All that is necessary to create lift is
to turn a flow of air. The airfoil of a wing
turns a flow, and so does a rotating cylinder.
A spinning ball also turns a flow and
generates an aerodynamic lift force.
The details of how a spinning ball creates lift are fairly complex.
Next to any surface, the molecules of the air
stick to
the surface, as discussed in the
properties of air slide. This thin
layer of molecules entrains or pulls the surrounding flow
of air. For a spinning ball the external flow is pulled
in the direction of the spin. If the ball is not
translating,
we have a spinning, vortex-like flow set up around
the spinning ball, neglecting three-dimensional and
viscouseffects in the outer flow.
If the ball is translating through the air at
some velocity, then on one side of the ball the entrained flow opposes
the free stream flow, while on the other side of the ball, the
entrained and free stream flows are in the same direction.
Adding the components of velocity for the entrained flow to the free
stream flow, on one side of the ball the net velocity is less
than free stream; while on the other, the net velocity is
greater than free stream. The flow is then turned by the
spinning ball, and a force is
generated. Because of the change to the velocity field, the pressure
field is also altered around the ball. The magnitude of the
force can be computed by
integrating
the surface pressure times the area around the ball.
The direction of the force is
perpendicular (at a right angle) to the flow direction.
On the figure at the left, we show the geometry of the spinning ball.
A ball of radius b rotates at speed s
measured in revolutions per second.
A black dashed line indicates the axis of rotation of the ball,
and the ball rotates clock-wise, when viewed along the axis from the
lower left. The ball has been sliced into a large number of
grey-colored sections along the axis of rotation.
The air with velocity V and density rho
strikes the ball from the upper left.
The resulting lift force L is perpendicular to the air velocity and
the axis of rotation.
To determine the ideal lift force on the ball,
we consider the spinning ball to be composed of an infinite number
of very small, grey-colored,
rotating cylinders.
Adding up (integrating) the lift of all of the cylinders along the axis
gives the ideal lift of the ball.
The Kutta-Joukowski lift theorem for a single cylinder
states the lift per unit length
L is equal to the density rho of the air times
the strength of the rotation Gamma
times the velocity V of the air:
L = rho * Gamma * V
The strength of rotation is directly related to the rotational speed
of the cylinder. For a cylinder of radius r rotating at angular
speed s, the surface of the cylinder moves at speed Vr
given by:
Vr = 2 * pi * r * s
Then the strength of rotation is equal to:
Gamma = 2 * pi * r * Vr
Combining these equations:
Gamma = 4 * pi^2 * r^2 * s
And substituting this value into the Kutta-Joukowski lift equation give:
L = 4 * pi^2 * r^2 * s * rho * V
This is the lift per unit length for a single small cylinder and is measured in
force per length (lbs/ft). The ball is composed of an infinite number of
cylinders with the radius of the cylinders changing along the axis of rotation.
r = f (l)
where l is a length measured along the axis of rotation.
For a ball of radius b we have to integrate the lift per unit length
along the axis of rotation from -b to +b. If we let S be
the symbol for integration and dl be an increment of l:
L = S (4 * pi^2 * r^2 * s * rho * V) dl
If we let phi be an angle from the center of the ball along the axis
of rotation:
r = b sin(phi)
l = b cos(phi)
dl = -b sin(phi) d(phi)
Then:
L = 4 * pi^2 * b^3 * s * rho * V * S (sin (phi))^3 d(phi)
with the limits of integration going from zero to pi.
Performing the integration:
L = (4 * pi^2 * b^3 * s * rho * V) * 4 / 3
Let's investigate the lift on a spinning ball by using a Java
simulator.
The left window shows a view of a ball placed in a flow of air.
The ball is a foot in diameter and it is moving 100 miles an hour.
You can spin the ball by using the slider below the view
window or by backspacing over the input box, typing in your new value and
hitting the Enter key on the keyboard. On the right is a graph of the lift
versus spin.
The red dot shows your conditions. Below the graph is the
numerical value of the lift. You can display either the lift value (in
English or Metric units) or the lift coefficient by using the choice
buttons surrounding the output box. Click on the choice button and select
from the drop-menu.
As an experiment, set the spin to 200 rpm (revolutions per minute) and
note the amount of lift. Now increase the spin to 400 rpm.
Did the lift increase or decrease?
Set the spin to -400 rpm. What is the value of lift?
Which way would this ball move?
You can further investigate the lift of a spinning ball, and a variety of
other shapes by using the
FoilSim II Java Applet.
You can also
download
your own copy of FoilSim to play with
for free.
There is also a Java Applet called CurveBall
to help you explore the aerodynamics of big league pitching.
It computes the path of a thrown curveball.
Be particularly aware of the simplifying assumptions that have
gone into this analysis. The type of flow field shown
on this figure is called an ideal flow field. We have produced
the ideal flow field by superimposing the flow field from an ideal
vortex centered on the ball with a uniform free stream flow. There is
no viscosity in this model, and therefore, no
boundary layer
on the ball, even though viscosity is the real origin of the
circulating flow!
The equation given above describes the ideal lift force generated
on a smooth, rotating ball. We can use this description as a first order,
or preliminary, estimate of the lift on a baseball.
But in reality, the flow around a spinning baseball is very complex
Viscosity generates a boundary layer on the ball and
the stitches used to hold the covering of the ball together
stick up out of the boundary layer and disturb both the boundary layer
and free stream flow.
On the ball, the boundary layer transitions to
turbulent flow
which affects the amount of
aerodynamic drag.
The stitches are not symmetrically distributed around the ball.
So the real flow around the ball is separated, unsteady, and
not uniform. To account for these real-world effects that we have neglected
in our ideal flow model, we define a
lift coefficient.
The lift coefficient is an experimentally determined factor that
is multiplied times the ideal lift value to produce a real lift value.
The ideal simplified model gives the first order
effects and tells us the relative importance of the factors that
affect the lift force on the baseball, while all of the complex factors
are modeled by the lift coefficient.
