lift measurement with 850RPM and velocity
of 5.01m/s, so the error is
B.2 Calibration:
Since the group before us already calibrated the sting balance, no calibration was needed except
to tare out the weight of the bracket-cylinder assembly.
B.3 Step-by-Step Computations:
Changing RPM to vortex strength Γ in units of
Where,
Appendix C
C.1
The main computer programs used were excel and Labview.
C.2 Charts, Materials, Schematics, Other Material, Etc.:
Figure 7 Schematic of sting balance used for experiment
19
Figure 8 Top View of Bracket
Figure 9 Front view of Bracket
Figure 10 Bracket used to hold cylinder
20
Figure 11 Breadboard with electronical components used to control motor speed
Figure 12 Final 3-D printed model of bracket used to hold cylinder
21
Figure 13 Top view of bracket holding cylinder and attached to sting balance by a screw and
plate.
Figure 14 Final setup of breadboard on table with wires connected from breadboard to the motor
inside of holder in wind tunnel test section.
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Figure 15 Experimental Plot of Coefficient of Lift per unit Length from Aoki and Ito (2001)