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

Kelvin's circulation theorem — where it appears

The result that the circulation round a material loop in an inviscid barotropic fluid does not change. It says circulation cannot be created, only separated, which is why a wing that starts to lift must leave an equal and opposite vortex behind it.

Named by 13 essays across 4 fields — each of them below, with the objects they name alongside it.

The starting vortex, and the circulation it pays for. A wing that has just begun to lift, and the vortex it shed as it started. The circulation round the wing and the circulation round the shed vortex are equal and opposite, so a circuit large enough to contain both has no circulation at all — which is what Kelvin's theorem requires of a circuit that began at rest.

The vortex a wing leaves behind

A wing at rest has no circulation. A wing in flight has a great deal. Circulation round a circuit of fluid particles cannot change, so the difference had to come from somewhere — and it did, as an equal and opposite vortex dropped on the runway.

circulation · Lift 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 The one place the stretching argument closes. Burgers' vortex: an axisymmetric strain carrying vorticity inwards at exactly the rate viscosity spreads it outwards. The vorticity profile is a Gaussian of radius √(4ν/α), the swirl velocity peaks at 1.12 core radii rather than at the core radius itself, and the circulation reaches its full value by about two. The steady vorticity equation is evaluated on this profile by differencing it, not by re-deriving it.

The spin that feeds itself

Stretch a vortex tube and its spin rises in exact proportion, because the circulation round it cannot change and its area has fallen. Nothing in that argument sets a limit — and the one flow where the limit can be written down exactly puts it at a length of √(4ν/α).

kinematics · Vorticity The lift does not arrive when the incidence does. Bound circulation against distance travelled, in units of the settled value. The wing starts far short of its final lift and takes tens of chords to collect the rest, because every scrap of circulation it takes has to be paid for with an opposite vortex shed behind it, and that vortex's own downwash holds the wing back until it is far away. Wagner's exact 1925 answer is drawn beside the model in the colour this site keeps for a borrowed claim: the shapes agree and the model is slower, by about a fifth at its worst.

The lift that arrives late

A wing set into motion does not have its lift. Every scrap of circulation it takes has to be paid for by shedding an equal and opposite vortex behind it, and until that debt is far downstream its own downwash holds the wing back — for tens of chords, not for an instant.

circulation · Unsteady lift The wall makes vorticity at a rate with no viscosity in it. At a stationary wall the momentum equation collapses to ν ∂²u/∂y² = (1/ρ) ∂p/∂x, and the left-hand side is the diffusive flux of vorticity out of the surface. So the pressure gradient along the wall is the vorticity source, and the viscosity that made the no-slip condition necessary has cancelled out of what the condition produces. The curve is that flux across the Falkner–Skan family, computed from profiles solved by shooting and differenced at the wall; the straight line is the pressure gradient each of those flows has. They agree to 2.3e-14. At zero pressure gradient the flux is exactly zero: a flat plate creates no vorticity at all after its leading edge, and everything in its layer arrived from there.

Where vorticity comes from

Every scrap of vorticity in a flow past a body entered through its surface, and the rate at which it enters contains no viscosity at all — it is the pressure gradient along the wall. A flat plate makes none, and a closed body makes exactly as much of each sign.

kinematics · Vorticity Wound up, and worth exactly what it started with. A material loop in a steady cellular flow — an exact solution of Euler's equations — drawn at four times. Each streamline in the cell has its own period, so the loop is stretched steadily into a spiral: by the last frame its perimeter is 5.7 times what it started as. The circulation round it is 0.903741 at the start and 0.903666 at the end. Nothing about the curve survives except the number.

What survives being wound up

Draw a loop of marked fluid particles and let the flow carry it. It will be stretched, folded and wound into a spiral until nothing about its shape is recognisable, and the circulation round it will not have moved at all — provided three conditions hold, each of which can be broken on purpose.

inviscid · Kelvin The core spreads and the outside never notices. The swirl velocity at four times a factor of four apart, with the free vortex Γ/2πr drawn behind them. Every curve leaves the free-vortex line at its own core radius and turns over into solid-body rotation inside it; outside the core all four are the same curve, to the precision of the plot. Viscosity has rounded off the singularity and changed nothing else. The peak swirl falls from 52.4 to 6.6 m/s across the four, and the circulation is identical for all of them.

What viscosity cannot take away

Leave a vortex alone in a viscous fluid and every local measure of it falls — the peak spin, the peak velocity, the enstrophy. The circulation round a large loop does not move at all, ever, and the far field is identical to the line vortex it started as.

viscous · Diffusion Flow net — a stream past a cylinder. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

The mirror that is a circle

A flat wall is made by reflecting everything in it. A round one is made the same way, except that the mirror is an inversion — the image of a point at distance d sits at a²/d, and a vortex acquires a second image at the centre that nothing about the wall requires.

inviscid · Images A cylinder in a uniform shear, K = 0.4. A stream whose velocity increases with height, meeting a circular cylinder. The oncoming profile is drawn at the left. The flow carries uniform vorticity −K, so it is a solution of Euler's equations and not of Laplace's, the pattern is no longer symmetric top to bottom, and the body feels a lift towards the fast side with no circulation anywhere.

Inviscid does not mean irrotational

Dropping viscosity gives Euler's equations. Assuming nothing is spinning gives Laplace's — one scalar, linear, unique. The second step is a separate hypothesis about the flow's history, and a flow that fails it is still an inviscid flow with exact solutions of its own.

inviscid · Euler rotational Wagner's function and Küssner's, from one solver and two inputs. Lift as a fraction of its steady value, against distance travelled in semichords. The step in incidence and the sharp-edged gust are the same unsteady problem with two different right-hand sides, and Jones's exponential fits to both are drawn over the solve. From four semichords on they are nearly the same curve — which is why they get interchanged.

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.

circulation · Unsteady lift A spun cylinder carries its whole circulation at once, and hides it until the vorticity has left. The circulation round circles of radius r about a cylinder of radius a started spinning at once, as a share of the circulation of its own surface, 2πa²Ω, against r/a on a logarithmic axis, at νt/a² = 0.01, 0.1, 1, 10 and 100. At the surface it is the whole of it from the first instant, because no slip makes the fluid there turn with the cylinder. Just outside, the spin-up has laid down an equal and opposite ring of vorticity, so the circulation round a larger circle is only what has diffused past it: at two radii 0.000, 0.035, 0.611, 0.936 and 0.993 of the surface's at the same five times. The circulation a Magnus rotor needs is in the fluid the moment it spins; the far field learns of it only as fast as the counter-vorticity moves out.

A wall puts in exactly its own speed

A wall sliding in its own plane makes vorticity at a rate equal to its acceleration, with no viscosity in the rate. So however a wall is started, the vorticity it has put into the fluid is its speed, to the last digit; a wall that stops takes all of it back and leaves the fluid moving; and a spinning cylinder carries its whole circulation from the first instant, hidden behind an equal and opposite ring until viscosity carries the ring away.

kinematics · Vorticity 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 Six flows past one cylinder, all of them legal. The tangential speed on the surface for six values of the circulation. Every one of them solves the same equation and lets nothing through the wall; the fastest point on the surface runs from twice the free stream to eight times it.

Nothing in the present picks the flow

Six flows past one cylinder satisfy the same equation and let nothing through the wall, to the last bit of double precision. Their lifts run from zero to 37.7 and their peak suctions differ by a factor of twenty-one. The equations do not choose between them, and the thing that does is the history.

inviscid · Multiply-connected

Named alongside it

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

CirculationVorticityBoundary conditionStarting vortexConservationModel limitThe no-slip conditionPotential flowBound vortexBoundary layerIrrotationalKutta condition

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