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Re: [Phys-l] football orientation in flight



Regarding why the football tends to point in the direction of it's
velocity: I want to resurrect John Denker's initial explanation (more
specifically, the way I recall his explanation) with a small but crucial
modification. We start with a football thrown and viewed by a
right-hander and view the football at an instant when the ball has a
small positive angle of attack. My understanding of John's argument was
that there would be a leftward Magnus force on the ball for which the
center of pressure would be forward (as measured along the axis of
symmetry) of the center of mass of the ball because the air flow would
be cleaner on the nose half of the football and where the flow is
cleaner, the Magnus effect is greater. This would result in a
counterclockwise torque, as viewed from above, which would cause the
angle of attack to increase further-the opposite of what is observed.

Ruprecht Nennstiel, in his document "How Bullets Fly" at:
<http://www.nennstiel-ruprecht.de/bullfly/index.htm>
indicates that for a spinning bullet, the center of pressure can be
forward or aft of the center of mass (depending on the velocity and spin
rate of the projectile) and that the torque associated with the Magnus
force (said torque is referred to as the Magnus Moment by Nennstiel)
will be stabilizing (tending to make the bullet point in the direction
of its velocity) if the center of pressure is aft of the center of mass.

So, assuming that John was right and it is the Magnus effect that
provides the torque that tends to keep the football pointing in the
direction of its velocity, the question is, what could lead to the
Magnus effect being greater on the tail end of the football than it is
on the nose? For that, it is helpful to turn to one of the three
editions of the tome, Flow around circular cylinders : a comprehensive
guide through flow phenomena, experiments, applications, mathematical
models, and computer simulations by M M Zdravkovich, Oxford ; New York :
Oxford University Press, 1997-<2003 >. On page 912 (which Google Book
Search will display) of that book there is a set of 16 curves, each for
a different value of the Reynolds number, on a graph of the Magnus force
vs. the ratio V_r/V where V_r is the rotational velocity and V is the
translational velocity. For most values of V_r/V, the higher Reynolds
number curve is above the lower Reynolds number curve (in fact, at all
values of V_r/V the curves are either the same or the higher Reynolds
number curve is above the lower Reynolds number curve) meaning that the
greater the Reynonlds number, the more positive the Magnus force. (For
many values of V_r/V you actually have an inverse Magnus force
represented by a negative value of the ordinate on the graph.)

In the case of the football, we have a wide variety of V_r/V at any
instant in time because of the variation of the radius of the football
along its axis of symmetry. However, that doesn't seem to matter; it
doesn't even matter whether we are dealing with the inverse Magnus
effect or the Magnus effect-the higher the Reynolds number the more
positive the Magnus force. (Evidence would suggest that we are not far
from the zero average Magnus effect since the football doesn't seem to
curve left or right in flight). I associate the higher Reynolds number
with "dirtier" flow. If the flow is indeed more turbulent on the tail
end of the football, this would mean there is a more positive Magnus
force on the tail of the football. All three of the possible cases
(Inverse Magnus force on nose and tail end that is stronger on the nose;
inverse Magnus force on nose and Magnus force on tail, Magnus force on
nose and tail that is stronger on the tail) result in a torque that is
clockwise as viewed from above-a torque that would cause the football
with its positive angle of attack to nose down.