Stability — where it appears
Named by 27 essays across 7 fields — each of them below, with the objects they name alongside it.
Where the lift acts
Lift gets drawn as an arrow, and an arrow has to start somewhere. The pressure is spread over the whole surface, so the choice of where to put the arrow is free — except that exactly one point on the chord has the property of not changing its answer when the wing is pitched.
Five numbers, one name
The Richardson number has a threshold at a quarter, and the quarter belongs to one of the five quantities that go by the name. On a single tanh shear layer they run from J to seventeen J, and the largest of them grows without limit as the measurement is extended further from the layer.
The shake that is not resonance
A steady airstream contains no oscillation at any frequency, so nothing is driving anything. What happens instead is that the aerodynamic forces couple two structural motions that were independent, drag their frequencies together, and turn one damping negative — and the wing takes the energy out of the air itself.
The film that heats itself
The oil in a bearing is not at the temperature of the metal around it. It is tens of kelvin hotter, the rise contains no length whatever, and whether the heating runs away or settles down depends on something that is not a property of the oil at all — whether the machine driving it holds the speed or holds the force.
The surface in the wake
A wing has no opinion about its own incidence, which is why it needs a second surface behind it. How much that surface is worth depends on how much of the wing's downwash it is sitting in, and the factor of two everybody quotes for that is the value at infinity — where no tailplane has ever been put.
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.
A stall that is a place
A stall is an event, and the usual accounts treat it as one — a boundary reached, a damping lost. A high tailplane makes it something else. Swept into the wing's wake, the tail loses the download that held the nose up, and the aircraft finds a second stable trim point thirty degrees past the stall from which the elevator cannot bring it back.
A puff that does not know how old it is
A patch of turbulence in a pipe below the critical Reynolds number dies at random, and its chance of dying in the next second does not depend on how long it has already lasted. The flow that contains it has a memory anyway, because the patches multiply — and where multiplying overtakes dying is a Reynolds number.
The control that works backwards
Divergence is the static aeroelastic failure everybody names, and a wing with its elastic axis at its aerodynamic centre cannot diverge at any speed. It can still reverse — deflect the aileron down above a certain dynamic pressure and the aeroplane rolls the other way — because the aileron's own nose-down moment is there whatever the elastic axis is doing.
A wake told what to do
Force a shedding wake near its own frequency and it abandons its own and adopts the forcing's. The band over which it will do so is proportional to how hard it is pushed — and near the edge of the band it takes six times as long to make up its mind.
The lag that makes flutter possible
This site's own flutter model set the lift deficiency to one and recorded in its notes that doing so throws away the lag which stabilises the torsion mode. Putting the lag back moves the flutter speed from 80.8 metres a second to 131, and removes the need for the structural damping that was covering the artefact.
Hexagons remember how the heat was turned up
A layer heated from below convects in rolls, unless its top and bottom are not mirror images of each other. Then three sets of rolls at 120° can feed one another through a term the symmetry used to forbid, hexagonal cells appear before the layer is formally unstable, and there is a range of heating in which rolls and hexagons are both stable — so the pattern a layer shows depends on whether the heat was turned up or down to get there.
Past three, an ellipse is a shear layer
Kirchhoff's elliptical vortex turns for ever without changing shape, and Love showed in 1893 that it stops being stable at an aspect ratio of exactly three. Computed, that threshold turns out to be the first of a sequence — a new way of coming apart every one and a half aspect ratios — and the sequence ends somewhere recognisable. A long enough ellipse is a strip of vorticity, and it comes apart the way a shear layer does, at a rate Rayleigh found for the strip.
A lost steady state still holds the film
Push a self-heating oil film one per cent past the stress at which it can no longer settle and it does not run away at once. It warms to the temperature the vanished steady state would have had, sits there as if nothing were wrong, and only then goes — after a delay that grows as one over the square root of the overload, and during nearly all of which the load could still be taken back.
A motor turns the runaway into a jump
A self-heating oil film has a fold at fixed stress and none at fixed speed, and a real motor is neither: its torque falls along a line as its speed rises. Put that line across the film's own torque–speed curve and the runaway disappears for any motor at all. What replaces it depends on the line's slope. A drive softer than one fourteenth of the cold oil's resistance jumps to a hot state and keeps a memory of the load; a stiffer one does neither. And the boundary between the two falls exactly where the motor is turning at a quarter of its no-load speed, which no motor near its rated speed can reach.
A strained vortex holds until it has no shape to hold
A patch of vorticity in a strain has two steady shapes for every strain below 0.150 of its vorticity, a rounder one and a longer one, and none above. The longer one comes apart at the slightest nudge. The rounder one, computed with disturbances of two, three, four and five lobes, never does: it nods and holds right up to the strain at which it ceases to exist. So the existence limit is the real limit, and past it a vortex is not shattered but stretched — lingering first near the shape it has lost, for a time that grows as the fourth root of how close the strain is to the limit.
The plenum decides whether a compressor surges
Throttle a compressor past the peak of its characteristic and it does one of two things. It settles into rotating stall, a steady state with a cell of dead flow running round the annulus, or it surges, the whole flow through the machine collapsing, reversing and recovering over and over. Which one is not decided by the blades. It is decided by the volume the compressor discharges into, against the inertia of the air in its duct — one number, Greitzer's B, which grows with the blade speed, so the same machine stalls at part speed and surges at full speed.
A rotor's inertia slows the jump and cannot make it ring
Give the motor driving a self-heating oil film a rotor that has to spin up, and there are two clocks: the film's diffusion time and the rotor's. Two clocks are what an oscillator is usually made of, and this pair cannot make one. Every eigenvalue stays real at every inertia, because the film and the rotor only ever push each other the same way. What inertia does instead is add its own delay to the film's, and in a real machine, where the rotor is hundreds of times slower, the delay past the fold is almost entirely the rotor's.
A slow sensor makes the bearing cycle
Derate the motor driving a self-heating oil film on the temperature of its housing, which warms over minutes, and a protection meant to hold the bearing cool instead makes it cycle for ever: jump hot, warm the housing until the hot state is lost, drop cool, let the housing cool until the cool state is lost, jump again. The period belongs to the housing, the amplitude belongs to the film, and the derating strength decides only whether the cycle happens at all.
A warm bush stops the bearing hunting
A bearing whose motor is derated on its housing's temperature can hunt, jumping between a cool film and a hot one on the housing's clock. Let the bush warm with the housing, and a second feedback joins the first, of the kind that usually makes things run away. It does the opposite: a warm wall is a stiffer drive, the stiffening closes the S the cycle runs round, and past a bush coupling of about a third the bearing settles, warm and derated, however the protection is set.
A throttle can hold the peak only in a small plenum
A compressor makes its most pressure at the peak of its characteristic, and at the peak it either surges or stalls. A throttle moved by feedback can fight both. Driven by the plenum's pressure it damps surge, with a gain that grows as the square of Greitzer's B until it meets a ceiling B does not move; driven by the flow, it cannot. Driven by the stall cell's amplitude it turns the sudden jump into deep stall into a gradual one and closes the hysteresis loop — but only while the plenum is small. Above B of about a third the same feedback turns the jump into a cycle.
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.
A strained vortex loses its ring at the limit of what it holds
A uniform patch of vorticity survives a strain up to 0.150 of its vorticity and no further. A real vortex is not uniform: its vorticity falls off outward, and in a strain its weak outer layers are drawn off while its core survives. The obvious way to predict how much survives is to apply the uniform limit layer by layer, each layer to its own vorticity. It is wrong by up to a factor of three. Built from a core inside a ring and followed by contour dynamics, the ring is stripped not at the limit of its own vorticity but at the limit of all the vorticity inside its edge, and a little after it.
A slow bush quickens the hunt it cannot start
A bearing's bush warms with its housing and makes the drive stiffer, and past a coupling of about a third it stops a derated bearing hunting. Give the bush a time of its own and the obvious fear is that the lag reopens the hunt. It cannot: whether a bearing hunts is decided by where its equilibrium sits, which no lag can move, and a lag on a positive link cannot make a settled state oscillate. What a slow bush does change is the hunt itself, which runs a sixth faster.
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.
A trip deeper than the film's window flips the bearing twice a cycle
A smooth derating acts on a bearing's housing temperature as it stands. Most real protections trip instead: they cut the motor at one temperature and restore it at a lower one, a hysteresis of their own laid on top of the film's. Whether the oil then jumps between its cool and hot states is decided by one comparison — the cut's depth against the width of the film's window — and past it the protection's own band sets the rhythm. A trip never lowers the film's peak at all; it only protects the housing, and a deep one buys that with a plunge to the cool branch twice in every cycle.
A compressor needs one actuator for each way it can fail
A throttle alone cannot hold a compressor left of its peak in a large plenum: it has one degree of freedom and the machine has two ways to fail, and the throttle's own surge gain runs into a ceiling that the plenum's size does not move. Give the stall cell its own actuator — air injected at the rotor tips against the cell — and give the throttle a second signal, the flow, and a machine at B = 2 is held at ninety per cent of its peak flow, recovering from a kicked cell nearly three times the injector's capacity. Let the injection lag the cell by a radian and the line still holds, but almost no disturbance is survived.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitBifurcationHysteresisLubrication filmThermal runawayViscous heatingDynamical systemRelaxation timeMeasurementOscillationStallUnsteady