How Fast Can a Sailboat Go: A Constrained Optimization Approach
Physics Behind a Sailboat
In some areas Hong Kong, to name a few, Repulse Bay, Sai Kung, etc., you can often witness many sailboats hovering around, ranging from those single-handed sports sailboats, to some ultra luxury 30-footer yachts. While the latter is way beyond the limit of we ordinary people, the former enables one to enjoy the fun of operating an unpowered vessel upwind, and is of particular interst of myself as an aerospace engineer.
It is not strange that a sailboat can sail downwind, just like anything will do in the wind. But it could be harder to imagine why a sailboat can go against the wind which in turn powers the vessel. In fact, this has something to do with the lift and drag produced by an airfoil, In this case, the sail. Once you understand the physics behind a sailboat, you will realize that a wind turbine can harvest wind energy with the principle.
First and foremost, the sail sees apparent wind, instead of the real wind. If you are driving 60 miles per hour in a windless day, you still experience a headwind of 60 miles per hour, which has the same speed as you travel but in the opposite direction. When there is some wind, apparent wind ($\vec{V}_ a$) is the vector sum of the velocity of true wind ($\vec{V}_ w$) and that of a moving object, i.e., the sailboat ($\vec{V}_ b$). $$ \vec{V}_ a=\vec{V}_ w+\vec{V}_ b. $$
The figure below shows the apparent wind seen by the sailboat.
In addition, the sail experiences an aerodynamic drag in the same direction as $V_ a$, and a lift perpendicular to apparent wind. The orientation of lift and drag is, by convention, perpendicular and parallel to the freestream, respectively. Both lift and drag is a function of $\vec{V}_ a$ and $\alpha$, known as the angle-of-attack.