Propeller theory is the science governing the design of efficient propellers. A propeller is the most common propulsor on ships, and on small aircraft.
History In the second half of the nineteenth century, several theories were developed. The momentum theory or disk actuator theory – a theory describing a mathematical model of an ideal propeller – was developed by W.J.M. Rankine (1865), Alfred George Greenhill (1888) and Robert Edmund Froude (1889). The propeller is modelled as an infinitely thin disc, inducing a constant velocity along the axis of rotation. This disc creates a flow around the propeller. Under certain mathematical premises of the fluid, there can be extracted a mathematical connection between power, radius of the propeller, torque and induced velocity. Friction is not included. The blade element theory (BET) is a mathematical process originally designed by William Froude father of Robert Edmund Froude (1878), David W. Taylor (1893) and Stefan Drzewiecki to determine the behaviour of propellers. It involves breaking an airfoil down into several small parts then determining the forces on them. These forces are then converted into accelerations, which can be integrated into velocities and positions.
Theory of operation
A propeller imparts momentum to a fluid which causes a force to act on the ship. The ideal efficiency of any propulsor is that of an actuator disc in an ideal fluid. This is called the Froude efficiency and is a natural limit which cannot be exceeded by any device, no matter how good it is. Any propulsor which has virtually zero slip in the water, whether this is a very large propeller or a huge drag device, approaches 100% Froude efficiency. The essence of the actuator-disc theory is that if the slip is defined as the ratio of fluid velocity increase through the disc to vehicle velocity, the Froude efficiency is equal to 1/(slip + 1). Thus a lightly loaded propeller with a large swept area can have a high Froude efficiency. An actual propeller has blades made up of sections of helicoidal surfaces which can be thought to 'screw' through the fluid (hence the common reference to propellers as "screws"). Actually the blades are twisted airfoils or hydrofoils and each section contributes to the total thrust. Two to five blades are most common, although designs which are intended to operate at reduced noise will have more blades and one-bladed ones with a counterweight have also been used. Lightly loaded propellers for light aircraft and human-powered boats mostly have two blades, motor boats mostly have three blades. The blades are attached to a boss (hub), and while it can be a big boss, it should be as small as the needs of strength allow – with fixed-pitch propellers the blades and boss are usually a single casting. An alternative design is the controllable-pitch propeller (CPP, or CRP for controllable-reversible pitch), where the blades are rotated normally to the drive shaft by additional machinery – usually hydraulics – at the hub and control linkages running down the shaft. This allows the drive machinery to operate at a constant speed while the propeller loading is changed to match operating conditions. It also eliminates the need for a reversing gear and allows for more rapid change to thrust, as the revolutions are constant. This type of propeller is most common on ships such as tugs where there can be enormous differences in propeller loading when towing compared to running free. The downsides of a CPP/CRP include: the large hub which decreases the torque required to cause cavitation, the mechanical complexity which limits transmission power and the extra blade shaping requirements forced upon the propeller designer. For smaller motors there are self-pitching propellers. The blades freely move through an entire circle on an axis at right angles to the shaft. This allows hydrodynamic and centrifugal forces to 'set' the angle the blades reach and so the pitch of the propeller. A propeller that turns clockwise to produce forward thrust, when viewed from aft, is called right-handed. One that turns anticlockwise is said to be left-handed. Larger vessels often have twin screws to reduce heeling torque, counter-rotating propellers, the starboard screw is usually right-handed and the port left-handed, this is called outward turning. The opposite case is called inward turning. Another possibility is contra-rotating propellers, where two propellers rotate in opposing directions on a single shaft, or on separate shafts on nearly the same axis. Contra-rotating propellers offer increased efficiency by capturing the energy lost in the tangential velocities imparted to the fluid by the forward propeller (known as "propeller swirl"). The flow field behind the aft propeller of a contra-rotating set has very little "swirl", and this reduction in energy loss is seen as an increased efficiency of the aft propeller. An azimuthing propeller is a propeller that turns around the vertical axis. The individual airfoil-shaped blades turn as the propeller moves so that they are always generating lift in the vessel's direction of movement. This type of propeller can reverse or change its direction of thrust very quickly. Fixed-wing aircraft are also subject to the P-factor effect, in which a rotating propeller will yaw an aircraft slightly to one side because the relative wind it produces is asymmetrical. It is particularly noticeable when climbing, but is usually simple to compensate for with the aircraft's rudder. A more serious situation can exist if a multi-engine aircraft loses power to one of its engines, in particular the one which is positioned on the side that enhances the P-factor. This power plant is called the critical engine and its loss will require more control compensation by the pilot. Geometric pitch is the distance an element of an airplane propeller would advance in one revolution if it were moving along a helix having an angle equal to that between the chord of the element and a plane perpendicular to the propeller axis.
Forces acting on a foil The force (F) experienced by a foil is determined by its area (A), fluid density (ρ), velocity (V) and the angle of the foil to the fluid flow, called angle of attack ( α {\displaystyle \alpha } ), where:
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