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

Regularisation — where it appears

The deliberate reintroduction of a length that an idealisation removed, so that a singular model can be computed. A vortex sheet needs one before it can be integrated past its own singularity, and the answer then depends on the length chosen rather than on anything physical.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

The growth rate a discretised sheet has, at every wavelength it can carry. Kelvin–Helmholtz gives a growth rate proportional to the wavenumber and without bound. A sheet represented by N point vortices has pi m (1 − m/N) instead — the same rate at long waves and half of it at the shortest wave the grid carries, with the fastest-growing mode at the grid scale itself. Smoothing the kernel over a length delta moves that mode back to a wavelength the physics chose.

A sheet that cannot stay a sheet

Let a shear layer's thickness go to zero and it becomes a surface across which the velocity jumps. The model is used everywhere in this subject, it is unstable at every wavelength, and the thing it does next is worse: it develops a singularity in its own shape, at a finite time, from a smooth start.

inviscid · Vortex sheet 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 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 Four wakes carrying exactly the same drag. Four velocity-deficit profiles behind a body, each normalised so that the integral of the deficit across the wake is exactly the same. That integral is the drag: the far-wake momentum balance says so with no assumption about the shape of anything. The four are a narrow Gaussian, a wide top hat, the two-lobed wake a body with a splitter plate leaves, and a profile with heavy tails.

Exact in the total, free in the profile

A constraint is one number imposed on a function. Four wakes built to carry exactly the same drag differ by a factor of four and a half in their peak deficit, and the general statement behind that is a question about angles: how much of the wanted answer survives being projected off the constraints, and how much does not.

regimes · Dimensional One body, three interior representations. An ellipse four fifths as tall as it is long, in a uniform stream, with three sets of singularities inside it: a point doublet at the centre, a ring of them at four tenths of the semi-major axis, and a uniform disc of them. Outside the ring the three produce the identical flow to fifteen figures. Inside, they are not remotely the same field.

The inside a flow does not decide

Every body in potential flow is built out of singularities that are not there. The exterior flow does not merely fail to determine them — it leaves an infinite family, whose members produce the identical field to the last bit outside and are nothing alike inside, and whose coefficients span five orders of magnitude for one unit free stream.

inviscid · Singularities A vortex sheet rolling up, at five stages. One period of an initially flat vortex sheet with a small perturbation on it, drawn at equal intervals. The perturbation grows, the sheet steepens, and the ends wind into a spiral. Nothing is added to the sheet after the first instant.

A spiral is a legible record

When a vortex sheet rolls up, the fluid in it can never change places: two points on a sheet cannot pass one another. So the arms of the spiral are a map of the initial sheet, in order, and the picture is a record of the roll-up rather than a snapshot of it.

inviscid · Vortex sheet Four views, four guesses at what they missed. The earlier essay's field — two puffs and an oblique streak — and its reconstructions from the same four views, each choosing the part of the field the views cannot see by a different assumption. With none it is set to zero: an error of 42.8 per cent. The smoothest field the views allow: 39.8. A non-negative field: 24.7. A sum of Gaussian puffs of one width: 37.6. The first three meet every one of the 96 rays; the dictionary's does not.

A prior the pictures can refute is the one worth having

A few views of a flow with no symmetry leave most of the field unseen, and a reconstruction has to fill that part in by assuming something. Three common assumptions were tried on the same field and the same views. Non-negativity saves three views of ten, and five on a field of puffs; smoothness saves one; a dictionary of known shapes saves none unless it is exactly right, and when it is nearly right it can be wrong by a factor of a hundred. The prior that helps most is also the one the data can catch being wrong.

misconceptions · Visualisation With the weight chosen from the data, non-negativity still saves views, and the saving shrinks as the noise grows. The fewest views that bring each prior's reconstruction of a field of two puffs within fifteen per cent of the truth, against the noise on the rays as a share of the rms ray: solid with each prior's weight chosen by the discrepancy principle, from the data alone; dashed with the weight that knows the answer. Points drawn at 25 never get there. At one per cent noise no prior needs 8 views and non-negativity 4; at three per cent, 8 and 6, against 4 with the answer known; at ten per cent, more than 24 and 10, against 7.

A prior chosen from noisy pictures keeps most of its gain, and a wrong one loses its alibi

With exact rays, a reconstruction's prior could be judged by its error, and a constraint that was wrong gave itself away by missing its own rays. With noise every prior needs a weight — how far to trust the rays against the assumption — and the weight has to be chosen without the answer. Chosen where the misfit falls to the noise, it costs the smooth and unassuming priors almost nothing and non-negativity two or three views, so the ranking survives. What does not survive is the refutation: a dictionary that misses a field's rays by four per cent is caught at three per cent noise and invisible at seven, where it fits the noise as well as any prior and is about as accurate as they are.

misconceptions · Visualisation

Named alongside it

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

Model limitCirculationMeasurementNull spacePoint vortexUniquenessContour dynamicsDiscretisationEquilibriumFlow visualisationIll-posedInverse problem

All concepts