The borrowed mass the boundary decides

Three essays have now followed the cushion under a body near the ground. The cushion that is not there disposed of it for a wing in steady flight; the cushion that is there after all found it again in the added mass of a body accelerating towards a plane; and a cushion that changes its physics found where that inertial cushion hands over to a viscous one.

All three were about a solid wall, and all three reached for the same picture: fluid trapped between a body and a surface, resisting being got out of the way. The picture invites a generalisation — that nearness itself is what makes the cushion, and that any boundary close to an accelerating body will make it harder to accelerate.

That generalisation is false, and by a sign rather than by a margin.

What a boundary is allowed to say #

The flow a body sets moving in an ideal fluid has a velocity potential that satisfies Laplace’s equation, and Laplace’s equation takes one condition on each boundary — the property that makes the ideal flow’s solutions exact and unique once the boundaries are stated. Which condition is the whole of the difference between boundaries, and there are two that matter.

A solid wall says that nothing passes through it: the velocity normal to the plane is zero, $\partial\phi/\partial n = 0$. The potential that obeys that is the body’s own flow plus a mirror image of it on the other side of the plane, with the same sign — every source reflected as a source — so that the two normal velocities cancel on the plane by symmetry.

A boundary held at constant pressure says something different. Linearised, the unsteady pressure in a potential flow is $p' = -\rho\,\partial\phi/\partial t$, and a boundary on which the pressure never changes is one on which $\phi$ never changes — so a flow started from rest has $\phi = 0$ there. The image that obeys that has the opposite sign, every source reflected as a sink, so that the two potentials cancel on the plane.

Two real boundaries are of the second kind, and both are ordinary. A free water surface struck suddenly — the underside of a hull as it hits the sea, a plate slapped onto a pond — is held at atmospheric pressure, and for a motion too quick for gravity waves to form it behaves exactly as $\phi = 0$. The edge of an open-jet wind tunnel, where the test section is a free stream surrounded by still air in a chamber, is also at constant pressure, which is the reason its steady corrections have the opposite sign to a closed tunnel’s.

The convention is worth naming because the result depends on it entirely. Everything below is an impulsive, high-frequency, gravity-free potential flow. That is the right model for an impact and for a start from rest, and it is not the right model for a body oscillating slowly under a free surface, where waves carry energy away and the answer changes with the frequency.

The plate, twice #

The plate of the previous essays, closing broadside on a boundary, gives the two answers side by side.

Against a wall the added mass is 1.97 times its free value at a tenth of a chord and 3.97 times at 0.035 chords, and it grows as the inverse of the gap, since the fluid in the gap has to be squeezed out sideways faster and faster. That is the inertial cushion, exactly as before.

Against a free boundary it is 0.677 at a tenth of a chord and 0.584 at 0.035 chords, and it falls towards one half. The limit is exact and has a one-line reason. A plate lying on a free surface and set moving downwards disturbs only the fluid below it; the fluid above has been replaced by the constant-pressure boundary. The flow below is precisely the lower half of the flow a plate sets up in an unbounded fluid — the image makes it so — and so it carries exactly half the kinetic energy, and the plate borrows exactly half the mass.

So the same plate, closing on a boundary at the same speed, borrows three times as much from a wall as from a free surface at a tenth of a chord, and seven times as much at 0.035 chords, and the ratio grows without limit. Nearness is common to both. What differs is what the boundary will let the fluid do.

Watching the fluid choose #

Drawn as streamlines, the two flows show the mechanism rather than the number. Ahead of a circle closing on a wall, the fluid cannot continue downwards, so it turns and runs along the wall through the narrowing gap, and it has to run fast to get out of the way in time. Fast fluid is kinetic energy, and kinetic energy the circle has to supply is mass it has to accelerate.

Ahead of the same circle closing on a free boundary, nothing turns. The fluid continues straight through the plane, because the plane holds no position and resists no crossing — it holds only its pressure. The fluid moves less than it would with no boundary at all, since the half of the flow that would have been below the plane is not there to be fed, and so the circle borrows less than its free mass: 0.805 of it at half a radius, against 1.259 from the wall.

The two conditions, read back off the flow #

A potential flow built from images is only as good as its claim to satisfy the boundary condition, so it is worth reading the condition back off the computed field rather than trusting the construction.

Along the wall, the velocity through the plane is exactly zero at every point, because the image is a mirror of the body with the same sign and the two normal velocities cancel identically rather than approximately. The velocity along the wall is not zero at all: the fluid slides sideways beneath the circle at up to 0.703 of the circle’s own speed. Along the free boundary the potential is exactly zero, for the same symmetry with the opposite sign, and the fluid crosses the plane at up to 0.752 of the circle’s speed.

Each image satisfies its own condition exactly and breaks the other’s plainly, and the two violations are of comparable size. That is the cleanest statement available of why the answers differ: neither boundary is a weaker version of the other. A wall lets fluid slide and forbids it to cross; a free boundary lets it cross and forbids its potential to change. A flow cannot have both freedoms, and the borrowed mass is a report of which one it was given.

A direction that drops out #

The plate moved towards its boundary. A circle can move towards the boundary or along it, and in a three-dimensional world those would be different problems with different answers.

For a circle one radius clear of a wall the added mass is 1.1351 times $\rho\pi a^2$ moving towards the wall, and 1.1351 moving along it — the two agree to every figure computed, fifteen of them. Near a free boundary the circle borrows 0.8836 in both directions. At a tenth of a radius the wall gives 1.636 and the free boundary 0.686.

In a plane flow, a circle’s borrowed mass near a straight boundary does not depend on which way the circle moves. That is not a coincidence of the numbers chosen. Two-dimensional potential flow is the real and imaginary parts of one analytic function of $x + iy$, and a translating circle’s flow is the same function multiplied by a complex constant whose argument is the direction of motion; the kinetic energy depends on the constant’s modulus and not its argument, so it cannot know the direction. The image changes nothing about that, since reflecting in a straight line preserves it.

A sphere near a wall has no such structure, and it borrows measurably different amounts moving towards the wall and along it. The equality is a privilege of the plane, of the same kind that lets a two-dimensional aerofoil be mapped from a circle — and it is a reminder that a two-dimensional result can be exactly right and still say nothing about the three-dimensional body it was meant to stand for.

A body in a tunnel #

The unsteady version of the wind-tunnel problem is the one the cushion that is there after all named as a question: does a body accelerating in a closed tunnel borrow more mass, and one in an open jet less? With two boundaries the images reflect back and forth between them without end, each carrying the boundary’s sign once per reflection, and the series converges.

It does, and by more than the single boundary suggested. A circle filling most of a closed tunnel — a half-width of 1.25 radii, a quarter of a radius clear of each wall — borrows 3.25 times its free mass moving along the tunnel and 1.75 moving across it. The same circle in an open jet of the same width borrows 0.59 along and 0.32 across. At a quarter-radius clearance the closed and the open tunnel differ by a factor of five and a half.

The direction matters here where it did not for a single boundary, because two boundaries break the symmetry the plane’s analytic structure relied on: moving along the tunnel, the fluid displaced ahead must pass the body through the gaps beside it; moving across, it must be pushed towards one wall and drawn from the other.

At a half-width of five radii every case is within seven per cent of free air — 1.067 and 1.034 for the closed tunnel, 0.968 and 0.938 for the open jet. That is the unsteady companion to the steady blockage correction, and it has the same property: the sign of the interference is set by the kind of boundary, and a closed tunnel and an open jet err in opposite directions by amounts that are not small for a large model. An oscillating model in a closed tunnel, measured to learn its unsteady loads, carries a borrowed mass the aircraft will not have; the same model in an open jet carries less than the aircraft will.

The mass a hull brings to the water #

The free-surface limit of one half has a practical form older than any of these calculations.

A flat-bottomed hull or a seaplane’s float striking the sea is a plate arriving at a free surface at speed. At the instant of contact the water it sets moving is the half-space below it, and the mass it borrows is exactly half of what the same plate would borrow in the open — the limit the plate figure tends to. As the hull sinks in, the wetted width grows, the borrowed mass grows with it, and the impact force is the rate at which that borrowed mass is being set moving.

That is the added-mass account of slamming, von Kármán’s for seaplane floats in 1929 and Wagner’s refinement of 1932, and it is still how a hull’s slamming load is first estimated. What it takes from this essay is the sign. A hull meeting water borrows less than the same hull submerged, and a model tested in a tank with a solid floor too close beneath it borrows more — so an impact test that is too shallow overstates the load by exactly the kind of interference the tunnel figure shows.

Where the borrowed momentum goes #

A body set moving from rest gives the fluid an impulse equal to its added mass times its speed, and that impulse has to be delivered somewhere. The two boundaries deliver it in opposite ways, and following it explains the numbers better than the images do.

Against a wall, the impulse is carried by pressure. The fluid ahead of a body started towards a wall cannot move through it, so in the instant of the start the pressure along the wall rises, over a patch a few body sizes wide, and the wall itself receives a push. The impulsive pressure is $\rho\phi$ on the plane, and the image that makes $\phi$ symmetric about the wall doubles it there. The fluid in the gap is squeezed out sideways by that pressure, fast, and its kinetic energy is the extra borrowed mass. A body started towards a wall is, in a precise sense, pushing the wall away with the fluid, and paying for the fluid that has to escape.

At a free boundary, the impulse is carried by motion. The pressure there is not allowed to change, so the boundary receives no push at all. What it does instead is move: the fluid ahead of the body crosses the plane, and a free surface bulges upward in the instant after a body beneath it is started towards it — the beginning of a splash. The fluid that would have been beyond the plane, and would have had to be set moving, is simply not there. Less fluid is accelerated, so less mass is borrowed.

That accounting makes the one-half limit feel less like a coincidence of images. A plate lying on a free surface has no fluid on one side of it, and the boundary does not even resist the fluid on the other side being pushed through the plane where the plate is not. Every piece of the borrowed mass comes from the half-space the plate faces, and a half-space of ideal fluid holds exactly half the energy of the whole.

It also says which boundaries are walls. A wall is anything the impulse cannot move in the time available. The bed of a shallow sea is a wall to a hull slamming into it at a metre a second; the surface of a lake is a wall to the air above it, since water is eight hundred times denser than air and hardly moves when air presses on it — which is why a wing skimming a lake behaves as if the lake were a runway. A boundary is free only to a fluid of comparable or greater density on the near side of it, and the question to ask of any surface near an accelerating body is whether that surface gives way before the fluid does.

What the two images leave out #

The free surface is struck, not waved. The condition $\phi = 0$ is the high-frequency limit, correct for an impact and for an abrupt start. A body oscillating near a free surface at a frequency at which gravity waves can form radiates energy in them, and its added mass then depends on the frequency, falling between the two limits here and at some frequencies becoming negative. None of that is computed.

The open jet’s edge is idealised. A real open-jet boundary is a shear layer, growing and unsteady, and it holds its pressure only on average. The image of the opposite sign represents it as a sharp constant-pressure line.

The flow is plane and the fluid is ideal. Every number is per unit span of a two-dimensional body in an inviscid fluid started from rest. Viscosity adds the squeeze film of the previous essay at small gaps against a wall, and has no analogue against a free boundary, where the fluid is not sheared against anything.

The panels are panels. The method converges at first order and each value is extrapolated from two resolutions; an isolated circle comes out at π to within five parts in ten thousand, an ellipse along each axis within two parts in a thousand, and the plate’s thin-gap term within a few per cent of $\rho c^3/12h$ at a twentieth of a chord.

The two boundary conditions are read back off each solution rather than assumed, as the figure above shows, and both signs are required to fade as the clearance grows: at eight radii every case must have returned to within a tenth of its deviation at one radius.

Images, and the people who reflected them #

The method of images belongs to Kelvin, who used it for electrostatics in the 1840s, and it entered hydrodynamics through Stokes and Lamb, whose treatise has the circle near a wall — and, in Milne-Thomson’s circle theorem, the image of a whole flow in a circle rather than in a line. The added mass of a body in an ideal fluid is Green’s and Stokes’ from the 1830s and 1840s. The wind-tunnel corrections that distinguish a closed test section from an open jet are Glauert’s, from the 1930s, and they were written for steady flow.

The unsteady corrections came later and for a practical reason: flutter and gust testing required a model to oscillate in a tunnel, and the borrowed mass of a model in a test section had to be removed from the measurement before the loads meant anything. The rule that emerged is the one drawn here — walls add, open jets subtract — and it is a rule about boundary conditions rather than about air.

Still open: what a free surface does between the two limits #

The two images are the two ends of a free surface’s behaviour. Struck quickly, it is a boundary at constant pressure and the borrowed mass halves; left alone for long enough under a slow enough motion, gravity holds it nearly flat and it behaves almost like a lid, and the borrowed mass rises back towards a wall’s. Between the two, the surface radiates waves, and the added mass becomes a function of the frequency, with a damping beside it that carries the energy away.

That intermediate case is the real problem of every body oscillating in or under the sea, and it is the question the two images here leave open: how the borrowed mass passes from one limit to the other as the frequency falls, and at which frequency it does the strange thing of changing sign.

Beside it is the steady case the cushion argument started from, seen from the boundary’s side. A wing in steady ground effect is also a body with an image, a vortex rather than a source, and a wing over a lake sees the lake as a runway because to the air the water hardly moves. The case where the steady image reverses is the wing on the other side of the surface: a hydrofoil running just beneath the water at speed has a constant-pressure boundary above it rather than a solid one below, and where a wing over the ground gains lift from its image, a fast hydrofoil near the surface loses it.