# What exactly is the 'lift' of a sailboat as explained by Bernoulli principle

Bernoulli's Principle is one very small piece of a large mathematical theory that explains lift. Unfortunately, most attempts to explain lift using Bernoulli's principle without the overall mathematical context (i.e. vector calculus, partial differential equations, boundary conditions, the Navier-Stokes equation, etc.) are either so confusing that few readers can follow them, or they are just out and out wrong. Often both.

From a layman's perspective, Bernoulli's principle does more to obscure the issue than it does to enlighten. US Sailing does not use Bernoulli's principle in their instruction materials to explain how sailboats move and we do not use it in our instruction program (I am the manager of a learn to sail program with ten US Sailing Certified Instructors). It's a red herring that confuses students without giving them any insight into how to sail.

To answer your question, "lift" is a defined term in aerodynamics with a different meaning than usual usage. When a solid object is immersed in a moving fluid, the fluid exerts a force on the object and the object exerts a force on the fluid that is equal in magnitude but opposite in direction. "Lift" is the component of this force perpendicular to the fluid flow. For a sailboat sail, the fluid is air and the "lift" is a horizontal force that propels the boat forward (and also makes it lean). So "lift" doesn't push the boat up, it pushes it forwards and sideways.

As for Bernoulli, there's nothing wrong with Bernoulli's principle, it just doesn't do a very good job of explaining why sailboat sails and airplane wings develop lift. Lift is the result of unequal air pressure on one side vs the other. An airplane wing has lower pressure on the top side of the wing and higher pressure on the bottom. A sail has lower air pressure on the front (leeward) side and more pressure on the rear (windward) side. If you look at the derivation of Bernoulli's equation, it concerns itself with pressure gradients *parallel* to the air flow, while lift is the result of air pressure gradients *perpendicular* to the air flow. So Bernoulli's equation doesn't get at the reason for the lift - to understand lift you need to understand why there is a pressure difference perpendicular (not parallel) to the air flow.

Fortunately there *is* an equation that deals with pressure gradients perpendicular to the air flow. It doesn't have a name, but was derived at the same time as Bernoulli's equation so it's been around for centuries. It is:

$$\frac{dp}{dz} = \frac{\rho v^2}{R}$$

where $p$ is the pressure, $dp/dz$ is the pressure gradient perpendicular to the air flow, $\rho$ is the density of the fluid, $v$ is the velocity and $R$ is the radius of curvature of the airflow. (BTW, the derivation of this formula is just a straightforward application of Newton's 2nd law to the kinematics of centripetal acceleration - Physics 101 stuff)

This equation shows that whenever air follows a path that is curved there are pressure gradients perpendicular to the airflow, the tighter the curvature the higher the pressure gradients, and for straight flow (R -> infinity) there is zero pressure gradients and therefore no lift.

Lift comes from curved air flow. Sails are curved. When air follows the curve of the sail, you get pressure differences. That's what drives the boat. Bernoulli's principle tells you that the air speeds up as it flows through the region of reduced pressure, but it doesn't tell you *why* the low pressure is there, and while the air speeding up may be an interesting side effect of lift it's not very important to understanding how a sail works.

If you know a little bit of calculus, Babinsky's paper "How Do Wings Work?' (http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf) is a good overview. Or check out NASA's site:

http://www.grc.nasa.gov/WWW/K-12/airplane/wrong1.html

An important point that is often overlooked is that sails do not just generate 'lift', but generate lift in a direction. The pressure differential across the sail's surface is typically much larger near the leading edge, which results in more forward force than you might expect. The two sails (with a sloop) also interact in a way that generates more forward force.

You can see some simulations with different sail configurations, and comparison to the performance of a real sailboat at this site:

https://sites.google.com/site/sailcfd/