Download Air drag

Air drag
Bestämning av luftmotståndskoefficienterna
för två olika bollar
Laboratory at Umeå University, Department of Physics
Instruction by Florian Schmidt (2004-03-22)
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The aim of this laboratory is to give you knowledge and understanding of the phenomenon "air
drag" and how this entity 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).
7KH7DVN 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
resistance as a function of velocity. The measurement results combined with the theory, should
enable you to give a proper explanation for why a well-hit golf ball flies further than a ball with
smooth surface.
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Why does a feather experience lower acceleration than a stone when falling? Why does an airplane fly
and a boomerang come back? Why do car producers perform wind tunnel tests on their cars? Why is
golf such a successful sport?
It all connects to the fact that those objects move in the earth’s atmosphere respectively in air. They
then experience a force due to DLUGUDJ (or DLUUHVLVWDQFH) that depends on their velocity, geometry and
material properties. These parameters determine if the airflow around the object is ODPLQDU or
WXUEXOHQW. Laminar flow is characterized by straight layers of air that move at the same velocity and do
not interact with each other (Fig. 1a). This type of flow is often assumed when the relative speed of
an object in a medium is "low". In turbulent flow, however, the flow lines are disorganized in many
eddies. The molecules in the turbulent wake bump into each other changing their speed constantly
(Fig. 1b).
In this lab you will derive an expression for a parameter called the GUDJFRHIILFLHQW, CD . You will use this
expression to experimentally determine CD for two different balls for a range of velocities. This
experiment will help to explain, among other things, why a well-hit golf ball (with a rough surface)
flies much further than a ball with a smooth surface.
Figure 1: A ball moving through air with the boundary layer shown as flow lines. Panels (a) and (b) show a smooth ball
with low respective high velocity. Panels (c) and (d) show a ball with dimples (rough surface) at high speed without
respectively with spin. Picture taken from [1].
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In the course of this lab we want to describe a ball travelling in air. A good way to describe the
movement of solid objects through a viscous media such as air is the OD\HU PRGHO. Specifically of
interest is the ERXQGDU\OD\HU that is the layer of air closest to the ball. This layer is subject to friction
from the surface of the ball and moves with it. The type of flow the ball is in depends on the
roughness of the ball’s surface. The boundary layer reaches until the first layer of air at rest.
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If a VPRRWKball moves relatively slow the boundary layer stays intact until behind the ball and the flow
is called ODPLQDU (Fig. 1a). The air pressure behind the ball is then almost equal to the pressure in the
front wherefore there is little air drag exerted on the ball.
The force on a sphere in laminar flow is called Stoke’s law (fluid dynamics required for derivation)
) = 6p h UY ,
(1)
where h is the dynamic viscosity of air, Y is the velocity and U the radius of the sphere. Viscosity is
the resistance of a liquid or gas to flow due to intermolecular friction. There are two related measures
of the viscosity, which are known as G\QDPLF (or DEVROXWH) and NLQHPDWLF viscosity. The latter is defined
as the ratio of dynamic viscosity and density, e.g. n = h / r .
Figure 2: Common shapes and car designs and their drag coefficients. Picture from [2].
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With increasing velocity of the ball a phenomenon called ODPLQDUVHSDUDWLRQ takes place. The laminar
boundary layer separates approximately at point B in Fig. 1 and a turbulent wake builds up behind
the ball (Fig. 1b). The flow is then called WXUEXOHQW. Following %HUQRXOOL
VSULQFLSOH the pressure is now
lower behind the ball than in front of it since there is fast stirring of the air and a part of the ball’s
kinetic energy is dissipated. The air drag is therefore large and directed against the movement of the
ball.
The exact expression for the force on a sphere in turbulent flow is
)=
1 2
rY & $ ,
2
(2)
where r is the density of air, $ is the cross section area of the ball and & is the drag coefficient.
As for the Stoke’s law, in order to rigorously derive this expression a full fluid mechanics analysis is
required.
The drag coefficient, & , is a function of the geometry and the material properties of the ball and is
always determined experimentally. Figure 2 shows some objects and their drag coefficients.
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At very high velocities, the boundary layer of the smooth ball itself will become turbulent and
experience WXUEXOHQWVHSDUDWLRQ. This separation takes place further back at the ball (between points B
and C in Fig. 1), decreasing the size of the turbulent wake (Fig. 3 and Fig. 1c). In that case, WKHDLUGUDJ
LVVLJQLILFDQWO\UHGXFHG.
Laminär
separation
Laminär
separation
Turbulent
reattachment
Figure 3: Turbulent flow with laminar and turbulent separation.
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Turbulent
separation
Turbulent
separation
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As we have seen, normally, turbulent flow means increased drag. Only when the critical velocity for
turbulent separation is reached the air drag gets smaller. This critical velocity is very high for a
smooth sphere and is seldom reached. 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 no thoroughly (mathematical) explanation for the effects of dimples yet. It just seems
that the small eddies of the turbulent boundary layer lie in the dimples and help to cling the layer to
the surface a longer time, nearly to the rear of the ball (Fig. 3 and Fig. 1c) [3, 4].
Another possibility to reduce the air resistance, often used in other contexts, would be to make the
object more VWUHDPOLQHG, but that is obviously not applicable for a golf ball.
Historically, the effect of 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
[1]. À 6SLQ
Yet, the dimples and Bernoulli’s principle cannot satisfactory explain why a golf ball flies so much
further than a smooth ball. One also has to take into account the VSLQ of the ball. The spin (let’s
consider a backspin) has two consequences: First, the turbulent wake behind the ball becomes
asymmetric. It is shifted underneath the ball (Fig. 1d). This would counteract a far flight of the ball
since the air drag has now a downward component. But, secondly, the spinning ball drags a layer of
air around with it and due to superposition there is higher air velocity where the spin adds to the
velocity of the streaming air of the translational motion. In our case there is then lower pressure
above the ball, which leads to a lifting force. The lift is usually larger than the additional drag. This
phenomenon is called 0DJQXVHIIHFW[1].
The Magnus effect also explains for example why a properly shot football has a curved trajectory [5].
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A parameter to determine which flow an object is in is the dimensionless Reynolds number, 5 . It is
proportional to the ratio between the turbulent force and the laminar force, i.e., for a sphere with
diameter, G ,
5 =
r YG
.
h
(3)
For 5 < 1 the flow is definitive laminar and for 5 > 104 the flow is definitive turbulent. In other
words, for low 5 the viscosity of the medium is the dominant factor and the drag depends on Y ,
whereas for high 5 kinetic effects, such as turbulence, rule and the drag depends on Y 2 .
For example, a ball with a diameter of 5 cm that travels with a speed of 5 m/ s has a Reynolds
number of the order of 10 4 which indicates turbulent flow.
5
In fact, for the velocities you will use in your experiment the flow can always be considered
turbulent.
At room temperature the dynamic viscosity of air is h = 1.86 ¿10 5 Ns/ m2 and its density is
r = 1.21 kg/ m3.
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You have by now learned a great deal about different types of flow, the drag coefficient, layers and
layer separation, the Reynolds number as well as the effects of dimples and spin. In the lab you can
now investigate whether dimples on the surface of a ball really have an effect on the air resistance.
Since we lack a mathematical description of all the described effects together, we can restrict
ourselves to a study of the air drag. In fact, the drag coefficient, & , is a function of 5 and depends
on the velocity. You can therefore study how & changes as a function of velocity. You will,
however, not observe spin or lift phenomena.
Based on your results and your theoretical knowledge you should be able to give a proper
explanation for the far flight of a golf ball.
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The experimental setup (Fig. 4) provides two different balls with a thread attached. One is a table
tennis ball and has a smooth surface. The other one has a rough surface, resembling a golf ball. With
a thread you can mount them on an axle that is driven by an electrical motor and can rotate with up
to 800 rpm (see the instruction for the motor at the setup). Fig. 4 shows the setup at one instant in
time. As the velocity of the axle respectively the ball changes, the distance 5 will change accordingly
(because the thread is not fixed at point B) to keep the force balance.
Figure 4: A ball rotates with velocity Y around the axle and experiences a drag force ) due to air resistance. With )
being the centripetal force, ) the force due to tension in the cord, 5 the distance between the center of the ball and
the center of the axle, 7 the distance between the center of the ball and point B, D the radius of the axle and q the
angle between the cord and a virtual line from the center of the ball to the center of the axle.
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À Based on the experimental setup (Fig. 4) and equation (2), GHULYH an expression for the DLU GUDJ
FRHIILFLHQW& .
À 'HWHUPLQH& for the two balls as a function of their YHORFLW\ (e.g. 7-30 m/ s). Plot your results in the
same figure and analyze them.
À Plot your results as a function of the 5H\QROGVQXPEHU. In Figure 5, draw an additional curve that
could correspond to the measurement of a rough sphere (like a golf ball). Determine the
Reynolds-number and velocity for which the boundary layer of the golf ball in your experiment
starts to experience WXUEXOHQWVHSDUDWLRQ. Compare your results even with Figure 9.27, which will be
handed out by the supervisor.
After presenting the results of the three tasks above, solve the following additional, theoretical task.
À For a UHDOJROIEDOO (weight: 45.73(3) gram and diameter: 42.69(6) mm), determine the IUHHIDOOYHORFLW\
in the earth’s atmosphere. How does this free fall velocity change when the size of the ball
changes but not its density?
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Figure 5: The drag coefficient as a function of the Reynolds number for a smooth sphere. Y is here the kinematic
viscosity. Picture from [6].
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drag coefficient
dimple
spin
laminar
turbulent
flow
rough
smooth
boundary layer
thread
friction
force
separation
wake
luftmotstånds koefficient
liten fördjupning
spin
laminär
turbulent
flöde
skrovlig
slät
gräns skikt
tråd
friktion
kraft
separation
kölvatten
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[1]
The Physics of Golf, http:/ / services.golfweb.com/ library/ books/ pog/ pog1.html.
[2]
Air Resistance: Distinguishing Between Laminar and Turbulent Flow,
http:/ / academic.reed.edu/ physics/ courses/ Phys100/ lab2/ Air.Resistance.doc.
[3]
Why does a golf ball slice or draw?,
http:/ / www.physlink.com/ Education/ AskExperts/ ae423.cfm.
[4]
Why do dimples on a golf ball allow it to travel farther?,
http:/ / www.physlink.com/ Education/ AskExperts/ ae39.cfm.
[5]
Takeshi Asai, Takao Akatsuka and Steve Haake, The physics of football, Physics World, June
1998.
[6]
Andreas Alexandrou, Principles of Fluid Mechanics, Prentice Hall, 2001.
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