Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Velocity made good — where it appears

The component of a boat's velocity along the direction it wants to go, usually straight upwind or downwind to a mark. Maximising it rather than boat speed puts the best course to windward well off the wind, where a line perpendicular to the wind touches the boat's polar.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

The fastest course is never the one towards the mark. Boat speed and speed made good against the course sailed. Speed rises steadily as the boat bears away, but what counts is the component along the direction wanted, and that has a maximum well off the straight line — at 55.0 degrees going upwind and 145.0 going down. Both optima are found here by search and agree with 45° + λ/2 and 135° + λ/2 to six figures.

The fastest way is not the straight one

A boat racing to a mark dead upwind sails seventy per cent further than the distance to it, and arrives first. The best angle is forty-five degrees plus half the apparent wind angle, it comes out of one line of trigonometry, and the same argument says to gybe downwind rather than run.

applied · Sailing A force ceiling ends the similarity: a breeze makes the boat slower as a share of the wind. Speed made good to windward as a fraction of the true wind, on the best course for each wind, for a rig that is flattened once the righting moment binds and for one that is reefed, with the hull's drag angle held at 6°. Below 3.70 m/s the two are the same boat, and the fraction does not depend on the wind — 1.122 at every speed, which is the similarity the two-angle polar rests on. Above it the flattened rig's fraction falls: 0.918 at 6 m/s, 0.572 at 10 m/s, 0.321 at 15 m/s and 0.169 at 20 m/s. The reefed rig's stays at 1.122, because a reefed sail keeps its least drag angle and the hull's angle is held. Once a force is limited, the triangle is no longer the same shape in every wind.

A breeze the boat cannot use

The two-angle polar makes a boat's speed a fixed fraction of the wind, in any wind. A righting moment ends that at 3.7 metres a second on the beat. Past it the crew must spill force, a flattened sail's drag angle climbs, and the best course to windward moves closer to the wind rather than away from it — which 45° + λ/2 cannot say, because λ now depends on the course.

applied · Sailing A keel pulls the best beat 12 degrees closer to the wind. Speed made good as a fraction of the wind against the course, in 3 m/s: for the boat with its keel solved as a wing, and for a boat whose λ is fixed at 25.69°, the value the keeled boat has on its own best course. The search puts the keeled boat's best beat at 45.83°; the fixed-angle rule puts it at 45° + λ/2 = 57.84°. Pointing higher loads the keel towards its best lift coefficient — 0.130 at 40° against 0.076 at 60° — and lowers its drag angle, so the curve peaks early and falls away faster on the far side. The fixed-angle curve promises 0.653 of the wind; the keeled boat can make 0.553, and only by sailing twelve degrees higher than the rule says.

A keel flies wherever the course puts it

A keel is a wing whose lift is whatever side force the rig happens to make, so its lift coefficient is chosen by the course and the wind rather than by its designer. Solved that way, the hull's drag angle stops being a property of the boat: it pulls the best beat twelve degrees closer to the wind, halves the reaching speed a fixed angle predicts, and finally charges for reefing.

applied · Sailing

Named alongside it

The objects these essays reach for when they reach for this one.

Drag polarModel limitApparent windInduced dragLift coefficientOptimisationRighting momentAspect ratioConstraintDimensionlessEfficiencyGlide ratio

All concepts