Free to rise and fall, a ground-effect craft rides its own cushion over the swell

A swell lifts a ground-effect wing more at its crests than it takes in its troughs held a five-metre wing at 1.2 metres over a head swell and let the sea move under it. The gap closed over each crest and opened over each trough, and because ground effect strengthens faster as the gap closes than it weakens as the gap opens, the lift rose more at the crests than it fell in the troughs: the swing was convex, and its mean was above the calm-water lift. The waves’ own air, moving with their orbital motion, added a swing in quadrature. And a craft that climbed out of the cushion to escape the swell spent the cushion’s benefit.

Every number there held the wing still. It ended on the obvious objection: a real craft is not held. It has a mass, its weight sits on its lift, and ground effect gives it a height stiffness of its own — the lift rises when it sinks and falls when it rises. The question was whether a craft free to move follows the long swell and flies level through the short, as a foil in a random sea follows it up to its peak found a foiling boat should, and whether the stiffness, which grows as the gap closes, makes the following automatic or makes the motion unstable near the crests.

The craft is an oscillator whose spring is its own cushion. It follows long swells, flies level over short ones, and is well damped, so there is no instability to find. The convex swing the held wing felt becomes height: a free craft over a short swell rides above its calm height, and over every swell it keeps more clearance than the held wing would.

A wing with a weight on it #

The wing is the earlier essay’s: a flat plate of five metres’ chord at four degrees and fifty metres a second, its leading edge 1.2 metres above the mean sea, solved as forty lumped vortices each mirrored in the surface directly beneath it — the image system the wall that pushes back built, applied panel by panel to a moving sea. The swell runs against it, half a metre in amplitude unless stated, and the air above the waves carries their potential flow. In calm water at its height the wing has a lift coefficient of 0.684, and its weight per metre of span is set equal to that lift: about 534 kilograms a metre.

Now the wing is free to move vertically. Its attitude is held, as if a perfect autopilot or a long tail kept its pitch, which is the stated simplification; heave is the motion the gap acts on. The equation is Newton’s: mass times vertical acceleration equals the lift’s excess over the weight. The lift depends on the gap, which now changes with both the sea and the craft, and on the angle of attack, which the craft’s own vertical speed reduces — a wing rising at a metre a second meets the air a little more nose-down, by that speed over the flight speed. The lift coefficient is computed once per swell on a grid of heights and phases, at two incidences for its slope, and interpolated; the interpolated value matches a direct solve to one part in a thousand.

The cushion is a spring #

Two numbers decide the motion, and both come from the calm wing. The gap law’s slope is −0.298 per metre: lowering the wing by a centimetre raises its lift coefficient by 0.003. Times the dynamic pressure and the chord that is a stiffness, a restoring force per metre of descent, and with the mass it gives a natural frequency of 2.07 radians a second, a period of three seconds. The craft’s vertical speed changes its incidence, and the lift slope of 9.9 per radian turns that into a force proportional to the speed: a damper, worth 0.69 of critical damping.

That is the whole of the craft’s dynamics in calm air: a mass on a spring, the spring being the cushion and the damper being the wing’s own lift slope. It is stable for the same reason ground effect is said to make a craft fly “on rails” — a craft that sinks gains lift and one that rises loses it — and well damped, because a wing’s lift slope is large and its vertical speed changes its incidence directly. Neither number is special to this craft; both follow from flying low with a wing, and a craft flown higher, or slower, would have a softer spring and less damping.

Long swells lift it, short ones pass under it #

The first figure is the response. In head seas the craft meets each swell at its encounter frequency, the wave’s own frequency plus the rate at which the craft runs through its crests, and the figure plots the heave, as a fraction of the swell’s amplitude, against that frequency over the craft’s own. A 30-metre swell is met nearly six times faster than the craft’s natural frequency, and the craft heaves 4 per cent of the swell’s amplitude: it flies level and the swell passes under it. An 800-metre swell is met at a third of the natural frequency, and the craft heaves the swell’s whole amplitude: it rides up and down with the sea. Near resonance, a 200-metre swell, it heaves 72 per cent.

The dashed curve is the textbook oscillator driven through its spring — the motion of a mass whose spring is attached to a moving ground — with the damping just computed. At resonance it predicts $1/2\zeta$, 0.73, and the craft heaves 0.723; at a third of resonance it predicts 0.97, and the craft 1.00. The craft is that oscillator. Its ground moves with the sea under it, and the cushion transmits the motion as a spring would: fully when the sea moves slowly, hardly at all when it moves fast. Above resonance the craft heaves a little more than the curve, a few hundredths of the swell, because the waves’ own air pushes on it in addition to the gap — the quadrature swing the earlier essay found.

The two limits answer the question the foiling boat raised. A ground-effect craft follows the long swell and flies level over the short one without any control, simply because its cushion is a spring of a definite stiffness and its mass sets a frequency against which the swells are long or short.

It never comes as close as the held wing #

The second figure is the clearance, which is what matters to anyone flying one. Held at its calm height over a half-metre swell, the wing’s trailing edge comes within 0.358 metres of the crests whatever the swell’s length. Free, its least gap is never that small. Over a 30-metre swell it is 0.48 metres: the craft is flying nearly level, but higher than its calm height. Over an 800-metre swell it is 0.74 metres: the craft follows the sea, and the gap barely changes. In between it dips to 0.42 metres over a 130-metre swell, met at 1.4 times the natural frequency.

That dip is where the oscillator is out of step with the sea. Just above resonance the craft’s motion lags the swell by about a third of a cycle: it is still settling from the last trough as the next crest arrives, and the two motions add in the gap rather than cancelling. Below resonance the craft rises with the crest; well above it the craft does not move and the crest arrives at a craft that has risen on average. Near resonance neither protection is complete. The claim that a ground-effect craft is most at risk over the shortest swells, because it cannot follow them, has the wrong swell: the short ones are where its mean height protects it, and the dangerous ones are those it meets near its own frequency.

Through three encounters #

The third figure draws the motion itself, at the dial’s first setting a 150-metre swell, near resonance. The craft heaves about half the swell’s amplitude, lagging it, about a mean almost ten centimetres above its calm height; the least gap under the wing is 0.424 metres. The dial changes the swell. Over a 50-metre swell the craft’s line is almost straight, above its calm height, while the sea rises and falls beneath it. Over a 400-metre swell the craft’s line follows the sea’s, nearly in step, a little higher.

The convex swing becomes height #

The fourth figure is the earlier essay’s convexity turned into motion. A held wing over a swell has a mean lift above its calm value, because the crests add more than the troughs take away. A free craft cannot carry more lift than its weight on average, so it climbs until the mean lift over the swell equals the weight again — to a height where the cushion is weaker and its convex swing is centred on the calm lift. Over a 30-metre swell of half a metre that height is 12.3 centimetres above the calm one. Over an 800-metre swell, which the craft follows, the gap hardly changes and nor does the mean: 0.6 centimetres.

The rise grows roughly as the square of the swell’s amplitude — 4.7 centimetres at 0.3 metres, 12.3 at 0.5 and 22.8 at 0.7 — which is the signature of a convex law: a swing of amplitude $a$ about a curved law shifts its mean by an amount proportional to $a^2$ times the curvature. The craft reads the curvature of its own gap law and moves to where that curvature’s push is balanced. It is a small, automatic increase in clearance exactly where the crests come fastest, and nothing about it needs a pilot.

What the ride feels #

The fifth figure is the cost of flying level. The dashed curve is the held wing’s lift swing as a share of its weight — the acceleration it would give the craft if all of it went into motion. Over an 800-metre swell the free craft follows the sea and accelerates only 0.016 g rms, against a lift swing of 0.19: the cushion’s change is absorbed by changing height, slowly. Over a 30-metre swell the craft flies level and accelerates 0.22 g rms against 0.26: above resonance a mass on a spring cannot avoid the force, it can only avoid moving far, and a small motion at a high frequency is a large acceleration. Over a 150-metre swell, near resonance, 0.15 g against 0.20.

So flying level over a short swell is not a smooth ride. The craft’s height is steady, but its occupants feel nearly the whole of the cushion’s swing at the encounter frequency, several times a second. A foiling boat’s ride control, which the foil essay studied, faces the same choice in the same terms: a stiff, slow-following craft rides the long waves smoothly and is jolted by the short ones, and no stiffness gives both.

The same spring, under other names #

The cushion’s spring is the oldest idea in ground-effect flight, and it has been explained wrongly as often as it has been used. The cushion that is not there showed that no pocket of compressed air holds the wing up: the extra lift is the image vortex’s, a change in the circulation, not a rise in pressure trapped underneath. The cushion that is there after all found the regime, very close to the surface, in which the trapped-air picture becomes partly right. The spring computed here belongs to the first of those, at a gap of a quarter of the chord: its stiffness is the slope of the image system’s lift with height, and no air is being compressed.

A hydrofoil boat has the opposite spring. A hydrofoil loses most of its lift on the way up, as the free surface above it approaches, so a foil that rises loses lift and one that sinks gains it — the same sign of stiffness, for the image of a different boundary. Both craft are therefore self-levelling in heave, and both, as the oscillator says, follow long waves and fly level over short ones. What differs between them is where on the oscillator’s curve a given sea falls, since each craft’s spring, mass and speed set its own natural frequency, and the same swell can be long for one and short for the other.

The craft’s mass should, strictly, include some air. A plate accelerating vertically carries the fluid around it with it, and the borrowed mass the boundary decides found that a nearby wall increases that borrowed mass. For a five-metre plate in air it is of the order of $\rho\pi(c/2)^2$, about 24 kilograms a metre in free air, a twentieth of the craft’s 534, and somewhat more near the surface; including it would lower the natural frequency by a few per cent and move every curve here slightly to the left, and nothing else.

The convention: heave, encounter and gap #

Heave is the craft’s vertical displacement from its calm-water height, measured at the leading edge, as a fraction of the swell’s amplitude when given as a ratio. The encounter frequency is the swell’s own frequency plus its wavenumber times the craft’s speed, for head seas. The gap is the least vertical distance between any point of the wing and the sea beneath it over a steady encounter — the trailing edge’s, since the wing is pitched nose-up. A steady encounter is the last third of a run of at least thirty encounters, or twelve natural periods, from rest at the calm height. Accelerations are root-mean-square, in units of the acceleration due to gravity.

What the calculation leaves out #

Pitch. The craft’s attitude is held. A real ground-effect craft also pitches, and its pitching stability in ground effect is the subject’s hardest problem: the wing’s centre of lift moves aft as the gap closes, so a craft that is stable in height can be unstable in pitch without a large tailplane far behind it. Freeing the pitch couples it to the heave computed here.

The wake’s memory. The lift is quasi-steady. At the shortest swells the reduced frequency, the encounter frequency times half the chord over the speed, is 0.6, and the wing’s shed vorticity would delay and reduce the lift’s response; the earlier essay reported it beside every result, and it matters most where the craft flies level.

Two dimensions and a regular sea. The wing has no span, so its ground effect is the two-dimensional section’s, stronger than a finite wing’s at the same height; and the sea is a single regular swell, where a real sea has a spectrum of lengths and directions.

Spray, and the surface’s response. The water does not respond to the wing’s pressure here; a wing over the sea barely touches it found the dent it makes to be millimetres, which justifies the omission at this height.

Who flew them #

Large ground-effect craft were built and flown in the Soviet Union from the 1960s, the Caspian Sea Monster among them, and smaller ones since in Germany, the United States, Australia and China; their designers learned, often expensively, that height stability is easy and pitch stability hard. The analysis of a craft’s heave over waves as a spring-mass system with the gap law as the spring is standard in the field’s design literature. The particular numbers here — the natural frequency, the damping, and the clearance’s least value near resonance — are this calculation’s, for the stated wing and swell.

How the craft was checked #

On a calm sea the free craft, started at its calm height, stays there to a third of a millimetre over the run, which checks that its weight and its lift balance in the tabulated law as they do in the direct one; the table’s lift matches a direct solve to one part in a thousand at a height and phase between its grid points. The heave response matches the linear oscillator built from the calm wing’s own stiffness and damping to within the waves’ air’s small addition. The underlying swell model is the earlier essay’s, checked there against the calm wing and against a very long swell acting as a raised floor.

Still open: the pitch that ground effect destabilises #

The heave here is stable because the cushion is a spring; the pitch need not be, because the cushion also moves the wing’s centre of pressure. As the gap closes, more of the lift comes from the wing’s rear, where the image system acts most strongly on the trailing edge, so a nose-down disturbance that lowers the trailing edge can shift the lift aft and pitch the craft further nose-down. The next calculation would free the craft in pitch as well as heave, give it a tailplane of stated area and arm out of ground effect, and ask what tail volume keeps the coupled motion stable over the swells drawn here, and whether the swell near the heave’s natural frequency is also where the coupled motion first loses stability.