Re: drag coefficient - a postscript.

[brian whatcott <inet@INTELLISYS.NET>]

I see that an earlier note of mine may contribute to the
..ah.. delinquency of students
(It's not what you don't know, it's what you think you know...).

Please look at this drag calculation.

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> > Using figure 7.3 on page 89 of Wegener, we have
> >   0.47 -- sphere -- representative of point-first flight
> >   1.17 -- cylinder -- representative of broadside flight
>
> and take the 1.17 value for the drag coefficient and use the weight and
> dimensions for the bullet reported by Brian W. (i.e. 0.073N and
> 9 mm x 27 mm) and take the air density to be 1.29 kg/m^3 we get a value
> of V = 20 m/s.

///
Fd = Q A Cd
Q dynamic pressure  (1/2 rho v squared)
A representative area
Cd Coeff of Drag

For the Luger round:
 0.073 = 0.5 1.29 V^2  2.43E-4  1.17

V^2 = 398
V = 20 m/s as David noted.
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I *should* have noted a couple of caveats:

 When you see a value for coefficient of drag,
it's important to ask yourself, how was drag defined?

   You can see that I implicitly defined it above by
the equation
 Fd = Q A Cd        in other words, drag force is a
product of dynamic pressure q, a 'representative area' A,
and a coefficient of drag.

If a *mathematician* derives a coefficient of drag, for example
by the method of dimensional analysis, he might well conclude
from a tentative

F = K A^a V^b rho^c

that the sensible form is

F=K A rho V^2

and associate the constant K
with a coefficient of drag that varies from 0.7 for
a flat disk, to 0.03 for a pear shape body
(See ref 1 for instance)

On the other hand a physicist might wish to make explicit
the contribution of dynamic pressure q which is half rho v squared
and she will associate the coefficient of drag with a different number:
perhaps 1.4 for a flat disk to 0.06 for a streamline body.
(see ref 2 for instance).

In the same (confusing) way you need to verify the representative
area in use: it can be the *plan* area for wings - or alternatively
the area perpendicular to the undisturbed flow direction.

  For a flat disk normal to the flow, this would greatly
affect ones results. Luckily, we had a cylindrical model in mind
which has the happy property of equal areas or another unwanted
disparity could have entered.

Having said that, the hobby horse that I have been riding is this:
Reynolds Number.

I see that the sumptuous introductory college physics text
by Crummett and Western, "University Physics" (Ref 3) handles this
topic in an appealing way, touching on the impact of Re
on page 143.

 Re = L rho V/viscosity

where L is a representative length,
rho is air density
V is speed
dynamic viscosity is 1.8E-5 for air.

for the jumper we have
 Re = 0.6  1.29  53/1.8E-5   = 2E6  (about)

for the luger round, we have
 Re = 9E-3 1.29   20/1.8E-5 =  1E4  (about)

Unfortunately, fig 6.8, p 143, ref 3 makes it plain that
Cd drops dramatically (for a sphere at least) at Re > 300,000
to a fraction of its prior value.

In conclusion, assuming the original Cd from Wegener (ref 4)
refers to a q derived value, we err by scaling from the
known values for a jumper, to an object as small as a pistol round
because we ignore the discontinuity in Cd due to Re between them.
 The figure of Cd = 1.17 is appropriate to the broadside bullets,
(as suggested earlier)  but not to the jumper.

Hence, to Ludwik's question as to the value of a careful estimate
for a 9 mm bullet, I would respond that David Bowman's value
of 20 m/s is reasonable. It would be good to see an experimental
value, I agree.

Brian W.

Ref 1: Mechanics & Properties of Matter  Stephenson, Wiley
pp320, 3rd ed

Ref 2:  Engineering Aerodynamics   Diehl,   Ronald.
as abstracted in Marks Handbook, Aerodynamics section.

Ref 3:  University Physics, Models & Applications,
Crummett & Western  Wm C Brown Pub.

Ref 4: What Makes Airplanes Fly?
P. Wegener,  Springer Verlag NY Inc 1990
brian whatcott <inet@intellisys.net>
Altus OK