A breeze the boat cannot use

Faster than the wind that drives it reduced a sailing boat to two angles — the rig’s drag angle and the hull’s — and found that the boat’s speed is the true wind times $\sin(\beta-\lambda)/\sin\lambda$, with $\lambda$ their sum and $\beta$ the course. The fastest way is not the straight one took that polar and found the best course to windward in one line of trigonometry: $45° + \lambda/2$.

Neither result contains a force. The polar is a statement about the shape of a velocity triangle, and the triangle has the same shape in two metres a second of wind as in twenty. That is why a boat’s performance could be quoted as a fraction of the wind at all, and it is also the first essay’s own admission about where the model stops: the side force a rig makes is large, it acts high up, and a boat sailed to that model’s optimum in a real breeze would be knocked flat.

This essay puts the one missing force back. It gives the boat a righting moment — the most its hull, ballast and crew can resist before it heels past where it can be sailed — and solves the same triangle subject to it, on every course and in every wind. The boat is a seven-metre keelboat of ordinary proportions: 25 square metres of sail with its centre of effort 3.6 metres above the keel’s, a righting moment of 4.0 kilonewton-metres, a sail polar of $C_D = 0.08 + 0.14\,C_L^2$, and a hull whose own drag angle is held at 6° throughout, so that everything that follows belongs to the rig.

The wind in which the full rig stops fitting #

Hold the triangle’s shape fixed and the heeling moment goes as the square of the apparent wind, which goes as the true wind. So on any course there is one true wind in which the full rig, set at its best, heels the boat with exactly its righting moment.

The ceiling arrives first across the wind, where the apparent wind is strongest, because a boat reaching at three times the wind’s speed is making most of its own breeze. On the best beat it arrives in 3.70 metres a second. Six or seven knots is a light breeze; a sailing club’s afternoon is usually above it. The regime in which the two-angle polar holds is the minority of real sailing, for this boat at least, and the regime above it is not a correction to the one below.

Above the ceiling the crew has a choice. They cannot carry the full rig at its best, so they must make it carry less force, and there are two ways to do that which are quite different aerodynamically.

Two ways to carry less force #

Reefing reduces the sail’s area and leaves its shape alone. The lift coefficient stays where it was and so does the rig’s drag angle; the force is smaller because there is less sail.

Flattening leaves the area alone and changes the shape — less camber, more twist, the mainsheet eased. The lift coefficient falls, and the drag angle does not stay put.

The sail’s drag polar says why. Its drag coefficient is a constant plus a term in the square of the lift coefficient, and those are the two drags any wing pays: a profile drag for having a surface and wetted rigging at all, and an induced drag for making lift with ends — a rig’s foot at the deck and its head at the masthead. The drag angle, $\arctan(C_D/C_L)$, is least where the two are equal, at $C_L^* = \sqrt{C_{D0}/k}$, which is the same condition that puts a glider at the speed at which its drag is least.

The algebra is two lines. The drag angle is least where $C_D/C_L = C_{D0}/C_L + kC_L$ is least; setting its derivative to zero gives $C_{D0} = kC_L^2$ — the profile and induced terms equal — and a least drag-to-lift ratio of $2\sqrt{C_{D0}k}$. For this rig that is a lift coefficient of 0.756 and an angle of 11.95°. Everything about depowering follows from which side of that minimum the crew moves the sail to, and flattening moves it to the side where the constant term wins.

To the left of the minimum the profile drag is the larger term, and it does not fall when the lift does. Halving the lift coefficient from its best value leaves nearly all the profile drag in place while halving the lift, so the force leans back from the perpendicular to the apparent wind. In eight metres a second the flattened rig’s drag angle is half as large again as its best; in twelve it has doubled.

This is the mechanism in the whole essay, and it is worth stating plainly because the words mislead. A flattened sail makes less drag, and it makes much less lift, and the ratio is what drives the boat.

The triangle stops being the same shape #

In the two-angle model a boat’s speed made good to windward is a fixed fraction of the wind: 1.122 for this rig and hull, in any wind at all. That similarity is what the model rests on, and the ceiling ends it.

Below the ceiling the two lines are the same line, flat at 1.122, which is the similarity drawn. Above it they part. The flattened rig’s fraction falls steadily: by ten metres a second the boat is making barely half the fraction of the wind it made in light air, and by twenty, less than a sixth. A boat that was sailing at more than the wind’s speed towards the mark in four metres a second is sailing at a sixth of it in twenty.

That is the qualitative fact every sailor knows — boats are relatively slow in a gale — given a mechanism and a number. The part of the wind the boat can use is capped by what its hull can resist, and the cap does not grow with the wind.

The wind worth most to windward #

Multiplying the flattened rig’s fraction by the wind it is a fraction of gives the boat’s actual speed made good, and that has a maximum. In four metres a second the boat makes 4.46 metres a second towards a mark dead upwind. In six it makes 5.51, and in eight and a half 5.79 — the most it ever makes. Past that the number falls: 5.72 in ten, 4.81 in fifteen, 3.39 in twenty.

So there is a best wind for this boat to windward, and it is about eight and a half metres a second, a moderate breeze of sixteen or seventeen knots. Past it, every further metre a second of wind has to be spilled, and spilling it by flattening costs more in drag angle than the stronger wind returns. A boat in twenty metres a second is not merely slower as a share of the wind than in four; it is slower outright, by a quarter.

That turn does not depend on the fine detail of the polar. It needs only a force that grows as the square of the wind, a ceiling on its moment, and a drag angle that rises as the lift coefficient falls below its best — the last of which is the two drags of any wing read from the far side of their minimum. A rig with less profile drag would move the best wind upwards and soften the fall after it; it would not remove either.

What a fixed hull angle hides #

The reefed rig’s line two figures back is flat for the whole range, and that deserves suspicion rather than admiration.

The flat line says that a reefed boat, with its sail always at its best lift coefficient, keeps exactly the same triangle in any wind — and the triangle then says how fast it is going.

The flattened boat behaves like a boat: its speed rises with the wind, levels, and falls slightly as the sail is flattened further than helps. The rig can push no harder than the righting moment allows, and flattening makes the push lean further back, so there is a best wind for this boat’s upwind speed and it is about ten metres a second.

The reefed boat does not behave like anything. Seventy-four knots to windward on a sail the size of a tablecloth is not a prediction, and nothing about the rig is responsible for it. It is the hull’s drag angle being held fixed. A hull whose drag is a fixed fraction of whatever side force it carries can be driven at any speed by any force, however small, because its drag shrinks with its load. No hull has that property: a hull’s friction goes with the square of its speed whether or not it is carrying a side force, and the keel that makes the side force is a wing with its own polar. Reefing looks free here only because the hull cannot charge for it, and the essay that solves the keel is where the charge is made.

So the reefed line is the limit of this model rather than a result of it, and what follows uses the flattened rig, whose behaviour belongs to the rig and survives a better hull.

λ becomes a property of the course #

The earlier essays treated $\lambda$ as a number a boat has, like a length. With the ceiling in place it is a number the boat has on a particular course in a particular wind.

In three metres a second the line is flat, because the ceiling has not been reached on any course drawn, and $\lambda$ is the constant the model assumed. In a breeze it rises as the boat bears away, and the reason is the first figure’s again: bearing away raises the apparent wind, a stronger apparent wind needs more flattening to hold the heeling moment, and more flattening means a larger drag angle.

That single change — $\lambda$ depending on $\beta$ — is enough to undo the closed form the best-beat argument rested on, and it undoes it in a definite direction.

Why the best beat moves the wrong way #

The derivation of $45° + \lambda/2$ maximised $\sin(\beta-\lambda)\cos\beta/\sin\lambda$ over $\beta$ with $\lambda$ held still. If $\lambda$ rises with $\beta$, the derivative gains a term: moving to a larger $\beta$ now also makes $\lambda$ larger, which lowers the speed made good, so every degree of bearing away costs more than the fixed-angle calculation charges. The optimum is pushed towards the wind.

Written out, the condition for the best course is

$$\cot(\beta-\lambda) - \tan\beta \;=\; \frac{d\lambda}{d\beta}\,\bigl[\cot(\beta-\lambda) + \cot\lambda\bigr]$$

With $d\lambda/d\beta = 0$ the right-hand side vanishes, the left gives $\beta - \lambda = 90° - \beta$, and that is $45° + \lambda/2$. With $\lambda$ rising as the boat bears away, the bracket is positive on every course that can be sailed, so the left-hand side must be positive too — and it is positive only on courses closer to the wind than $45° + \lambda/2$. The direction of the shift is decided by the sign of one derivative before any number is computed; the figures say how large it is.

The frozen-$\lambda$ curve is what a navigator gets by measuring the boat’s drag angle on its best course and then using $45° + \lambda/2$. It touches the true curve at exactly one course and diverges from it on both sides: to leeward of it the real boat is worse, because it is more heavily flattened there; to windward it is better, because it is less. The real peak is six and a half degrees higher than the frozen one, and the frozen curve promises 0.756 of the wind at its peak where the boat can make 0.723.

The whole wind range tells the same story with a turn in it. From the ceiling up to about six and a half metres a second the best beat moves closer to the wind by more than four degrees. Beyond that the rig is flattened so far that the drag angle is dominated by its profile drag on every course, $\lambda$ grows large everywhere, and the optimum drifts back out — though from six metres a second up it stays five degrees or more inside the rule.

The essay that derived the rule said, in its account of where its model stops, that an overpowered boat’s drag angle worsens and so its optimum moves off the wind exactly when the wind is strongest. The first half of that is right and the second does not follow from it. Over the range of wind in which a crew is actually flattening sail, the best course moves towards the wind.

Sailors have a word for doing that. Feathering a boat to windward in a breeze — sailing a little higher than normal with the sails eased, spilling force rather than fighting the heel — is ordinary upwind technique in dinghies and keelboats, and it has always sounded like a compromise between speed and comfort. The calculation says it is neither. It is the speed-made-good optimum of a boat whose drag angle depends on its course, and the rule it appears to break was derived for a boat whose drag angle does not.

What the ceiling model leaves out #

The ceiling is a constant. A real boat’s righting moment grows as it heels, peaks, and falls, and its sail’s effective area falls as the rig leans. A crew sails at some heel angle the boat tolerates, and the moment available there is the ceiling used here; the shape of the curve around it would round the corner at 3.7 metres a second without moving where it is.

Flattening moves along one polar. Real depowering changes twist, camber and angle together, and its effect on the profile drag is not zero. Modelling it as a slide along a fixed parabola is the simplest assumption that gives a drag angle rising as lift falls, which is all the argument needs.

Reefing lowers the rig. A reefed mainsail’s centre of effort is lower than the full sail’s, so the same righting moment allows more force. The model keeps the height fixed, which makes reefing look worse than it is — and it already looks free, for the reason above.

The hull has no speed limit. There is no wave drag. A displacement hull’s resistance rises steeply as its speed approaches the speed of the waves it makes, a wave pattern whose angle does not depend on the boat, and a model with that in it would level the flattened boat’s speed earlier.

The wind is the same at every height. Over water the wind strengthens with height, so the head of a rig sees a stronger apparent wind, arriving from further aft, than its foot. That is one reason sails are twisted, and it puts more of the force high up for a given drive, so the ceiling here would arrive a little earlier with it included.

The rig does not stall. At the lift coefficients a beating rig uses, the sail is well below its stall; the stall matters on a reach with the sheets eased too far, which is not what this essay is about.

A force balance as old as rating rules #

Solving a boat’s velocity triangle subject to its forces and its heeling moment on every course is what a velocity prediction program does, and the programs developed for handicapping ocean racers from the late 1970s onwards do exactly this, with a far more detailed hull, rig and stability model than the one here. Their output is a polar that depends on the wind, with a best beat and a best run that depend on the wind, and the fact that the optimum does not follow $45° + \lambda/2$ is built into every polar they print.

What the calculation here adds is the reason, in a form small enough to see. A single ceiling on a single moment is enough to make $\lambda$ a function of the course, and a $\lambda$ that is a function of the course is enough to move the optimum the other way from the one the closed form suggests.

Still open: what the keel charges #

Every figure here held the hull’s drag angle at six degrees, and the reefed rig’s impossible speeds are the price of that. A keel is a wing whose lift is the rig’s side force, whose lift coefficient is therefore set by the course and the wind rather than by its designer, and whose drag angle is least at one lift coefficient and grows on either side of it.

That is a keel that flies wherever the course puts it: the hull’s drag angle solved as a wing on every course, the best beat pulled a further twelve degrees towards the wind in light air for a reason that has nothing to do with the rig, and the charge for reefing made at last — because a keel carrying a capped force at a rising speed flies further and further from its best. Beside it is the question that both rig and keel share, which is how much of a wing’s drag is the price of its span rather than its area, and why a tall rig and a deep keel are each worth more than the same area spread lower.