A keel flies wherever the course puts it

A breeze the boat cannot use opened up the rig’s drag angle and held the hull’s at six degrees. That was enough to show a righting moment ending the similarity of the two-angle polar, and it ended with one result that was plainly not a result: a reefed boat, its sail always at its best, keeping its light-air triangle in any wind and reaching seventy-four knots to windward in a gale on a scrap of sail. Nothing about the rig does that. A hull whose drag angle is fixed does.

The hull and keel’s drag angle has been fixed in every calculation of a sailing boat so far, and it is a strange thing to fix. A keel is a wing. Its job is to make a sideways force equal and opposite to the rig’s, and like any wing it makes that force at a lift coefficient, pays a drag that depends on the lift coefficient, and has one lift coefficient at which the ratio of the two is least. What is unusual about it is who chooses the lift coefficient. The designer chooses the keel’s area and span. The rig, the course and the wind choose its load, and the keel flies at whatever lift coefficient that load and the boat’s speed demand.

The boat is the same seven-metre keelboat, with its 25 square metres of sail, its 4.0 kilonewton-metre righting moment and its sail polar. It is given a keel of 0.9 square metres and 1.3 metres of span, and the hull’s own resistance is carried as part of the keel’s profile drag.

A wing reflected in its own hull #

The keel hangs from the bottom of the hull, and at its root the hull is a wall that water cannot cross. A wall like that is equivalent to a mirror image of the wing on the far side of it: the root of the keel sheds no tip vortex, because the image cancels it, and the keel behaves as half of a wing twice its span. So the aspect ratio that matters is not the keel’s own span squared over its area but twice that — 3.76 for this keel, against a geometric 1.88.

That doubling is worth a great deal, because a wing’s induced drag is set by its span and very little else. A keel’s depth below the hull is its most valuable dimension for exactly that reason, and it is why keels are deep and narrow wherever the water allows, and why a boat that heels and so loses depth pays for it in induced drag.

Even doubled, 3.76 is a low aspect ratio, and a low aspect ratio has a shallower lift curve than the two-dimensional $2\pi$ per radian of the thin-aerofoil result. Helmbold’s form for a low-aspect-ratio wing gives 3.77 per radian for this keel — sixty per cent of $2\pi$ — so a given lift coefficient needs a larger angle of attack. For a keel that angle is leeway: the angle between where the boat points and where it goes.

The best lift coefficient, and where the boat actually flies #

The polar has the same shape as the rig’s, for the same reason: a profile drag and an induced drag, the first constant and the second growing as the square of the lift coefficient. The drag angle is least where they are equal, at a lift coefficient of 0.519, and there the hull and keel together cost 5.58° — better than the six degrees the earlier essays assumed.

The profile drag here is large, and that is the hull’s doing. A hull’s wetted surface drags whether or not the keel is lifting, and folded into the keel’s coefficient it adds more than the keel’s own section drag does. So the constant term is big, the best lift coefficient is high, and any lift coefficient below it is expensive.

Every point the boat actually sails at is below it. In three metres a second of wind, on a course forty degrees off it, the rig’s side force divided by the keel’s dynamic pressure and area comes to 0.130 — a quarter of the best value — and the drag angle is 11.67°, twice the minimum. Across the wind the lift coefficient is 0.042 and the drag angle 31.54°. The keel is on the steep side of its own polar on every course, and how far up that side it sits depends on the course.

The hull’s angle changes with every course #

That is the whole departure from the earlier essays, and it can be drawn as one curve.

The mechanism has two parts, and both push the same way. As the boat bears away the rig’s force swings round towards the direction of travel, so the sideways part of it — the load the keel must carry — shrinks. And the boat goes faster, so the keel’s dynamic pressure grows. Less load over more dynamic pressure is a lower lift coefficient, further down the steep side of the polar, where the hull’s constant friction is a larger share of a smaller lift.

Leeway tells the same story from the other side. Close-hauled at 35°, the keel is working at nearly two and a half degrees of leeway; on a broad reach it is doing almost nothing at a seventh of a degree. The keel works hardest where it is cheapest per unit of its work, and idles where its cost is all friction.

This curve is what the earlier essays’ single number was standing in for. Six degrees is a fair value for a hull and keel near their best, and this hull and keel are never near their best.

Where the boat points, and where it goes #

Leeway also changes what a course means. Every course in these figures is the boat’s track through the water — the direction it actually moves, which is what the velocity triangle needs. The boat’s bow points to windward of that track by the leeway angle, because the keel can make lift only by meeting the water at an angle.

On the best beat in light air the track is 45.83° off the true wind and the leeway is 1.64°, so the bow points 44.19° off it. A crew watching the compass through a tack sees the heading swing through 88 degrees; the track swings through 92. The difference is small for this keel, and it is not small for a keel with half the span, which needs four times the angle of attack for the same load at the same lift slope — or for a hull whose keel has been raised, which slides to leeward on a heading that looks close-hauled.

Below the righting moment’s ceiling, the keel’s lift coefficient on a given course does not depend on the wind at all. The side force goes as the square of the apparent wind, the keel’s dynamic pressure as the square of the boat’s speed, and on a given course both speeds are fixed fractions of the true wind — so their ratio, which is the lift coefficient, is fixed too. That is the two-angle model’s similarity surviving the keel: the drag angle now depends on the course, and still not on the wind. The ceiling is what breaks the second, and the friction coefficient’s slow fall with Reynolds number, left out here, is the only other thing that would.

Pointing higher is cheaper than the rule allows #

The essay before found that a $\lambda$ rising with the course pulls the best beat closer to the wind than $45° + \lambda/2$, because the condition for the optimum gains a term in $d\lambda/d\beta$ whose sign fixes the direction. There it took a righting moment and a flattened sail to make $\lambda$ rise. The keel makes it rise in any wind at all.

Twelve degrees is not a refinement. A boat beating at 57.84° to the true wind tacks through 116°; one beating at 45.83° tacks through 92. Keelboats tack through something close to ninety degrees, and the two-angle model with the drag angle a boat actually has on its best course would have them tack through a hundred and sixteen. The difference is the keel: pointing higher loads it towards its best lift coefficient, which lowers its drag angle, which makes pointing higher worth more than a constant angle says.

The speed made good is the other half of the correction. The fixed-angle curve says this boat should make 0.653 of the wind to windward, and the solved boat makes 0.553 — fifteen per cent less. Measured on the beat and extrapolated along the rule, the calculation would have the boat pointed twelve degrees too low and credited with fifteen per cent more speed made good than it has — a combination a navigator cannot see from the helm, because each error hides the other.

A polar measured on the beat is wrong on a reach #

If the drag angle depends on the course, then a polar drawn from one value of it is right on one course, and the question is how wrong it is elsewhere.

On the beat the two nearly coincide, as they must. Across the wind the fixed-angle polar overstates the boat’s speed by a factor of two, and on a broad reach by three. The reason is the same as before: across the wind the keel is asked for almost no lift, the hull’s friction is nearly the whole of the hydrodynamic drag, and a drag angle that was modest on the beat, where the keel’s lift was large, becomes very large once that lift is gone.

That is the correction faster than the wind that drives it needed and could not make. Its $1/\sin\lambda$ on a beam reach is the right answer for a boat whose drag angle on a reach equals its drag angle on the beat. A keelboat’s does not, and the high reaching speeds of the two-angle polar belong to craft whose drag stays small when their side force does — ice yachts and foiling boats, whose runner or foil friction is tiny beside the lift they make.

The charge for reefing #

With the hull’s angle solved, the reefed boat of the essay before can be run again.

Reefing keeps the rig at its least drag angle, exactly as before, and the reefed boat still gets relatively slower as the wind rises. The keel is the reason. Once the righting moment binds, the side force is capped at the moment divided by the rig’s height — 1.11 kilonewtons for this boat — whatever the wind. The boat keeps getting faster, and a capped force at a rising dynamic pressure is a falling lift coefficient.

So a boat in a breeze sails with its keel ever further from its best, and pays for it in drag angle whether the crew reefs or flattens. The charge for a force ceiling that a fixed hull angle could not make, the keel makes, and it is not small: a reefed boat in twenty metres a second makes 0.222 of the wind to windward, where the fixed hull angle said 1.122.

The second figure also says something about the keel’s size that the steady calculation cannot explain. On the beat this keel never carries more than a fifth of the lift coefficient at which it is most efficient, and in a breeze it carries an eighth. A smaller keel would fly closer to its best. Something other than steady sailing to windward must be deciding how large a keel is.

Sized for the slowest moment #

The steady calculation has a clear answer about keel size, and it is the wrong one.

In steady sailing a smaller keel is faster at both winds, all the way down to the smallest drawn, because it sheds the friction of keel area it does not need and flies nearer its best lift coefficient. Nothing in a steady force balance says stop.

What says stop is a moment the steady calculation does not contain. Straight after a tack the boat has lost speed, the sails fill on the new side, and the rig’s side force arrives before the boat has accelerated. A keel’s lift goes as the square of the speed, so at half the speed the same side force needs four times the lift coefficient, and a keel stalls at an angle rather than a speed. Taking a stalling lift coefficient of 0.9 — a stated value, not a computed one — a keel smaller than 0.4 square metres cannot carry the load at half speed, slides sideways instead of accelerating, and has to be sailed free until the boat is moving. The 0.9 square metre keel carries the same moment at 0.43, with room to spare.

It is worse than that, because a keel set moving does not have its lift at once: lift arrives late, over tens of chords of travel, while the wake’s starting vortex moves away. So a keel’s size is set by the slowest moment it has to work in, and the steady beat that follows each tack finds it oversized. That is not a design error to be optimised away; it is what a single fixed foil has to be, and it is why boats that can adjust their foils do.

What the wing model leaves out #

The keel is also the ballast. A real keel carries the lead that makes the righting moment. A smaller keel with the same lead is deeper or heavier at the tip, and a larger one can carry the ballast lower; the model treats the two jobs as unrelated, and the trade between them is most of keel design.

The hull makes lift too. A hull moving at a leeway angle is itself a very low-aspect-ratio lifting body, and the rudder behind the keel is a second lifting surface flying in the keel’s downwash. Both share the side force here assigned to the keel alone.

Heel costs span. A heeled keel is shallower and leans its lift away from the horizontal, so it must make more lift for the same side force with less effective span. The righting moment’s ceiling and the keel’s induced drag are coupled through the heel angle, and the model keeps them apart.

The hull has no waves. The hull’s resistance is a friction drag area with no wave drag in it, which is fair at low speed and not near the speed at which a displacement hull’s own waves begin to hold it back — a wave pattern whose angle does not depend on the boat, and whose drag rises steeply as the boat approaches its length’s natural wave speed.

The friction does not depend on Reynolds number. A keel’s section drag and a hull’s friction both fall with speed as a fraction of dynamic pressure, more steeply once the layer is turbulent. A constant coefficient keeps the similarity below the ceiling exact where a real boat would drift slightly.

Keels that stopped being fins #

The idea that a keel is a wing, with a span that matters and a tip that sheds a vortex, reached yacht design late and then all at once. The winged keel under Australia II in 1983 put endplates on the tip of a keel to recover some of the span a shallow keel lacks, and won the America’s Cup with it. Modern offshore racers split the keel’s two jobs outright: a canting keel carries the ballast and swings it to windward for righting moment, while separate daggerboards or foils carry the side force, and can be raised when a reach asks for almost none of it.

That division is the argument of this essay turned into hardware. A fixed keel is a wing whose lift coefficient it does not choose, sized for its worst moment and flown everywhere else far from its best. Taking the side force away from the ballast lets each be sized for its own job, and lets the foil be retracted on exactly the courses where the figures here show the hull’s friction being paid for a lift nobody is using.

Still open: what the rudder flies in #

The keel here carries the whole side force alone. On a real boat the rudder behind it carries a share, and the rudder is a wing flying in the downwash the keel has already turned — the air a wing pushes down, here water pushed sideways — so its angle of attack and its drag depend on how hard the keel is working ahead of it. Two lifting surfaces in tandem have a best division of load between them, and it is not the one that minimises the drag of either alone.

Beside it is the case that removes the hull’s friction from the problem altogether: a foil that lifts the hull clear of the water, leaving only foil drag against foil lift, and restoring on a reach the small drag angle a keelboat loses there. The two-angle polar’s reaching speeds were always the right answer for that craft, and the question worth computing is what the foil’s own limit — its lift coefficient, its depth below a free surface, and the pressure its suction side can hold — does to them.