Free surface — where it appears
Named by 25 essays across 4 fields — each of them below, with the objects they name alongside it.
What a jet cannot push sideways
A jet striking a plate divides in two, and how it divides is fixed by a single sentence — an inviscid fluid exerts no force along a surface. That one statement, plus mass, gives the split exactly — and the same sentence turns a flat plate into a bucket worth twice as much.
The section that decides the river
A reach of open channel has a normal depth and a critical depth, and the water has neither. What it has is a profile obeying a first-order equation, which needs exactly one condition — and whether that condition belongs at the upstream end or the downstream end is not a choice, because the equation is stable in one direction and unstable in the other.
The drag that is made of waves
D'Alembert's paradox says a body in a steady, irrotational, incompressible, inviscid flow feels no drag. Put a free surface above it and every one of those words still holds — and the drag is not zero. It is the energy walking away in the wave train behind.
A tank that turns a hammer into a swing
Shut a turbine at the end of a two-kilometre tunnel in two seconds and the valve takes a rise of 256 metres of head. Put a shaft open to the air beside it and the rise is 51, the tunnel never carries the closure as a wave at all, and its water slows instead against a level that climbs for a minute and a half — to a height that is a closed form with the tank's area under a square root.
The borrowed mass the boundary decides
A body accelerating near a solid wall has to squeeze out the fluid between them, and borrows more mass than it would in the open. The same body accelerating near a free surface, or inside an open-jet wind tunnel, borrows less. The fluid, the body and the speed are identical, and what reverses the answer is the one thing each boundary is allowed to tell the flow.
The vorticity a clean surface cannot refuse
A clean bubble's surface cannot hold a shear stress, and it is easy to conclude that it makes no vorticity. On a curved surface it must carry exactly 2κu — three times the speed over the radius at a sphere's equator, whatever the Reynolds number. That is so much weaker than a rigid wall's that the flow stays irrotational to leading order, and the bubble's drag is the dissipation of that irrotational flow: 48/Re, three to ten times below a rigid sphere's.
Twice as slippery along as across
A surface of alternating gas and solid stripes lets a liquid slip, and a flow far above it sees one number in place of the pattern — but the number depends on which way the flow goes. Along the stripes it is Philip's logarithm; across them it is exactly half, for a reason that takes one substitution to show. And the logarithm means that the slip is bought by the pattern's period rather than by how much of it is gas.
The teapot effect is a tension, not a pressure
A slow pour runs back under a spout and down its outside, and the name it is usually given is the Coandă effect. The Coandă effect borrows the ambient pressure, and a liquid in air has none to borrow. A liquid sheet is held to a lip by its own surface tension, and it lets go at the one speed that tension cannot carry — the speed of waves along the sheet — whatever the lip's radius.
The theory with no memory in it
Laplace's equation has no time in it, so an ideal flow's response to a body being jerked into motion is instantaneous and complete. Its indicial kernel is a spike and nothing afterwards. Beside it sit the two kernels that are not, and the comparison says which ingredient every memory in this collection came in through.
The current that runs across the river
The standard channel calculation is one-dimensional, and a river is not. Put a bend in it and the surface tilts by a few centimetres, which is invisible — and a helix forms, a few per cent of the main flow, which decides where the river will be in a hundred years.
The borrowed mass that goes negative
A body's added mass is taught as the water it carries with it, fixed by its shape. Put the body under a surface that can make waves and the amount depends on how fast it is shaken, runs from a wall's value to a free boundary's and past both, and for a circle close enough to the surface falls below zero.
An edge holds any angle it is given
A liquid's contact angle is taught as a property of the liquid and the solid. On a smooth face it is. On a sharp edge the line stops, the angle is free to take any value in a band as wide as the edge is sharp, and that freedom is how a coin carries a dome of water and a glass stands full above its brim.
A hydrofoil loses most lift on the way up
A wing near the ground gains lift, and a hydrofoil near the surface is often described as the same thing upside down. At low speed it is: the surface acts as a lid and the foil gains lift. At high speed the surface is a boundary that cannot hold a pressure, and the foil loses lift instead. But the lift does not pass smoothly from one to the other. Between them, where the foil's waves are a few depths long, it falls well below both — to 43 per cent of its deep-water value half a chord down — and a foil boat meets that speed in the middle of its take-off run.
A rounded edge spills before its line goes round
Gibbs's band says a sharp edge lets a liquid stand at any angle across a range as wide as the edge's turn. No real edge is sharp, and on a rounded one the line does not stop; it slides round the curve, meeting it at the Young angle everywhere. The band survives the sliding. What the rounding costs is height, because the line drops as it goes round, and past a point the drop outweighs the steeper angle and the liquid spills before the line has reached the far face. An edge is worth its sharpness measured against the capillary length, and nothing smaller.
A foil under the surface flies with a phantom
At foiling speed the water's surface cannot hold a pressure, and the image that condition requires is not the reversed one a wall makes. For a horizontal foil it is an identical wing, equally loaded, above the surface — a biplane partner that takes lift and gives nothing back, doubling the induced drag as the foil rises. For the board that pierces the surface it is a reversed copy, which makes the board pay two and a half times what a keel under a hull pays. It is the board, not the foil, that decides how deep a foiling boat rides.
A coast sends the drift back, or sends it along
A shelf sea under a swell keeps part of the Stokes drift's transport, turned to the right by the rotating Earth, and near a coast some of it points at the land. The sea answers with a slope in its surface. In water shallower than an Ekman depth the slope drives the water straight back as an undertow, as it would without rotation; a few Ekman depths down the coast turns the transport into a current along the shore instead, and there the sea can slope down towards the land it faces.
A foil flies level through a short sea and follows a long one
A foiling boat in waves has two ways to fly. It can hold its height and let the surface rise and fall over its foil, or it can follow the surface and heave with every wave. Flying level costs a deeper ride and a few per cent of drag whatever the wavelength; following costs the square of the heave, which grows with the frequency the boat meets the waves at. Into a head sea the two cost the same at a wavelength of about sixteen metres, running with the waves at about six, and the wave's height hardly moves either.
A sloping shelf puts the swell's current three Ekman depths down
A swell's Stokes transport arrives at a coast and the sea must send it back or turn it aside. On a flat shelf one depth decided which. On a real shelf, deepening from the beach to its edge, every depth is present at once, and the sea does both: it sets up a few centimetres against the beach, dips a tenth of a millimetre where the depth passes two Ekman depths, and runs a current along the shore that is fastest about three Ekman depths down — wherever the friction puts it.
A foil in a random sea follows it up to its peak
In a regular wave a foiling boat chooses between holding its height and following the surface. A real sea is a spectrum, and a wand that filters its signal can do both at once: follow the long waves and fly level through the short. The best place to divide them is close to the frequency of the sea's own peak, the divided control beats either pure strategy, and in a rough sea it is the only one of the three that keeps the foil a safe distance under the surface and the drag near its calm-water value.
Water climbs a falling wedge, and the load comes from the climbing
A hull that strikes the sea sets the water under it moving, and the force is the rate at which it does so. The obvious estimate measures the wetted width where the hull crosses the undisturbed surface. But the water does not wait: pushed aside, it rises up the hull and wets it sooner, a factor π/2 wider for a wedge and √2 for a round bottom. The force carries that factor twice, and the peak pressure, where a thin jet leaves the hull, carries it squared on a cotangent that grows without bound as the bottom flattens.
A wing over the sea barely touches it
A wing skimming the water presses its whole weight onto the surface beneath it, spread over a width about its own height. The sea is a deformable ground, and one might expect it to give way: a trough under the wing, waves behind, a drag to pay. It barely notices. Air is eight hundred times lighter than water, and that one ratio sets the dent at millimetres, the waves at a few millimetres high, and the wave drag at less than a two-thousandth of the lift even at the worst speed.
A light hull slams softly, and a round one does not
A hull striking water slows as it sets the water moving, and the two share one momentum. For a V-shaped bottom that caps the slam: the force peaks when the water recruited weighs a fifth of the hull, and a hull as heavy as the water under its whole width feels a quarter of the force a constant-speed calculation predicts. For a round bottom it caps nothing, because the peak comes at the instant of contact, before the hull has slowed at all.
A swell lifts a ground-effect wing more at its crests than it takes in its troughs
A wing skimming the sea gains its lift from the sea's reflection of it, and the reflection grows sharply as the gap closes. Over a swell the gap opens and closes with every wave, so the lift swings — further up at each crest than down at each trough, which raises the mean lift as well as shaking it. Climbing out of the swing is possible, but the height that calms the lift spends the cushion the craft was built for, and in a sea about two metres high it has spent nearly all of it.
A wand that lags its sea makes flying level the better bet
A foiling dinghy whose height control divides the sea sharply follows the long waves, flies level through the short and beats both pure strategies. Give the division the lag a real wand or filter has and the gain disappears in a moderate sea; in a rough one no crossover both keeps the foil under and spares it from pulling down. A second-order filter, which cuts more sharply, does worse rather than better, because it flies through more of the sea near its crossover than the sea contains.
Plating that gives way slows the water that slams it, but only on a flat bottom
The slamming pressure on a hull's bottom is a ridge that rides at the edge of the wetted width, and how fast that edge moves depends on how far the water has to climb. A panel of plating that bends inward under the load gives the water further to climb, and while it is moving away it slows the climb itself. Fed back into Wagner's condition, that takes a seventh off the peak stress of a panel on a four-degree bottom loaded to the edge of what aluminium can bear, and almost nothing on a deep V — because what matters is the plating's deflection against the bottom's own rise across the panel.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitMeasurementAdded massFroude numberMisconceptionBoundary conditionGround effectHydrofoilImage vortexInduced dragPotential flowSurface tension