S/V Mathematical Abstraction
Every now and then, I put on
my skeptic hat and stop believing in aerodynamics. Forget Bernoulli
and Reynolds and Mach; forget all of that foolishness about
pressure differentials and laminar flow; airplanes stay in the
air because they beat air molecules with a blunt instrument,
expend kinetic energy, and transfer momentum.
While in one of these moods recently, I designed
a boat, the S/V "Mathematical Abstraction", which
has a frictionless hull, a 100% efficient zero-leeway keel,
zero windage, and a flat, frictionless sail. The question: Can
the "Abstraction", without reference to any aerodynamic
eyewash, still sail to windward? I would certainly hope so,
but have never actually seen the calculation laid out before...
I should probably point out, before we go any
further, that just because we have eliminated friction losses
in this little mental experiment, the boat is NOT free to move;
there is still the matter of the boat's mass to deal with, and
even more important, the mass of the water that the boat displaces.
Every time the boat travels its own length forward, it needs
to smash a mass of water equal to the boat's weight out of the
way. (Yes, I know, the water behaves elastically, and actually
pushes the boat forward to some extent as it returns to fill
the hole left by the boat's passing, but the important thing
is that the boat is held in place by forces other than friction.)
Suppose the "Abstraction" is sailing
to windward on a course that is "I" (for "incidence",
as in "angle of incidence") degrees off the wind,
and the sail is trimmed to "I/2" (which, it turns
out, is the best place for a flat, frictionless sail to be trimmed).
The force of the wind can then be divided into a "working"
component, which acts on the sail, and is at 90 degrees to the
sail, and a "waste" component, which is parallel to
the sail. The "working" component of the wind is proportional
to the Sine of the angle at which the wind stikes the sail,
or in this case "I/2".
So what does the boat do? Force is being applied
to it at 90 degrees to the sail, and this force can then be
broken into a "heeling" component, at 90 degrees to
the direction of travel, and a "driving" component,
the "driving component being, once again, equal to the
Sine of the angle between the sail and the boat, which is once
again "I/2". In other words, the force available to
drive the boat is proportional to the square of the Sine of
one half the angle between the wind and the boat's heading.
("Driving force is proportional to the square of the Sine
of one half the angle of incidence.")
All of this suggests another calculation: since
the speed of the boat directly into the wind is equal to the
Cosine of the angle of incidence, we can wonder if there is
a specific angle of incidence at which the "Abstraction"
makes maximum progress to windward (as opposed to simply maximum
speed on current heading). It turns out that there is; the function
(Cosine of I times the square of the Sine of I/2) reaches a
maximum value at 60 degrees.
So... What does any of this have to do with those
of us living in a world with friction, windage, and leeway (to
say nothing of Bernoulli, Reynolds, Mach, and company...)? Probably
not very much, though it is interesting to find mathematical
evidence for the old advice against "pinching the wind".
And of course, it serves as a reminder that there is usually
at least one more way to look at any problem.
August 28, 2004