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Referring to my comments about wind forces and current forces, which I said
would vary as the square of the velocity, Dan Allen responded-

>On Thursday, April 17, 2003, at 07:16 AM, George Huxtable wrote:
>
>> The force will vary as the square of the velocity
>
>Why the square?

and Marvin Sebourn commented-

I may misunderstand something
though--I thought that the power or force of the wind is proportional to the
cube of the velocity, not the square.

and Dave Weilacher asked-

...so would a 2 knot current be the sqrt((2^2) * 830) = 58kt wind?

It's gratifying to find such an interest in the science of hydrodynamics.

Dave's question first. The answer is yes (other things being equal, which
they usually are not, exactly: see below).

Now for Dan's fair question: why the square?

Let's consider something like a dustbin-lid on a long bridle, towed far
behind an immersed submarine. It makes no difference whether it's the
bin-lid moving through the water, or a water-current impinging on the lid.
If the speed doubles, then in a given time twice as much water impinges on
the lid. And with the speed doubled, every drop (or every molecule or every
gram) of that water has twice as much momentum (which is mass x velocity)
which it gives up to the lid when it's brought to a stop. Force is given by
the rate of transfer of momentum, so we get the force increased by 2 x 2 =
4. Is that good enough to satisfy Dan, or does he wish to get more
fundamental?

Now for Marvin's statement-

I thought that the power or force of the wind is proportional to the
cube of the velocity, not the square.

Well, power and force are NOT the same thing. Power is a measure of the
rate at which work is done, which is the product of force and speed. If, as
we said above, force goes as the square of speed, then power, being force
multiplied by the speed again, goes as the cube of speed.

All this is rather an oversimplification. At very low speeds the forces get
so low that the viscosity of water has an appreciable effect, and that
component of force doesn't follow a square-law (but don't ask me what law
it DOES follow!)

There's an additional effect, a really important one. A displacement vessel
floating on the surface is supported by displacing a pit in the water into
which the hull nestles. As the vessel moves, that pit has to move with it,
and it has to elbow water out of the way to do so. This process feeds
energy into surface waves, and at certain higher speeds this wavemaking
becomes the dominant energy-loss mechanism, contributing greatly to the
drag. That component of drag doesn't follow a square-law either (which is
why we specified our bin-lid as being towed by a submarine, to avoid making
surface waves).

These effects were spelled out by a fellow called William Froude in the
mid-1800s in order to interpret the data from model ships towed in
test-tanks, and to decide what speeds to tow them at.

George.

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contact George Huxtable by email at george@huxtable.u-net.com, by phone at
01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
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