Joshua Beckerman Lab 4 – Fluid Mechanics Final Write-Up

Lab 4 – Fluid Dynamics
Final Write-Up
Joshua Beckerman
Dr. Glauser
MAE 315 – Mechanical and Aerospace Engineering Lab
Date of Experiment, Part 1: November 12, 2015
Date of Experiment, Part 2: November 19, 2015
Date of Submission: December 20, 2015

Beckerman Lab 4 – Fluid Dynamics
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Table of Contents
1.
Abstract................................................................................................................................... 4

2.
Introduction............................................................................................................................ 5

2.1 – Conservation of Energy................................................................................................................. 5

2.2 – Conservation of Mass .................................................................................................................... 7

2.3 – Conservation of Momentum ......................................................................................................... 8

2.4 – Forces on a Body Due to Fluid Flow ............................................................................................ 8

𝐶𝐷 = 2𝐷𝜌 𝑈02 𝑆.................................................................................................................................... 11

[2.13]....................................................................................................................................................... 11

2............................................................................................................................................................... 11

2.5 – Flow Characteristics .................................................................................................................... 11

2.5 – Airfoil Theory............................................................................................................................... 13

2.6 – Vortex Shedding........................................................................................................................... 16

2.7 - Uncertainty.................................................................................................................................... 18

3.
Procedure.............................................................................................................................. 19

Part 1: Force Balance Calibration, Boundary Layer, and Cylinder Wake..................................... 19

3.1.0
– Equipment.............................................................................................................................. 19

3.1.1 – Force Balance Calibration ....................................................................................................... 20

3.1.2 – Boundary Layer Test................................................................................................................ 21

3.1.3 – Cylinder Wake Test.................................................................................................................. 21

Part 2: Airfoil Test, Cylinder Drag Test, Vortex Shedding .............................................................. 22

3.2.0 – Equipment ................................................................................................................................. 22

3.2.1 – Free Stream Velocity ................................................................................................................ 22

3.2.2 – Airfoil Test (LabVIEW Program: Force_Balance_Airfoil.vi).............................................. 23

3.2.3 – Cylinder Drag Test ................................................................................................................... 23

3.2.4 – Vortex Shedding........................................................................................................................ 23

4.
Results................................................................................................................................... 24

4.1 – Force Balance Calibration .......................................................................................................... 24

4.2 – Boundary Layer Test................................................................................................................... 25

4.3
– Cylinder Wake Test.................................................................................................................. 26

4.4 – Airfoil Test.................................................................................................................................... 27

4.4
– Cylinder Drag Test................................................................................................................... 29

4.5
Vortex Shedding .......................................................................................................................... 30

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5.
Conclusion ............................................................................................................................ 32

6.
References............................................................................................................................. 33

7.
Appendix............................................................................................................................... 34

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1. Abstract
The goal of this lab is to look at some of the most prevalent concepts in the study of Fluid
Dynamics and how they relate to and are applied to everyday life. These principles are then
harnessed by engineers everyday. Whether it is in the form of “fins” on tractor trailers, used to
eliminate pressure drag an increase fuel efficiency or aerodynamic downforce used to keep
Formula 1 cars “glued” to the pavement, an understanding of Fluid Dynamics can be
instrumental in any engineer’s career and success.
This laboratory will explain concepts such as Bernoulli’s Equation, the Reynolds
Number, Strouhal Number, various flow characteristics, and airfoil theory. The purpose of
understanding such Fluid Dynamics principles is in order to be able interpret the results of a
number of experiments that will be outlined in the procedure section. Much of the experimental
procedure was focused on obtaining Coefficients of Lift, Drag, and Moment, then comparing
them, as well as looking at multiple velocity profiles and Von Kármán Vortex Street Shedding.
It was found that the Vortex Shedding Frequency was dependent upon the Reynolds
Number, which in turn affected the Strouhal Number. In the results section, a comparison of
methods for measuring the shedding frequency allowed for analysis as to the accuracy of
measurement devices with respect to the ideal calculated frequencies. Some of the other
computations and measurements, for example the cylinder wake velocity profiles, were quite
accurate and helped to give a clear and concise justification for many assumptions made about
how velocity profiles are formed. Another important observation was that the Drag Coefficient
for bluff bodies, such as cylinders, were collected with values that were reasonably close to
published values.

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2. Introduction
The objective of this laboratory is to gain a greater understanding of key concepts in
Fluid Dynamics, how those concepts relate to everyday phenomena, and the application of the
more complex theories to engineering. Before any of this can be done, a working knowledge of
the basics of Fluid Dynamics must first be established. This section will cover all of the
proprietary knowledge necessary to interpret the results of each experiment.
To start, a fluid is a substance that deforms continuously when acted on by a shearing
stress of any magnitude1
. This definition comes from Fundamentals of Fluid Mechanics, a
textbook used in many introductory-level courses on the subject. The definition provided is a bit
vague and parsing it out may help in the understanding of its true meaning. A fluid can be a
liquid or a gas (i.e. air, water, maple syrup, etc.) and, while solid mechanics involves idealized
point forces that translate into normal stress and shear stress, fluid mechanics deals in pressures
and shear stress.
Because all of the experiments conducted in this lab will be performed in a wind tunnel,
the fluid in question will be air. This makes things fairly simple in terms of determining
important initial conditions because the Ideal Gas Law can be applied.
𝑃 = 𝜌 𝑅! 𝑇 [2.1]1
Where P is the absolute pressure in Pascals (Pa = N/m2
), 𝜌 is the density in kg/m3
, Rg is the gas
constant (Rg = 287 J/kg-K), and T is the temperature in Kelvin. In this lab, the temperature will
and pressure will often be collected and the Ideal Gas Law will be used to find the density of air
under a particular set of conditions.
Much that is necessary for the understanding of Fluid Dynamics can be derived from the
three conservation laws: Conservation of Energy, Conservation of Mass, and Conservation of
Momentum. These three laws are where the theoretical explanation of fluids begins:
𝐸𝑛𝑒𝑟𝑔𝑦 𝐼𝑛 = 𝐸𝑛𝑒𝑟𝑔𝑦 𝑂𝑢𝑡
𝑀𝑎𝑠𝑠 𝐼𝑛 = 𝑀𝑎𝑠𝑠 𝑂𝑢𝑡
𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝐹𝑖𝑛𝑎𝑙 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚
[2.2]
[2.3]
[2.4]
2.1 – Conservation of Energy

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One of the most fundamental concepts in the study of fluids, Bernoulli’s Equation, can be
derived from the Conservation of Energy. Three of the most common forms of energy make up
Bernoulli’s Equation, but instead of being in terms of energy they are in energy per unit volume.
The components of Bernoulli’s Equation are pressure (force per unit area or energy per unit
volume), kinetic energy per unit volume, and potential energy per unit volume.
𝑃! +
𝜌 𝑉!
!
2
+ 𝜌𝑔ℎ! = 𝑃! +
𝜌 𝑉!
!
2
+ 𝜌𝑔ℎ!
[2.5]1
Bernoulli’s Equation is incredibly versatile in ideal situations. The issue is that there are
no truly ideal cases in the real world. There are always going to be losses that cannot be perfectly
quantified and other error of the like, but Bernoulli helps us to develop a realistic approximation
of what is occurring in many fluids problems.
Of course, Bernoulli’s Equation requires several assumptions in order to be applicable –
many of which one cannot readily be made about a realistic system. These assumptions are:
• The fluid in question is incompressible – fluid density remains constant.
• The fluid in question is inviscid – viscous effects are assumed to be negligible1
.
• The path from Point 1 to Point 2 is along a streamline – lines that are tangent to the
velocity vectors throughout the flow field1
(along the path of flow motion).
• The flow is steady – the flow velocity does vary with time
!"
!"
= 0 .
Viscosity is an inherent property of a fluid to resist gradual deformation by shear or
tensile stress.5
For example, maple syrup is more viscous than ethanol or water. From the
Bernoulli Equation, many of the concepts used in this lab and their governing equations can be
derived. One such example of this is the Pitot-static tube, which is a device that measured the
pressure of given system in order to find the
velocity of a flow. The Pitot-static tube is pointed
in opposite the direction of the flow, creating a
stagnation point. The stagnation point is a
specific location where the flow comes to rest2
–
the flow velocity is equal to zero. Creating
stagnation at the entrance to a Pitot-static tube is
Figure 2.1: Diagram depicting Pitot-static tube,
Static/Dynamic Pressure, and Velocity
https://www.grc.nasa.gov/www/k-12/airplane/Images/pitot.jpg

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necessary because it is the difference between total pressure and static pressure that allows the
use of Bernoulli’s Equation for calculating the velocity of the flow (see Fig. 2.1). This gives the
equation used for calculating the velocity of the flow hitting the Pitot-static tube:
𝑉 =
2 𝑃! − 𝑃!
𝜌
[2.6]
Where PS is the static pressure at the entrance to the Pitot-static tube (if the device were
in ambient air, PS would be equal to atmospheric pressure or zero if measured in gage pressure),
PT is the total pressure, and 𝜌 is the density of the fluid. The difference between total
pressure and static pressure is called the dynamic pressure.
Another important equation for the following experiments is the one needed to
convert fan speed in hertz to free stream inlet velocity in meters per second:
𝑉 = 1.134𝑓 − 1.9793 [2.7]3
Where V is the test section airflow velocity (m/s) and f is the fan velocity (Hz). This is a
characteristic of the particular wind tunnel that was used for the experimental results provided
later in this lab. Other wind tunnels will require a similar conversion, but it will likely not be the
same.
2.2 – Conservation of Mass
In order to use the conservation of mass for the next part of Fluid Dynamics, the concept
of a control volume must first be understood. A control surface is an imaginary boundary placed
around a system of interest created for the purpose of analyzing the inputs and outputs of that
system. A control volume is everything enclosed
within the control surface. Fig 2.2 shows an inlet and
an outlet cross-sectional area, velocity, and density.
These are a few of the key properties that would be of
concern when studying a control volume.
Expanding upon Eq. 2.2, it can be said that
the change in mass of the system is equal to the sum
of the rate of change of the mass of the control
Figure 2.2: Diagram depicting Control Volume
http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/cvanalysis/onedm.gif

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volume and the rate of flow of mass through the control surface.1
𝐷
𝐷𝑡
𝜌
!"!
𝑑𝑉 =
𝜕
𝜕𝑡
𝜌
!"
𝑑𝑉 + 𝜌
!"
𝐕 ∙ 𝐧 𝑑𝐴 = 0 [2.8]1
The first part of this equation is a mathematical expression for the conservation of mass
(because density multiplied by volume is equivalent to the mass) and the second part of the
equation is called the Continuity Equation. The continuity equation is important because it can
be used to solve problems involving geometrical changes in an otherwise steady system. For
example, in a problem like the one shown in Fig. 2.2, the continuity equation justifies the
intuitive notion that, in order to maintain a constant flow rate, a flow with a larger inlet cross
sectional area than its outlet will increase its velocity proportional to the decrease in area
(assuming incompressible – density remains constant). This is a powerful statement that will
soon be used for more profound applications.
2.3 – Conservation of Momentum
Newton’s second law of motion for a system says that the rate of change of linear
momentum of a system is equivalent to the sum of the forces on that system:
𝐷(𝑚 𝐕)
𝐷𝑡
= 𝑚 𝐚 = 𝐅!"! [2.9]
𝐷
𝐷𝑡
𝐕 𝜌
!"!
𝑑𝑉 =
𝜕
𝜕𝑡
𝐕 𝜌
!"
𝑑𝑉 + 𝐕 𝜌
!"
𝐕 ∙ 𝐧 𝑑𝐴 = 𝐅!"#$%#$& !" !"#
!"#$%"& !"#$%&
[2.10]1
Using the Linear Momentum Equation (Eq. 2.10) and the Continuity Equation (Eq. 2.8), it is
possible to solve for the forces involved in a given system. This will be shown in the following
section.
2.4 – Forces on a Body Due to Fluid Flow
In one of the experiments performed in this lab, a cylinder is placed into the wind tunnel.
The inlet flow has a uniform velocity profile, which means that the velocity has the same
magnitude at every location normal to the plane of motion (see Fig. 2.3a). In Fig. 2.3b, the
velocity profile is non-uniform.

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Though not shown in the figure, the only way for this to occur is if there is force acting in
the direction opposing the flow, which can be proved by the Linear Momentum Equation.
Evaluating Eq. 2.10 for this scenario, it can be determined that the force on the bottom of the
tunnel due to the flow is equivalent to:
𝐅!"! = 𝜌 𝑈!
!
− 𝜌
!
!
𝑢 𝑦 !
𝑑𝑦 [2.11]
As shown in the figure above, the velocity at the surface of the wind tunnel is equal to
zero. This is the no slip condition, which is an effect of viscous force interactions between the
fluid and the surface thus creating a shear effect (much like in the frictional forces seen in solid
mechanics). The deformation of the flow creates the parabolic velocity profile, seen in Fig. 2.3b.
The thin layer of fluid that comes in immediate contact with the surface of an object is called the
boundary layer.
In terms of flow around a cylinder, the force with which this lab concerns itself most is
Drag Force. Drag is the force on an object due to a fluid that acts in the direction opposite the
direction of flow motion. When skydiving, a person can only fall so fast because of the effects of
Figure 2.3: Example of (a) Uniform Velocity Profile and (b) Boundary Layer Effects (Unit Depth)
Altered from http://d2vlcm61l7u1fs.cloudfront.net/media%2Fefb%2Fefbc192a-c662-4e89-84c8-c9a9b90be243%2FphpfVq2mp.png

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drag. There is a drag force that increases as the person accelerates (due to gravitational forces).
When the drag force is equivalent to the weight force of a skydiver, they reach terminal velocity
(see Fig 2.4).
Drag force is important for any number of practical applications. For example, all
production and racecars are tested for their aerodynamic drag coefficient, a non-dimensional
quantity that is an inherent property of the object. For the purposes of this lab, Aerodynamic
Drag will be calculated for a much simpler object, the cylinder. Using Eq. 2.8, Eq. 2.10, and the
velocity profile shown in Fig. 2.5, the drag on a cylinder of unit length, L, due to a uniform
velocity profile of unit length, L, reduces to:
𝐷 = 𝜌 𝑈!
!
1 −
𝑢 𝑦
𝑈!
!
!
𝑑𝑦 [2.12]
Where U0 is the inlet free stream
velocity and u(y) is the non-uniform
velocity profile behind the cylinder. Eq.
2.12 can then be non-dimensionalized to
find the coefficient of drag, as
previously mentioned. The coefficient
of drag is useful because it allows for
Figure 2.5: Cylinder Wake and Non-Uniform Velocity Profile
http://www.ifh.uni-karlsruhe.de/science/envflu/research/shallow-flows/img/defnew.gif
Figure 2.4: Description of Drag Force
http://skydivin.wdfiles.com/local--resized-images/blog:_start/ctn.gif/medium.jpg

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the comparison of how two different objects move through the air based on their shape and
ability to “cut” through the air. This coefficient of drag can be incredibly important in everyday
life. The 2016 Toyota Prius (marketed as one of
the most efficient production cars) has a
coefficient of drag of 0.24, while the 2003
Hummer H2 (infamously one of the least
efficient production cars) has a coefficient of
drag of 0.574
(Fig 2.6).
The equation for the Coefficient of Drag for a
cylinder is as follows:
2
.
Where S is the frontal area of the object. In the
case of the cylinder, S is equivalent to the
product of the diameter and the length.
2.5 – Flow Characteristics
In Fluid Dynamics, it is quite difficult to characterize a flow simply by looking at the
geometry, the velocity, or physical properties concerning the fluid in question. For this reason,
there is a dimensionless quantity known as the Reynolds Number, which is a ratio of the inertial
forces to the viscous forces of the flow.
𝑅𝑒 =
𝜌 𝑉 𝐷
𝜇
[2.14]
Where 𝜌 is the density of the fluid, V is the velocity, D is the characteristic dimension,
and 𝜇 is the viscosity of the fluid. The Reynolds Number can tell a lot about the kind of motion
the flow is experiencing. The two main types of flow are Laminar Flow and Turbulent Flow. A
flow is considered laminar if the Reynolds Number is around 2100 or less. If the flow is laminar,
𝐶! =
2𝐷
𝜌 𝑈!
!
𝑆
[2.13]
Figure 2.6: (a) Hummer H2 – CD = 0.57
(b) Toyota Prius – CD = 0.24
http://car-pictures.cars.com/images/?IMG=CAB30HUS011A0101.png&WIDTH=624&HEIGHT=300&AUTOTRIM=1
http://yoursinglesourcefornews.com/wp-content/uploads/2015/09/2016-toyota-prius1.jpg

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this means that the fluid flows in parallel layers with no disruption between the layers. This can
be thought of as geese flying in formation. All of
the individual geese are flying in the same direction
relative to the flock (see Fig 2.7).
If the Reynolds Number is greater than
about 4000, this indicates a turbulent flow. If the
flow is turbulent, it means that, although the fluid is
moving in a particular direction, the movement of
the individual particles of fluid is far more chaotic.
Turbulent flow can be imagined as a swarm of bats flying through the sky. There is no order to
the motion of any particular bat, but the swarm still flies in a specific direction (see Fig 2.8). In
most cases, turbulence is additionally characterized
by swirling vortices, but this is not necessarily the
case. Vortices will be discussed in more depth soon.
Figure 2.7: Migrating Geese –Laminar Flow
https://karthijaygee.files.wordpress.com/2011/01/flok-of-birds-v-formation.jpg
Figure 2.8: Swarm of Bats – Turbulent Flow
https://iamryshel.files.wordpress.com/2015/02/img_0646.jpg

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Another non-dimensional number used to characterize the nature of a flow is called the
Strouhal Number. The Strouhal Number is specific to unsteady systems that have some sort of
oscillation associated with the fluid flow.
𝑆𝑡 =
𝑓 𝑑
𝑉
[2.15]
Where f is the frequency of oscillation, d is the characteristic dimension, and V is the inlet fluid
velocity. The right hand side of Eq. 2.14 only holds true for a specific range of Reynolds
Numbers (250 < Re < 20,000). The technical interpretation of the Strouhal Number is that it is a
ratio of the local inertial force to the convective inertial force. This technical jargon will be
cleared up once the types of flows characterized by a Strouhal Number are explored: unsteady,
oscillatory fluid motion.
2.5 – Airfoil Theory
Until this point, everything in the Introduction as been quite theoretical and, with the
exception of a few concepts, are difficult to apply to real-life scenarios. All of the preceding
information has been critical, yet useless to engineers without proper implementation. The
purpose for all of this is to now be able to analyze an airfoil – an object possessing the shape of a
wing, blade, sail, or other similar component designed for aerodynamic uses. Airfoils can either
be symmetric or asymmetric in design and are shaped to provide a scale model for whatever
usage it is meant to imitate.
Building onto the theory discussed earlier in this section, it was shown that forces on a
body could be calculated using the Continuity Equation (Eq. 2.8) and the Linear Momentum
Equation (Eq 2.10) in conjunction with the characteristics of the object being subjected to a
flow. This was proved for a cylinder, but the same can be done for an airfoil. The calculations
necessary to do so are beyond the scope of this lab so a special device that directly measures
forces and moments will be used during parts of the experiment. For now, it is easiest to continue
under the assumption that the forces needed for analysis will be measured directly during the
procedure. This being said, what are the forces and moments that act on an airfoil?

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In a wind tunnel, an airfoil can be
adjusted to various angles with respect to
the direction of the flow. This angle is
often denoted with the Greek letter, α,
and is called the Angle of Attack. When
the leading edge of the airfoil (the part
of the wing that comes in contact with
the air first) is pointed directly into the
flow, the Angle of Attack is 0°.
At this Angle of Attack, the force
on the airfoil due to the flow is all part of the drag force. However, if the Angle of Attack were to
be changed (either up or down), this would no longer be the case. The resultant force due to the
flow would not change its direction, but the force must be broken up into components: Lift Force
and Drag Force, as depicted in Fig. 2.9. The Pitching Moment is the final component to this
statics problem and is generated due to the flow as well, causing an overturning effect in the
airfoil.
Just as the Drag force can be translated into a non-dimensional parameter of the object
and conditions (i.e. Angle of Attack), so can the Lift force and Pitching moment:
𝐶! =
2𝐿
𝜌 𝑈!
!
𝑆
[2.16]3
𝐶! =
2𝑀
𝜌 𝑈!
!
𝑆 𝑥
[2.17]3
Where CL is the Coefficient of Lift and is defined quite similarly to the Coefficient of
Drag. CM is the Coefficient of Moment and is calculated with a moment arm of one quarter of the
total chord length (x = c/4). Also, in the case of the airfoil, the frontal surface area, S, is equal to
the product of the chord and the span. In this lab, there will be experiments using a NACA 0012
Airfoil. This particular airfoil is characterized by its symmetry, making it the simplest to study
and see the effects of various phenomena. The characteristic dimensions of the NACA 0012 are
as follows:
Figure 2.9: Lift, Drag, and Pitching Moment
http://www.dymoresolutions.com/dymore4_0/UsersManual/AerodynamicProperties/figures/AirTable_fig0.png

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NACA 0012 Airfoil Dimensions
Span [in] Chord [in]
14.875 ± 0.0313 6.000 ± 0.0156
Table 2.1: Airfoil Dimensions
The basic shape of an airfoil is a simple concept to grasp, but this shape was neither
chosen at random nor by accident. The shape of the airfoil is critical to how it behaves in a flow.
When an airfoil is positioned at a positive Angle of Attack, it creates a pressure difference
between the flow above the top surface and below the
bottom surface of the wing. This pressure difference,
justified by Bernoulli’s Equation, gives rise to a
difference in velocity between the upper and lower
flows such that the upper boundary layer grows
increasingly turbulent with increasing α, while the
lower boundary layer remains laminar (see Fig. 2.11).
As the Angle of Attack is increased, the boundary
layer on the top surface lifts up and away from the
trailing edge (the last part of the airfoil to come in contact with the air), moving up towards the
leading edge. Once the boundary layer completely detaches from the leading edge of the airfoil,
stall occurs. The resulting effect of stalling is a massive increase in aerodynamic drag, as well as
a dramatic decrease in lift.
When boundary layer separation occurs on any surface, two significant things arise:
pressure drag (drag force due to a difference in pressure between the two surfaces of the airfoil)
and trailing vortices (a phenomenon illustrated in Fig. 2.11 where turbulent flow creates swirling
fluid vortices, this will be covered in more depth in the next part). Some of the most significant
Figure 2.11: Turbulent and Laminar Flow on Airfoil
http://www.pilotfriend.com/training/flight_training/aero/images/15.jpg
Figure 2.10: Airfoil Geometry
https://www.grc.nasa.gov/www/k-12/airplane/Images/geom.gif

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differences in forces on a 2-Dimensional object and a 3-Dimensional object come from boundary
layer separation. For a 2-D object, boundary layer separation only occurs on the top and bottom
because the sides of the object span the length of the flow. A similar 3-D object (with boundary
layer separation on the top, bottom, and both sides) will have more drag than its 2-D counterpart
for the reasons previously stated. This concept will be explored in greater depth during the
experimental portion of the lab.
2.6 – Vortex Shedding
Vortex shedding, which was glossed over in earlier sections of the Introduction, is a
complex phenomenon that forces engineers to step outside of their assumption-riddled, steady
state “comfort zones”. A bluff body is a body that, as a result of its shape, has separated flow
over a substantial part of its surface and a very strong interaction between the viscous and
inviscid regions.4
On a bluff body, vortex shedding is an extremely prominent occurrence. This
was constantly happening on the hypothetical cylinders being examined in the velocity profile
discussion because circular and spherical objects are, in general, bluff bodies.
When vortex shedding exists on a bluff body, it occurs in an oscillatory manner. The
vortices are formed and shed at a frequency determined using the Strouhal Number. As the fluid
flows around the bluff body, the early boundary layer separation causes the fluid to turn in
towards the center of the body. The low-pressure zone behind the body then pulls in the fluid to
create a swirling shape that develops until it is pushed away from the body by a developing
vortex on the opposite end of the body. This alternating pattern of vortex shedding is what gives
rise to the harmonic nature.
Figure 2.12: Vortex Shedding on a Bluff Body
http://colonius.caltech.edu/fluidparticles.png

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This type pattern of alternating vortices is called a Von Kármán Vortex Street, which can
occur anywhere between 47 < Re < 107
. At some point downstream the viscosity of the flow
absorbs the energy lost by the creation of vortices. It is important to remember that vortex
shedding is a 3-D effect, which can be seen in Fig. 2.13.
It will be seen in this lab that these vortices do more than just
swirl and go away, in fact the oscillation in pressure due to boundary layer separation creates
temporary forces that act on the upper and lower surfaces of the bluff body. This periodic force
becomes a forcing frequency on the bluff body, which must be considered thoroughly in
engineering applications. For example, if the forced frequency at any operating speed on an
airfoil or other object were to approach the natural frequency, the object could be subjected to
resonance – thus ensuing a potentially dangerous situation. For a cylindrical bluff body, Vortex
Shedding Frequency and the Strouhal Number can be determined:
𝑆𝑡 =
𝑓 𝑑
𝑉
= 0.198 1 −
19.7
𝑅𝑒
[2.18]3
Figure 1.13: 3-D Von Kármán Vortex Street
http://www.jens-kasten.de/assets/PaperImages/FTLE-Cy2d-Combined-Height7-lowres.png

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2.7 - Uncertainty
The final measure of this report, which must be discussed, is the concept of uncertainty.
Uncertainty is the quantifiable error inherent in the devices and methods used to measure and
calculate result data. In this lab, uncertainty will be determined using a method known as the
Zeroth Order of Uncertainty. This method is crucial for understanding the accuracy of the
experiment and for those reviewing the results to assess the overall scope of the claims that can
be made when analyzing those results. Zeroth Order of Uncertainty is found using the following
equation:
𝑢!,! = ± (
𝜕𝑦
𝜕𝑥!
𝑢!,!!
)! [2.19]3
Here, u0 is the Zeroith Order of Uncertainty of y if y is a function of variables x through
xi. The ways in which Zeroith Order of Uncertainty plays a role can be seen quantitatively in the
Appendix section and qualitatively below in the Results section.

Page | 19
3. Procedure
During the second part of the lab procedure, the majority of the raw data collected will
not be in Force (N) or Moment (Nm), but instead measured in Voltage. This is because those
experiments rely on data collected from sensors and transducers. The analog signal is converted
into a digital signal that is read into the computer in terms of volts. It is up to the engineer to take
the data and establish a conversion into quantities that are useful in an analysis. This is why the
Force Balance and Calibration is performed first.
Part 1: Force Balance Calibration, Boundary Layer, and Cylinder Wake
In this part of the lab, the calibrations necessary for the second part will be performed.
Then the concepts of boundary layer and cylinder wake will be explored experimentally. Some
goals of this part of the lab are to understand how edges of the flow can differ from the laminar
flow and how objects can obstruct a flow.
3.1.0 – Equipment
• Closed Loop Wind Tunnel
• Pitot-Static Tube
• Aerolab Pyramidal Force Balance
o 3 Load Cells
• NACA 0012 Airfoil
• Daytronic System 10 DataPac
• Pressure Systems 9010
Optomux
• 3x 2” O.D. Cylinder
• Pitot Tube Rake
• Thermometer
• LabVIEW
Positive
Moment
Negative
Moment
Drag
Force
Lift
Force
Figure 3.1: Lift, Drag, and Moment Calibration Diagram

Page | 20
3.1.1 – Force Balance Calibration
1. Calibrating Lift:
a) Connect the Carriage Chord to the middle of the hole on the force balance.
b) Place the weight on the Carriage.
c) Run the LabVIEW Software.
d) Incrementally add 5 lbs. to the Carriage and collect each voltage reading until the
Carriage is holding 25 lbs.
2. Calibrating Drag:
a) Connect the Carriage Chord to the end hole on the force balance.
b) Place the weight on the Carriage.
c) Runt the LabVIEW Software.
d) Incrementally add 5 lbs. to the Carriage and collect each voltage reading until the
Carriage is holding 25 lbs.
3. Calibrating Moment:
a) Hang the 1 lb. weight by a string around the
first notch on the left side of the top metal
cylinder. Measure the distance away from the
center of the cylinder, as depicted in Fig. 3.2.
b) Run the LabVIEW Software.
c) Move the weight to the next notch, note the
distance away from the center of the cylinder,
and collect the measured voltage.
d) Repeat step (c) until the last notch on the
right side is measured.
-6 -5 -4 -3 -2
2 3 4 5 6 7
-1
1
Figure 3.2: Moment Calibration Setup

Page | 21
3.1.2 – Boundary Layer Test
1. Set the Wind Tunnel Speed to 25 Hz.
2. Secure the Pitot-Static Tube in the 1st
hole in the bottom of the test section of the tunnel,
which should be about 14” (35.56 cm) from the wall. Be sure to align the tube opening along
the direction of the free stream flow (parallel to this section of the tunnel).
3. With each position of the Pitot-Static Tube, collect one reading of temperature and five
readings of pressure (these pressures will be averaged in the analysis of a more reliable
reading due to constant variations in pressure).
4. Once the readings are collected for the initial position, lower the Pitot-Static Tube to the
positions in Table 3.1 and collect the necessary data.
Tube Height [in] 16 13 10 7 4 3 2 1 0.5 0.0625
Tube Height [cm] 40.64 33.02 25.4 17.78 10.16 7.62 5.08 2.54 1.27 0.15875
Table 3.1: Pitot-Static Tube Positions, Boundary Layer Test
3.1.3 – Cylinder Wake Test
1. This section of the laboratory will be run twice: once at a wind tunnel speed of 25 Hz and
once at a speed of 45 Hz.
2. Measure the distance between each tap on the rake. In this lab, the data collected from every
other tap on the rake will be used in the analysis. The measurement of the spread should be
similar to the following:
Pressure Rake Tap Position [cm]
(measured from left)
Tap # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Position 0 2 4 6 8 10 12 13 14 15 16 17 18 19 20 22 24 26 28 30
Table 3.2: Pressure Rake Tap Positions, Cylinder Wake Test
*Note that there are 40 total taps along the 30cm rake, but only 20 are used in this test.
3. Place rake within the wind tunnel (without the cylinder in place).
4. Run the LabVIEW software and record the pressure measurements 20 times (again, these
will be averaged for accuracy).

Page | 22
5. Secure the cylinder 8 diameters ahead of the rake in the wind tunnel.
6. Run the LabVIEW software and record the pressure measurements 20 times, this time with
the cylinder in place.
7. Repeat the Cylinder Wake Test for the other wind tunnel speed.
Part 2: Airfoil Test, Cylinder Drag Test, Vortex Shedding
3.2.0 – Equipment
• Closed Loop Wind Tunnel
• Pitot-Static Tube
• Aerolab Pyramidal Force Balance
o 3 Load Cells
• NACA 0012 Airfoil
• Daytronic System 10 DataPac
• Pressure Systems 9010 Optomux
• Five 4.8 cm O.D. Cylinders of Length:
o 10.5” (26.67 cm)
o 18” (45.72 cm)
o 20” (50.8 cm)
o 22” (55.88 cm)
o 24” (60.96 cm)
• Pitot Tube Rake
• Thermometer
• LabVIEW
• Accelerometer
• Two High Frequency Pressure Transducers
3.2.1 – Free Stream Velocity
Use the pressure readings from part 1 to calculate the free stream velocity for a given fan speed.

Page | 23
3.2.2 – Airfoil Test (LabVIEW Program: Force_Balance_Airfoil.vi)
1. Place the NACA 0012 Airfoil on the top of the force balance inside the wind tunnel.
2. Set the wind tunnel to 30 Hz fan speed.
3. Set the angle of attack (AoA) on the airfoil to 0° (this is when the lift force equals zero).
4. Adjust the angle of attack to -6° (sign convention is negative is nose down). At this position,
record the lift, drag, and pitching moment data, as well as the temperature.
5. Increase the angle of attack in 2° steps (20 increments on the dial) until the airfoil is at an
AoA of +18° and record the data at each incremental position.
6. Increase the wind tunnel fan speed to 45 Hz and perform steps 4 and 5 again.
7. Measure the airfoil chord length and depth.
3.2.3 – Cylinder Drag Test
1. Record the length and outside diameter of each of the cylinders.
2. Before recording data with cylinders in the flow, run LabVIEW at 25 Hz fan speed without a
cylinder for a control (label this data “Cylinder 0”).
3. Put Cylinder 1 into the wind tunnel, ensuring it is secured and placed perpendicular to flow.
4. Run the wind tunnel at 25 Hz fan speed, record drag force and temperature.
5. Repeat steps 3 and 4 for each of the other cylinders, recording the data.
3.2.4 – Vortex Shedding
1. Measure the length and diameter of the cylinder.
2. Attach the cylinder and accelerometer to the first hole of the test section of the wind tunnel.
3. Set the wind tunnel fan speed to 30 Hz, the sampling frequency to 6000 Hz and the number
of samples to 16384 samples.
4. Collect data from the accelerometer, then perform the same test with a fan speed of 45 Hz.
5. Remove the cylinder/accelerometer rig and replace with cylinder/pressure transducer rig.
6. Record pressure data for several seconds – first at 30 Hz, then at 45 Hz.

Page | 24
4. Results
In this section, the data collected from the experimental procedure will be provided and
interpreted. This will ultimately be done in order to make a greater analysis about how
experimental data relates to the theoretical concepts discussed in the Introduction. All graphs
and tables in this section have only been added for quick reference in order to identify trends.
Key figures have been labeled and a larger copy can be found in the Appendix as well.
4.1 – Force Balance Calibration
During the Force Balance Calibration, the
conversions for Lift, Drag, and Pitch Moment
were found such that all measured voltages from
the transducers could be interpreted in terms of
Newtons and Newton-meters. It was found that
the conversion equations were all linear and that
there was a certain amount of drift, or load offset
in each of the three calibrations. For each of the
three conversions, there was approximately 3.25
Newtons of drift (positive for Drag and negative
for Lift and Moment). It is possible that this could
be due to minor compliance in the calibration
device as it is loaded with increasing weight.
This drift could potentially skew the
measurements collected during the experiments.
In order to account for this drift, it may be helpful
to “zero” the conversion such that it creates a line
of best fit that passes through the origin, as it
would in reality.
Figure B1: Lift Calibration
Figure B2: Drag Calibration
Figure B3: Moment Calibration

Page | 25
Calibration Equations
Lift Equation 𝐿𝑖𝑓𝑡 = −63.87 𝑉 + 3.26 [N]
Drag Equation 𝐷𝑟𝑎𝑔 = 71.26 𝑉 + 3.10 [N]
Moment Equation 𝑀𝑜𝑚𝑒𝑛𝑡 = 10.14 𝑉 − 3.46 [N m]
Table B1: Calibration Equations
4.2 – Boundary Layer Test
In the Boundary Layer Test, there was a
visible difference between the upstream velocity
profile (35.56 cm from inlet) and the downstream
velocity profile (96.52 cm from inlet). In both
cases, very little changed as the Pitot-static tube
got closer to the wall until it was less than 5 cm
away from the surface. This implies that the flow
in the wind tunnel is almost completely uniform
for the majority of the test section. This is a very good thing when attempting to do tests in a
wind tunnel because if there were a large amount of variation in the velocity profile, it would
make obtaining usable data quite difficult.
There was some notable difference between the upstream and downstream velocities (as
seen in Fig. B4). There was a gradual decrease in velocity over the majority of the data range,
with the exception of couple of points that do not agree with the rest of the trend. The velocity
approaches zero at a height of zero because the bottom shear layer of the fluid is where
stagnation occurs (this was discussed in depth in the Introduction). The comparative lack in
uniformity of the downstream velocity can be explained by the fact that the free stream has a
Reynolds Number twice that of the upstream velocity, indicating an increase in turbulence (see
Table B2).
Reynolds Number (Re) Uncertainty in Re (uRe)
Upstream 5.367 E 5 ± 533.2
Downstream 1.480 E 6 ± 1455
Figure B4: Free Stream Velocity
Table B2: Reynolds Numbers for Upstream and Downstream Free Stream Flow

Page | 26
The difference in upstream and downstream velocities can be attributed to the
Conservation of Mass (Continuity Equation) in that the flow passing through the center of the
test section has had more of an opportunity to change from a uniform velocity profile to a
parabolic one (shown in Fig. 2.3).
4.3 – Cylinder Wake Test
The Cylinder Wake Test results can be
seen in Fig. B5. On the left, the 25 Hz Fan Speed
showed a moderate wake generated by the
cylinder, while the 45 Hz Fan Speed showed a
dramatic wake generated by the cylinder on the
right.
It can also be seen that even when the wake runs
parallel to the free stream velocity profile, there is
still a difference in flow velocity. This discrepancy between the two “free streams” in each
tunnel run can be attributed to the loss of momentum in the flow that was expended in the form
of force on the cylinder. If there were no losses in momentum, the edges of each graph would
have orange and
blue lines overlaid. It also follows that the momentum gap is greater in the 45 Hz run
because more force was applied to the cylinder in this case.
Fan Speed 25 Hz 45 Hz
Drag Force [N] 15.960 ± 0.315 67.574 ± 0.928
Coefficient of Drag (CD) 1.7680 ± 0.00133 2.1648 ± 0.00186
Reynolds Number (Re) 7.286 E 4 ± 66.3 1.355 E 5 ± 117.3
Table B3: Cylinder Wake Velocity Profile Data
At a similar Reynolds Numbers for the Cylinder Wake Test, some published data for the
Coefficient of Drag are as follows:
• Re = 7.3 E 4 à CD = ~ 1.6
• Re = 1.4 E 5 à CD = ~ 2.1
Figure B5: Cylinder Wake Profiles, 25 Hz (right) and 45 Hz (left)

Page | 27
Comparing the values found
to the ones published, it is clear that
the data is quite close. The graph
used from the resource made it quite
difficult to obtain accurate numbers,
giving the graph reading an
uncertainty of almost 0.5. Even with
this wide range, the trend can be
confirmed and helps to confirm the
obtained values.
4.4 – Airfoil Test
Slope (CL / AoA) 1.419 E -2 ± 1.82 E -4 1.495 E -2 ± 1.84 E -4
Table B4: Airfoil Data
Thin airfoil theory states that the
coefficient of lift can be approximated as:
CL = 2 π α
Thus, the Lift-Curve Slope would be equal to 2 π
while the calculated values are on the order of 10-2
there is obviously a discrepancy here, perhaps in
the calculation of the coefficient of lift.
Tuffs began to separate at about 16°
Angle of Attack from the surface of the airfoil.
This makes sense because, once the data reaches this value for α, the Coefficient of Lift drops
and the Coefficient of Drag increases by a large amount. The tuffs separate from the airfoil
because this is the point at which the airfoil stalls. The concept of stalling is explained in the
Introduction as well as why it occurs, but noteworthy observation from the experiment is that,
Figure 4.1: CD vs Re for a Cylinder
http://www.disasterzone.net/projects/docs/mae171a/water_tunnel_experiment.pdf
Figure B9: Airfoil Coefficient of Lift versus Angle of Attack

Page | 28
at stall, the airfoil began to shake and move
around in unexpected ways. The tuffs separate at
stall because, when the boundary layer separates
from the top surface of the airfoil, turbulent
vortices flow over the top surface. The turbulence
on the top of the wing is what picks up the tuffs.
This is confirmed by observation during
the lab: before reaching total boundary layer
separation, the tuffs closest to the trailing edge
began to lift up farther than those closest to the
leading edge. Since this is consistent with the
theory discussed earlier, it supports this analysis.
Reynolds Number also affects the Lift,
Drag, and Moment in this situation because, with
all else equal, an increase in Reynolds Number
indicates an increase in flow velocity and
therefore inertial forces. As the inertial forces
increase, the forces and moments on the airfoil
become amplified.
Another contributor to the overall system
is the Angle of Attack. As the angle of attack
increases, the amount of space through which the
fluid can flow decreases (more of the airfoil is
open to the flow). Looking back at the Continuity
Equation and Fig. 2.2, when the cross-sectional
area of the free stream flow decreases, the flow
velocity must increase at an inversely proportional
rate.
Figure B10: Airfoil Coefficient of Drag versus Angle of Attack
Figure B11: Airfoil Coefficient of Drag versus Coefficient of Lift
Figure B12: Airfoil CL / CD versus Angle of Attack
Figure B13: Airfoil Coefficient of Moment versus Angle of Attack

Page | 29
4.4 – Cylinder Drag Test
From the Cylinder Drag Test, it was
recorded that Cylinder 1, the longest specimen,
had the greatest amount of drag force. This is
because, even though it was the only one that
represented a 2-D flow, it had the most frontal surface area, which ended up making the greatest
difference in this test. However, when the Drag force was non-dimensionalized for each cylinder,
it turned out that Cylinder 5, the shortest specimen, had the greatest coefficient of drag. This is
because it had a large amount of boundary separation at the sides, as well as at the top and
bottom. These findings remain consistent with the expectations set forth in the Introduction.
Cylinder Number 1 2 3 4 5
Reynolds Number
Re
8.36E4 8.35E4 8.38E4 8.34E4 8.34E4
Uncertainty in Re
uRe
± 79 ± 79 ± 79 ± 79 ± 79
Drag Force
D [N]
10.24 9.27 8.55 8.03 5.98
Uncertainty in Drag
uD [N]
± 0.12 ± 0.08 ± 0.08 ± 0.07 ± 0.06
Coefficient of Drag
CD
0.881 0.840 0.854 0.895 1.15
Uncertainty in CD
u CD
± 0.0007 ± 0.0009 ± 0.0008 ± 0.0009 ± 0.0012
Table B5: Cylinder Drag Test Data
The CD found in for Cylinder 1 was decently smaller than that of the CD found for the
wake measurements in Part 1 (by about .6), which was quite similar to that of the published data.
Figure B14: Airfoil Coefficient of Moment versus Coefficient of Lift

Page | 30
However, it is important to note that the CD found for Cylinder 1 was for a fan speed of 25 Hz,
rather than 30 Hz, which certainly accounts for a portion of this descrepency. This can be said
because the CD for the wake measurements in Part 1 changed by .4 with an increase of 15 Hz.
These findings were even greater in the published data.
4.5 Vortex Shedding
Velocity [m/s] 32.04 ± 0.286 49.05 ± 0.532
Strouhal Number 0.197962 ± 0.00021 0.197975 ± 0.00034
Table B6: Vortex Shedding Data
Shedding Frequency
Method Accelerometer Pressure Transducer Calculation
30 Hz Fan Speed 125 ± 2.5 115 ± 1.7 131.4 ± 0.98
45 Hz Fan Speed 153 ± 4.9 174 ± 3.2 201.2 ± 2.27
Table B7: Shedding Frequency Data
Looking at the discrepancies between the calculated and measured shedding frequencies,
the results beg the question: Why are these numbers so different? It should be assumed from the
beginning that, in cases such as this, measured values would not coincide with the calculated
values. This is due to energy and momentum
losses that cannot be accounted for in the basic
calculations performed. The unsteady nature of
the flow also adds to the uncertainty in final
values because the only way to eliminate such
unpredictability is by taking a numerical
average of a relatively small sample size.
Figure B15: Vortex Shedding, 30 Hz Pressure Transducer, Time and Fourier

Page | 31
From the pressure transducer at 30 Hz, the
previous statement can be proven most easily by
looking at the semi-visible alternating pattern in
the time domain. The amount of noise in both
plots lends itself to the fact that it is not possible
to get a “perfect” looking graphical representation
from the pressure transducer. From the initial
point, the pressure signature seems to increase
slightly in amplitude, then every so often it will
create large “clumps” of data that makes the pattern
harder to see. Fig. 4.2 gives a better look at the
difference in pressure signatures between the two
fan speeds. It can be seen that the 45 Hz fan speed
not only provides a greater shedding frequency, but
also an increase in amplitude. It is seen from the
accelerometer plots that the cylinder vibrated
more quickly at 45 Hz. This can be assumed, as stated in the Introduction, that the shedding
frequency at 45 Hz is closer to the resonant frequency of the cylinder.
Figure 4.2: Comparison of Pressure Signatures, 30 Hz and 45 Hz
Figure B17: Vortex Shedding, 30 Hz Accelerometer, Time and Fourier
Figure B18: Vortex Shedding, 45 Hz Accelerometer, Time and Fourier

Page | 32
5. Conclusion
The entirety of this lab has been focused on various concepts of Fluid Dynamics. Much
of the introduction was required in developing the knowledge necessary to calculate the various
values, but not as much towards understanding what was going on in the final analysis of the
results. The difficulty in establishing an analysis of data relating to concepts such as vortex
shedding are that the unsteady nature of the oscillating vortices proves quite difficult to get data
that is accurate. This being said, the resulting measured values were close enough to the
calculated shedding frequencies that these methods proved to be more or less sufficient.
Some of the most well validated results came from the comparison of 2-D and 3-D flows
(Cylinder Drag Test), velocity profile development, and boundary layer separation. These
sections of the results held a strong correlation with the theory as stated in the introduction. The
accuracy of the results was qualitatively quite close (i.e. trends and cause/effect relationships),
which helps to confirm many of the intangible assumptions one must make when establish all of
the theory in Fluid Dynamics.
The most obvious section where discrepancy occurred was undoubtedly in the
comparison between the Lift-Curve Slope that was determined experimentally and that of the
Lift-Curve Slope established from Thin Airfoil Theory. Further study of the sources of error as
well as additional experimentation on the subject would be a logical next step considering, not
only the inconsistency, but also the fact that this was the only major inconsistency between
theory and results.

Page | 33
6. References
1. Fundamental of Fluid Mechanics Munson
2. https://www.princeton.edu/~asmits/Bicycle_web/Bernoulli.html
3. http://ecs.syr.edu/faculty/glauser/mae315/Fluids/MAE315Lab4Week1.htm
4. http://www.nap.edu/read/5870/chapter/40
5. https://en.wikipedia.org/wiki/Viscosity

Page | 34
7. Appendix
Table of Contents
A.
Key Equations .................................................................................................................... A1

B.
Simulated and Experimental Results Tables................................................................... A2

Table B1: Calibration Equations........................................................................................................... A3

Table B2: Reynolds Numbers for Upstream and Downstream Free Stream Flow............................... A4

Table B3: Cylinder Wake Velocity Profile Data .................................................................................. A5

Table B4: Airfoil Data .......................................................................................................................... A6

Table B5: Cylinder Drag Test Data .................................................................................................... A11

Table B6: Vortex Shedding Data........................................................................................................ A12

Table B7: Shedding Frequency Data .................................................................................................. A12

C.
MATLAB Code ................................................................................................................ A15

Page | A1
A. Key Equations
Ideal Gas Law 𝑃 = 𝜌 𝑅! 𝑇 2.1
Conservation of Energy 𝐸𝑛𝑒𝑟𝑔𝑦 𝐼𝑛 = 𝐸𝑛𝑒𝑟𝑔𝑦 𝑂𝑢𝑡 2.2
Conservation of Mass 𝑀𝑎𝑠𝑠 𝐼𝑛 = 𝑀𝑎𝑠𝑠 𝑂𝑢𝑡 2.3
Conservation of Momentum 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝐹𝑖𝑛𝑎𝑙 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 2.4
Bernoulli’s Equation 𝑃! +
𝜌 𝑉!
!
2
+ 𝜌𝑔ℎ! = 𝑃! +
𝜌 𝑉!
!
2
+ 𝜌𝑔ℎ!
2.5
Velocity at a Pitot-Static Tube 𝑉 =
2 𝑃! − 𝑃!
𝜌
2.6
Continuity Equation
𝐷
𝐷𝑡
𝜌
!"!
𝑑𝑉 =
𝜕
𝜕𝑡
𝜌
!"
𝑑𝑉 + 𝜌
!"
𝐕 ∙ 𝐧 𝑑𝐴 = 0 2.8
Linear Momentum Equation
𝜕
𝜕𝑡
𝐕 𝜌
!"
𝑑𝑉 + 𝐕 𝜌
!"
𝐕 ∙ 𝐧 𝑑𝐴 = 𝐅!"#$%#$& !" !"#
!"#$%"& !"#$%&
2.10
Drag Force 𝐷 = 𝜌 𝑈!
!
1 −
𝑢 𝑦
𝑈!
!
!
𝑑𝑦 2.12
Drag Coefficient 𝐶! =
2𝐷
𝜌 𝑈!
!
𝑆
2.13
Reynolds Number 𝑅𝑒 =
𝜌 𝑉 𝐷
𝜇
2.14
Lift Coefficient 𝐶! =
2𝐿
𝜌 𝑈!
!
𝑆
2.16
Pitching Moment Coefficient 𝐶! =
2𝑀
𝜌 𝑈!
!
𝑆 𝑥
2.17
Strouhal Number 𝑆𝑡 =
𝑓 𝑑
𝑉
= 0.198 1 −
19.7
𝑅𝑒
2.18
Zeroith Order Uncertainty 𝑢!,! = ± (
𝜕𝑦
𝜕𝑥!
𝑢!,!!
)! 2.19

Page | A2
B. Simulated and Experimental Results Tables
Calculations Part 1: Calibrations
𝐿𝑖𝑓𝑡 = −63.87 𝑉 + 3.26
Figure B1: Lift Calibration
Figure B2: Drag Calibration
𝐷𝑟𝑎𝑔 = 71.26 𝑉 + 3.10

Page | A3
Calibration Equations
Lift Equation 𝐿𝑖𝑓𝑡 = −63.87 𝑉 + 3.26 [N]
Drag Equation 𝐷𝑟𝑎𝑔 = 71.26 𝑉 + 3.10 [N]
Moment Equation 𝑀𝑜𝑚𝑒𝑛𝑡 = 10.14 𝑉 − 3.46 [N m]
Table B1: Calibration Equations
Figure B3: Moment Calibration
𝑀𝑜𝑚𝑒𝑛𝑡 = 10.14 𝑉 − 3.46

Page | A4
Calculations Part 1: Boundary Layer Test
Reynolds Number (Re) Uncertainty in Re (uRe)
Upstream 5.367 E 5 ± 533.2
Downstream 1.480 E 6 ± 1455
Table B2: Reynolds Numbers for Upstream and Downstream Free Stream Flow
Figure B4: Free Stream Velocity

Page | A5
Calculations Part 1: Cylinder Wake Test
Drag Force [N] 15.960 ± 0.315 67.574 ± 0.928
Coefficient of Drag (CD) 1.7680 ± 0.00133 2.1648 ± 0.00186
Table B3: Cylinder Wake Velocity Profile Data
Figure B5: Cylinder Wake Velocity Profiles, 25 Hz Fan speed (right) and 45 Hz Fan Speed (left)

Page | A6
Calculations Part 2: Airfoil Test
Slope (CL / AoA) 1.419 E -2 ± 1.82 E -4 1.495 E -2 ± 1.84 E -4
Table B4: Airfoil Data
Figure B6: Airfoil Lift Force versus Angle of Attack

Page | A7
Figure B7: Airfoil Drag Force versus Angle of Attack
Figure B8: Airfoil Moment versus Angle of Attack

Page | A8
Figure B9: Airfoil Coefficient of Lift versus Angle of Attack
Figure B10: Airfoil Coefficient of Drag versus Angle of Attack

Page | A9
Figure B11: Airfoil Coefficient of Drag versus Coefficient of Lift
Figure B12: Airfoil CL / CD versus Angle of Attack

Page | A10
Figure B13: Airfoil Coefficient of Moment versus Angle of Attack
Figure B14: Airfoil Coefficient of Moment versus Coefficient of Lift

Page | A11
Calculations Part 2: Cylinder Drag Test
Cylinder Number 1 2 3 4 5
Reynolds Number
Re
8.36E4 8.35E4 8.38E4 8.34E4 8.34E4
Uncertainty in Re
uRe
± 79 ± 79 ± 79 ± 79 ± 79
Drag Force
D [N]
10.24 9.27 8.55 8.03 5.98
Uncertainty in Drag
uD [N]
± 0.12 ± 0.08 ± 0.08 ± 0.07 ± 0.06
Coefficient of Drag
CD
0.881 0.840 0.854 0.895 1.15
Uncertainty in CD
u CD
± 0.0007 ± 0.0009 ± 0.0008 ± 0.0009 ± 0.0012
Table B5: Cylinder Drag Test Data
Cylinder Length:
1 – 23 3/16”
2 – 22 1/16”
3 – 19 15/16”
4 – 17 15/16”
5 – 10 7/16”

Page | A12
Calculations Part 2: Vortex Shedding Test
Velocity [m/s] 32.04 ± 0.286 49.05 ± 0.532
Strouhal Number 0.197962 ± 0.00021 0.197975 ± 0.00034
Table B6: Vortex Shedding Data
Shedding Frequency
Method Accelerometer Pressure Transducer Calculation
30 Hz Fan Speed 125 ± 2.5 115 ± 1.7 131.4 ± 0.98
45 Hz Fan Speed 153 ± 4.9 174 ± 3.2 201.2 ± 2.27
Table B7: Shedding Frequency Data

Page | A13
Figure B16: Vortex Shedding, 45 Hz Pressure Transducer, Time Domain (left) and Fourier Domain (right)
Figure B15: Vortex Shedding, 30 Hz Pressure Transducer, Time Domain (left) and Fourier Domain (right)

Page | A14
1.
Figure B17: Vortex Shedding, 30 Hz Accelerometer, Time Domain (left) and Fourier Domain (right)
Figure B18: Vortex Shedding, 45 Hz Accelerometer, Time Domain (left) and Fourier Domain (right)

Page | A15
C. MATLAB Code
%Josh Beckerman
%Dr. Glauser
%MAE 315 - Mechanical and Aerospace Engineering Lab
%December 9, 2015
%Lab 4 - Fluid Mechanics
clear all; close all; clc
Part 1: Force Calibration, Lift
P1Lift = dlmread('fenton_lift_Part1.txt','t',1,0);
P1LiftV = P1Lift(:,4) + 0.003242;
P1LiftM = P1Lift(:,1);
P1LiftF = (P1LiftM * 0.45359) * 9.80665002864; %(lbm to kg) to N
P1LiftFit = polyfit(P1LiftV,P1LiftF,1);
P1Liftk = P1LiftFit(1);
P1Liftb = P1LiftFit(2);
P1Lifteq = P1Liftk*P1LiftV + P1Liftb;
fprintf('Part 1 Lift Equation t Lift = %f * Voltage +',P1Liftk)
fprintf(' %f t[N] n',P1Liftb)
figure(1);
plot(P1LiftV, P1LiftF, P1LiftV, P1Lifteq)
title({'Part 1: Calibrations'; 'Lift'})
xlabel('Voltage [volts]')
ylabel('Lift Force [N]')
axis on
grid on
legend('Raw Lift Data Correlation', 'Linear Lift Fit')
Part 1 Lift Equation Lift = -63.865775 * Voltage + 3.262236 [N]
Part 1: Force Calibration, Drag
P1Drag = dlmread('fenton_drag_Part1.txt','t',1,0);
P1DragV = P1Drag(:,2) + 0.008728;
P1DragM = P1Drag(:,1);
P1DragF = (P1DragM * 0.45359) * 9.80665002864; %(lbm to kg) to N
P1DragFit = polyfit(P1DragV,P1DragF,1);

Page | A16
P1Dragk = P1DragFit(1);
P1Dragb = P1DragFit(2);
P1Drageq = P1Dragk*P1DragV + P1Dragb;
fprintf('Part 1 Drag Equation t Drag = %f * Voltage +',P1Dragk)
fprintf(' %f t[N] n',P1Dragb)
figure(2);
plot(P1DragV, P1DragF, P1DragV, P1Drageq)
title({'Part 1: Calibrations'; 'Drag'})
ylabel('Drag Force [N]')
axis on
grid on
legend('Raw Drag Data Correlation', 'Linear Drag Fit')
Part 1 Drag Equation Drag = 71.257981 * Voltage + 3.095526 [N]
Part 1: Force Calibration, Moment
P1Moment = dlmread('fenton_moment_Part1.txt','t',1,0);
P1MomentV = P1Moment(:,6) + 0.009597;
P1MomentD = P1Moment(:,1);
P1Momentmass = 5; %lbm
P1Momentmass = (P1Momentmass * 0.45359) * 9.80665002864; %(lbm to kg) to N
P1MomentF = (P1MomentD * 0.0254); %(in to m)
P1MomentM = P1MomentF * P1Momentmass; %N*m
P1MomentFit = polyfit(P1MomentV,P1MomentM,1);
P1Momentk = P1MomentFit(1);
P1Momentb = P1MomentFit(2);
P1Momenteq = P1Momentk*P1MomentV + P1Momentb;
fprintf('Part 1 Moment Equation t Moment = %f * Voltage +',P1Momentk)
fprintf(' %f t[N m] n',P1Momentb)
figure(3);
plot(P1MomentV, P1MomentM, P1MomentV, P1Momenteq)
title({'Part 1: Calibrations'; 'Moment'})
ylabel('Moment [N*m]')
axis on
grid on
legend('Raw Moment Data Correlation', 'Linear Moment Fit')
Part 1 Moment Equation Moment = 10.141806 * Voltage + -3.464799 [N m]

Page | A17
Part 1: Boundary Layer Test
P1mu = 0.00001789; %kg/(m-s)
P1BLupstream = xlsread('Upstream_BoundaryLayerData.xlsx');
P1BLupH = P1BLupstream(1:5:41,1);
P1BLupT = P1BLupstream(1:5:41,2);
P1BLupP = P1BLupstream(1:45,3);
P1BLupP = P1BLupP * 133.322368; %Torr to Pascals
P1BLupP = [mean(P1BLupP(1:5));mean(P1BLupP(6:10));mean(P1BLupP(11:15));...
mean(P1BLupP(16:20));mean(P1BLupP(21:25));mean(P1BLupP(26:30));...
mean(P1BLupP(31:35));mean(P1BLupP(36:40));mean(P1BLupP(41:45))];
P1BLupH = P1BLupH * 0.0254; %Inches to Meters
P1BLupPatm = 101325*ones(9,1); %Pascals
P1BLupT = (P1BLupT + 459.67) * (5/9); %Fahrenheit to Kelvin
GasConst = 287; %J/(Kg*K)
P1BLupRho = P1BLupPatm ./ (GasConst * P1BLupT);
P1BLupU = sqrt((P1BLupP * 2) ./ P1BLupRho);
P1BLupD = 14 *0.0254; %meters
P1BLupRe = (P1BLupRho(1) * P1BLupU(1) * P1BLupD) / P1mu;
fprintf('Part 1 Boundary Layer Up Stream t Re = %f n',P1BLupRe)
P1BLdownstream = xlsread('Downstream_BoundaryLayerData.xlsx');
P1BLdownH = P1BLdownstream(1:5:41,1);
P1BLdownT = P1BLdownstream(1:5:41,2);
P1BLdownP = P1BLdownstream(1:45,3);
P1BLdownP = P1BLdownP * 133.322368; %Torr to Pascals
P1BLdownP = [mean(P1BLdownP(1:5));mean(P1BLdownP(6:10));mean(P1BLdownP(11:15));...
mean(P1BLdownP(16:20));mean(P1BLdownP(21:25));mean(P1BLdownP(26:30));...
mean(P1BLdownP(31:35));mean(P1BLdownP(36:40));mean(P1BLdownP(41:45))];
P1BLdownH = P1BLdownH * 0.0254; %Inches to Meters
P1BLdownPatm = 101325*ones(9,1); %Pascals
P1BLdownT = (P1BLdownT + 459.67) * (5/9); %Fahrenheit to Kelvin
P1BLdownRho = P1BLdownPatm ./ (GasConst * P1BLdownT);
P1BLdownU = sqrt((P1BLdownP * 2) ./ P1BLdownRho);
P1BLdownD = 38 *0.0254; %meters
P1BLdownRe = (P1BLdownRho(1) * P1BLdownU(1) * P1BLdownD) / P1mu;
fprintf('Part 1 Boundary Layer Down Stream t Re = %f n',P1BLdownRe)
figure(4);
plot(P1BLupH, P1BLupU, P1BLdownH, P1BLdownU)
title({'Part 1: Boundary Layer Test'; 'Free Stream Velocity'})
xlabel('Height [meters]')
ylabel('Velocity [m/s]')
axis on
grid on
legend('Upstream Velocity', 'Downstream Velocity')

Page | A18
Part 1 Boundary Layer Up Stream Re = 536666.242299
Part 1 Boundary Layer Down Stream Re = 1479510.633116
Part 1: Cylinder Wake Test
P1PRakeX = [0,2,4,6,8,10,12,13,14,15,16,17,18,19,20,22,24,26,28,30]; %cm
P1CylLength = 2 * 0.3048; %meters
P1CylDia = 1.9 * 0.0254; %meters
P1CylArea = P1CylDia * P1CylLength; %meters^2
P1NoCyl25HzT = (76.6 + 459.67) * (5/9); %Fahrenheit to Kelvin
P1NoCyl25HzRho = 101325 ./ (GasConst * P1NoCyl25HzT);
P1NoCyl25HzP = dlmread('thurs930_25Hz_nocylinder_Part1.txt','t',[0 1 19 20]);
P1NoCyl25HzP = mean(P1NoCyl25HzP);
P1NoCyl25HzP = [P1NoCyl25HzP(1:10),(P1NoCyl25HzP(10)+P1NoCyl25HzP(12))/2,P1NoCyl25HzP(12:20)];
P1NoCyl25HzP = P1NoCyl25HzP * 6894.76; %atm to pa
P1NoCyl25HzV = sqrt((P1NoCyl25HzP * 2) / P1NoCyl25HzRho);
P1Cyl25HzT = (75.0 + 459.67) * (5/9); %Fahrenheit to Kelvin
P1Cyl25HzRho = 101325 ./ (GasConst * P1Cyl25HzT);
P1Cyl25HzP = dlmread('thurs930_25Hz_cylinder_Part1.txt','t',[0 1 19 20]);
P1Cyl25HzP = mean(P1Cyl25HzP);
P1Cyl25HzP = [P1Cyl25HzP(1:10),(P1Cyl25HzP(10)+P1Cyl25HzP(12))/2,P1Cyl25HzP(12:20)];
P1Cyl25HzP = P1Cyl25HzP * 6894.76; %atm to pa
P1Cyl25HzV = sqrt((P1Cyl25HzP * 2) / P1Cyl25HzRho);
P125HzVfunc = 1 - (P1Cyl25HzV./P1NoCyl25HzV);
P1Cyl25Drag = (P1Cyl25HzRho * P1CylLength * mean(P1NoCyl25HzV)^2) *
trapz(P1PRakeX*.01,P125HzVfunc);
fprintf('Part 1 Wake Test 25 Hz t Drag = %f t[N]n',P1Cyl25Drag)
P1Cyl25DragCo = P1Cyl25Drag / (.5 * P1Cyl25HzRho * mean(P1NoCyl25HzV)^2 * P1CylArea);
fprintf('Part 1 Wake Test 25 Hz t Cd = %f n',P1Cyl25DragCo)
P1Cyl25Re = (P1Cyl25HzRho * mean(P1NoCyl25HzV) * P1CylDia) / P1mu;
fprintf('Part 1 Wake Test 25 Hz t Re = %f n',P1Cyl25Re)
P1NoCyl45HzT = (76.6 + 459.67) * (5/9); %Fahrenheit to Kelvin
P1NoCyl45HzRho = 101325 ./ (GasConst * P1NoCyl45HzT);
P1NoCyl45HzP = dlmread('thurs930_45Hz_nocylinder_Part1.txt','t',[0 1 19 20]);
P1NoCyl45HzP = mean(P1NoCyl45HzP);
P1NoCyl45HzP = [P1NoCyl45HzP(1:10),(P1NoCyl45HzP(10)+P1NoCyl45HzP(12))/2,P1NoCyl45HzP(12:20)];
P1NoCyl45HzP = P1NoCyl45HzP * 6894.76; %atm to pa
P1NoCyl45HzV = sqrt((P1NoCyl45HzP * 2) / P1NoCyl45HzRho);
P1Cyl45HzT = (75.0 + 459.67) * (5/9); %Fahrenheit to Kelvin
P1Cyl45HzRho = 101325 ./ (GasConst * P1Cyl45HzT);
P1Cyl45HzP = dlmread('thurs930_45Hz_cylinder_Part1.txt','t',[0 1 19 20]);
P1Cyl45HzP = mean(P1Cyl45HzP);

Page | A19
P1Cyl45HzP = [P1Cyl45HzP(1:10),(P1Cyl45HzP(10)+P1Cyl45HzP(12))/2,P1Cyl45HzP(12:20)];
P1Cyl45HzP = P1Cyl45HzP * 6894.76; %atm to pa
P1Cyl45HzV = sqrt((P1Cyl45HzP * 2) / P1Cyl45HzRho);
P145HzVfunc = 1 - (P1Cyl45HzV./P1NoCyl45HzV);
P1Cyl45Drag = (P1Cyl45HzRho * P1CylLength * mean(P1NoCyl45HzV)^2) *
trapz(P1PRakeX*.01,P145HzVfunc);
fprintf('Part 1 Wake Test 45 Hz t Drag = %f t[N]n',P1Cyl45Drag)
P1Cyl45DragCo = P1Cyl45Drag / (.5 * P1Cyl45HzRho * mean(P1NoCyl45HzV)^2 * P1CylArea);
fprintf('Part 1 Wake Test 45 Hz t Cd = %f n',P1Cyl45DragCo)
P1Cyl45Re = (P1Cyl45HzRho * mean(P1NoCyl45HzV) * P1CylDia) / P1mu;
fprintf('Part 1 Wake Test 45 Hz t Re = %f n',P1Cyl45Re)
figure(5);
subplot(1,2,1)
plot(P1PRakeX, P1NoCyl25HzV, P1PRakeX, P1Cyl25HzV);
title({'Part 1: Cylinder Wake Test'; '25 Hz Fan Speed'})
xlabel('Pitot Tube Location [centimeters]')
ylim([0,60]);
axis on
grid on
legend('Without Cylinder (Free Stream)', 'With Cylinder')
subplot(1,2,2)
plot(P1PRakeX, P1NoCyl45HzV, P1PRakeX, P1Cyl45HzV);
title({'Part 1: Cylinder Wake Test'; '45 Hz Fan Speed'})
xlabel('Pitot Tube Location [centimeters]')
ylim([0,60]);
axis on
grid on
legend('Without Cylinder (Free Stream)', 'Hz With Cylinder');
Part 1 Wake Test 25 Hz Drag = 15.960103 [N]
Part 1 Wake Test 25 Hz Cd = 1.767963
Part 1 Wake Test 25 Hz Re = 72856.416447
Part 1 Wake Test 45 Hz Drag = 67.574396 [N]
Part 1 Wake Test 45 Hz Cd = 2.164786
Part 1 Wake Test 45 Hz Re = 135478.431106
Part 2: Airfoil
P2FreeStreamV30 = (1.134 * 30) - 1.9793; %m/s
P2FreeStreamV45 = (1.134 * 45) - 1.9793; %m/s
P2AFChord = 6 * 0.0254; %m
P2AFSpan = 14.875 * 0.0254; %m
P2AF_S = P2AFChord * P2AFSpan; %m^2
P2AF30Hz = dlmread('Thursday_930_30hz_2airfoil_Part2.txt','t',2,0);
P2AF30HzAoA = P2AF30Hz(:,2);

Page | A20
P2AF30HzDrag = P1Dragk*(P2AF30Hz(:,3)+0.001518) + P1Dragb;
P2AF30HzLift = P1Liftk*(P2AF30Hz(:,5)-0.000450) + P1Liftb;
P2AF30HzMoment = P1Momentk*(P2AF30Hz(:,7)+0.003893) + P1Momentb;
P2AF30HzT = [72.8,73.5,73.7,74.0,74.4,74.6,74.8,75.2,75.4,75.6,75.8,76.0,76.4].';
P2AF30HzT = (P2AF30HzT + 459.67) .* (5/9); %Fahrenheit to Kelvin
P2AF30HzRho = 101325 ./ (GasConst * P2AF30HzT);
P2AF30HzCL = P2AF30HzLift ./ (.5 * P2AF30HzRho * P2FreeStreamV30^2 * P2AF_S);
P2AF30HzCd = P2AF30HzDrag ./ (.5 * P2AF30HzRho * P2FreeStreamV30^2 * P2AF_S);
P2AF30HzCm = P2AF30HzMoment ./ (.5 * P2AF30HzRho * P2FreeStreamV30^2 * P2AF_S * P2AFChord);
P2AF30HzRe = (mean(P2AF30HzRho) * P2FreeStreamV30 * P2AFChord) / P1mu;
P2AF30HzSlope = (P2AF30HzCL(3)-P2AF30HzCL(2))/(P2AF30HzAoA(3)-P2AF30HzAoA(2));
fprintf('Part 2 Airfoil Test 30 Hz t Re = %f n',P2AF30HzRe)
fprintf('Part 2 Airfoil Test 30 Hz t Slope = %f n',P2AF30HzSlope)
P2AF45Hz = dlmread('Thursday_930_45hz_airfoil_Part2.txt','t',2,0);
P2AF45HzAoA = P2AF45Hz(:,2);
P2AF45HzDrag = P1Dragk*(P2AF45Hz(:,3)-0.000917) + P1Dragb;
P2AF45HzLift = P1Liftk*(P2AF45Hz(:,5)+0.000853) + P1Liftb;
P2AF45HzMoment = P1Momentk*(P2AF45Hz(:,7)+0.005980) + P1Momentb;
P2AF45HzT = [77.3,78.1,78.8,79.5,79.9,80.4,80.9,81.5,82.0,82.4,82.8,83.2,84.0].';
P2AF45HzT = (P2AF45HzT + 459.67) .* (5/9); %Fahrenheit to Kelvin
P2AF45HzRho = 101325 ./ (GasConst * P2AF45HzT);
P2AF45HzCL = P2AF45HzLift ./ (.5 * P2AF45Hz * P2FreeStreamV45^2 * P2AF_S);
P2AF45HzCd = P2AF45HzDrag ./ (.5 * P2AF45HzRho * P2FreeStreamV45^2 * P2AF_S);
P2AF45HzCm = P2AF45HzMoment ./ (.5 * P2AF45HzRho * P2FreeStreamV45^2 * P2AF_S * P2AFChord);
P2AF45HzRe = (mean(P2AF45HzRho) * P2FreeStreamV45 * P2AFChord) / P1mu;
P2AF45HzSlope = (P2AF45HzCL(3)-P2AF45HzCL(2))/(P2AF45HzAoA(3)-P2AF45HzAoA(2));
fprintf('Part 2 Airfoil Test 45 Hz t Re = %f n',P2AF45HzRe)
fprintf('Part 2 Airfoil Test 30 Hz t Slope = %f n',P2AF45HzSlope)
figure(6);
plot(P2AF30HzAoA,P2AF30HzLift,P2AF45HzAoA,P2AF45HzLift);
title({'Part 2: Airfoil Test'; 'Lift vs AoA'})
xlabel('Angle of Attack [degrees]')
ylabel('Lift Force [N]')
axis on
grid on
legend('30 Hz Fan Speed','45 Hz Fan Speed')
figure(7);
plot(P2AF30HzAoA,P2AF30HzDrag,P2AF45HzAoA,P2AF45HzDrag);
title({'Part 2: Airfoil Test'; 'Drag vs Aoa'})
ylabel('Drag Force [N]')
axis on
grid on
figure(8);
plot(P2AF30HzAoA,P2AF30HzMoment,P2AF45HzAoA,P2AF45HzMoment);
title({'Part 2: Airfoil Test'; 'Moment vs AoA'})

Page | A21
ylabel('Moment [N m]')
axis on
grid on
figure(9);
plot(P2AF30HzAoA,P2AF30HzCL,P2AF45HzAoA,P2AF45HzCL);
title({'Part 2: Airfoil Test'; 'Cl vs AoA'})
ylabel('Coefficient of Lift')
axis on
grid on
figure(10);
plot(P2AF30HzAoA,P2AF30HzCd,P2AF45HzAoA,P2AF45HzCd);
title({'Part 2: Airfoil Test'; 'Cd vs AoA'})
ylabel('Coefficient of Drag')
axis on
grid on
figure(11);
plot(P2AF30HzCL,P2AF30HzCd,P2AF45HzCL,P2AF45HzCd);
title({'Part 2: Airfoil Test'; 'Cd vs Cl'})
xlabel('Coefficient of Lift')
ylabel('Coefficient of Drag')
axis on
grid on
figure(12);
plot(P2AF30HzAoA,P2AF30HzCm,P2AF45HzAoA,P2AF45HzCm);
title({'Part 2: Airfoil Test'; 'Cm vs AoA'})
ylabel('Coefficient of Moment')
axis on
grid on
figure(13);
plot(P2AF30HzCL,P2AF30HzCm,P2AF45HzCL,P2AF45HzCm);
title({'Part 2: Airfoil Test'; 'Cm vs Cl'})
xlabel('Coefficient of Lift')
ylabel('Coefficient of Moment')
axis on
grid on
Part 2 Airfoil Test 30 Hz Re = 324544.786630
Part 2 Airfoil Test 30 Hz Slope = 0.014191
Part 2 Airfoil Test 45 Hz Re = 491288.434560
Part 2 Airfoil Test 45 Hz Slope = 0.014952

Page | A22
Part 2: Cylinder Drag Test
P2FreeStreamV25 = (1.134 * 25) - 1.9793; %m/s
P2CDTemp = [79.7,79.4,79.2,79.2,78.9].';
P2CDTemp = (P2CDTemp + 459.67) .* (5/9); %Fahrenheit to Kelvin
P2CDLength = [23.1875,22.0625,19.9375,17.9375,10.4375].';
P2CDLength = P2CDLength * 0.0254; %meters
P2CDDia = [1.896,1.892,1.899, 1.889,1.888].';
P2CDDia = P2CDDia * 0.0254; %meters
P2CD = dlmread('Thursday_930_25hz_cylinder_Part2.txt','t',[1 0 5 3]);
P2CDNum = P2CD(:,2);
P2CDdrag = P1Dragk*(P2CD(:,3)-0.013728) + P1Dragb;
P2CDRho = 101325 ./ (GasConst * P2CDTemp);
P2CDRe = (P2CDRho .* P2FreeStreamV25 .* P2CDDia) ./ P1mu;
P2CD_Cd = P2CDdrag ./ (.5 .* P2CDRho .* P2FreeStreamV25^2 .* (P2CDLength .* P2CDDia));
for i = 1:5;
fprintf('Part 2 Cylinder Drag Test t Cylinder %f Re =',P2CDNum(i))
fprintf(' %f n',P2CDRe(i))
end
for i = 1:5;
fprintf('Part 2 Cylinder Drag Test t Cylinder %f Drag =',P2CDNum(i))
fprintf(' %f [N]n',P2CDdrag(i))
end
for i = 1:5;
fprintf('Part 2 Cylinder Drag Test t Cylinder %f Drag Coeff =',P2CDNum(i))
fprintf(' %f n',P2CD_Cd(i))
end
Part 2 Cylinder Drag Test Cylinder 1.000000 Re = 83638.031132
Part 2 Cylinder Drag Test Cylinder 1.000000 Drag = 10.235290 [N]
Part 2 Cylinder Drag Test Cylinder 1.000000 Drag Coeff = 0.880860

Page | A23
Part 2: Vortex Shedding
P2VSCylDia = 1.9 * 0.0254; %meters
P2VS30Temp = (78.8 + 459.67) .* (5/9); %Fahrenheit to Kelvin
P2VS45Temp = (80.3 + 459.67) .* (5/9); %Fahrenheit to Kelvin
P2VS30Rho = 101325 ./ (GasConst * P2VS30Temp);
P2VS45Rho = 101325 ./ (GasConst * P2VS45Temp);
P2VS30Re = (P2VS30Rho .* P2FreeStreamV30 .* P2VSCylDia) ./ P1mu;
fprintf('Part 2 Vortex Shedding 30 Hz t Re = %f n',P2VS30Re)
fprintf('Part 2 Vortex Shedding 30 Hz t Velocity = %f n',P2FreeStreamV30)
P2VS45Re = (P2VS45Rho .* P2FreeStreamV45 .* P2VSCylDia) ./ P1mu;
fprintf('Part 2 Vortex Shedding 45 Hz t Re = %f n',P2VS45Re)
fprintf('Part 2 Vortex Shedding 45 Hz t Velocity = %f n',P2FreeStreamV45)
P2VS30Strouhal = .198 * (1 - (19.7 / P2VS30Re));
fprintf('Part 2 Vortex Shedding 30 Hz t Strouhal = %f n',P2VS30Strouhal)
P2VS45Strouhal = .198 * (1 - (19.7 / P2VS45Re));
fprintf('Part 2 Vortex Shedding 45 Hz t Strouhal = %f n',P2VS45Strouhal)
P2VS30ShedFreq = (P2VS30Strouhal * P2FreeStreamV30) / P2VSCylDia;
fprintf('Part 2 Vortex Shedding 30 Hz t Shedding F = %f n',P2VS30ShedFreq)
P2VS45ShedFreq = (P2VS45Strouhal * P2FreeStreamV45) / P2VSCylDia;
fprintf('Part 2 Vortex Shedding 45 Hz t Shedding F = %f n',P2VS45ShedFreq)
P2VorShedPres30 = dlmread('VortexSheddingPressureTranducer30Hz.txt','t',1,1);
P2VorShedPres30time = linspace(0,(16384/6000),16384); %seconds
P2VS_P30fft = fft(P2VorShedPres30);
P2VS_P30Trans = P2VS_P30fft .* conj(P2VS_P30fft) .* (1/(6000^2));
P2VS_P30freq = [0:(length(P2VorShedPres30)-1)];
P2VS_P30freq = (P2VS_P30freq .* 6000) ./ length(P2VorShedPres30);
P2VorShedPres45 = dlmread('VortexSheddingPressureTranduce45Hz.txt','t',1,1);
P2VorShedPres45time = linspace(0,(16384/6000),16384); %seconds
P2VS_P45fft = fft(P2VorShedPres45);
P2VS_P45Trans = P2VS_P45fft .* conj(P2VS_P45fft) .* (1/(6000^2));
P2VS_P45freq = [0:(length(P2VorShedPres45)-1)];
P2VS_P45freq = (P2VS_P45freq .* 6000) ./ length(P2VorShedPres45);
P2VorShedAcc30Amp = dlmread('Thursday930_accelcyl30Hz_Part2.txt','t',1,0);
P2VorShedAcc30time = linspace(0,(16384/6000),16384); %seconds
P2VS_A30fft = fft(P2VorShedAcc30Amp);
P2VS_A30Trans = P2VS_A30fft .* conj(P2VS_A30fft) .* (1/(6000^2));
P2VS_A30freq = [0:(length(P2VorShedAcc30Amp)-1)];
P2VS_A30freq = (P2VS_A30freq .* 6000) ./ length(P2VorShedAcc30Amp);
P2VorShedAcc45Amp = dlmread('Thursday930_accelcyl45Hz_Part2.txt','t',1,0);
P2VorShedAcc45time = linspace(0,(16384/6000),16384); %seconds
P2VS_A45fft = fft(P2VorShedAcc45Amp);

Page | A24
P2VS_A45Trans = P2VS_A45fft .* conj(P2VS_A45fft) .* (1/(6000^2));
P2VS_A45freq = [0:(length(P2VorShedAcc45Amp)-1)];
P2VS_A45freq = (P2VS_A45freq .* 6000) ./ length(P2VorShedAcc45Amp);
figure(14)
subplot(1,2,1);
plot(P2VorShedPres30time,P2VorShedPres30)
title({'Part 2: Vortex Shedding'; 'Time Domain 30 Hz Fan Speed'})
xlabel('Time [s]')
ylabel('Pressure [V]')
axis on
grid on
subplot(1,2,2);
plot(P2VS_P30freq,P2VS_P30Trans)
title({'Part 2: Vortex Shedding'; 'Frequency Domain 30 Hz Fan Speed'})
xlabel('Frequency [Hz]')
xlim([0 300])
axis on
grid on
figure(15)
subplot(1,2,1);
plot(P2VorShedPres45time,P2VorShedPres45)
xlabel('Time [s]')
axis on
grid on
subplot(1,2,2);
plot(P2VS_P45freq,P2VS_P45Trans)
xlim([0 300])
axis on
grid on
figure(16)
subplot(1,2,1);
plot(P2VorShedAcc30time,P2VorShedAcc30Amp)
xlabel('Time [s]')
ylabel('Voltage Amplitude [V]')
axis on
grid on
subplot(1,2,2);
plot(P2VS_A30freq,P2VS_A30Trans)
ylim([0 4e-6])

Page | A25
xlim([0 500])
axis on
grid on
figure(17)
subplot(1,2,1);
plot(P2VorShedAcc45time,P2VorShedAcc45Amp)
xlabel('Time [s]')
axis on
grid on
subplot(1,2,2);
plot(P2VS_A45freq,P2VS_A45Trans)
title({'Part 2: Vortex Shedding'; 'Fourier Domain 45 Hz Fan Speed'})
ylim([0 4e-6])
xlim([0 500])
axis on
grid on
Part 2 Vortex Shedding 30 Hz Re = 102005.756527
Part 2 Vortex Shedding 30 Hz Velocity = 32.040700
Part 2 Vortex Shedding 45 Hz Re = 155725.514949
Part 2 Vortex Shedding 45 Hz Velocity = 49.050700
Part 2 Vortex Shedding 30 Hz Strouhal = 0.197962
Part 2 Vortex Shedding 45 Hz Strouhal = 0.197975
Part 2 Vortex Shedding 30 Hz Shedding F = 131.430447
Part 2 Vortex Shedding 45 Hz Shedding F = 201.218607
Uncertainties
syms w Tf Dm P Fs L F chordm spanm visc
bits = 16;
range = 10;
u_V = range/(2^(bits - 1)); %volts
u_W = .01 * ((w* 0.45359) * 9.8066500286); %kg
u_tapemsr = (1/32) * 0.0254; %meters
u_caliper = 0.0005 * 0.0254; %meters
u_Temp = 0.05 .* (5/9); %Kelvin
u_span = (1/32) * 0.0254; %meters
u_chord = (1/64) * 0.0254; %meters
u_FanSpeed = .05; %Hz
u_Viscosity = P1mu * .001; %kg/(m-s)
u_Forces = F * .02; %N
u_Sc = u_caliper * u_tapemsr; %m^2
u_Sa = u_span * u_chord; %m^2
T = (Tf + 459.67) .* (5/9);

Page | A26
rho = 101325 / (287 * T);
D = Dm * 0.0254;
Vp = sqrt((P * 2) ./ rho);
Vf = (1.134 * Fs) - 1.9793;
Re_p = (rho * Vp * D) / visc;
Re_p_vars = [Tf Dm P visc];
u_Re_p_vars = [u_Temp u_tapemsr u_V u_Viscosity];
[u_Re_p] = Uncertainty_Eval(Re_p,Re_p_vars,u_Re_p_vars);
Re_f = (rho * Vf * D) / visc;
Re_f_vars = [Tf Dm Fs visc];
u_Re_f_vars = [u_Temp u_tapemsr u_FanSpeed u_Viscosity];
[u_Re_f] = Uncertainty_Eval(Re_f,Re_f_vars,u_Re_f_vars);
Sc = D * L;
chord = chordm * 0.0254;
span = spanm * 0.0254;
Sa = chord * span;
Cpc = F / (0.5 * rho * Vp^2 * Sc);
Cpc_vars = [F Tf visc P Dm L];
u_Cpc_vars = [u_Forces u_Temp u_Viscosity u_V u_caliper u_tapemsr];
[u_Cpc] = Uncertainty_Eval(Cpc,Cpc_vars,u_Cpc_vars);
Cfc = F / (0.5 * rho * Vf^2 * Sc);
Cfc_vars = [F Tf visc Fs Dm L];
u_Cfc_vars = [u_Forces u_Temp u_Viscosity u_FanSpeed u_caliper u_tapemsr];
[u_Cfc] = Uncertainty_Eval(Cfc,Cfc_vars,u_Cfc_vars);
Ca = F / (0.5 * rho * Vf^2 * Sa);
Ca_vars = [F Tf visc Fs chordm spanm];
u_Ca_vars = [u_Forces u_Temp u_Viscosity u_FanSpeed u_chord u_span];
[u_Ca] = Uncertainty_Eval(Ca,Ca_vars,u_Ca_vars);
Strouhal = .198 * (1 - (19.7 / Re_f));
Strouhal_vars = [Tf Dm Fs visc];
u_Strouhal_vars = [u_Temp u_tapemsr u_FanSpeed u_Viscosity];
[u_Strouhal] = Uncertainty_Eval(Strouhal,Strouhal_vars,u_Strouhal_vars);
Shedf = (Strouhal * Vf) / D;
Shedf_vars = [Tf Dm Fs visc];
u_Shedf_vars = [u_Temp u_tapemsr u_FanSpeed u_Viscosity];
[u_Shedf] = Uncertainty_Eval(Shedf,Shedf_vars,u_Shedf_vars);
%Part 1 Boundary Layer Up
Tf = subs(Tf,P1BLupstream(1:5:41,2));
Dm = subs(Dm,14);
P = subs(P,P1BLupP);
visc = subs(visc,P1mu);
u_P1BLupRe = double(eval([u_Re_p]));
u_P1BLupRe = mean(u_P1BLupRe);
fprintf('Part 1 Uncertainty Boundary Layer Up Stream t Re = %f n',u_P1BLupRe)

Page | A27
syms Tf Dm P
%Part 1 Boundary Layer Down
Tf = subs(Tf,P1BLdownstream(1:5:41,2));
Dm = subs(Dm,38);
P = subs(P,P1BLdownP);
visc = subs(visc,P1mu);
u_P1BLdownRe = double(eval([u_Re_p]));
u_P1BLdownRe = mean(u_P1BLdownRe);
fprintf('Part 1 Uncertainty Boundary Layer Down Stream t Re = %f n',u_P1BLdownRe)
syms Tf Dm P L F
%Part 1 Cylinder Wake Test 25 Hz
Tf = subs(Tf,75);
Dm = subs(Dm,1.9);
P = subs(P,P1Cyl25HzP);
u_P1Cyl25HzRe = double(eval([u_Re_p]));
u_P1Cyl25HzRe = mean(u_P1Cyl25HzRe);
fprintf('Part 1 Uncertainty Cylinder Wake 25Hz t Re = %f n',u_P1Cyl25HzRe)
L = subs(L,24);
chordm = subs(chordm,6);
spanm = subs(spanm,14.875);
F = subs(F,P1Cyl25Drag);
u_P1Cyl25HzDrag = double(eval([u_Cpc]));
u_P1Cyl25HzDrag = mean(u_P1Cyl25HzDrag);
fprintf('Part 1 Uncertainty Cylinder Wake 25Hz t Drag = %f n',u_P1Cyl25HzDrag)
syms Tf Dm P L F
%Part 1 Cylinder Wake Test 45 Hz
Tf = subs(Tf,78);
Dm = subs(Dm,1.9);
P = subs(P,P1Cyl45HzP);
u_P1Cyl45HzRe = double(eval([u_Re_p]));
u_P1Cyl45HzRe = mean(u_P1Cyl45HzRe);
fprintf('Part 1 Uncertainty Cylinder Wake 45Hz t Re = %f n',u_P1Cyl45HzRe)
L = subs(L,24);
F = subs(F,P1Cyl45Drag);
u_P1Cyl45HzDrag = double(eval([u_Cpc]));
u_P1Cyl45HzDrag = mean(u_P1Cyl45HzDrag);
fprintf('Part 1 Uncertainty Cylinder Wake 45Hz t Drag = %f n',u_P1Cyl45HzDrag)
Part 1 Uncertainty Boundary Layer Up Stream Re = 533.202036
Part 1 Uncertainty Boundary Layer Down Stream Re = 1454.656732
Part 1 Uncertainty Cylinder Wake 25Hz Re = 66.278893
Part 1 Uncertainty Cylinder Wake 25Hz Drag = 0.001330
Part 1 Uncertainty Cylinder Wake 45Hz Re = 117.349164
Part 1 Uncertainty Cylinder Wake 45Hz Drag = 0.001859

Page | A28
function [ u_S ] = Uncertainty_Eval( S,vars,u)
%Calculate the Uncertainty of any Function
%This loop symbollically differentiates a function S with respect to each
%of it's variables then adds those differentials to a matrix
for i = 1:length(vars);
partial(i) = diff(S, vars(i));
end
%This takes the differentials found in the loop and applies them to the
%function for uncertainty
u_S = sqrt(sum((partial.*u).^2));
end
Published with MATLAB® R2015a

Joshua Beckerman Lab 4 – Fluid Mechanics Final Write-Up

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