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

Vortex — where it appears

A concentrated region of vorticity, which moves with the fluid and induces a velocity on everything around it. An isolated one in an inviscid fluid never decays, and its circulation is the same on every loop that encloses it.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Ideal flow past a cylinder. A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.

Flows add up

The equations of ideal flow are linear, so solutions can be laid on top of one another. A uniform stream plus a doublet produces a cylinder that nobody put there, and almost every classical result is built this way.

inviscid · Superposition Stokes' theorem on a solved wake at Re 40. A rectangle drawn in a viscous flow that was solved on a grid. The circulation round its boundary is computed by walking the four sides and adding up the velocity along them; the vorticity inside it is computed by adding up the stored vorticity cell by cell. The two computations share no sample point and the theorem says they must agree.

Circulation is vorticity, added up

One of these two quantities is measured by walking round a loop and one by summing over the area inside it, and a theorem says they are the same number. That reconciles the site's most confusable pair — and explains how a flow with circulation can have no spin in it anywhere.

kinematics · Vorticity Q and the vorticity along a radius of one vortex. Two candidate measures of where the vortex is, along a radius of a Lamb–Oseen vortex. The vorticity is a Gaussian: positive at every radius, so a threshold on it puts the edge wherever the threshold is put. Q — the excess of rotation over strain — changes sign exactly once, at 1.121 core radii, and that radius is a property of the flow rather than of the person drawing it. Inside it, 71.5 per cent of the circulation.

Where a vortex stops

Four criteria decide where a vortex ends, and in two dimensions three of them are the same criterion. The fourth is a knob. And the one that is not a knob is not objective: a co-rotating pair of vortices occupies two per cent of a window to one observer and twenty-five to another.

kinematics · Coherent structures Elliptic loading rolls up to πb/4, and it is π that puts it there. Three loadings on the same span carrying the same lift, with the rolled-up core positions marked below each. The core sits at the centroid of the shed vorticity, which is a quadrature over the loading and needs nothing about the roll-up itself. Elliptic loading gives πb/8 from the centreline, so the pair ends up 0.7854 of the span apart — the number every wake-separation rule is written against, and one of the few places in this subject where π turns up in an answer an engineer uses directly. The bell rolls up to 0.586 and a nearly rectangular loading to 0.978, because it sheds at the tips.

Where the wake ends up

The sheet a wing sheds rolls up into two cores within a few spans, and nobody can compute the roll-up cheaply. Nobody has to: what the cores conserve is fixed before they form, and for an elliptically loaded wing the answer contains π and comes out at 78.5 per cent of the span.

circulation · Tip vortex Hill's spherical vortex. A sphere of rotating fluid travelling steadily through fluid at rest, drawn in the frame that moves with it. Outside the sphere the flow is the ordinary potential flow past a sphere; inside, the vorticity is proportional to the distance from the axis and the fluid recirculates. The two solutions match in value and in slope across the surface, and there is no body anywhere — the boundary is a streamline and nothing else.

The one rotational solution anybody can write down

A sphere of spinning fluid travelling steadily through fluid at rest, with no body anywhere in it — the boundary is a streamline and nothing else. It is exact, it is two lines long, and the reason it is the famous one turns out to be the reason it is the only one a real fluid can settle into.

inviscid · Euler rotational One material loop, at five stages of being drawn out. A circle of fluid particles carried by four point vortices, drawn at equal intervals over fourteen time units. Its length grows by a factor of six and its shape becomes unrecognisable; the circulation round it does not move at all.

The drift was the instrument

Kelvin's theorem was checked on this site by carrying a loop and watching its circulation move by eight parts in a hundred thousand. That drift is not the flow forgetting. The exact number is a count of what is inside the loop, it does not move by anything at all, and the drift belongs entirely to the two instruments used to measure it.

inviscid · Kelvin

Named alongside it

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

CirculationVorticityConserved quantityIrrotationalKelvin's circulation theoremMeasurementModel limitWakeAngular momentumBoundary conditionBoundary layerCentroid

All concepts