Air drag
Laboratory at Umeå University, Department of Physics
Aim
The aim of this laboratory is to give you knowledge and understanding of the phenomenon of air drag and
how it can be measured with a simple setup. Furthermore, you will learn how to analyze measurement
results and draw conclusions from them. Last but not least, this laboratory gives you an example of how
different physical effects combine to describe, though only qualitatively, a rather complex problem - such
as the trajectory of a golf ball.
Task
Your task in this laboratory will be to understand the theory behind air drag and with this knowledge
figure out how the provided experimental setup could be used to investigate the air drag as a function of
velocity. The measurement results, combined with the theory, should enable you to give a proper
explanation for why a dimpled golf ball flies farther than a ball with a smooth surface. You will, along the
way, also gain insight in how to use already established knowledge, in order to evaluate your attained
results.
Department of Physics
Jenny Enevold
2014-03-24
Klassisk Mekanik, 7,5 or 9 credits
Air Drag
Contents
1
Introduction .......................................................................................................................................................... 3
2
Theory .................................................................................................................................................................... 3
2.1
Different types of flow ............................................................................................................................... 3
Laminar flow ......................................................................................................................................................... 3
Turbulent flow ...................................................................................................................................................... 3
2.2
Interaction between flow and object ....................................................................................................... 3
Drag force in different regimes .......................................................................................................................... 4
Reynolds number ................................................................................................................................................. 4
Quantitative expressions describing air drag on a sphere .............................................................................. 5
Laminar and turbulent separation...................................................................................................................... 5
2.3
Dimples......................................................................................................................................................... 7
3
Experiment ............................................................................................................................................................ 8
4
Tasks to accomplish before coming to the laboratory class.......................................................................... 9
5
Tasks to accomplish in class ............................................................................................................................... 9
6
Some key words in Swedish..............................................................................................................................10
7
References ...........................................................................................................................................................10
2
Department of Physics
Jenny Enevold
2014-03-24
Klassisk Mekanik, 7,5 or 9 credits
Air Drag
1 Introduction
Why does a feather experience lower acceleration than a stone when falling? How come an airplane stays
in the air and a boomerang comes back? Why do car producers perform wind tunnel tests on their cars? It
all connects to the fact that these objects move in the earth’s atmosphere. An object that is immersed in a
flowing gas or liquid, experiences a force from the flowing medium. In the case of an object travelling
through the atmosphere, the force is called air drag. The impact from the air drag on the object depends
on the object’s velocity, geometry and surface properties.
In this laboratory you will derive an expression for the force exerted on a moving ball by the air, in terms
of parameters and variables whose values are given or can be measured in your experimental setup. For
comparing how different objects interacts with an air flow, you will make use of a parameter called the
drag coefficient, . You will experimentally determine
for two similar balls with different surfaces,
travelling through air at a range of velocities. This experiment will help to explain, among other things,
why a golf ball, which has a rough surface, flies much farther than a ball with a smooth surface.
2 Theory
A good way to describe what happens in a viscous medium when a solid object moves through it is the
layer model, first developed by Ludwig Prandtl [1]. In the course of this laboratory you will examine the
air drag on two sphere-shaped balls with different surfaces travelling through stationary air. The entire
layer of air that is significantly affected by the movement is called the boundary layer. Of special interest is
the part of the boundary layer closest to the ball, which is subject to friction from the surface of the ball
and will move along with it. Due to the viscosity of air, every layer between the layer that travels with the
ball and the first layer at rest, where the boundary layer ends, will experience friction from its neighboring
air layers. The layer closest to the ball will thus tend to drag the adjacent layer along with it, while that one
is in turn slowed down by the next neighboring layer.
2.1 Different types of flow
The flow of a gas or liquid is characterized by its behavior.
Laminar flow
In laminar flow, the streamlines are well defined and stable as a function of time. This means that the
pressure, speed and direction at a given point in the flow will be the same at all times – one can choose a
spot in the flow, look away, have a sandwich and come back, and the velocity vector at the chosen spot
will be unchanged [2].
Turbulent flow
In turbulent flow, the streamlines are disorganized and fluctuate with time. Eddies are typically randomly
created, and soon disappear again. The behavior of the flow is chaotic on a small scale; the path of a
molecule released somewhere in the flow cannot be predicted [2].
2.2 Interaction between flow and object
When characterizing the interaction between a flow and an object, the concepts of turbulent and laminar
flow are used in a somewhat different way.
3
Department of Physics
Jenny Enevold
2014-03-24
Klassisk Mekanik, 7,5 or 9 credits
Air Drag
Drag force in different regimes
In this context, laminar flow means that the force that the medium exerts on the object is only due to the
viscosity of the flow. This is truly fulfilled only if no turbulence arises anywhere around the object, as
depicted in Fig. 1.
Figure 1. Laminar flow around a sphere. The arrow
indicates the velocity of the ball relative to the medium.
Laminar flow is thus a idealized concept, but is often successfully assumed when the relative speed of an
object in a medium is low. Experiments show that viscous forces are proportional to velocity, , and so is
therefore also drag force on an object in laminar flow;
( )
( )
.
When the relative velocity between the object and the flow increases, a turbulent wake will appear behind
the object. The molecules in the wake bump into each other, changing their speed constantly. With
increasing turbulence, a resultant pressure fall is observed. The drag force on the object becomes more
and more dominated by the pressure difference between the front side and the back side of the ball. This
alters the velocity dependence of the drag force. When the turbulence is well developed, the velocity
dependence becomes quadratic;
( )
.
( )
Reynolds number
The Reynolds number
is a dimensionless parameter that characterizes the flow with respect to an
object. A small
implies that the drag force on the object is exerted by a laminar flow, and a large
implies that the flow is turbulent. It is defined as
,
( )
where is the density of the medium, is the relative velocity of the object and the medium, is the
dynamic viscosity of the medium, and is the “typical cross section diameter” of the object. If the object
is sphere-shaped with radius , we have that
.
The interaction between flow and object thus depends on whether the flow is laminar or turbulent. In
many real cases, the flow must be described by a mixed behavior of both types.
4
Department of Physics
Jenny Enevold
2014-03-24
Klassisk Mekanik, 7,5 or 9 credits
Air Drag
Quantitative expressions describing air drag on a sphere
In the true laminar case, mathematical analysis is able to provide an exact expression that quantitatively
determines the drag force on a sphere with radius . The derivation is beyond the scope of this text, but
results in the celebrated Stokes’ law [3]
( )
,
where is the dynamic viscosity of the medium. This expression is only applicable when the drag is
dominated by viscous forces and
is very small. As a rule of thumb,
is required for Eq. 4 to be
valid.
Turbulence on the other hand means chaos, why any exact mathematical analysis would be too
complicated to be usable for quantitative characterization of the drag on an object in turbulent flow.
Instead, the drag force is expressed with the help of the drag coefficient .
is a dimensionless
parameter that depends on the geometry and surface properties of the object, and can only be determined
experimentally. For
, the drag force is completely dominated by turbulence in the flow and can
be described by the expression
,
( )
where is the density of the medium and is the “typical cross section area” of the object. When the
object at hand is a sphere of radius , means the largest cross section area of the sphere, i.e.
[2].
From Eq. 1 and 2, it is clear that increased velocity normally means increased air drag, no matter whether
the flow is laminar or turbulent. Note, however, that is not constant with respect to velocity, although
its value normally varies slowly with speed.
Laminar and turbulent separation
The symmetrical appearance of the streamlines in Fig. 1 is disrupted when the speed is increased. At high
enough velocities a large turbulent wake is created behind the ball, as shown in Fig. 2. The boundary layer
is laminar all the way along the surface of the ball, until the point where the innermost stream lines
separate from the surface. This is called laminar separation. The pressure in the turbulent area comprising
the wake is significantly lower than the pressure in the laminar area in front of the ball. Pressure
differences give rise to a resulting force that is proportional to the area on which the pressure acts; the
larger the cross section area of the wake, the larger the drag force due to the pressure difference.
Figure 2. Laminar separation occurs when the velocity
is high enough to tear the laminar stream lines off the
surface of the object, creating a wake of turbulent flow
behind it. The drawing is a schematic illustration of the
wake at Reynolds numbers well above
. The arrow
indicates the speed of the ball relative to the medium.
At even higher velocities, turbulence creeps along the surface of the ball towards the front. Interestingly
enough, the outer stream lines in the boundary layer cling to the turbulence and preserve laminar flow to
5
Department of Physics
Jenny Enevold
2014-03-24
Klassisk Mekanik, 7,5 or 9 credits
Air Drag
follow the round of the ball further to the rear, before separation takes place. The resulting wake thus has
a smaller cross area facing the rear of the ball, than the wake created by laminar separation. This is
schematically shown in Fig. 3. As a consequence, the pressure difference decreases so much so that the air
drag actually decreases, despite the increased velocity. Because of the turbulence in the boundary layer,
this is called turbulent separation.
Figure 3. At very high velocities, the flow around the
sphere undergoes turbulent separation. The laminar
stream lines in the outer part of the boundary layer clings
to the eddies in the turbulence close to the surface and
are able to follow the shape of the ball further back to its
rear. As a consequence, the cross section area of the
wake decreases, resulting in a decreased drag force. The
arrow indicates the speed of the ball relative to the
medium.
Turbulent separation followed by a strong enough decrease in to decrease the drag is a rather specific
phenomenon for sphere-like objects, and should not be taken for a general feature observed for all objects
travelling through a viscous medium. Fig. 4 presents a graph of as a function of , where the sudden
steep decrease in at
reveals the onset of turbulent separation [4] for a smooth sphere.
6
Department of Physics
Jenny Enevold
2014-03-24
Klassisk Mekanik, 7,5 or 9 credits
Air Drag
Figure 4. The graph shows the drag coefficient
as a function of Reynolds number
for a smooth sphere. Note
the logarithmic scale. The picture is taken from the work of Niloofar Moradian et. al. [4].
2.3 Dimples
The critical velocity for obtaining turbulent separation is very high for a smooth sphere moving in air. For
a ball with dimples on the surface, however, turbulent separation takes place at much lower velocities,
velocities that are easily reached by a well-hit golf ball. There is, as of today, no thoroughly mathematical
explanation for the effects of dimples. It just seems that small eddies arise in the dimples, that help the
onset of turbulence in the boundary layer [5].
Historically, the effect of equipping a golf ball with dimples has been discovered accidentally, when, in the
19th century, golf players observed that old, slightly damaged balls flew further than new smooth balls.
7
Department of Physics
Jenny Enevold
2014-03-24
Klassisk Mekanik, 7,5 or 9 credits
Air Drag
3 Experiment
The experimental setup provides two different balls with a thread attached. One is a table tennis ball and
has a smooth surface. The other ball, which will be referred to as the dimpled ball, is similar in size and
weight, but has a rough, dimpled surface that resembles that of a golf ball. There will also be a stand, in
the form of a low table, with a vertical, rotatable axle sticking up through its top. The axle is driven by an
electric motor and can rotate with up to 800 rpm (read the instruction for the motor at the setup). With
the thread you can mount a ball on the axle in order to accelerate it into a horizontal orbit. From the
reading on the display of the motor, you will be able to determine the angular velocity, , of the ball. Fig.
5 shows the setup at an instant in time when the experiment is running.
Figure 5. The experimental setup while running the experiment.
As the angular velocity of the axle with respect to the ball changes, the thread will wind or unwind on the
axle and the radius of the orbit will change accordingly, in order for the system to equilibrate into a new
state of force balance. A schematic drawing of the experimental setup, as seen from above, is given in Fig.
6.
Figure 6. A ball rotates with a constant velocity around an axis and experiences a drag force
due to air
resistance. is the distance between the center of the ball and the center of the axle and is the distance between
the center of the ball and the point where the wound thread separates from the axle. Note that must vary with .
is the radius of the axle. is the tension in the thread, i.e. the force by which the thread acts on the ball. is the
angle between the thread and a virtual line from the center of the ball to the center of the axle.
is the total force
acting on the ball.
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Department of Physics
Jenny Enevold
2014-03-24
Klassisk Mekanik, 7,5 or 9 credits
Air Drag
Note that when the orbital movement of the ball is uniform, its speed is constant, which means that there
is no tangential force on the ball. Hence the total force,
, must be orthogonal to the tangential
direction. As the ball is kept in the orbit, the direction of must be inwards, towards the center of the
circular orbit.
In the laboratory classroom you will access facilities such as a spirit level, a scale and a vernier caliper.
4 Tasks to accomplish before attending the laboratory class
1. Make sure you grasp the concepts of layer model, boundary layer, air drag, drag coefficient and
Reynolds number. Formulate a written definition in simple words.
2. Formulate, in your own words, the meaning of laminar and turbulent flow with or without an
object to relate to, and the difference between laminar and turbulent separation.
3. Study the experimental setup in Fig. 6. There are two triangles in the schematic drawing, one
depicting the relation between the forces acting on the ball, and one relating geometrical
distances. How do these two triangles relate to each other? We can be certain about both of them
being right-angled. How come?
4. Identify which quantities you are supposed to be measuring during the laboratory class.
5. Given that is known, start from Eq. 5 and rewrite the expression for the drag coefficient in
terms of what can be measured in the experimental setup. When you have arrived to an
expression of as a function of , rewrite it again as a function of . can also be taken to be
known.
5 Tasks to accomplish in class
At room temperature, the dynamic viscosity of air is
kg/m3.
Ns/m2 and its density is
1. Perform the measurements that you have identified before attending the laboratory class and
determine
( ) for both balls, for the largest range of possible. How is this interval of
identified? Discuss within your group. Also discuss how to select an appropriate set of data
points. Motivate for you laboratory supervisor before starting the experiments.
2. Plot your results, i.e.
( ) for both balls, in the same figure, for example in Origin. Do the
same for
( ). Analyze your results with respect to the theory given in the theory section.
Have you made any assumptions on the way? Were these assumptions valid, regarding the results?
3. Compare your results to the graph in Fig. 4. Do your results reproduce the plot? If not, why?
A valid experimental result should give correct predictions about real events. If a golf ball falls freely
through air, it will eventually reach force balance, when the gravity force and the air drag cancel each
other.
4. Express the equilibrium velocity (the velocity reached at force balance) of the golf ball. Choose an
appropriate value of based on your experimental results and calculate the equilibrium velocity,
given that a real golf ball weighs
( ) g and has a diameter of
( ) mm. Evaluate your
result by searching for a measured free fall velocity for some comparable object on the internet.
How does the equilibrium velocity change when the size of the ball increases, but not its density?
9
Department of Physics
Jenny Enevold
2014-03-24
Klassisk Mekanik, 7,5 or 9 credits
Air Drag
6 Some key words in Swedish
Air drag coefficient
Luftmotståndskoefficient
Boundary layer
Gränsskikt
Dimple
Liten fördjupning, grop
Flow
Flöde
Laminar
Laminär
Reynolds number
Reynoldstal (böjs Reynoldstalet i bestämd form)
eller
Reynolds tal (kan inte böjas i bestämd form)
Turbulent
Turbulent
Wake
Kölvatten
Eddy
Virvel
7 References
1
John D. Andersson, Jr., Ludwig Prandtl’s Boundary Layer, Physics Today 2005, Volume 58, Issue 12.
2
Pijush K. Kundu, Ira M. Cohen, David R Dowling, Fluid Mechanics, Academic Press 2011.
3
D. J. Acheson, Elementary Fluid dynamics, Oxford University Press, 1990.
4
Niloofar Moradiana, David S.-K. Tinga, Shaohong Chengb, The effects of freestream turbulence on the drag coefficient of
a sphere, Experimental Thermal and Fluid Science 2009, Volume 33, Issue 3.
5
Jin Choi, Woo-Pyung Jeon, and Haecheon Choi, Mechanism of drag reduction by dimples on a sphere, Physics of
Fluids 2006, Volume 18, Issue 4.
10