The cushion that is there after all

The rung below this one disposes of the cushion. A wing near the ground makes more lift, the mechanism is an image vortex, there is no mattress of compressed air anywhere in the physics, and the word is a metaphor for a boundary condition.

It also states, in its closing paragraph, exactly where its own argument runs out: the unsteady case, a wing descending towards a plane, where the added-mass term appears and where the cushion becomes a genuine part of the physics rather than a metaphor for it.

That is the case a wheel touches down in, and it is the case a helicopter hovers in.

The term the steady argument has none of #

A body accelerating through a fluid needs a force beyond its own mass times its acceleration, because the fluid has to be accelerated too. That surplus is the mass a body has to borrow, and in free air it is exactly $\rho\pi c^2/4$ per unit span for a plate of chord $c$ moving broadside — the mass of the circle the chord spans.

Near a wall it is larger, and the reason is the one the word cushion was reaching for.

Fluid in the gap between the plate and the wall cannot simply move aside; it must leave sideways, through a passage of height $h$, at a speed of order $(c/h)$ times the plate’s own. Its kinetic energy therefore goes as $(c/h)^2$, and the added mass with it. As the gap closes the borrowed mass diverges.

That is a cushion in the ordinary sense of the word — fluid that has to be got out of the way and resists being — and it is entirely absent from the steady problem, where nothing is accelerating and nothing is borrowed.

What it costs a wing arriving #

A wing entering ground effect at a metre and a half a second, with the steady lift gain the rung below computes included in both cases, arrives at the ground four per cent more slowly with the near-wall added mass in the model than without it.

Four per cent is a modest number and the calculation is worth having anyway, for two reasons.

The steady lift gain is in both curves, so none of that four per cent is the image-vortex effect. Running the same descent with the lift gain removed from both leaves the same gap between them, which is the check that separates the two mechanisms rather than assuming they are separable.

And the term grows without limit while the lift gain does not. The steady lift gain saturates: a wing very close to the ground has an image very close to it, and the lift approaches a finite value. The added mass does not saturate — it goes as $1/h$ — so at a small enough gap the unsteady term is the larger of the two, and the crossover is inside the range a wheel actually passes through.

The crossover, computed #

The claim above that the unsteady term overtakes the steady one deserves a number rather than an assertion, and the two have different shapes so the crossing is well defined.

The steady lift gain saturates. As the gap closes the image approaches the wing, the effective camber change approaches a limit, and the extra lift approaches a finite fraction of the free-air value — thirty-odd per cent for the model drawn in the rung below.

The added-mass reaction does not saturate. It is $V^2\,dm_a/dh$, and with $m_a$ going as $1/h$ that reaction goes as $V^2/h^2$ — so it grows with the square of the descent rate and with the square of the closeness, and there is no gap at which it stops growing.

Both of those dependences matter and they matter differently. A wing settling gently has a small $V$ — and how gently is a reduced frequency, which is what decides whether anything is quasi-steady —, so the unsteady term is small at every gap and the steady one dominates all the way down. A wing arriving fast has a large $V$, and the unsteady term overtakes the steady one at a gap that falls as the reciprocal of the descent rate.

That is the honest form of the pilot’s intuition. The cushion is felt when the arrival is firm and not when it is gentle, which is exactly what a term proportional to $V^2$ does and exactly what a steady lift gain does not — and it is a testable difference between the two accounts that requires no instrumentation beyond a rate of descent.

What the same term does on the way up #

An added mass opposes acceleration in either direction, and the consequence for departure is worth naming because it is the opposite of what the word cushion suggests.

A wing accelerating away from the ground has to accelerate the gap fluid too, so the added-mass term resists the climb exactly as it resisted the descent. There is no assistance on the way up. A cushion in the everyday sense pushes back and gives some of it back; this one takes energy on the way in and takes it again on the way out, because it is an inertia rather than a spring.

That is the sharpest test between the two pictures available, and it is one anybody can perform. A compressed mattress of air would return energy: a body pressed onto it and released would bounce. A wing in ground effect does not bounce off the cushion, and the reason is that the term is $m_a\,\dot{V}$ rather than $k\,x$ — a mass, not a spring.

There is one qualification and it belongs to the viscous case rather than to this one. A squeeze film of oil does return some energy, because the film’s pressure builds as the surfaces approach and relaxes as they separate; but the return is incomplete and the film is dissipative overall. Air at these gaps is neither — it has too little viscosity to build the film and too little compressibility to store the energy.

Three mechanisms, one phrase #

What has happened over these two rungs is that “ground effect” has turned out to name three different things, and it is worth setting them beside each other because they act at different times and in different directions.

The image system. A steady wing near a plane has a mirror-image circulation that changes its effective incidence and induced drag. It needs forward speed, it does not need acceleration, and it is what the rung below computes. No air is carried and no air is compressed.

The added mass. An accelerating body near a plane borrows more fluid than in the open, because the gap flow is fast. It needs acceleration, it does not need forward speed, and it opposes whatever the body is doing. This one is genuinely about carried air.

And the rotor’s. A hovering rotor near the floor has its wake constrained and its induced velocity reduced, so it makes more thrust for the same power. It needs neither forward speed nor acceleration.

The rotor’s is the one a pilot feels most sharply and it is the one furthest from the rung below’s argument. A helicopter can hover in ground effect on power it cannot hover on out of it, the difference is several per cent of thrust at a rotor radius, and the transition as the machine climbs through is a real event with a name.

Its mechanism is momentum theory rather than potential flow, so it is a control-volume argument of the kind the applied field is built on rather than an image argument at all. The rotor’s own downwash is turned by the floor, the wake’s expansion is prevented, and by the same disc argument that prices every rotor a smaller induced velocity at a given thrust means less induced power. The expression drawn is Cheeseman and Bennett’s correlation, and it is drawn in the colour this site reserves for a claim taken from elsewhere.

The three tested against one measurement #

Since three mechanisms share a name, it is worth asking which of them a given observation is evidence for — and there is a measurement that separates all three at once.

Fly level at a fixed height and measure the lift. Only the image system contributes: nothing is accelerating, so the added mass borrows nothing, and there is no rotor. A lift increment here is evidence for the boundary condition and for nothing else.

Descend at a fixed rate through the same heights and measure the force. Now the added mass contributes too, and the difference between this and the level case is the unsteady term alone. Double the descent rate and the difference should quadruple, because the reaction goes as $V^2$; the image system’s contribution should not move at all.

Hover a rotor at the same heights and measure the thrust at fixed power. Neither of the other two is present. Whatever is measured is the wake constraint.

Those three experiments are separable, they are not difficult, and the vocabulary has survived anyway — which says something about how a phrase attaches to a phenomenon. The name went to the sensation, the sensation belongs to the second experiment, and the explanation in the textbook belongs to the first.

There is a fourth case worth naming because it is the one where all three are present and nothing can be separated: a helicopter making a running landing. It has forward speed, so the image system acts; it is descending, so the added mass acts; and it has a rotor near the floor, so the wake constraint acts. That is the flight condition the phrase is used about most often and it is the one in which it means least.

Why the metaphor survives, which is a better story now #

The rung below concludes that the cushion is a metaphor. With this rung in hand the conclusion can be sharpened, and the sharpening is more interesting than the refutation.

The word is right about the phenomenon a person experiences and wrong about the one usually being discussed. What a pilot notices in the flare is a wing that becomes reluctant to descend the last few feet — and that reluctance is genuinely a resistance to motion towards the ground, genuinely proportional to how fast the descent is, and genuinely about fluid that must be pushed out of a narrowing gap. Every part of the intuition is correct.

What the intuition is not about is the lift of a wing in level flight near the ground, which is the quantity the textbook explanation attaches it to and which the image system explains completely.

So the mistake is a misattribution rather than an invention, which puts it in this field’s commonest category: something real given a job it does not do — except that here the real thing and the job are adjacent, both are called ground effect, and both happen within a few feet of the runway ten seconds apart.

What a designer of a ground-effect vehicle is using #

The three mechanisms above are all about a body that is passing near the ground. There is a class of machine that stays there, and it is worth saying which of the three it lives on because the answer is not the obvious one.

A wing-in-ground-effect craft — an ekranoplan, a lift-generating hull skimming a metre above water — cruises steadily, so the added mass borrows nothing and the rotor mechanism is absent. It runs entirely on the image system, which is the mechanism the rung below computes and the one with no cushion in it.

That is worth stating because the vehicles are the thing most people picture when the phrase is used, and they are the case in which the metaphor is most completely wrong. What they exploit is a reduction in induced drag: the image cancels part of the wing’s own downwash, the effective aspect ratio rises, and the lift-to-drag ratio at cruise can be twice a comparable aircraft’s. There is no air being compressed under them and there is no cushion.

A hovercraft is the opposite case and the only vehicle in this essay for which the everyday word is correct: it genuinely maintains a plenum of air at a pressure above ambient, held in by a skirt, and the pressure times the area is the weight. That is not aerodynamics in the sense of this collection at all — there is no wing, no circulation and no wake — it is a pressure vessel with a leak, and its power is what the leak costs.

So the phrase covers a mechanism that is a boundary condition, one that is an inertia, one that is a wake constraint, and one that is a literal cushion — and only the last of the four is what the word means, and it is the one that is not ground effect.

What the picture cannot show #

The added mass near the wall is a blend rather than a solution. The free-air value is exact and the squeeze-film term is exact as the gap closes, and adding them is a construction that has both limits right and is not a solution of anything in between. A real potential-flow calculation of a plate near a plane would give a single expression; this gives two that are each right where the other is not. That calculation has since been made for the same plate, and the blend is within about ten per cent of it at every gap drawn here.

The descent is a lumped model. One degree of freedom, no pitch, no wing shape, and a lift that is a prescribed function of height rather than a solved one. It compares two cases that differ in one term, which is what it is for.

The rotor expression is borrowed. Cheeseman and Bennett’s correlation is a source-image argument on the inflow, fitted; nothing here derives it and nothing here solves a rotor. It refuses to evaluate below about half a rotor radius, because its own denominator approaches zero there and what it returns stops being a thrust.

And viscosity is absent throughout. A real gap at a small height has a boundary layer occupying a substantial fraction of it, and the squeeze-film term computed inviscidly is the wrong limit once that happens — the viscous squeeze film is a different and much larger force, and it is a lubrication problem rather than an inertial one.

The assertion behind these figures makes the separation the essay depends on: the descent is run twice, once with the near-wall growth and once without, and then twice more with the steady lift gain removed from both — so a reader can be sure that the difference attributed to added mass is not the steady effect wearing a different label. It also requires the added mass to exceed the free-air value at every gap and to diverge, and it refuses a plate resting on the plane.

Who computed it, and when #

The added mass of bodies near boundaries is classical hydrodynamics and is in Lamb; the squeeze-film limit belongs to lubrication theory and to Reynolds. Neither was written about aircraft.

The aeronautical version arrived with the helicopter, and it arrived as a measurement rather than as a theory: rotors in the hover were found to lift more near the ground, by amounts that mattered to whether a machine could take off with a given load, long before anybody had an expression for it. Cheeseman and Bennett’s correlation is from 1955 and it remains what a performance manual uses.

The fixed-wing side is later still and is mostly about the flare, where the unsteady term and the steady one act together over a few seconds and are difficult to separate in flight data — which is part of why the two have stayed conflated in the vocabulary. A separation that is hard to make experimentally is a separation that will not be made linguistically, and the phrase “ground effect” has been doing three jobs for seventy years for that reason.

Where the ladder goes next #

The rung above is the case this one keeps deferring: the viscous squeeze film, where the gap is small enough that the boundary layers fill it and the inertial term computed here is replaced by a much larger viscous one. That is the regime a body actually touching down enters, its force goes as $\mu c^3 V/h^3$ rather than as $\rho c^3 V^2/h^2$ — the same cube of a gap a bearing film’s load depends on, and the two have different powers of the gap and of the speed — so which one dominates is a Reynolds number of the gap flow, and computing where the crossover sits is a rung with a real number in it.

The one beside it is the tunnel case the rung below already named. The same image system with the wall further away and all round is a wind tunnel, and the sign of the interference reverses if the boundary is an open jet rather than a solid wall — which is as clean a demonstration as the subject offers that the effect belongs to the boundary condition rather than to the air. That is a steady problem and it belongs to the ladder below rather than above; what would make it a rung here is asking the same question of the unsteady terms, where a body accelerating in a closed tunnel borrows more mass and one in an open jet borrows less.