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

Sailing — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Three times the wind, on a reach. The polar diagram: boat speed in every direction, as a multiple of the true wind speed, for drag angles of 14 and 6 degrees. The shaded wedge at the top is the no-go zone, whose half-angle is exactly the sum of the two drag angles. Everywhere outside about twice that angle the boat is faster than the wind, and the maximum is 2.92 times the wind at 110 degrees — which is 1/sin λ at 90° + λ, both checked.

    Faster than the wind that drives it

    An ice yacht in a fifteen-knot breeze does forty. That is not a trick and it does not need a special sail — it follows from two drag angles and a triangle, and the best speed a boat can reach is one over the sine of their sum.

    part 1 · applied
  2. 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.

    part 2 · applied
  3. 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.

    part 3 · applied
  4. 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.

    part 4 · applied

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