A hydrofoil loses most lift on the way up

The borrowed mass that goes negative puts a circle under a free surface and oscillates it. At low frequency the surface cannot keep up and acts as a rigid lid; at high frequency it cannot resist and acts as a surface of constant pressure; and the added mass does not pass from one image to the other but overshoots, going negative in between. That essay closes by naming the steady counterpart: a hydrofoil running just beneath the surface, whose lift should move between a lid at low speed and a pressure-release surface at high, and asks where the loss sits in speed and depth for a foil of realistic proportions.

This essay answers it, and the answer is the same shape as the added mass’s. The lift does not move from one image to the other. It falls below both.

Two images, and why they have opposite signs #

Everything about a boundary beneath a wing begins with a cushion that is not there: a wing near the ground is computed by reflecting its bound vortex in the ground, and the reflection has the opposite sign, because a rigid wall must stop the flow crossing it. For a lifting wing the reflected vortex beneath it reduces the downwash at the wing, and the wing gains lift.

A hydrofoil beneath a surface has a reflection too, and its sign depends on the speed. Written in the linearised form, the surface condition for steady flow at speed $U$ is

$$U^2\phi_{xx} + g\,\phi_y = 0 \quad \text{on } y = 0 .$$

When the foil is slow, $g$ wins: $\phi_y = 0$, no flow crosses the surface, and the surface is a lid with an opposite-signed image exactly like the ground’s. When the foil is fast, $U^2$ wins: $\phi$ is constant along the surface, so the surface cannot sustain a pressure difference, and the image has the same sign as the foil’s own vortex. A same-signed image above a lifting vortex slows the flow past it, and the foil loses lift.

That is the reflection the usual comparison misses. A foil boat at cruise is not a wing in ground effect upside down; it is the opposite of one. And between the two limits the surface is neither: it is a wave-maker, and the image is replaced by a train of waves.

The Green function, and the foil #

The velocity a single vortex adds beneath the linearised surface can be written in closed form apart from one integral. With $\nu = g/U^2$ and the vortex at depth $h$,

$$u - iv = -\frac{i\Gamma}{2\pi}\,\frac{1}{z - \bar z_j} + \frac{\nu\Gamma}{\pi}\left[\mathrm{PV}\!\!\int_0^\infty \frac{e^{-k\zeta}}{k - \nu}\,dk - i\pi\,e^{-\nu\zeta}\right], \qquad \zeta = i(z - \bar z_j),$$

the first term being the same-signed image and the bracket the waves, taken with the radiation condition so that they appear only downstream. As $\nu \to 0$ the bracket vanishes and the image is all there is. As $\nu \to \infty$ it grows to twice the image with the opposite sign, and the image flips. The principal value was evaluated numerically with its pole subtracted, and checked against a brute-force sum to six figures.

At the vortex itself the bracket gives a vertical velocity, and Kutta–Joukowski turns that into a drag: for one vortex of strength $G$ it is

$$D = \frac{\rho g G^2}{U^2}\,e^{-2gh/U^2},$$

the classical wave resistance of a submerged vortex. The calculation reproduces it to rounding.

The foil is a flat plate of one chord carrying twelve vortices at its panels’ quarter points, with the flow made tangent to the plate at their three-quarter points and the Kutta condition built in by that choice. In an unbounded stream this returns thin-aerofoil theory’s $2\pi\alpha$ exactly, for any number of panels. Under the surface, each vortex feels every other one’s image and waves, the circulation readjusts, and the lift and wave drag follow from Kutta–Joukowski on each.

The lift, from standstill to cruise #

The first figure is the answer. At a chord Froude number $U/\sqrt{gc}$ below about 0.6 the foil sees a lid and gains lift — 1.22 times its deep-water value half a chord down in the slow limit, rising to a peak of 1.89 near a Froude number of 0.52 as the waves it makes lengthen towards its own size. Then the lift falls steeply, through its deep-water value near a Froude number of 0.7, to a minimum well below anything either image predicts: 0.43 of the deep-water lift half a chord down, near a Froude number of 1.2. From there it climbs slowly back towards the fast limit, 0.84, which it approaches only at Froude numbers above ten.

Deeper foils do the same thing more gently. One chord down, the peak is 1.17, the dip 0.61, the fast limit 0.94. Two chords down, 1.04, 0.76 and 0.98. The dip is the dominant feature at every depth, and it is far deeper than the fast-foil loss that a hydrofoil designer’s rule of thumb is built around.

Worse than either image #

The second figure sets the three numbers against depth. The rigid lid gives a gain that grows as the foil rises, exactly as a wing’s does over the ground. The pressure-release surface gives a loss that is the lid’s image, and which also grows as the foil rises. The worst speed lies below both, at every depth: at a quarter of a chord the foil keeps 0.30 of its lift, at half a chord 0.43, at four chords still only 0.87 against a fast-foil 0.99.

The dip is the steady counterpart of the added mass’s overshoot, and its reason is the same. Between its two limits the surface is neither a lid nor a hole; it moves, and it moves in phase with the foil in a way that neither image allows. When the wavelength is a few depths, the surface falls away over the foil towards the first trough of its wave train, which stands just behind the trailing edge — the next figure shows it — and the wave’s own motion adds a downwash along the foil that neither image contains. No single image can reproduce that, because an image’s influence has one sign along the whole foil, and the wave’s changes with its phase.

The boundary decides, and so does the speed #

The borrowed mass the boundary decides made the point on which this whole comparison rests: what a body near a boundary feels is set by what the boundary allows — a wall that stops the normal flow, a surface that cannot hold a pressure — and not by how close the body is. The ground gives a wing one kind of boundary at every speed, so the ground’s effect is a function of height alone. The sea gives a foil a boundary whose kind is set by the Froude number, so its effect is a function of height and speed together, and it cannot be read off a ground-effect chart by turning it over.

That is the practical content of the first two figures. A hydrofoil designer who sizes a foil for its cruise lift using the fast-foil loss has sized it correctly at cruise and at no other speed; the same foil at a third of that speed has a quarter less lift than the designer assumed, and on the slow side of the dip a gain that would carry it upwards. Neither surprise is a failure of the image method: each image is exactly right in its limit. The error is in believing that the lift moves monotonically between the two limits.

The waves themselves #

The third figure draws the surface. The wavelength is $2\pi U^2/g$, which in chords is $2\pi F^2$: 2.3 chords at a Froude number of 0.6, 6.3 at one and 25 at two. The slow foil’s waves are short and small, because a short wave’s motion decays within a fraction of its length below the surface and barely reaches a foil a chord down. At a Froude number of one they are five times higher, with a trough already forming above the foil’s trailing edge. At two they are higher still — a long, shallow swell — but their slopes are gentle, and a steady wave carries energy in proportion to the square of its slope times its length, not of its height alone.

That is the balance the wave drag strikes, and it has been met once before: the drag that is made of waves found that a submerged body in an otherwise ideal flow has a drag, and that it is exactly the energy carried off in the wave train behind it.

Where the wave drag peaks #

The fourth figure is the wave drag, divided by the square of the lift so that the three depths can be compared at the same load. Against the depth Froude number $U/\sqrt{gh}$, a single vortex’s wave drag peaks at exactly $\sqrt2$: that is where $e^{-2gh/U^2}$ and the prefactor $g/U^2$ trade off. The foils peak just above it, at 1.44 two chords down, 1.45 one chord down and 1.55 half a chord down, where the foil’s own length begins to count against the depth. The foil one chord deep lies almost on the point vortex’s curve.

The wave drag was checked independently. The waves far behind the foil have an amplitude that follows from the same vortex strengths, and the energy they carry off, $\tfrac14\rho g A^2$ per unit width, equals the drag the forces on the vortices give, in four cases, to rounding. The two are the same statement written at the foil and far behind it, and the calculation keeps them in step.

The factor of a quarter is itself a piece of wave physics. A steady wave train behind a moving body is a region of wave energy that grows longer every second by the body’s speed, and the energy needed to fill it is supplied by the drag. But in deep water the energy of a wave travels at half its phase speed, so half of what each new wavelength needs is brought forward by the waves already there, and the body pays for only the other half: the energy per unit length of the train, $\tfrac12\rho g A^2$, times one half. The same slowness is why no wave appears ahead of the foil — energy never outruns the wave crest that carries it — and why a parcel of water beneath the train is left drifting slowly forward by a wave that, at any fixed point, has no mean flow in it at all.

The lift’s dip and the drag’s peak sit at nearly the same speed. They are one phenomenon: the speed at which the foil’s waves are a few depths long is the speed at which it is doing the most work on the surface and receiving the most interference back from it.

A foil boat on its take-off run #

The fifth figure puts the answer in the question’s own terms: a foil of 0.3 metres chord, 0.3 metres below the surface, the proportions of a small foiling boat’s main foil. At ten metres a second, cruise, it keeps 0.86 of its deep-water lift, and that is the loss a designer plans for. At 2.7 metres a second it keeps 0.61, and its wave drag over lift squared is 0.089 — at a lift coefficient of half, a wave-drag coefficient of 0.022, about twice the foil’s skin friction.

That speed is inside the take-off run. A foil boat accelerating from rest is carried by its hull until the foils lift it, and the foils pass through their worst lift and largest wave drag on the way. The same foil has a quarter more lift at twice that speed. Part of what foilers call the take-off hump — the speed at which the boat must be driven hardest — is this, and it is the counterpart of the hump a displacement hull meets, which is not at hull speed either.

The depth dependence has a practical side too. At high speed the loss grows as the foil rises towards the surface, so a foil that rises loses lift and sinks back: the depth is self-stabilising, which is what fully submerged foils without active control rely on. Below a chord Froude number of about 0.75, one chord down, the dependence reverses — a foil that rises gains lift and rises further. A foil boat is depth-stable at cruise and depth-unstable on the slow part of its take-off run, before the hull has let go. The obvious remedy is depth, and the first figure prices it: two chords down the dip is 0.76 and the fast value 0.98, so a deeper foil trades a worse structure — a longer strut, more of it in the water — for a take-off that barely notices the surface.

Where the linear theory stops #

The sixth figure is the calculation’s own warning. A foil only a quarter of a chord below the surface, slow, shows a spike in lift of seven times its deep-water value near a chord Froude number of 0.4. The waves that spike would leave behind have a steepness $kA$ of 1.2 at five degrees incidence. No water wave is steeper than about 0.44; beyond that it breaks. The spike is where the linear theory has stopped describing water, and the same check applied to the depths in the first figure keeps every case there below breaking at five degrees, though the half-chord foil’s peak, at 0.385, comes within fifteen per cent of it.

What was checked #

The seventh figure is the ledger: thin-aerofoil theory’s $2\pi\alpha$ unbounded, for one, four and twelve vortices; the lid and the pressure-release image recovered at very low and very high speed, to 0.14 and 0.64 per cent; the single vortex’s wave drag; and the foil’s drag against its waves’ energy.

What the picture cannot show #

Two dimensions. A real foil has a span and tips, and its trailing vortices interact with the surface too; a foil whose span is several times its depth is closest to this picture. The cushion that is there after all found how much a finite span changes a wing’s ground effect, and the change here should be of the same kind.

A flat plate. Camber moves the lift but not, at small incidence, the ratios; a thick foil adds a thickness wave that is not here.

Linear waves. The surface condition is linearised about a flat surface, which the sixth figure shows is safe at a chord’s depth and suspect at a quarter of one. Nonlinear waves break, and a foil close enough to the surface draws air down along its suction side — ventilation — which ends its lift altogether.

No viscosity. Skin friction and the boundary layer are not in the forces; the wave drag is compared with a typical skin-friction coefficient only to give it scale.

The convention the numbers depend on #

The foil has unit chord and incidence $\alpha$, its depth $h$ is measured from the undisturbed surface to the plate, and lift is quoted as a ratio to its deep-water value $2\pi\alpha$ so that it does not depend on the incidence. The chord Froude number is $U/\sqrt{gc}$, the depth Froude number $U/\sqrt{gh}$. The wave drag is divided by the lift coefficient squared so that it too is independent of incidence.

Who found it, and when #

The wave resistance of a submerged vortex is Lamb’s, and the linearised theory of a submerged hydrofoil was developed through the 1950s and 1960s — by Kochin’s methods, and in the discrete-vortex form used here by Plotkin and others — for the fully submerged foils of the fast hydrofoil craft of the time. That the lift passes through a minimum at intermediate Froude numbers is in those calculations; the comparison with the added mass’s overshoot is new here.

Still open: a wing over water that the wing moves #

The surface here is pushed by a foil beneath it. The other half of the question is a wing above water, whose own pressure deforms the surface below it — the ground-effect craft that skims a sea rather than a runway. There, a cushion that changes its physics and the essays around it took the ground as rigid. The next calculation lets it move: the wing’s pressure depresses the water, the depression is a wave that travels, and the question is whether a wing over water keeps the lift gain the ground gives it, and at what speed a wing flying low over the sea pays its own wave drag.