Propellers are usually described as pushing against water in order to propel a ship forward. In fact, this isn't quite the case. What a propeller does is apply an acceleration to a mass of water. According to Newton's Law of Action and Reaction, the action of increasing the velocity of a mass of water in a given direction generates an equal and opposite reaction in the propeller/shaft assembly. This is described as the thrust of the propeller and it is this thrust that drives the ship forward.

The basic theory of how a propeller works was put together by three eminent Victorians, Rankine, Greenhill and RE Froude (son of our old friend William Froude) between 1865 and 1889. They envisaged an idealized propeller called a Rankine Disk Actuator which imparts a sudden uniform acceleration to all the fluid passing through it, the flow being frictionless and the water being present in unlimited quantities and the Rankine Disk working with 100 percent efficiency. In this idealized system, the energy imparted to the water by the Rankine Disk Actuator is

This is, of course, a slightly modified standard kinetic energy equation. Unfortunately, that's the last easy bit of mathematics. Because adding any energy into the system changes the values of both V1 and V0, the equation has to be integrated between zero and t seconds where t is the time taken for the system to come to equilibrium.

Now, (V1-V0), the increase in velocity of the water, is determined by the design of the screw. Each turn of the screw accelerates a package of water from V0 to V1. Increasing the rate of revolution increases the number of those packages that goes through the Rankine Disk but does not increase the speed at which they leave the disk. This is important; it doesn't matter how fast the screw turns or how large it is, it the design of the screw and that design only that determines the acceleration of the water. A good comparison is a road with a 55 mph speed limit - improving the quality of the road or widening it to include more lanes will increase the volume of traffic the road can handle but the speed of the traffic will only be increased by raising the speed limit.

M, the mass of the water passing through the Rankine Disk Actuator, is equivalent to the density of water times the volume of water passed. Increasing the volume (and thus the mass) of the disk can be achieved by using a larger disk and/or increasing the revolutions per minute of the propeller. In mathematical terms, the water passing through the Rankine Disk is a cylinder, the diameter of which is the diameter of the disk and the length of which (the number of packages transiting the disk) is determined by the speed at which the disk is turning. From an energy point of view, it doesn't matter very much whether the cylinder is long and thin (a small prop running at high speed) or short and fat (a large prop running at slow speed). As long as the two cylinders contain the same volume of water being accelerated the same amount, they'll demand the same amount of energy and yield the same level of thrust. (Remember these cylinders are mathematical constructs not physical reality).

This treatment gives us one very important lesson which takes some complex mathematics to prove because it seems so outrageous. Since we are accelerating a cylinder of water through a disk, fully half the thrust developed by the acceleration of that cylinder is delivered before the water ever touches the disk! In short, we seem to get the thrust before the water gets the acceleration. This is outrageous, ridiculous, unbelievable and perfectly correct - it is a major consideration in designing underwater hull forms.

Unfortunately, when we leave the idealized world of the Rankine Disk Actuator and enter the real world, life starts to get complex. Firstly, the cylinder isn't a cylinder. Before the water hits the propeller it is being drawn along at speed above that of water outside the cylinder. Bernoulli's law dictates that water will be drawn into the cylinder from outside, causing the cylinder to bulge outwards. The other side of the prop, the fact that acceleration is constant by the volume of water being pushed through is increased as the revolving speed of the prop goes up causes increased pressure areas aft of the prop. This causes a high-pressure bulge here too. (A simple experiment illustrates this - take a garden hose and set it running full blast. Now put your thumb over the nozzle). Eventually, this high-pressure region reaches the proportions where it breaks the surface, giving the famous rooster-tail effect. (It can also have a forward vector that has a propulsive effect on the ship). These two factors mean that the propeller isn't at the center of a cylinder but a complex shape rather like an hourglass with the propeller at the thin neck. Again, the equations have to be integrated in order to get the "volume" (i.e. the energy content) of the system. If we were following the maths in detail, we would now be dealing with several layers of integrated equations.

Another problem is induced rotation. In the Rankine Disk Actuator, no axial rotation is applied to the water flow. At low transiting volumes, this is almost true, but as volumes get larger and the ratio of prop diameter to speed of rotation reaches critical values, the water leaving the propeller (the race) becomes more and more spiral in shape. This is purely awful - every drop of energy that goes into rotating the water instead of accelerating it is wasted (in effect it shortens our mathematical-construct cylinder). In mathematical terms the pitch of the spiral shortens as speed of prop rotation increases and the loss of energy is proportional to the square of that pitch.

Increasing prop size and speed of rotation are both good in that they increase the volume of water the prop accelerates. However, there are limits on both. Propeller size has physical limitations (we really do not want the blade tips hitting the hull plating), material restrictions (having the prop fly apart from metal fatigue is usually quite depressing) and also hydrodynamic restrictions which we'll come to later. If speed of rotation is pushed too high, the propeller starts to hit the axial rotation problem described above and also starts to cavitate. This is by way of being an upper limit - reductions in propeller efficiency from cavitation quickly get so high that adding extra power will actually slow the ship down.

The inefficiency of a small, fast running propeller is murderous. For example, if the efficiency of a prop was really that of a Rankine Disk Actuator, halving the diameter of a propeller could be compensated by increasing the speed of the propeller by a factor of four - the energy contents would be the same. In reality, the efficiency of the half-diameter quadruple-speed propeller would be only 61.8 percent of the full-size, slow speed version - it would provide less that 2/3 the thrust. So, mathematically, the Rankine Disk Actuator equations eventually show us that a large, slow-turning propeller is a better deal than a small, fast-turning one. As an insight into a science nobody had thought of a few years earlier, the Rankine Disk momentum theory isn't bad for a group of Victorian gentlemen who had virtually nothing to work with except slide-rules and their own perceptive brilliance.

In effect we have a cycle by which the engines generate power, that power is used by the screws to accelerate water, the reaction to which is thrust which pushes the ship forward. Unfortunately, the limitations on prop size and speed of rotation plus the fact that the acceleration applied to the water by the props is fixed by the design of the props, means there is a limit to the energy the props can use (in mathematics, to the size of our hour-glass or cylinder depending on whether we are looking at reality or theory). Any extra power generated by the engines above that limit is so much deadweight. In reality, of course, this limit isn't a sharp point but an area in which the efficiency by which the screws convert energy into thrust quickly drops to zero. Nonetheless, adding 500 tons of machinery to a ship that is already overpowered will not achieve anything at all.

We can't do much about the density of water (well actually we can. The effects of pressure from depth are quite important - a propeller running at 45 feet will give measurably more thrust than the same prop at 15 feet due to water pressure. Also, the compressive effect of a heavy hull will have a beneficial effect on the effective mass of water going through the prop. These are, however, relatively minor effects in terms of the sort of gains we are looking for). If we are going to get a major gain in energy utilization out of the power train we have to improve the amount by which the propeller accelerates the water and design the prop so that cavitation is delayed as long as possible. Unfortunately, here the Rankine Disk ceases to be of help since the mechanism by which it accelerates the water is not considered. There are two theories that do deal with this, the Blade Element Theory (which evolved shortly after the pioneering work of Rankine, Greenhill and Froude) and the Circulation Theory (evolved by F.W. Lanchester for aircraft in 1907 and applied to ships by Betz and Prandtl some years later). Both involve mathematics of extreme complexity. The Circulation Theory in particular allows the acceleration applied to water by a blade of given shape to be calculated by a thing called the Kutta-Joukowski Equation. The fun question is, what is the ideal shape?

What makes this question difficult is the fact that the propeller works in the ship's wake. What is normally called a wake isn't; its a combination of the ship's real wake and the race from the screws. Differentiating between the two is easy - the race travels backwards relative to the ship, the wake travels in the same direction as the ship but at a lower speed. The wake results from (a) the frictional drag of the hull which produces a following current, maximizing around the stern (b) the streamline flow past the hull causing increased pressure where the hull lines close also creating a following current and © the wave pattern formed by the ship on the surface in which the water packages have an orbital motion, the top being in the same direction as the movement of the ship and the bottom being in the opposite direction. The forward speed of the wake in proportion to that of the ship is called the Wake Fraction. This will significantly reduce the (V1-V0) value (by half for a wake fraction of 50 percent) with obvious effects on thrust. The three factors that create the wake give a hydrodynamic picture of unsurpassed complexity. Newton and Hadler did a whole series of studies back in 1960 on the performance of propellers using Fourier Analysis to create mathematical constructs of wakes using the computers then available. They produced a series of flow diagrams of single and twin-screw ships sections aft and of the props working in those flow conditions. These clearly showed that the twin-screw environment was much less chaotic than the single-screw situation. In a single centerline screw, the relative intensity of the wake/prop interaction (which should be constant for maximum efficiency) varied from 0.10 at the tip to 0.67 at the root of the blade. In a twin-screw, the same figures were 0.02 at the tip to 0.04 at the root. The study also showed that the effect on the wake form from a centerline screw was enough to badly disrupt the more favorable environment surrounding the wing screws. These experiments at last provided a reasonable explanation of why twin screws work better than single centerline props and lead to a concerted effort to relate hull design to wake characteristics.

In 1965, Van Manen produced a series of conclusions based on his extrapolation of Newton and Hadler's work. These were that the wake pattern is largely a product of the aft body of the ship, that harmonic amplitudes (transverse vibration) are inherently more severe on ships with centerline screws, the finer the stern, the more efficient the props, that blade geometry has a significant effect on induced shaft vibration, that transom sterns are less prone to cavitation, that the rudder has little effect on the wake and that minor changes in speed, displacement, hull form and trim have major and completely unpredictable effects on the wake pattern and, therefore, screw performance. In 1972, Van Oossanen et al investigated these areas but failed to come up with meaningful answers as did Holden in 1980 (although he did have some success in predicting effects on wakes with low peak values). This whole area is still largely a mystery although Chaos Theory may provide some clues as to what is happening back there.

So, having gotten the theory out of the way, how do we design a prop to convert more power into thrust and, thus, to make bigger battleships possible? If the ship design isn't pushing the limits of practical, it's quite simple. We take the desired prop diameter from the hull design, take the desired speed of revolution from the machinery design people and put the two figures into a pre-calculated graphical projection that will give us the "optimum" prop design for those conditions. This optimum isn't really that, its the best commercial approximation that can be mass produced for those conditions. If we want anything better, its has to be custom-designed for that specific ship.