# Abstract

We explain how the sail and keel of a sailing boat, both acting like wings, together pull/drive the boat forward

in beating at 35-45 degrees against the wind. We explain the somewhat different action of a sail and a symmetric wing like a keel, with the angle of attack of a sail 15-25 degrees and of a keel 5-10 degrees.

We show that classical explanations are incorrect.

Sail action in a nutshell: Windward Hi pressure and leeward Lo(w) pressure from counter-rotating low-pressure rolls of streamwise vortices at leeward separation (sideview left), and resulting lift L and drag D (topview middle) with angle of attack

aoaindicated. Compare with keel action according to New Mathematical Theory of Lift. Also compare with Windtunnel Movies. Beating at 45 degrees (right).

#### Key Fact 1: Sail and keel act like wings

**lift**and

**drag**, as explained by New Mathematical Theory of Lift, but the action, geometrical shape and

**angle of attack (aoa)**of the sail and the keel are somewhat different. The effective

**aoa**of a sail in tacking is

**15-25**degrees and that of a keel

**5-10**degrees, for reasons explained below. The

**aoa**of the keel is also referred to as the

**l**

**eeway**, the difference between the direction the boat is pointed and the actual direction of travel.

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 small friction force boundary condition [3].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).

#### Key Fact 2: Sail gives forward drive at the price of heeling

The boat is pulled forward by the sail, assuming **aoa = 15** with the boom inclined **5** degrees to the direction of the boat on a close-hauled beat, by the forward drive component **sin(20)L** ~ **0.3L** of the lift **L** counted perpendicular to the effective wind direction, which is the usual for a wing. There is also a side (heeling) force **cos(20)L** from the sail, which tilts the boat and needs to be balanced by lift from the keel. A sail has less lift than a symmetric wing because the strong concentration of lift at the upper rounded leading edge of the wing, is missing for the sail.

#### Key Fact 3: L/D of sail > 6-10

The forward pull/drive form the lift **L** is reduced by a backward pull/drag from a component of the drag **D** counted parallel to the effective wind direction, with similar contributions from the leeward and windward side of the sail because the shape is the same. This makes an important difference with a symmetric wing for which the backward pull/drag is larger from the windward side because of the high pressure at the lower leading edge of the wing, which is displayed in the above pressure plot and in Fig. 4 of Why It Is Possible to Fly.

**L/D**

**> 6-10**at

**aoa**=

**15-20**for a sail, which reduces the forward pull/drive to

**~ 0.2L**. Compare with the following plot showing that

**L/D**for a sail peaks at

**aoa = 15 (c.f. [4]):**

**L/D**~

**3**for a wing at

**aoa =**

**20**as shown in Fig. 3, which would reduce the forward pull/drive to

**0.1L,**which is too small according to:

#### Key Fact 4: Keel balances heeling at the price of drag

The heeling force from the sail is balanced by lift from the the keel in the opposite direction. Assuming the lift/drag ratio for the keel is **10** at **aoa = 5-10**, the forward drive is then reduced to **(0.2 – 0.1)L = 0.1L**, which is used to overcome the drag from the hull minus the keel.

**L/D < 3**for the sail, the net forward drive would disappear. Replacing the sail by a wing thus does not seem to be a good idea for a standard boat, because an

**aoa > 15**is required to get sufficient drive, while it may work better for a very slender fast hull allowing a smaller angle of attack. But a keel like a wing works fine, because an

**aoa**of

**5-10 is sufficient.**

#### Key Fact 5: Sail area vs keel area

**10 m/s**at

**aoa = 15**and the speed of the keel/boat through the water is

**3 m/s**at

**aoa = 5**, we find (using that the density of water is about 800 times that of air) that the sail area can be up to

**25**times the keel area. In practice, the ratio is typically

**7-9**with a traditional full-keel and 1

**0-15**with a standard modern fin-keel, while a modern Americas Cup boat reaches beyond

**25**.

Classes for 32nd Americas Cup 2007 (right) and 33rd 2009 (left).

#### Summary

The shape of a sail is different from that of a wing which gives smaller drag from the windward side and thus improved forward pull/drive, while the keel has the shape of a symmetric wing and acts like a wing. A sail with **aoa = 15-20** degrees gives maximal pull/drive forward at high heeling/lift with contribution also from the rear part of the sail, like for a wing just before stall, while the drag is smaller than for a wing with **L/D ~ 3 **at **aoa = 20**, with the motivation given above.

**L/D**curve for a sail is thus different from that of wing: at

**aoa = 15-20 L/D > 6-10**for a sail, while

**L/D < 3-4**

**for a wing. On the other hand, a keel with**

**aoa = 5-10**degrees has

**L/D > 6-10**. A sail at

**aoa = 15-20**thus gives maximal pull at strong heeling force and small drag, which together with a keel at

**aoa = 5-10**with strong lift and small drag, makes an efficient combination. This explains why modern designs combine a deep narrow keel acting efficiently for small aoa, with a broader sail acting efficiently at a larger aoa.

**aoa.**

#### Confusion and Desinformation

- How a wing/sail does not work.
*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*[7]*.**The wind moving around the “upper,” or downwind, side of the sail is forced to take the longer path*[8]

*There are all kinds of controversies about sails…**Physics of Sailing.**A wing is designed to have a longer path along the top surface…the flow of air over the top will have a larger speed…which results in a decrease of pressure alnog the top..; this is what is called lift [9]**It is difficult to explain the generation of lift for**laymen*[10]*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 [11].**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…*[12]

*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)*