Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Equilibrium — where it appears

A state in which the terms of a balance are steady, whether or not the flow is at rest. An elliptical vortex patch in a strain is one; a shear layer with production equal to dissipation is another; and calibrating a model in an equilibrium state is what limits where it can be used.

Named by 14 essays across 6 fields — each of them below, with the objects they name alongside it.

A high tail buys a second trim point, at 31.52 degrees. The pitching moment of two layouts against incidence, with the stable trim points marked. Both cross zero with a negative slope near 0.76°, which is the ordinary cruise trim. Past the stall the conventional layout's tail is caught only glancingly by the wake and its moment stays nose-down, so it has no second crossing; the T-tail's tail is swept into the wake and sees 12 per cent of the dynamic pressure, its download collapses, and the wing's own nose-up moment carries the curve back across zero at 31.52°. That second crossing is stable — the slope there is negative too — which means an aircraft that reaches it stays there.

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.

misconceptions · Stall speed The junction the minimisation is over. A parent vessel entering from the left and two daughters leaving to fixed points. The radii are settled by Murray's law; what is left free is where the branch point sits, and the cost of the junction depends on it. The point drawn is the one the minimisation finds.

The angle a junction chooses

Murray's law fixes the radii at a branching vessel and is where every account of it stops. The same minimisation fixes the angles completely — 74.93 degrees for a symmetric bifurcation, a right angle for a vanishing side branch — and it does so as a triangle of forces, with tensions proportional to the squares of the radii.

applied · Branching Kirchhoff's rotation rate, which a point vortex does not have. A patch of uniform vorticity bounded by an ellipse turns rigidly at omega a b/(a+b)², a rate that depends on the shape alone. It is largest for a circle, where it is unobservable, and falls away as the patch is drawn out. A point vortex has no shape and therefore no entry on this axis at all.

The shape a vortex keeps

Outside a circular patch of uniform vorticity the flow is exactly the point vortex's — not nearly, exactly — so replacing one by the other looks free. It is not. The patch has a shape, the shape has a rotation rate of its own, and there is a strain above which no shape exists at all.

inviscid · Vortex patch The temperature behind a shock, which is not the jump condition's. The static temperature along the flow behind a Mach 6 normal shock, with the frozen value the jump conditions give and the equilibrium value they give a long way behind. The gas arrives at 2382 K and settles at 2059, over about four tenths of a millimetre.

A gas that has not finished being shocked

The jump conditions give the state a long way behind a shock. Immediately behind it the molecules have not started vibrating yet, so the temperature is 2,382 K where the equilibrium answer is 2,059 — and the gas takes four tenths of a millimetre to get from one to the other.

compressible · Real gas The patches' centroids, against the point-vortex circle. Two circular patches of uniform vorticity, advected by nothing but the velocity their own boundaries induce, over one full co-rotation. Their centroids stay within one per cent of a separation of the exact point-vortex orbit — which they must, because the exterior field of a circular patch is the point vortex's and a harmonic function's area average over a disc is its value at the centre.

What a point vortex is not

Two circular patches of vorticity move exactly as two point vortices do — the centroid velocity is the point model's with no correction of any order, and that is a theorem rather than an approximation. The trouble starts the moment they stop being circular, which is immediately.

inviscid · Vortex dynamics The layer that grows from a change of surface. The internal boundary layer's height against distance downwind of a change in roughness, with the sublayer inside it that is genuinely in equilibrium with the new surface. The layer grows as the fetch to the four-fifths power and the equilibrium sublayer is a tenth of it.

How far downwind a surface is remembered

Walk from a field into a wood and the wind ten metres above your head is still the field's wind. It takes about a kilometre of trees before a ten-metre measurement is measuring the trees — a hundred times the height it is made at, and a great deal more than most masts are given.

turbulence · Roughness The production jumps and the dissipation does not. Production and dissipation against time, through a step change in the strain rate. The production follows the strain immediately — it is the strain squared times an eddy viscosity — and the dissipation moves by less than one per cent at the instant of the step, because it is set by a cascade that has not been told yet.

A dissipation that lags its production

Change the strain rate on a patch of turbulence and the production of energy follows instantly — it is the strain squared. The dissipation moves by less than one per cent, because it is the far end of a cascade that has not been told yet, and the two are out of balance by a factor of six for the next turnover.

turbulence · Dissipation Three lobes, then a filament. An ellipse of aspect ratio 4 with a three-lobed bump of three thousandths, as contour dynamics carries it, drawn in the frame turning with the undisturbed ellipse at t = 0, 30 and 42. By t = 30 the bump has grown to a visible three-fold asymmetry — one end fattened, the other thinned — and by t = 42, about a turn and a tenth of the ellipse, the thinned end is being drawn out into a filament. The march is stopped there, while the area is still conserved to a few parts in a thousand; resolving the filament needs a contour that adds nodes, which this one does not.

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.

inviscid · Vortex patch A drop fed into a tube, to scale, up to the largest one it can hold. Six members of the family of static drops a tube of radius half a capillary length holds, drawn to scale and hung from the rim, with their volumes in cubic capillary lengths. The drop grows from a shallow cap through a hemisphere to a bulb with a neck. The last is the fold: no static drop on this tube holds more, and it holds 76 per cent of what Tate's law says the tube can carry.

The drop falls at a fold

Tate's law says a drop leaves a tube when its weight equals the tension round the rim. No force balance decides it. A tube holds a family of static drops, the family has a largest member, and the drop falls because there is no static shape with more liquid in it — which Tate's balance overestimates by a quarter on a millimetre tube and underestimates on a wide one.

regimes · Bond Two eddies that stand still behind a cylinder in ideal flow. A uniform stream past a circular cylinder with a pair of opposite vortices standing 2 radii behind its centre and 0.859 either side of the axis, each of strength 10.31 in units of the stream speed times the radius — Föppl's equilibrium. The shading is the stream function: the fluid between the vortices and the cylinder circulates in two closed eddies and never leaves, while the stream divides round the whole assembly as if it were one longer body.

Two eddies can stand behind a cylinder, but not for long

Ideal flow, which has no viscosity and no wake, can still hold a pair of eddies standing behind a cylinder — at any distance behind it, with a strength fixed by the distance. They cost the cylinder nothing. And they cannot stay: nudged sideways by a thousandth of a radius, the pair grows its displacement exponentially and leaves, which is the first step of shedding a wake.

inviscid · Images Half the forced summit is gone in a fifth of an eddy time. The third moment of the velocity differences as a fraction of four-fifths εr, against separation in Kolmogorov lengths of the forced flow, at the instant the forcing is switched off and 0.1, 0.2, 0.5, 1 and 4 large-eddy times later. Forced, the curve peaks at 0.94 near 67 η. A tenth of an eddy time later the summit is 0.899, at a fifth 0.838, at a half 0.759, and after that it hardly moves: 0.742 at one and 0.72 at four, where the flow is simply decaying. The large separations fall first and farthest.

The summit forgets the forcing before the dissipation does

A flow forced at its largest scales carries the four-fifths law closer to exact than a decaying one. Switch the forcing off and the third moment has to pass from one value to the other. It does not wait for the cascade to drain: half the forced summit is gone in a fifth of an eddy turnover, while the dissipation has hardly moved, and the largest separations overshoot the decaying value before they settle on it.

turbulence · Decay A wake throws a following cylinder out to the side, and the pair still meets. One cylinder's centre relative to the other's, the stream from left to right, for pairs released six radii apart at 2°, 5°, 10° and 20° from tandem at Re = 200. Without wakes (faint) the near-tandem pairs drift apart, out to 13 radii, before the drag lets the turn bring them in. With wakes the follower first drafts in, is thrown sideways out of the leader's wake, and never gets beyond 6.26 radii. Every pair touches, 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.

inviscid · Dalembert An adverse gradient hollows the layer from the outside in. The velocity profile of an equilibrium layer, as a fraction of the edge velocity against y/δ, at Clauser's β = 0, 2, 10 and 50, with the wake strength from Das's fit. The inner part keeps the wall law; the outer part's wake grows — Π = 0.476, 1.59, 4.67 and 15.2 — and the profile sags until most of the layer is moving slowly over a thin fast wall region.

An adverse gradient grows the wake, not the log law

Push a turbulent boundary layer up a rising pressure and its profile changes from the outside in. The wall region keeps its law; the wake grows, the layer hollows, the shape factor climbs and the friction falls. Clauser found the layers that stay similar while this happens, and their arithmetic has a surprise in it: there is a steepest pressure rise any such layer can climb, close to a free stream falling as the inverse fourth root of the distance, and the friction approaches zero only as the layer becomes all wake.

turbulence · Wall law However long its forcing remembers, a stationary flow's averaged four-fifths summit stays at 0.94. The summit of −Dₗₗₗ/((4/5)εr) computed from the time-averaged spectrum and source — the third moment averaged, then divided by the averaged dissipation — against the forcing's correlation time in large-eddy times, for a forcing amplitude whose logarithm wanders with standard deviation 0.5 and 1, over runs of 200 to 400 eddy times. The steady forcing's summit is 0.94; across correlation times from 0.03 to 10 eddy times and both spreads the averaged summit lies between 0.9371 and 0.9438, within its batch errors of the steady value.

A wandering forcing leaves the averaged four-fifths law alone

A forced flow's third moment loses half its excess within a fifth of an eddy time of the forcing stopping, so a forcing that wanders ought to drag a simulation's four-fifths ratio about with it. Instantaneously it does: the summit swings from 0.8 to 1.4. Averaged it does not — the third moment averaged and then divided stays at the steady forcing's 0.94 for every correlation time from a thirtieth of an eddy time to ten. The reason is that the balance is linear and stationarity fixes its source; what the forcing's memory changes is only the answer to a different question, the average of a ratio.

turbulence · Decay

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitMeasurementDissipationMemory kernelCascadeClosureContour dynamicsModel validityPoint vortexRegimeStabilityVortex patch

All concepts