A foil under the surface flies with a phantom

The rudder pays for the keel’s wake divided a yacht’s side force between two foils hanging beneath its hull, each flying wherever the course puts it, and used one fact about the hull throughout: at a keelboat’s speed the hull’s bottom is a wall, and a wall reflects a foil that hangs from it into one of twice its depth. That essay’s last section named the craft that removes the hull from the water altogether. A foiling boat lifts itself on a horizontal wing a little under the surface, and the surface it runs under is not a wall.

A hydrofoil loses most lift on the way up worked out what that surface does to a foil in two dimensions, across the whole range of speeds. At low speed it is a lid and the foil gains lift, as a wing does over the ground; at high speed it is a boundary that cannot hold a pressure, its image of the foil’s vortex keeps the same sign, and the foil loses lift; between the two, where the foil’s own waves are a few depths long, the lift dips below both. A foiling boat at cruise sits at the high-speed end. This essay takes that end into three dimensions, where a foil has a span and a wake, and asks the questions a boat asks of it: what the surface does to the induced drag, what it does to the board that must pierce it, and how deep the foil should run.

The image a pressure-release surface makes #

At a boat’s cruising speed the waves a foil makes are many chords long and are left behind it, and to first order the surface over the foil can do only one thing: stay at atmospheric pressure. In the linearised theory that means the disturbance’s velocity potential vanishes along the surface. The image system that satisfies it is the foil’s reflection in the surface, and the rule for its sign is the one the two-dimensional essay found for a single vortex, now applied to every segment of a vortex line: a horizontal vortex segment is reflected with its sense unchanged, and a vertical one with its sense reversed. A wall does exactly the opposite.

For a horizontal foil every vortex is horizontal: the bound vortex along the span and the trailing legs streaming behind the tips. So its image is an identical foil, carrying the identical loading, at twice the depth above it. The foil and its image are a biplane — two equal wings, one above the other, with a gap of twice the depth — of which only the lower wing is in the water. That is a surprising object to meet under a boat. Two wings, and it does not matter where computed what a biplane partner does: each wing sits in the other’s downwash, and the pair’s induced drag exceeds two independent wings’ by a factor Prandtl called $\sigma$. A wall’s reversed image is the opposite partner, one whose downwash cancels, which is why a wing in ground effect sheds induced drag. The foil under a surface at speed has the partner that adds — the partner a box wing with a light spar becomes, except that here the upper wing carries a lift nobody receives.

The calculation is a vortex lattice. A flat rectangular foil of aspect ratio 9 is divided into forty strips across its span and six panels along its chord, each carrying a horseshoe vortex bound at its quarter-chord with its control point at three-quarters; the flow is made tangent to the foil at every control point, with every real horseshoe and every image horseshoe contributing. Lift is the density times the speed times the circulation summed along the real bound vortices. Induced drag is taken far downstream, in the plane where the wake is a pair of vortex sheets: the real sheet’s circulation times the downwash there from both sheets, which is the energy the wake leaves in the real half of space. Everything is linear, the foil flat, and its depth measured to its plane.

The lift in three dimensions #

The first figure starts where the two-dimensional essay’s fast limit stood, and adds the span. The faint curve is the simplest possible foil — one vortex at the quarter-chord, one control point at three-quarters — for which the image’s effect has a closed form, $(1 + 16h^2)/(2 + 16h^2)$ with the depth $h$ in chords. It falls to exactly a half at the surface, because there the image vortex sits almost on the foil’s own and doubles the downwash at the control point. The lattice reproduces that closed form to a part in $10^{16}$ when given one panel, which is the first check. With forty chordwise panels the flat plate falls the same way, to 0.85 of its deep-water lift slope at half a chord and 0.95 at one chord, within a per cent of the fast limits of 0.84 and 0.94 that the two-dimensional essay reached with its own twelve-vortex plate.

The foil of aspect ratio 9 loses more at every depth: 0.81 of its lift slope at half a chord where the plate keeps 0.85, 0.90 at one chord where the plate keeps 0.95, 0.60 at a tenth of a chord. The extra is the span’s contribution. The image’s trailing vortices lie above the real ones and add downwash along the whole span, on top of what the image’s bound vortex does to the chordwise loading, and the foil has to fly at a higher incidence for the same lift. On a long enough foil the extra vanishes: at an aspect ratio of 60 the lattice’s lift ratio is within 2.3 per cent of the plate’s at every depth tested, and the discrepancy falls further at 200. That is the second check, and it says the three-dimensional loss is the two-dimensional one plus something with its own cause.

The phantom’s bill #

The second figure is the something. It holds the lift fixed and asks what induced drag the foil pays, against its depth now measured in spans, since the trailing vortices’ images are separated from the real ones by twice the depth and it is the span they are compared with.

The real foil pays its own induced drag plus the mutual induced drag of the biplane — its lift times the downwash the phantom’s wake puts on it. For two equal elliptic wings a gap $G$ apart the mutual term is $\sigma(G/b)$ times the self term, so the foil pays $1 + \sigma(2h/b)$ times its deep-water value. The dashed curve is that, computed from the two elliptic loadings in the far-wake plane; it agrees with Prandtl’s fitted formula $(1 - 0.66r)/(1.055 + 3.7r)$ within 0.6 per cent at four gaps, and runs to one as the gap closes, so the factor runs to two. The solid curve is the lattice foil, which solves its own loading at each depth and follows the elliptic curve closely: 1.11 at four chords’ depth, 1.25 at two, 1.43 at one, 1.74 at a quarter, 1.87 at a tenth.

That is the answer to the question the ground-effect picture gets wrong. A wing descending towards a runway sheds induced drag, because the ground’s reversed image cancels its downwash. A foil rising towards the surface at speed gains it, because the surface’s image adds. At a chord’s height over a wall this foil would pay 0.56 of its free-air induced drag; at a chord’s depth under the surface it pays 1.43. The free surface is ground effect with the sign of every change reversed, and the phantom partner is what the reversal looks like in three dimensions: a wing that the foil cannot see, cannot use and has to pay for.

What the phantom does to the loading #

A natural guess is that the extra drag comes from a distorted loading — that the phantom’s downwash, strongest in some part of the span, pushes the foil’s loading away from the efficient shape. The third figure tests it. The three loadings carry the same lift, and they are almost the same shape. The phantom’s downwash, coming from a wake twice the depth away, is spread more evenly across the span than the foil’s own, which is concentrated near the tips where its trailing vortices roll off. So the phantom unloads the centre very slightly relative to the tips: nine-tenths of the way to the tip, the deep foil carries 45 per cent of its centre loading, the foil a quarter-chord down 50 per cent.

The shape has barely moved, and it has moved in a direction that does not matter much. Almost all of the drag rise is the mutual term, the phantom’s downwash acting on a loading that is nearly the deep-water one. That is also why the elliptic biplane’s $1 + \sigma$, which knows nothing of the lattice’s loading, tracks the lattice so closely. The phantom’s bill is a property of the gap, not of the foil’s shape, and no amount of twist or taper on the real foil can avoid it — the same statement Munk’s stagger theorem made about a biplane’s mutual drag, which depends only on the two wakes’ positions in the far-field plane.

The board goes through the surface #

A foiling boat has a second foil, and it meets the surface differently. The horizontal foil hangs from the foot of a vertical board — a centreboard or a strut — which pierces the surface and carries the sail’s side force, as a keel does. Its bound vortex runs vertically, and the surface reflects a vertical vortex with its sense reversed. So the board’s image is a copy of the board above the surface carrying the opposite loading, and the pair is a wing with antisymmetric loading across the surface — the loading of a wing that is rolling rather than lifting.

The fourth figure computes what that costs. It plots the induced-drag factor $K$ — the induced-drag coefficient divided by the square of the lift coefficient — times $\pi$ and the submerged aspect ratio $A$, so that a wing with elliptic loading and free ends sits at one. Under a hull, as the keel in the tandem essay did, the wall’s image extends the board into a wing of twice its depth and the factor sits near 0.5: the wall made by reflection doubles the span and halves the price. A board standing free, with two tips, sits near 1. A board through a surface at speed sits near 1.25, worse than a free board, because its loading must fall to nothing at the surface and its reversed image, right beside that end, sheds a second strong vortex at the waterline where a free board would shed a gentle one.

So a foiling boat pays the surface twice. The foil beneath it flies with a phantom partner that adds up to its whole induced drag again, and the board that carries its side force pays two and a half times what the same board would pay under a hull. The second is the larger bill in practice, because the side force is carried on the short span the board happens to have in the water.

How deep to fly #

The fifth figure puts both bills on one boat. It is a foiling dinghy of 110 kilograms, boat and sailor, lifted by a horizontal foil one metre in span and 11 centimetres in chord — an aspect ratio of 9.1 — with a profile-drag coefficient of 0.008; the foil hangs on a board 12 centimetres in chord with a profile-drag coefficient of 0.009, carrying a side force of 400 newtons. At 8 metres a second, just over 15 knots, the foil needs a lift coefficient of 0.30. Every coefficient comes from the lattice at the depth in question, the foil’s with its phantom and the board’s with its reversed image.

Four drags respond to depth in four ways. The foil’s profile drag, 28.9 newtons, does not respond at all. The foil’s induced drag rises as the surface nears, from 11.2 newtons deep to 16.3 at ten centimetres. The board’s profile drag grows with the wetted length, 3.5 newtons per decimetre. And the board’s induced drag, carrying 400 newtons on the span that is under water, grows as one over the square of that span: 7.4 newtons at half a metre, 45.7 at twenty centimetres, 183 at ten.

The total is least at 0.48 metres, 4.4 chords down, at 66.1 newtons: 28.9 in the foil’s profile, 12.2 in its induced drag, 17.1 and 7.9 in the board. The foil’s phantom is almost irrelevant at that depth — at 4.4 chords it has added only a newton — and the depth is set by the board, where a shorter wetted span would save profile drag and cost more induced. The foil does not choose how deep it flies; the board does.

Faster boats ride shallower, and the surface pushes them down #

The sixth figure repeats the search at every speed from 6 to 16 metres a second, and a second time with the surface removed: the foil given its deep-water coefficients and the board its under-a-hull ones, as though it had a wall at the waterline.

Faster boats ride shallower. The side force is fixed and the dynamic pressure grows as the square of the speed, so the board needs less span to carry it: the best depth falls from 0.72 metres at 6 metres a second to 0.48 at 8, 0.36 at 10 and 0.19 at 16. That is the direction foiling sailors adjust ride height as a boat accelerates, and here it has a mechanism.

The surface pushes every best depth down by about two-fifths — 0.48 against 0.35 metres at 8 metres a second — and at the deeper ride the boat pays between 7 and 13 per cent more drag than it would with no surface. At 8 metres a second, 6.5 of the extra 7.6 newtons are the board’s, and 4.8 of those are profile drag on the extra wetted length it takes to keep the board’s induced drag down. The surface’s cost is paid mostly as depth.

Set beside the keelboat of the tandem essay, the contrast is complete. A keelboat’s two foils are reflected in a wall and doubled; a foiling boat’s are reflected in a surface that cannot hold a pressure and are penalised instead, and the boat buys back part of the penalty by flying deeper than it would otherwise need to. The rig is the same kind of wing in both, and the drag angle that sets how fast either can sail is paid in the water.

What the calculation was checked against #

The seventh figure is the ledger. The lumped vortex matches its closed form at five depths to rounding. A foil ten thousand chords deep matches the same foil with no image to a part in $10^8$. A foil of aspect ratio 60 matches the two-dimensional plate’s lift ratio within 2.3 per cent at three depths, the shortfall being the finite span’s own trailing images. And the elliptic biplane’s $\sigma$, computed from the two loadings in the far-wake plane, matches Prandtl’s fitted formula within 0.6 per cent at four gaps and reaches 0.99 as the gap closes to a thousandth of a span, one within the quadrature’s resolution.

The lattice’s own convergence is worth a sentence. Its absolute numbers move with resolution — the deep foil’s lift slope from 4.86 to 4.76 per radian as the lattice goes from 16 by 4 panels to 60 by 8 — but the ratios the argument rests on move by a fraction of a per cent: the lift ratio at half a chord from 0.804 to 0.806 and the drag ratio from 1.62 to 1.60.

What the phantom picture leaves out #

Finite Froude number. Everything here is the high-speed limit, and the two-dimensional essay showed that the limit is approached slowly: at a depth of one chord the foil’s lift is still a few per cent off its fast value at chord Froude numbers near seven. The dinghy’s chord Froude number at 8 metres a second is 7.7 and its depth Froude number at its best depth 3.7. At the low end of the speed range, 6 metres a second with a depth Froude number of 2.3, the boat is nearer the waves than the image, and the wave drag and the lift dip that essay computed begin to matter. The phantom is a cruise picture.

Ventilation and spray. A board through the surface at speed can draw air down its low-pressure side and lose its side force abruptly, and it throws spray that costs drag this calculation does not have. Both limit how little of the board a boat can wet, and both act in the direction the image already points: deeper.

The junction. The foil meets the board at a T, and the interference drag of that corner and the vortex it sheds are not in a flat lattice of two separate foils. Nor is the heel of a real boat, which tilts the board and so its image.

Linear theory. The foil is flat and thin, the lift coefficient of 0.3 is small, and the image’s own horizontal velocity at the foil — which changes the lift at second order in the depth ratio — is left out.

The convention: depths to the foil’s plane #

Depths are measured from the undisturbed surface to the foil’s plane, in chords for the lift and in spans for the induced drag, because the lift’s image effect is chordwise and the drag’s is spanwise. Lift slope is per radian of incidence; the induced-drag factor $K$ is $C_{D_i}/C_L^2$ on the foil’s own planform. The surface is the pressure-release limit of the linearised free-surface condition. The board’s submerged aspect ratio is its wetted depth over its chord.

Who worked it out #

The images of a vortex under a free surface in the two speed limits go back to the early theory of hydrofoils: Kotchin in the 1930s and Keldysh and Lavrentiev set up the linearised problem, and Hough and Moran in the 1960s gave the fast-limit lift of a foil near a surface in closed form. The biplane factor is Prandtl’s, from 1918–19, and the stagger theorem Munk’s of 1921. Wadlin and his colleagues at the NACA measured submerged and surface-piercing foils in the 1950s and reported the loss of lift near the surface, and Breslin and Skalak studied how a surface-piercing strut ventilates at the end of the same decade.

Still open: a foil that rides the waves it meets #

Every number here is for a foil moving steadily under a flat surface. A foiling boat at sea flies through waves whose own orbital velocities change the foil’s incidence and whose troughs change its depth, and a boat whose best depth is half a metre is flying through the top half-metre of a wave’s orbital motion. The next calculation puts the foil under a regular train of waves of given height and length, lets the phantom partner follow the surface’s local depth, and asks how much the boat’s drag rises on average when the depth oscillates about its best value — since the drag curve is steeper on the shallow side than the deep, the answer should be a further push downward, and its size, for a wave the height of the foil’s depth, is what decides how deep a boat should fly in a seaway rather than on a lake.