The wall that pushes back

An aircraft in the last few feet before touchdown floats. It refuses to settle, the controls feel different, and pilots are taught to expect it. The explanation usually given is a cushion: air trapped and compressed between the wing and the runway, holding the aeroplane up.

The model here has no compressibility in it whatsoever. The density is a constant, there is nothing to compress, and the effect is still there at three quarters of its full size.

The ground is not a boundary condition #

A wall made by reflection sets out the trick this essay runs on. Rather than requiring that no flow crosses a plane — which is a condition to be enforced, and enforcing it is work — put a mirrored copy of everything on the far side of where the plane would be. The symmetry then guarantees that the vertical velocity vanishes on the plane of symmetry, so the plane is a streamline, and it is a streamline because of the arrangement rather than because anybody insisted.

For a vortex the mirror rule carries a sign flip: the image of a vortex of strength $\Gamma$ at height $h$ is a vortex of strength $-\Gamma$ at $-h$. That single rule is the whole of ground effect.

The check is direct: the vertical velocity is sampled along the ground plane, and the worst value anywhere is zero to the last bit of double precision. Not small — zero, because the two contributions cancel identically rather than nearly.

Why the naive version gets the sign wrong #

There is a version of this argument that is easy to reach for and gives the wrong answer, and it is worth working through because the correction is the physics.

Take the wing to be a single bound vortex of fixed strength $\Gamma$. Its image induces a velocity at the wing, and that velocity is horizontal and upstream: the image slows the local flow. Lift is $\rho \Gamma$ times the local speed, so the lift goes down.

That is the opposite of what happens, and the error is the phrase of fixed strength. The circulation of a wing is not a property it carries about; it is whatever the flow leaving the trailing edge requires. Change the flow around the section and the circulation changes with it, and it is that change — not the change in local speed at fixed circulation — that constitutes ground effect.

What the solver computed, and how it was checked #

The wing is modelled the way thin-aerofoil theory models it when it is discretised: the camber line is divided into sixty panels, each carrying a point vortex at its quarter point, with a flow-tangency condition imposed at each panel’s three-quarter point. That gives sixty equations in sixty unknown vortex strengths, solved directly. The ground is included by adding, for each vortex, a mirrored vortex of opposite sign, whose contribution appears in the same linear system.

The model is calibrated before it is trusted. In free air it must reproduce thin-aerofoil theory, and it does: the flat-plate lift-curve slope comes out at 6.2666 per radian against $2\pi = 6.2832$, a circular-arc section’s zero-lift angle at −4.522° against the theoretical −4.584°, and Glauert’s flap-effectiveness result to within half a per cent. None of those three was used in building the solver.

The sweep is asserted to be monotone — every step closer to the ground must produce more lift — and to reach free air at the far end, within 2%. The far point comes out at 0.9997 of the free-air value, which is the check that the sweep is wide enough to have converged rather than merely long.

What the extra lift is made of #

With the circulation free to adjust, the mechanism runs the other way round from the naive account.

The image vortex induces, at the wing, a velocity with a component that changes the effective incidence of each panel. Because the image is of opposite sign and below, the induced field over the section resembles a small upwash near the leading edge, so the section behaves as though it were at a slightly larger angle. The tangency conditions then demand more circulation, and more circulation is more lift.

Two consequences follow, and both are testable.

The first is that ground effect is not uniform along the chord. The image is closer to the trailing edge than to the leading edge for a section at incidence, so the load distribution changes shape as well as size — the rear of the section gains proportionally more.

The second is what the ground feels.

The pressure on the ground is genuinely raised: that part of the cushion story is correct. What is wrong is the causal order. The pressure is high because the flow between the wing and the ground has been slowed, and it has been slowed because the circulation rose, and the circulation rose because the boundary changed. The cushion is a consequence, not a cause, and calling it the cause predicts — wrongly — that the effect should weaken in a less compressible fluid.

Images stack, and that is why wind tunnels lie #

One wall gives one image. Two parallel walls give an infinite ladder of them: the image in the floor is itself reflected in the ceiling, that reflection is reflected in the floor, and so on forever. A closed rectangular working section gives a two-dimensional lattice.

That is not a curiosity. It is the reason wind-tunnel results need correcting before they can be compared with anything.

A model in a closed tunnel therefore sits in a flow that is not the free-air flow, and the difference has a definite sign. The walls constrain the streamlines that would otherwise spread outward, so the flow over the model is faster than it should be and the measured lift is too high — “solid blockage” and “wake blockage” in the tunnel-correction literature, both derived by exactly this image argument.

The corrections are of order the ratio of the model’s size to the tunnel’s, so they are small for a small model and a small model is exactly what cannot reach the right Reynolds number. That is the same squeeze the whole subject of scale testing lives in, arriving from a different direction.

Downforce is the same calculation, upside down #

Racing cars use this effect harder than aircraft do, and the arithmetic is identical with the sign of the incidence reversed.

An inverted wing close to the ground makes far more downforce than the same wing in free air, and the closer it runs the more it makes. That is why ground-effect cars were fast and why they were banned: the force depends steeply on a ride height that changes with load, bumps and speed, and a car whose downforce disappears when the floor is disturbed is a car that loses grip without warning.

The steepness is visible in the curve above. Going from one chord to a fifth of a chord multiplies the effect by nearly twenty, so the derivative of force with respect to height is large in exactly the region where the height is least controlled.

Where this model stops being about the ground #

Below about a fifth of a chord the numbers stop being trustworthy, and the site refuses to draw them.

A flat vortex sheet with no thickness, no wake displacement and no viscosity returns a lift ratio of 3.2 at a tenth of a chord. The single-vortex version of the same model returns 5.4 at 0.15 chords. Neither is a measurement of anything: as the gap closes, the assumptions that let a wing be represented by a line of vortices on its camber line — that the disturbances are small, that the wake leaves along the chord line, that the flow between wing and ground is not doing something structurally different — fail one after another.

So the assertion refuses a sweep that reaches below 0.2 chords, with a message saying why. The trend above that height is right and matches the classical two-dimensional results; the magnitude at extreme proximity is the model coming apart, and drawing it would be the site claiming a resolution it does not have.

The three-dimensional effect, which is the one that matters #

Almost everything above is two-dimensional, and the ground effect that keeps an aeroplane floating is mostly not.

For a real wing the dominant ground effect is on induced drag rather than on lift. The trailing vortices behind a wing with ends induce a downwash over the wing, which tilts the lift vector backwards; their images below the ground induce an upwash that partly cancels it. Less downwash means less induced drag at the same lift, and in the landing configuration — high lift coefficient, so induced drag dominating — that reduction is large.

An aircraft in ground effect is therefore not so much held up as unable to slow down. The float before touchdown is a drag effect at least as much as a lift effect, and the two-dimensional model here cannot show it at all, because a two-dimensional wing has no trailing vortices and no induced drag to reduce.

What a pilot actually experiences #

The two-dimensional lift increase and the three-dimensional drag reduction produce different symptoms, and separating them explains the handling.

The float. An aeroplane held off in the flare descends into a region of reduced induced drag, so at a fixed thrust and attitude it decelerates more slowly than expected. It keeps flying. This is the drag effect, and it is the dominant one on a wing of ordinary aspect ratio.

The pitch change. Ground effect does not act uniformly along the chord — the load moves aft, as the panel solve shows — and it acts on the tailplane as well as on the wing, at a different height and therefore by a different amount. The net is usually a nose-down pitching moment, which is why the control column comes back further in the flare than the same attitude would need in free air.

The trim change on take-off. The same effects in reverse: an aircraft that lifts off in ground effect and then climbs out of it finds its induced drag rising and its lift falling at fixed attitude. An aircraft that is marginal on power can get airborne in ground effect and then be unable to climb away from it, which is a known and lethal accident category.

None of the three is the cushion. All three follow from a mirror, a circulation free to adjust, and the fact that circulation cannot be carried off the end of a wing.

The same construction with the sign changed, and the effect reverses #

The image rule used throughout — a vortex reflects with its sign flipped — encodes one particular boundary condition: no flow through the plane. There is a second kind of plane in fluid mechanics, and it takes the other sign.

A free surface does not forbid flow through itself; it forbids a pressure difference across itself, since the air above cannot support one. Linearising that condition gives a reflection with the vortex’s sign unchanged rather than reversed — the potential is made antisymmetric where the wall made it symmetric — and every conclusion in this essay inverts with it.

So a hydrofoil running below a water surface loses lift as it approaches it, where a wing above a runway gains. The image now induces the opposite field at the foil, the tangency conditions demand less circulation, and the same panel solve run with one sign flipped would produce a curve falling towards the surface instead of rising towards the ground. That is why hydrofoil craft run their foils at a submergence of roughly a chord: shallower and the lift falls away, deeper and the strut is longer and heavier than it needs to be.

And which sign applies is not fixed by the surface — it is fixed by a Froude number. The linearised free-surface condition contains both terms, weighted by $g/U^2$, and the two limits are the two images. At low speed the surface has time to respond to gravity, deforms hardly at all, and behaves as a rigid wall — the negative image, and ground effect. At high speed gravity cannot keep up, the surface is a constant-pressure boundary, and the positive image applies. The same water, the same foil, opposite behaviour, decided by how fast it is going.

Between the two limits the surface does something neither image can represent: it makes waves, and the foil pays a wave drag for them that no image construction contains. That is the genuine limit of the method here — it converts a boundary condition into an arrangement only when the boundary is a fixed plane, and a free surface is a fixed plane only in its two limits.

Which is a useful thing to have found by pushing on one sign. The image trick is exact and it is not free: it works where the boundary is passive, and a boundary that can move has a life of its own that must be solved rather than mirrored.

What the picture cannot show #

The field pictures are steady, and a landing aeroplane is descending. The rate of approach to the ground introduces an unsteady term the model has no representation for, and near touchdown it is not negligible.

Nothing in the pictures is viscous, so the ground has no boundary layer on it. A real runway does, and in the very small gaps where this model already fails, that layer occupies a noticeable fraction of the gap.

Why the image argument is worth more than the answer #

The number this essay produces — 74% more lift at a fifth of a chord — is worth less than the method that produced it, and it is worth saying why before leaving the subject.

The image construction turns a boundary-value problem into a superposition problem. Solving the first means finding a flow that satisfies a condition on a surface, which in general requires iterating: guess, evaluate the residual on the boundary, correct. Solving the second means adding up known solutions, which requires nothing. The wall is not satisfied to within a tolerance; it is satisfied identically, and the check reports zero rather than a small number.

That is the same economy superposition buys everywhere in ideal flow, and it is available here only because the equations are linear. Every ground-effect result in this essay would be unobtainable in this form for a compressible flow, because the governing equation stops being linear before the Mach number gets interesting, and unobtainable for a viscous one, because the Navier–Stokes equations were never linear at all.

It also generalises further than it looks. Any plane of symmetry can be constructed this way — a wall, a free surface with the sign of the image reversed, the centreline of a symmetric aircraft in a symmetric flow — and each construction converts a condition into an arrangement. The pattern is worth recognising when it appears, because the alternative is always work.

Who found it, and when #

Wieselsberger gave the first proper analysis in 1921, working at Göttingen, and derived the reduction in induced drag by exactly the image argument used here — applied to the trailing vortices rather than to the bound one. The lift increase in two dimensions was worked out over the following decade by several people, Tomotika’s 1933 treatment being the one usually cited.

The practical history is older than the theory. Ground effect was noticed by pilots as soon as there were pilots, and the first aircraft designed deliberately to exploit it — the Soviet ekranoplans of the 1960s — are among the strangest machines ever built, and among the least successful, for reasons that have nothing to do with the aerodynamics being wrong.

Where the ladder goes next #

The rung below is the wall made by reflection, which establishes the image method on a single singularity. This rung applies it to something with a circulation that is free to respond, which is where the interesting behaviour comes from.

Above and beside it, the same panel solver takes a hinge in the camber line and answers what a flap does, and the same image argument applied to trailing vortices rather than bound ones is the three-dimensional effect this essay had to leave out.