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

Sailing — the series

9 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
  5. The rudder should carry an eighth of the side force. The drag of keel and rudder together, as a percentage of the keel carrying everything, against the rudder's share of the side force, for a rudder 0.809 of the keel's depth. The induced drag is least with the keel alone — a shallower rudder makes the combined loading less elliptic for any load it takes. The profile drag is least near equal lift coefficients. Their sum is least at 12.4 per cent, 0.97 per cent below the keel alone; for the yacht drawn here, a rudder 0.809 of the keel's depth, that is 12.4 per cent.

    The rudder pays for the keel's wake

    A rudder angled to carry side force feels drag out of all proportion to its share, and that is the sensation behind the rule that weather helm is slow. The pair's drag does not care who is billed. It is least with the rudder carrying an eighth of the load, at under a degree of helm, and that share does not change with how hard the boat is pressed.

    part 5 · applied
  6. A phantom biplane partner doubles the induced drag. Induced drag at a given lift, as a multiple of its deep-water value, against the foil's depth in spans. The surface's image of the foil is an identical wing, equally loaded, twice the depth above it, so the real foil pays its own induced drag plus the mutual drag of a biplane with a gap of twice its depth: 1 + σ, with σ Prandtl's biplane factor. The elliptic biplane's 1 + σ and the aspect-ratio-9 foil's lattice both run to two as the depth closes.

    A foil under the surface flies with a phantom

    At foiling speed the water's surface cannot hold a pressure, and the image that condition requires is not the reversed one a wall makes. For a horizontal foil it is an identical wing, equally loaded, above the surface — a biplane partner that takes lift and gives nothing back, doubling the induced drag as the foil rises. For the board that pierces the surface it is a reversed copy, which makes the board pay two and a half times what a keel under a hull pays. It is the board, not the foil, that decides how deep a foiling boat rides.

    part 6 · applied
  7. Fly level through short waves, follow long ones. The mean drag of the two strategies against wavelength, for a wave 0.4 m high at 8 m/s: flying level at the best mean depth for that sea, and following the surface at the calm boat's best depth, with the flat-water drag as the faint line. Level flight costs about the same whatever the wavelength; following costs the square of the heave it demands and falls away as the waves lengthen. Into the waves the two cost the same at 15.6 m; running with them, where the boat meets each wave slowly, at 5.9 m.

    A foil flies level through a short sea and follows a long one

    A foiling boat in waves has two ways to fly. It can hold its height and let the surface rise and fall over its foil, or it can follow the surface and heave with every wave. Flying level costs a deeper ride and a few per cent of drag whatever the wavelength; following costs the square of the heave, which grows with the frequency the boat meets the waves at. Into a head sea the two cost the same at a wavelength of about sixteen metres, running with the waves at about six, and the wave's height hardly moves either.

    part 7 · applied
  8. Heading into the sea, follow the long waves and fly level through the short. The dinghy's mean drag against the crossover encounter frequency below which it follows the surface, in units of the sea's peak frequency, heading into seas of significant height 0.2, 0.4 and 0.6 m. Following everything — the right-hand end — costs the heave's lift, and flying level through everything — the left — costs the foil's swinging depth and, in the roughest sea, breaches. Between them each sea has a best crossover: for 0.4 m at 2.92 times the peak frequency, 67.6 N against 67.9 flying level and 199 following.

    A foil in a random sea follows it up to its peak

    In a regular wave a foiling boat chooses between holding its height and following the surface. A real sea is a spectrum, and a wand that filters its signal can do both at once: follow the long waves and fly level through the short. The best place to divide them is close to the frequency of the sea's own peak, the divided control beats either pure strategy, and in a rough sea it is the only one of the three that keeps the foil a safe distance under the surface and the drag near its calm-water value.

    part 8 · applied
  9. A soft filter flies through most of the waves it is meant to follow, and follows them late. Against the encounter frequency over the crossover, for the ideal cut and the first- and second-order complementary pairs: left, the share of a wave's height the boat flies through rather than follows; right, how far the followed part lags the surface, in degrees. At the crossover itself the first-order pair flies through 0.707 of each wave and follows it 45° late; the second-order pair 1.22 and 90°.

    A wand that lags its sea makes flying level the better bet

    A foiling dinghy whose height control divides the sea sharply follows the long waves, flies level through the short and beats both pure strategies. Give the division the lag a real wand or filter has and the gain disappears in a moderate sea; in a rough one no crossover both keeps the foil under and spares it from pulling down. A second-order filter, which cuts more sharply, does worse rather than better, because it flies through more of the sea near its crossover than the sea contains.

    part 9 · applied

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