A wall made by reflection

There are two ways to make a flow that does not cross a plane. One is to write the condition down and solve the resulting problem. The other is to arrange things so that the condition is satisfied by accident, and then observe that it is.

The second is the method of images, and it is the cheapest trick in the subject.

The rule is one sentence. Reflect every singularity in the plane; reverse the sign of every vortex as it goes; leave the sign of every source alone. The plane is then a streamline, exactly, everywhere, forever.

Why it works #

Consider the mirrored pair by itself and ask what the velocity is at a point on the plane of symmetry.

Each singularity contributes something. The contribution from the real one and the contribution from its image are mirror images of each other, so their components along the plane add and their components across it cancel. That cancellation is the whole method: the flow at every point of the plane is parallel to the plane, and a flow parallel to a surface is a flow that does not cross it, which is exactly what a wall requires of an inviscid fluid.

The sign rules follow from what “mirror image” means for each object. A source pushes fluid outward in all directions, and the mirror image of pushing outward is pushing outward, so the sign stays. A vortex circulates in a definite sense, and a mirror reverses handedness, so the sign flips. Get the vortex sign wrong and the transverse components add instead of cancelling — the flow then crosses the plane at twice the rate it would with no image at all, which is the sort of error that is instantly visible once the right quantity is measured and completely invisible in the picture.

What the solver computed, and how it was checked #

The claim being made is precise enough to be measured, so the figures measure it. Along a line of eighty-one points spanning the plane, the component of velocity perpendicular to it is computed and divided by the local speed.

With the image present, the largest value found is exactly zero — not small, zero, because the cancellation happens between two floating-point numbers of identical magnitude and opposite sign, and that subtraction is exact in binary arithmetic. It is one of the very few places on this site where a computed residual is not a small number but nothing at all.

The gate holds both halves. assertWallIsStreamline passes on the image system at a tolerance of 10⁻¹², and it refuses the same vortex with the image removed, reporting that flow crosses the supposed wall at 16% of the local speed at x = −0.70. A check that only ever passed would be proving nothing; the pair proves the check can see.

That third figure is the one worth looking at twice. Nothing about it is visibly wrong. The streamlines are smooth, the vortex looks like a vortex, and the picture would pass any inspection that consisted of looking at it. What is wrong is a quantity, and the only way to know is to compute it.

Ground effect #

The construction is not a curiosity. Its first real use is what happens to a wing near the ground.

A wing is, to the outside world, a vortex. Bring it close to a plane and the image vortex appears below, circulating the other way, and the image induces a velocity at the wing. Working out the direction is worth doing carefully: for a wing lifting upwards, the image is below and turning the opposite way, and its influence at the wing is an upwash.

Upwash reduces the angle at which the wing meets the air, which reduces the induced drag — and that is ground effect. An aircraft in the last few metres before touchdown is flying in a flow field that has an extra wing in it, and the extra wing is helping.

The size of the effect is set by one ratio, height over span, and it falls away quickly. At a height of one span the induced drag is reduced by a few per cent; at a quarter of a span it is reduced by about half. This is why ground effect is a landing-and-take-off phenomenon rather than a cruise one, and why the aircraft designed to exploit it deliberately are all extremely low-flying.

How strong the image is #

The construction is exact, so the size of its effect can be written down rather than estimated.

A vortex of strength Γ at height h has an image of strength −Γ at depth h, and the velocity the image induces at the real one is Γ/4πh. Everything about ground effect follows from that single expression, and two features of it are worth reading off.

It falls off as one over the height, not one over the square. That is much slower than most people expect, and it is why the effect is still measurable at a height comparable to the span rather than dying away within a chord. It is also why the phenomenon is so hard to fly out of: an aircraft climbing away from the runway loses the benefit gradually and continuously rather than at some threshold.

And it is proportional to the circulation, which means proportional to the lift. A heavily loaded wing gets more of it. This is the source of the standard observation that an aircraft in ground effect feels as though it is being held up — a wing near its stalling angle, carrying the most circulation it ever will, is exactly the case where the image is strongest.

Putting numbers on it needs a little more than one vortex, because the real quantity of interest is the reduction in induced drag and that involves the whole trailing sheet rather than a single line. The standard result of doing it properly is that induced drag is multiplied by a factor that depends only on the ratio of height to span, reaching about 0.9 at half a span and about 0.5 at a quarter. A large aircraft at touchdown, with its wing perhaps six metres up and sixty across, sits at a tenth of a span and is saving something over half its induced drag.

The same arithmetic run the other way explains a hazard. An aircraft taking off in ground effect leaves it as it climbs, and the induced drag it was not paying arrives. If the climb was begun with barely enough thrust margin, the margin disappears at the worst possible moment, and the aircraft settles back. This is a recognised accident category, and its mechanism is one term in one equation that has been known since 1848.

The wind tunnel is the same problem #

A less obvious application, and the one that made the technique compulsory rather than decorative.

A model in a wind tunnel is a model between two walls, and the walls have to be dealt with. Each wall demands an image; but each image is itself a singularity near the other wall, which demands an image of the image, and so on. A closed tunnel therefore requires an infinite doubly-periodic array of images, and the corrections that come out of summing that array are the reason tunnel data has to be corrected before it means anything.

The corrections are not small. A model spanning a substantial fraction of the tunnel measures a lift curve slope noticeably steeper than the same model in free air, because the walls constrain the flow in a way that behaves like extra span. Every published tunnel result has been through this arithmetic, and the arithmetic is the method of images summed to convergence.

The arithmetic has one property worth noticing before leaving it. The images alternate in sign as they recede — the image of a reversed vortex is reversed again — so the series that has to be summed is alternating, and alternating series converge. Had the signs all been the same the corrections would have diverged and the whole technique would have been useless for a closed tunnel. That the tunnel correction exists at all is a consequence of the sign rule in the second paragraph of this essay.

There is a connection here that is worth pausing on, and it is this essay’s surprise. The image system for a wall is mathematically identical to the image system in electrostatics for a charge near a conducting plane — same reflection, same sign rules, same infinite arrays between parallel plates. That is not an analogy. Both problems are Laplace’s equation with a boundary condition on a plane, and once the equation and the condition match, everything that follows matches. The velocity potential is where that identity comes from.

What the wake does when it arrives at the ground #

The images have so far been used to change a force. They also change a motion, and the case where they do is the one that decides how closely aircraft may follow one another.

The pair of vortices left behind a wing induces a downward velocity on itself, so a wake sinks — a few hundred feet in the first minute or two. Near the runway it stops sinking, and the image system says exactly why. Give each vortex its mirror, and each now feels two influences: its partner, pushing the pair downwards, and the images, pushing it sideways. As the pair descends the images strengthen, the descent slows, and the lateral drift grows, until the two vortices are travelling almost parallel to the ground and moving steadily apart. In the inviscid construction they approach the surface asymptotically and never reach it.

So a wake does not conveniently sink into the tarmac and disappear. It flattens out, spreads sideways along the ground at a few metres a second, and lingers — which is the whole reason wake-turbulence separation is quoted in minutes as well as in miles, and why the hazard to a following aircraft is worst near the ground rather than at altitude.

The dangerous case is a light crosswind. It carries both vortices downwind, and if its speed happens to match the outward drift of the upwind vortex, that vortex is held stationary over the runway — the one place a landing aircraft is certain to pass through, at the one height where a roll it cannot correct is unrecoverable.

The model has a limit here worth naming, because it fails in a specific and observable way. A real vortex sliding along a real floor drives a boundary layer that separates and rolls up into vorticity of the opposite sign, and that secondary vorticity lifts the primary vortex back up. Wakes are seen to rebound, and the image system, which has no boundary layer in it, cannot produce that at all.

What the picture cannot show #

The wall in these figures has no thickness, no friction and no boundary layer. It is a plane of symmetry that has been relabelled, and relabelling is all that has happened.

A real wall does two things the image system does not. It sticks — the no-slip condition holds on it, so the fluid in contact with it is at rest, and there is a thin sheared layer next to it that this construction has no representation for. And the flow along it is being decelerated by friction, which changes the pressure distribution over it.

For ground effect this matters less than it sounds, because the ground under an aircraft is moving past at flight speed and there is no boundary layer on it in the aircraft’s frame. It matters a great deal in a wind tunnel, where the floor is stationary and a boundary layer grows along it, which is why tunnels used for ground-effect work have moving belts in the floor.

The figures also cannot show the reason to believe the image exists, because it does not exist. The image vortex is drawn below the wall as though something were there, and nothing is: the fluid below the plane in the second figure is fictional, and only the half above is a claim about anything. Drawing both is what makes the symmetry visible, and a reader who takes the lower half as a prediction has been misled by the very device that makes the method clear.

Where the model stops #

The method works for planes. It works, with more effort, for circles — a vortex outside a cylinder has an image inside it at the inverse point, plus a compensating vortex at the centre — and for a small number of other shapes with enough symmetry.

For a general curved boundary there is no image system, and that is not a gap waiting to be filled. The technique depends on the boundary being a set the reflection maps to itself, and most curves are not. For anything else, the honest route is to distribute singularities over the surface and solve for their strengths, which is what the inverse method inverted amounts to and is what panel methods do.

The other limitation is that the whole construction is inviscid. Images place a boundary condition on the normal velocity and say nothing about the tangential one, which in a real fluid must be zero at a stationary wall. So the method gets the outer flow right and the surface flow wrong, in exactly the way the ideal theory always does.

Who found it, and when #

The method is Kelvin’s, in 1848, and it was electrostatics before it was fluid mechanics — a charge in front of an earthed conducting plane, and the observation that an equal and opposite charge at the mirror position produces the same field. Kelvin was twenty-four and the paper is about three pages.

Its migration into fluid mechanics was almost immediate, because the equation was already known to be the same one. By the 1870s it was standard, and by the time wind tunnels needed wall corrections in the 1910s the mathematics was fifty years old and waiting.

The technique’s reputation has an odd shape. It is taught as a trick, in the sense of something clever that happens to work, and it is not a trick at all: it is a statement that a boundary-value problem with a symmetry can be replaced by a symmetric problem with no boundary. Put that way it is a completely general principle, and the reason it applies so rarely is only that most boundaries are not symmetric about anything.

What Kelvin was actually solving is worth a sentence, because it explains the shape of the argument. The question was what force a charge feels near an earthed conductor, and the difficulty is that the induced surface charge is unknown — it depends on the field, which depends on the charge distribution, which is what is being asked for. The image dissolves the circularity by replacing an unknown distribution with one known point. The fluid version does the same thing to an unknown pressure distribution on a wall.

There is a small historical irony in the migration. The fluid problem is older than the electrical one: people wanted to know what happened to a ship near a bank long before anybody had an earthed conductor to worry about, and the phenomenon — a vessel drawn sideways towards a nearby bank — had been a navigational nuisance for centuries. The mathematics arrived from the newer subject to explain the older one, which is the usual direction of travel for this equation.

Where the ladder goes next #

Next rungs on this anchor: the circle theorem, which gives the image system for a cylindrical boundary and turns any external flow into a flow round a circle; the infinite array between parallel walls, and the tunnel corrections that come out of summing it; a free surface, which reflects with the opposite sign convention from a solid one and is why a submarine near the surface behaves quite differently from one near the sea bed; and the ground-effect calculation done properly with a lifting line rather than a single vortex.

Then across to bodies made out of nothing, which is the same superposition used to make a body rather than a boundary, and to the starting vortex, where a pair of opposite vortices turns up again for a completely different reason.