The rudder pays for the keel's wake

A keel flies wherever the course puts it solved a yacht’s keel as a wing whose lift is whatever side force the rig makes. It gave the keel the whole of that force and let the rudder trail behind carrying nothing. It ended by noting that on a real boat the rudder carries a share, and that the rudder is a wing flying in water the keel has already turned. This essay divides the force between them.

The sailor’s version of the question is weather helm: the angle at which the tiller has to be held to keep a boat going straight. A boat that wants to turn into the wind has to be held off it, with the rudder angled so that it pushes to leeward on the water — which means it carries some of the side force. The standard advice is that a little weather helm is good and a lot is slow. The explanation usually given for the second half is that an angled rudder is a brake. Both halves turn out to be about the same calculation, and the brake is not where it seems to be.

Two wings, one wake #

The model is the one the keel essay used, with a second foil added. A hull moving at a yacht’s speed barely disturbs the water surface round it, so the hull’s bottom acts as a wall. Each foil hanging from it behaves as half of a wing twice its depth, reflected in the hull. Here the keel is 1.6 metres deep with a chord of 1.0 metre, and the rudder 1.3 metres deep with a chord of 0.4, three metres further aft — the foils of a ten-metre cruising yacht. Each is a lifting line: a row of horseshoe vortices bound across its span, trailing straight back. Every length below is in keel depths, and the water’s density and the boat’s speed are one.

Each foil’s loading is elliptic over its own depth. The rudder’s lift is a fraction f of the total side force and the keel’s the rest. For each foil the calculation gives what that foil feels: its circulation times the sidewash at its own bound vortex, summed over every vortex in the system. The keel feels its own trailing vortices, the rudder’s bound vortex behind it, and the rudder’s trailing vortices, which begin behind it. The rudder feels its own trailing vortices, the keel’s bound vortex ahead of it, and the keel’s whole trailing sheet, which it is sitting in.

That is the drag a foil would report if it were on a balance of its own, and it is the drag a helmsman feels through the tiller. What the boat pays is the sum.

Where the rudder sits changes nothing #

Two wings, and it does not matter where established Munk’s stagger theorem for a biplane. Move one lifting surface along the stream while holding every circulation fixed, and the total induced drag does not change. The drag is set entirely by the wake far behind, and far behind, both surfaces’ wakes lie in the same cross-section, whatever their order along the stream.

A keel and a rudder are the purest case of it. They are in one plane, so their wakes do not merely project onto the same cross-section, they coincide. The far wake of the pair is a single line carrying the keel’s circulation plus the rudder’s. The stagger figure shows the theorem at work. With the rudder half a keel depth behind, it feels 2.91 thousandths and the keel 2.22. With the rudder ten keel depths back, it feels 2.34 and the keel 2.80. At every distance their sum is 5.1339 thousandths, the far-wake drag of the combined loading, to one part in a million.

The division between them is bookkeeping. Close behind the keel, the rudder sits in the downwash of the keel’s bound vortex as well as its trailing sheet, and pays more. The keel, just ahead of the rudder’s bound vortex, sits in its upwash, and pays less. Move the rudder aft and the bound vortices’ influence fades on both. The rudder’s bill falls to what the trailing sheet alone charges, and the keel’s rises by the same amount. The total was never in question.

What the rudder is billed for #

Put the rudder where a yacht has it, 1.9 keel depths aft, and ask what it feels as its share of the load grows. The answer is out of proportion. With an eighth of the side force the rudder is billed for a quarter of the pair’s induced drag. With half the side force it is billed for four-fifths. Per unit of side force the rudder’s induced drag is about twice the keel’s, and a helmsman who feels it through the tiller is feeling something real.

It is the keel’s wake. A lifting foil works by pushing the fluid it passes across, here sideways rather than down. The keel’s trailing vortices turn the water to leeward, and by the time that water reaches the rudder it is moving sideways at nearly the keel’s own induced angle. The rudder carries its load against that sidewash and pays for having to. Under Munk’s accounting, though, the keel is relieved of exactly what the rudder is charged for its presence. The pair’s induced drag depends only on the combined loading. The question that matters is which combined loading is cheapest, and what else the division changes.

The division the total rewards #

Two things depend on how the load is split.

The first is induced drag, and for a given total side force it is least when the combined loading is elliptic over the deepest span available. The keel is deeper than the rudder, so the best is the keel carrying everything. Any load the rudder takes piles circulation onto the upper part of the span, and the combined loading becomes fuller near the hull than an ellipse. The penalty is small at first — the curve is flat at the keel-alone end — and grows as the square of the rudder’s share.

The second is profile drag, the viscous drag of each foil’s sections. It is modelled here by a stated polar for a symmetric section at a yacht keel’s Reynolds number, cd = 0.0065 + 0.012 cl². Its lift-dependent part is least when every section works at the same lift coefficient, which puts a quarter of the side force on the rudder — its share of the total foil area. A keel carrying everything flies at a lift coefficient of 0.40. Handing some load to the rudder brings the keel’s down, and the saving is quadratic in the keel’s lift.

The two pull in opposite directions, and the sum has a minimum between them: 12.4 per cent of the side force on the rudder. There the induced drag is 0.8 per cent above the keel-alone value and the profile drag 3.6 per cent below it. The pair’s drag is 0.97 per cent lower than with the keel carrying everything. The keel then flies at a mean lift coefficient of 0.35 and the rudder at 0.15.

A per cent of the foils’ drag is small, and the foils are only part of a yacht’s resistance; the hull’s friction and its waves are the rest. The saving is also the smaller of the two things the division buys. The other is leeway, and it comes below.

Why the best share is the same in any wind #

The best share does not move with the side force. Pressing the boat harder raises both the induced drag and the lift-dependent profile drag, and both rise as the square of the load. The load therefore factors out of the question of how to divide it. At side forces of 0.06, 0.125 and 0.2 — keel lift coefficients of 0.19, 0.40 and 0.64 — the best share is 12.43 per cent every time, the same to thirteen figures. What does change is what the division is worth. The constant part of the profile drag does not grow with the load, so it dilutes the saving in light going. The saving is 0.43, 0.89 and 1.11 per cent at the three loads, the middle figure measured against the keel alone at its own lift coefficient.

That is a statement a sailor can use: the right share of load for the rudder is a property of the boat’s foils, not of the conditions. What changes with the conditions is the helm angle it takes to deliver it.

A centred tiller already loads the rudder #

Each foil’s angle to the water is the angle its sections need, taken here from thin-aerofoil theory as the lift coefficient divided by 2π, plus the sidewash it sits in. For the keel that angle is the leeway, the boat’s crabbing angle. The rudder’s angle, less the leeway, is the helm.

With the keel carrying everything, the leeway is 5.93°. The rudder, though, sits in 4.71° of sidewash from the keel’s wake, so the water reaches it at only 1.22° even with the tiller central. To carry nothing it has to be turned 1.22° to leeward — lee helm. A centred tiller already loads the rudder, with 7.6 per cent of the side force, and the best share of 12.4 per cent takes 0.78° of weather helm. The load the rudder takes off the keel also takes leeway off the boat: 5.17° instead of 5.93°. A boat that points three quarters of a degree higher at the same speed makes more ground to windward — the fastest way is not the straight one prices exactly that trade — and it is a gain the drag figure does not include.

So the first half of the sailor’s rule is right, and the calculation says how little “a little” is. Loading the rudder until the helm is 4° — a figure often quoted as the sweet spot — puts a third of the side force on the rudder and costs 1.4 per cent over the keel alone. It is 2.4 per cent worse than the best. Where that larger figure comes from, this model cannot say. A heeled hull is asymmetric and makes side force of its own, the rig’s centre of effort moves, and the helmsman is balancing the boat’s turning moment as much as its drag. None of that is in two lifting lines. What is in them is that the drag-optimal rudder carries an eighth of the load at under a degree.

The rudder’s felt bill, drawn #

The felt bill explains the second half of the rule, and why sailors have long believed that weather helm is a brake. A rudder working in the keel’s sidewash is billed at twice the rate per unit of side force that the keel is. Any instrument or sensation that reads the rudder’s own drag will report the rudder as expensive. Munk’s theorem says that the keel’s relief is the other half of the same entry. The pair’s total, which is all the boat pays, falls as the rudder takes load until the eighth is reached.

the surface in the wake computes at 2.46 times the downwash at the wing where a tailplane actually sits, approaching its far-wake value from above as the rudder’s bill does here. A tail carrying lift is billed for induced drag far beyond its share, while the wing ahead is relieved of the same amount. Designers of canards and three-surface aircraft have argued about which surface should carry the load in exactly these terms. Munk’s answer — that the total depends on the combined loading and nothing else — is the one that settles those arguments too.

A deeper rudder earns a larger share #

The induced penalty exists only because the rudder is shallower than the keel. A rudder as deep as the keel adds load in the same elliptic shape over the same span, the combined loading stays elliptic, and the induced drag is the same at every division. The computation finds it identical to machine precision at three different shares. Profile drag then decides alone. The best share becomes 28.6 per cent, close to the rudder’s share of the area, and the saving 2.2 per cent. Halving the rudder’s depth drops the best share to 3.5 per cent and the saving to 0.28. The span effect the span is the whole story describes for wings decides how much work a rudder should do.

That is why modern racing yachts carry deep, high-aspect-ratio rudders, and some of them two rudders angled so that the leeward one stays deep when the boat heels. A deep rudder is worth loading, and a shallow one is best left nearly idle.

What the picture cannot show #

The hull is a flat wall. The reflection that doubles each foil’s depth assumes the hull’s bottom is a plane, level with the foils’ roots, and that the water surface does not move. A real hull is curved, the rudder often hangs further from it than the keel does, and a heeled hull puts both foils at an angle to the water and to each other.

The lifting line overstates a low-aspect-ratio foil’s lift slope. The keel here has a doubled aspect ratio of 3.2, and a lifting line treats its sections as if they were two-dimensional. A keel flies wherever the course puts it used Helmbold’s formula instead, which gives a slope about ten per cent lower at this aspect ratio. The leeway and helm angles here are correspondingly a little small. The drag and the division are induced-drag results, which the lifting line gets right.

The rudder flies in clean water. It is modelled as seeing the keel’s sidewash and nothing else. A real rudder also sits in the keel’s viscous wake and the hull’s boundary layer, both slower than the free stream, so it makes less lift for its angle than drawn.

The polar is stated. The section’s profile drag, cd = 0.0065 + 0.012 cl², is a plausible symmetric-section polar, not a measured one. A polar whose drag rises faster with lift pushes the best share up, towards equal lift coefficients; one that is flatter pushes it down, towards the keel alone.

The convention the numbers depend on #

The keel’s depth is the unit of length, the water’s density and the boat’s speed are one, and the total side force is 0.125 — a keel lift coefficient of 0.40 with the keel carrying everything. Each foil is half of a wing of twice its depth, reflected in the hull. The rudder’s depth is snapped to the nearest node of the keel’s grid, 0.809 keel depths, so that both foils share one set of vortex positions. The rudder’s share is its fraction of the total side force. Helm is positive to windward; leeway is the keel’s angle to its direction of motion. Drags are the real half of each reflected pair.

How each number was checked #

Each part of the calculation is checked against something that does not share its code. One foil alone gives the closed-form elliptic induced drag, both as felt at its bound vortex and in the far wake, to 10⁻¹⁵. The far wake here is the drag matrix used for the biplane, with the two foils’ loadings added on one trace. The pair’s felt drags summed at five staggers agree with that far-wake value to 10⁻⁶. A rudder as deep as the keel leaves the induced drag unchanged across the divisions. The best share at three side forces agrees to 10⁻¹³. Two further checks carry a direction the calculation could have got wrong: as the rudder moves aft the keel’s felt drag must rise and the rudder’s fall, and the best share must lie strictly between the keel alone and the rudder’s share of the area. The calculation refuses a rudder deeper than the keel, a stagger of zero, and a tolerance of zero.

The first version of the computation put each foil on its own grid of vortices, and the total drifted six per cent from the far-wake value and moved with the grid. The rudder’s control points sat at arbitrary distances from the keel’s discrete trailing vortices and sampled a singular kernel badly. Putting both foils on one grid, so that every control point lies midway between trailing vortices as on a single wing, brought the total to within one part in a million of the far wake.

Who found it, and when #

Munk’s stagger theorem is from his 1921 report on the minimum induced drag of aerofoils, which also contains the reciprocal theorem. Prandtl’s biplane theory, a few years earlier, had already treated two lifting surfaces through their shared wake. The keel as a low-aspect-ratio wing reflected in the hull is the framework of yacht hydrodynamics since the 1960s. Tank tests of keel and rudder together, including the rudder’s operation in the keel’s downwash, have been standard since the America’s Cup programmes of that decade. That the pair’s drag is least with the rudder carrying some of the load is well established in that literature. What is added here is the arithmetic of why, and that the share is independent of the load.

Still open: the foil that lifts the hull out #

Two continuations follow from this one. The first is the one the keel essay set beside this question: a foil that lifts the hull clear of the water. It removes the hull’s friction and leaves only foil drag against foil lift, and on a reach it restores the small drag angle a keelboat loses there. Faster than the wind that drives it already has the speed polar for a craft whose drag angle is small. What a foiling boat adds is a lifting surface running just under a free surface. At speed that surface is a constant-pressure boundary rather than a wall, it takes lift away rather than adding it, and it fixes a depth below which the foil must run. Computing how that surface’s image reduces the foil’s lift slope, and what it does to the polar, is the next step.

The second is the heeled boat. Heel tilts both foils, shortens their effective depth, and makes the hull itself asymmetric. The division computed here then has to share the job with the hull’s own side force and with the boat’s balance of turning moments. That is the calculation that would say whether the lore’s few degrees of weather helm are the drag optimum of a heeled boat or the price of holding a heeled boat straight.