Added mass — where it appears
Named by 35 essays across 6 fields — each of them below, with the objects they name alongside it.
The force of getting going
The exact theory says a body moving steadily through an ideal fluid feels no force at all. It does not say the fluid is free. Accelerating the body has to accelerate the fluid too, and the bill for that is exactly the mass of fluid the body displaces.
Three dimensions are kinder
The ideal flows usually drawn first are plane, and plane flow is the harsh case. Put the third dimension back and the fastest surface speed drops from twice the free stream to one and a half times, the disturbance dies as the cube of distance instead of the square, and the body cannot carry circulation at all.
The mean is not the flow
Average an unsteady flow and the result is a new object with its own properties, and it is not a solution of anything. An inviscid stream oscillating about zero has a mean velocity of exactly nothing everywhere, a mean pressure that reaches minus two dynamic pressures at the shoulders, and a missing term in its own momentum equation that can be written down in closed form.
A body with no lift, and a moment anyway
A fuselage in ideal flow carries no lift at any incidence and still tries to turn the aeroplane over. The couple is computable in one line, it is why tails are the size they are, and the line comes from applying the wall condition to a place where there is no wall.
Slow enough to be steady
A wing moving slowly enough is assumed to carry the lift its instantaneous angle asks for. The reduced frequency has two thresholds — one where the apparent-mass and circulatory lifts are equal, and one where the quasi-steady answer stops being right — and they are a hundred and seventy-eight apart.
The flow with the least energy in it
Draw a flow that conserves mass and does not go through the walls, and it will look exactly like a solution. There are infinitely many of them and one is the flow. What separates it from the others is not visible anywhere in the picture — it is a number, and the number is an energy.
The tracer that is not one
Every Stokes number is built on a relaxation time that counts the particle's own inertia and nothing else. Adding the two terms it leaves out gives a bubble a relaxation time where the usual formula gives zero, makes a neutrally buoyant tracer half as slow again as advertised, and sends bubbles into vortex cores that droplets are flung out of.
The momentum with no value
A cylinder is pushed from rest to a steady speed. Work was done, energy went into the fluid, something was pushed. How much momentum does the fluid carry? The integral converges, the answer is finite, and it is a different finite number for every shape of region it is summed over.
The pressure that depends on the past
Bernoulli's equation for an unsteady flow has a term nobody writes down and a right-hand side that is a function of time rather than a constant. The term is exactly zero once a flow has started and is the whole of the flow while it is starting, which is why it never appears in an answer and is never negligible in getting to one.
Long enough to make a wake
A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.
The cushion that is there after all
A wing near the ground makes more lift for reasons that have no cushion in them, and that account cannot reach two cases — a wing descending, and a rotor in the hover, where there is no steady flight for the image-vortex account to be about. Those cases have a term that really is about carried air, and it grows without limit as the gap closes.
What a fluid takes out of a swing
The damping a body feels from the air around it is not Stokes' drag, and stops being it far earlier than anybody expects — a millimetre sphere in air is already forty-six per cent above the steady answer at one hertz. Past that the damping rises as the square root of the frequency, and the fluid it is fighting is a shell a fraction of its own size.
One group, three exponents
Lubrication theory, shallow water and slender-body theory are taught in three places and are one expansion in one group — a ratio of two lengths, with no speed, no viscosity and no fluid in it at all. The error is supposed to be second order. In three problems that look alike it is first order, second order, and an exponent that does not exist.
Two answers to one question
Unsteady aerofoil theory collapses a wing's whole history onto one function of one variable, and every quasi-steady gust calculation convolves something with it. There are two such functions, not one: a wing that is pitched changes its boundary condition everywhere at once, and a wing flying into a gust has not met most of the gust yet.
A cushion that changes its physics
A plate closing on a plane is resisted by the fluid it has to squeeze out, and the resistance is two different forces with two different laws — one from the fluid's inertia and one from its viscosity. They hand over at a gap of twenty-four kinematic viscosities per unit of closing speed, which for a wing in air is a third of a millimetre, and the two films disagree about whether the plate ever lands at all.
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 drag that integrates a whole history
A particle released in still air does not stop exponentially. The unsteady drag on it carries a term that is an integral over everything the particle has already done, weighted by the inverse square root of how long ago — so there is no time constant, and five particle time constants later it is still moving four times faster than the quasi-steady answer allows.
Two forces, and only one of them remembers
Morison's equation adds an inertia term to a drag term and is usually presented as an empirical patch. It is not: the two terms are two different kinds of memory, one a function of the present acceleration and one a function of the wake left by the previous half cycle.
The mass a body has to borrow
Accelerate a sphere through water and it resists as though it were half again as heavy. The half is exact, it is a rational number rather than a measurement, and almost everything a reader infers from it about carried fluid is false.
Everything about the start, except one vector
Six ways of accelerating a body from rest to the same speed produce six force histories with nothing in common — peaks spanning a factor of thirty-nine, two of them negative for part of the journey. The impulse left in the fluid is the same ten-figure number in every case, and so is the energy.
The inflow that takes time to arrive
Momentum theory gives a rotor's induced velocity from its thrust, instantly. It does not arrive instantly: the air the disc has to accelerate has a mass, and the response is a first-order climb with a time constant of 33 milliseconds — a twentieth of the time the wake itself takes to convect a 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 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.
Thickness costs a fuselage far less than a wing
A wing section a tenth as thick as it is long speeds the air over its middle by ten per cent. A body of revolution with the same proportions speeds it by two. The difference is not a detail of shape: a plane section pays for its thickness linearly and a body of revolution pays quadratically, and that one exponent is why a fuselage reaches the speed of sound on its surface long after its wing.
A pair set free in a stream collides or parts
Two cylinders held in an ideal stream pull together side by side and push apart in tandem. Let them go and the forces become a motion, and the motion has a law of its own: the stream's force on each is exactly the slope of how large the pair looks from far away, so the pair moves to look bigger. Side by side that means closing, and the fluid squeezed out of the gap costs so little that nothing stops them: released four radii apart they collide in six and a half radii of stream. In tandem it means parting, for good.
A free pair turns, and forty-five degrees decides
Two cylinders in an ideal stream move to look bigger to it: side by side they close, in tandem they part. Free to turn as well, a staggered pair swings towards side by side — and does not stop there, because nothing in an ideal fluid damps the swing. Whether it then collides or flies apart is decided, far apart, by one angle and a theorem: the stream's pull on the pair falls as the inverse square of the spacing, and for such a force the sign of the energy alone decides, which changes at exactly forty-five degrees.
Drag makes every free pair meet
Two cylinders set free in an ideal stream swing towards side by side like a pendulum, and whether they collide or part is fixed, far apart, by forty-five degrees. Give each a drag on its motion relative to the stream and the pendulum stops swinging — but it does not stop the pair. A drag coefficient of order one turns the swing into a creep, lets no pair get far from where it started, and brings every one of them round to meet near side by side.
A trailing wake hurries a pair together
Two cylinders free in a stream and damped by their drag always end up meeting near side by side, and the obvious candidate for what holds real pairs apart is their wakes: a body beside another's wake is pushed towards the faster fluid, away from it. Give each cylinder a wake that carries its drag and the push is there — but it points the wrong way. It throws a following cylinder out of the leader's wake and round towards side by side, where neither is in the other's wake and nothing opposes the stream's pull. Near tandem the wakes make the pair meet seven to sixty times sooner, and the only balance they create is a saddle.
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 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 nose at incidence makes its vorticity along the flow
A wall makes vorticity along its isobars, and on a swept wing that sends part of it along the flow. A fuselage at a few degrees of incidence has no sweep anywhere, and still most of the vorticity its wall makes points along the flow — a fifth for a body twice as long as it is wide, four-fifths for one eight times — because at incidence its pressure varies round the body faster than along it. A sphere, at any incidence, makes none. The lee vortices begin at the wall.
A hull bottom feels the slam it has time to answer
The slamming pressure on a V-bottom is a narrow ridge at the spray root, a megapascal high, that sweeps across each panel of plating in a few milliseconds — about the panel's own period. So the panel neither feels the ridge's height nor ignores its speed: it overshoots the static stress by up to half, a rule that designs for the peak pressure asks for half as much plating again as it needs, and a light hull's slower, softer slam is gentler on its plating twice over.
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.
Free to rise and fall, a ground-effect craft rides its own cushion over the swell
A wing held still over a swell sees its gap change and its lift swing with it. A real ground-effect craft is not held: its weight sits on that lift, and ground effect, whose lift rises as the gap closes, gives it a height stiffness — a spring made of its own cushion — while its vertical speed changes its angle of attack and damps it. Free in heave it is an oscillator driven through that spring. Over long swells it follows the sea and keeps twice the held wing's clearance; over short ones it flies level and rides a dozen centimetres higher, because the cushion's swing is convex; and it comes closest to the crests over the swells it meets near its own natural frequency, never as close as the held wing does.
Below separation a cylinder's drag is a Stokes layer, and KC stops deciding the force
Morison's equation splits the force on a cylinder in an oscillating flow into inertia and drag, and the Keulegan–Carpenter number says which dominates — once the flow separates and leaves a wake. Below that, nothing separates, and the drag term describes the thin oscillating boundary layer on the cylinder's surface. Its coefficient then grows without limit as the motion shrinks, as one over the Keulegan–Carpenter number, and that exactly cancels the number's power to decide anything: the drag's share of the force is fixed by the frequency parameter alone, a few per cent at most, until Honji's instability ends the regime.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitPotential flowUnsteady flowMeasurementd'Alembert's paradoxKinetic energyRegimeDoubletFree surfaceImpulseMemory kernelModel validity