Vortex Theory for Bodtes Moving tn Water 
DISCUSSION 
John P. Breslin 
Stevens Instttute of Technology 
Hoboken, New Jersey, UsS.A. 
Admiral Brard has certainly given us an exhaustive treat- 
ment of vortex representations, and I think he has made an excellent 
point that this approach to ship problems has been largely neglected. 
I just wanted to make a point here of an intuitive character. 
He points out on page 1255, regarding Casal's theory - which, 
as I recall, is a vortex theory for the lateral force and moment ona 
ship, which is considered to be quite thin, that while the lift force ob- 
tained from this theory is quite reasonable, (and I believe was in agree- 
ment with low aspect ratio theory if one takes twice the draft divided 
by the length as the aspect ratio of an equivalent wing) there is a dis- 
appointment in that the position of this lateral force does not coincide 
with the observations obtained when one measures the lateral force 
and static yawing moments on ship models, In that case, when one 
divides the moment by the force the result is a lever arm which is 
greater than the half length of the ship. In any event, this so-called 
point of application of the lateral forces is often off the ship. It seems 
to me, most of us are aware that if you take a body revolution - the 
spheroid is the classic example - it has a zero lateral force in an in- 
viscid fluid but it has a very definite moment, say, about its mid- 
length called the Munk moment. It is my feeling that ships may be 
neither fish nor fowl, they are neither bodies of revolution, nor are 
they flat lifting surfaces to which the lifting vortex theory could be ex- 
pected to apply in all respects. In particular, if we think of the ship 
as a modification of a body revolution, that is that the central portion 
looking at it side on, together with its zero Froude number reflection 
on the water plane, can be described as a body of revolution plus flat 
surfaces fitted into the bow and stern representing the sharp bow and 
the sky in the region of the stern, then we should expect that it would 
be necessary to account for the combined effect of these fin areas or 
to treat the body as a body revolution with fins fore and aft. This pro- 
blem, then, has to be solved theoretically. This, I should think, would 
then exhibit some of the body revolution characteristics in its moment 
and without disturbing particularly the rather excellent agreement that 
one obtains from small angles with the vortex theory. 
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