Understanding the Prismatic Coefficient

By Stuart Slade
Updated 18 December 1998

OK, here goes.

The speed of a ship is determined by the point where the energy put into the water (by the engines) equals the energy lost to the water by drag. At this point, the ship is in dynamic balance with its surroundings and it will maintain speed. The ship's speed can be increased by injecting more energy into the water or by ensuring the ship looses less energy to the water. Eventually there comes a point where the ship's engines cannot add more power to the water so the ship cannot accelerate faster. This point, by the way, may not correspond to the maximum power of the engines - it may be that the props are threshing in a mass of cavitating, turbulent water and simply do not transmit power to the water. Thus, it is actually possible to increase power output and have the ship slow down.

The ship's energy loss to water is that of the resistance of the hull (drag). There are two primary causes of that drag. One is wavemaking resistance, the other is wetted area. In surface ships, the wavemaking resistance is by far the dominant factor at anything more than slow speed. On submarines, wavemaking resistance is pretty unimportant and wetted area dominates. We'll come back to that later.

Wavemaking resistance is determined by two factors. One (the primary factor) is the length of the ship. The longer she is in proportion to her beam, the less the wavemaking resistance and the faster she'll go. This is why lengthening the hull of a ship (as was done by the Japanese) will actually cause her to increase maximum speed without any additional modifications. The second factor in increasing speed is the beam to draft ratio. Deepening the hull causes the ship to lose resistance thus increase speed.

What is actually crucial here is the rate at which the cross sectional area of the ship changes. A ship with a slow change (increase or decrease) in cross-sectional area will have less wavemaking resistance than a ship with a rapid change. This is where prismatic coefficient comes in - it actually is a pretty good rough measure of how quickly the cross sectional area changes.

Now we come to dodges. The Iowa's have a pronounced bulbous bow. At slow speeds (where wetted area resistance predominates), this increases resistance due its wetted area and is a disadvantage. However, at higher speeds, it tricks the water into thinking that cross-sectional area is changing much more slowly than is the case and therefore reduces wavemaking resistance. Thus a bulbous bow benefits at higher speeds and is a liability at low speeds. You'll note that modern tankers have very large bulbous bows - this is because they have very high prismatic coefficients and the bulbous bow tricks the water into thinking that the hull is less bluff than is the case.

The problem with all this is, as prismatic coefficient drops, so does usable volume within the hull. Worse, so does buoyancy. At this point, the great trick is to maximize usable internal volume while reducing rate of cross-sectional area change.

The best hull form for usable internal volume is a box - which has zero unit cross-sectional change along its length and thus zero wavemaking resistance. Unfortunately at its ends, its rate of cross-sectional change is infinite and so is its wavemaking resistance. It is possible to move it through the water but the energy output is dreadful for any given speed.

In the good old days of ship design, ships had constantly changing cross sectional area along their length so had minimized wave-making resistance. This caused immediate penalties in internal volume and also in cost (curved things cost more to make than straight things). So, the great idea was to use a nice, box-like centersection and gently-tapering ends.

Now we have yet more problems. A ship with fine ends (those that have a gently-changing cross-sectional area) have little buoyancy fore and aft. They will dig their bows into the water in heavy weather and generally be very wet and unpleasant. This can be countered (partially) by giving the ships greater freeboard forward. Battleships have another problem that is pretty much peculiar to them. Their guns and armor ensure that most of their weight is amidships and this is where their bouyancy is needed. This causes much grief.
About 1935, in the US design offices, somebody had what we call in The Business a BFBO - a "Blinding Flash of the Bloody Obvious" - a sudden realization of a very simple concept that had eluded everybody else. Since a rectangular centersection is essentially wavemaking resistance-free, if it is extended as far fore and aft as possible it should be possible to have a relatively rapid change in cross-sectional area fore and aft of that block that averages out to a reasonable value over the ship's length. This gives internal volume and buoyancy just where it is needed while keeping resistance at a minimum.

You can see the effect of this BFBO by comparing the hull forms of the North Carolina and South Dakota. With Iowa, they took the game one stage further. By extending the bow out forward, they gave the ship a very fine entry and only a very slow increase in cross-sectional area. By restricting the area of rapid cross-sectional area to a limited section of the front, the average wave-making resistance over the length of the bow was very small.

At the same time, they had a large box-section amidships that provided a lot of volume to support the weight of the guns and armor and to accommodate the engines. Even better, by building the hull with a deeper draft, they were able to use larger screws and absorb power more easily.

This left the stern to smooth out and skegs and a transom did nicely there, the water is thoroughly fooled and thinks the hull is much finer than it is. By the way, this is why skegs didn't work very well on European BB designs - they still used continuous curvature hulls so the water-fooling properties of skegs didn't show any advantages but their wetted area drag did.

And that is why the Iowa is shaped the way she is. You can get the same performance out of a more conventional hull design but that doesn't give you the weight-lifting capacity and volume amidships.

Oh, by the way, since subs have wetted area as their primary resistance cause, their key determinant is usable volume versus surface area. This is why the ideal shape for a submarine is short and fat while that for a surface ship is long and thin.

There will be a quiz Monday Morning!


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