# Abstract

We explain how the blade of a rotating propeller by acting like a wing generates lift and drag with a forward axial component giving thrust at the expense of tranversal angular components of lift and drag requiring torsion of the propeller axis.

## A Propeller Acts Like a Wing

**wing**generating

**lift**with a forward component giving

**thrust**, but also a transversal angular component from lift and

**drag**requiring

**torsion**of the

**propeller axis.**

**pitch**angle to a plane of rotation and with a certain

**angle of attack aoa**to the apparent direction of the incoming flow (relative wind

**), which depends on the rotational velocity of the propeller and the forward speed. Increasing the rotational velocity with the same pitch, increases the aoa and increasing the forward speed decreases the aoa. A propeller blade acting like a wing gives lift L counted perpendicular to the incoming flow and drag D in the direction of the incoming flow as indicated in the figure below. The component of L in the forward direction (minus a component from drag D) gives thrust. The transversal components of L and D create torsion on the propeller axis which needs to be balanced. The lift/drag ratio L/D is crucial for the performance of the propeller, since L is crucial for**

**D**for

**turbulent losses**. For a symmetric wing

**L/D = 10-20**for normal

**aoa = 3-15,**with however only

**L/D = 3 at maximall lift for aoa = 20,**as discussed below.

## Lift and Drag of a Wing

## Mechanism

**potential flow**arising from a mechanism of instability at rear separation, which modifies the pressure distribution so as to give both lift and drag. Notice that large lift comes from suction on the leeward surface resulting from the fact the flow

**does not separate on the crest.**This is

**because the fluid (air or water**

**) has very small viscosity which means that the boundary layer is turbulent with small skin friction closely approximated by the slip boundary condition of potential flow. Viscous flow with laminar boundary layer separates on the crest and gives poor lift.**

Sideview of velocity and pressure, and topview of streamwise vorticity of Naca0012 wing at

aoa = 14. Observe the turbulent streamwise vorticity emanating from top separation, as sketched above. Computed solution of the Navier-Stokes equations with slip boundary condition [1].Principle of action of a wing: Potential flow (upper left) with zero lift/drag modified by low-pressure counter-rotating rolls of streamwise vorticity from instability mechanism at separation (upper right), switching the pressure on rear wing (bottom) to give both lift and drag (H high, L low pressure).

### Lift/Drag Ratio L/D

We see (larger figs) that for a symmetric airlfoil like a Naca0012, **L ~ 2.5 **is** maximal **for **aoa = 20 **with **L/D ~ ****3** **small,** while for unsymmetric airfoil like a sail, **L/D > 6 **for **aoa = 20 **at maximal lift. This means that a sail works efficiently at maximal lift for **aoa = 20,** while a symmetric airfoil has a satisfactory L/D only for **aoa < 15 **with non-maximal lift (as discussed in more detail in Why It Is Possible to Sail). A propeller normally is rather thin and is more similar to a sail than a symmetric airfoil.

### Variable Pitch

Since the speed of the propeller blade depends on the distance to the axis, and a constant angle of attack is desirable, the pitch should increase (linearly) with the distance to the axis, that is, the blades should be twisted (as in the above picture).

### Efficiency

**W**

**L**performed by a propeller scales with the pressure drop, which scales with the lift

**L**, while

**the turbulent loss WD scales with drag D. Energy balance gives with WT the work performed by torsion of the propeller axis:**

**WL + WD = WT**

**from which follows that a first estimate of the efficiency E (output work WL divided by input work WT) is given by**

**E = (1-D/L).**

**L/D = 10**

**,**between

**14**for a Naca0012 and

**6**for a sail, the efficiency would be

**0.9**which is comparable to the maximal efficiency of a McCauley 7557 propeller on a Cessna 172:

### The Difference between a Screw and a Wing

A screw gives forward thrust by high pressure on the the windward side, while for a wing the low pressure on the leeward side contributes three times as much. A screw-propeller thus requires up to three times as large area as a wing propeller.

## Confusion and Desinformation

- How a wing/sail does not work.
*There is a long-held and still-continuing argument, particularly in newsgroups and other internet venues, about the pros and cons of the various lift generation theories. None of the arguments put forward (often ill-informed) affect in any way how an aircraft flies, how it should be safely and economically operated, or how it should be built; so it is best to ignore them unless you are particularly interested in the science of aerodynamics and skilled in calculus [1].**When the wind flows over one side it fills the sail while the air flowing on the other side is moving faster and cannot push as hard and thus the sail recieves a force that is perpendicular to the direction of the wind*[2]*.**The wind moving around the “upper,” or downwind, side of the sail is forced to take the longer path*[3]

*There are all kinds of controversies about sails…**Physics of Sailing.**It is difficult to explain the generation of lift for**laymen*[4]*The wind passes around the sail and because the distance is greater on the leeward side of the sail, the wind must travel faster**QUB Sailing Club.**The air being also deflected by the upper side of the wing, by the Coanda effect, is harder to understand**sailtheory**.**The air traveling over the leeward surface of the cambered sail creates the second force. It has to travel a longer way to reach the end of the sail (the leech), and as a consequence goes faster. This is causing a pressure differential in accordance with Bernoulli’s principle**University of Hawai**.**The sails propel the boat by redirecting the wind coming in from the side towards the rear. In accordance with the law of**conservation of momentum**, air is redirected backwards, making the boat go forward**Sail, Wikipedia**.**The fact that, after all these years, there is still any question about how sails work suggests that somewhere we’ve started with some wrong assumptions…**Sailing World Magazine.**Air will follow the curved shape of an airfoil due to Coanda effect. Why is this important? As long as airflow is laminar or in contact with the airfoil surface, it will continue to be turned in the same direction of the airfoil’s shape. This ensures the change in wind direction needed to drive the boat forward [5].**The fundamental problem as to how a surface such as a sail generates lift is rather difficult to understand for the average non-technical sailor. The fact that it is the viscosity of air which make lift possible is even more difficult to grasp…Although the circulation about the airfoil as generated in theoretical aerodynamics and as simulated by potential flow programs seems like just a mathematical trick, this is not the case…*[6]

*A good introduction to sail theory can be obtained in the work of Arvel Gentry**fx sails.**Aerodynamics is a difficult subject, and all attempts to simplify it for the average person leads to wrong interpretations. The facts are that lift comes about because air has viscosity, which leads to the starting vortex. This is followed by the formation of a circulation field about the airfoil necessary to meet one of Helmoltz’s theorems of vortex motion. Then the Kutta condition is satisfied at the trailing edge, and bingo — we have lift. These principles, together with knowledge of boundary layer theory, lead to a correct understanding of the interaction between the jib and the mainsail. (Arvel Gentry)*