1

Analyzing Lift on a Rotating Cylinder

Department of

Mechanical and Aerospace Engineering

University of California: Irvine

by

Nicholas Cordero

December 9, 2014

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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

15

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.

22

Figure 15 Experimental Plot of Coefficient of Lift per unit Length from Aoki and Ito (2001)