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

Uniqueness — where it appears

The question of whether a problem's solution is the only one. A potential flow in a region with a hole in it has a one-parameter family of them, one for each circulation, and nothing in the equations chooses — which is why the Kutta condition has to be an additional physical statement.

Named by 9 essays across 6 fields — each of them below, with the objects they name alongside it.

A Joukowski aerofoil at 8°. A cambered aerofoil in a uniform stream. The circulation is not chosen: it is whatever value makes the flow leave the sharp trailing edge smoothly, and that single condition fixes the lift.

The sharp edge decides

Ideal flow round a wing admits infinitely many solutions, each with a different lift, and all of them exact. One extra requirement — that the air leaves the trailing edge instead of whipping round it — picks a single one.

circulation · Lift The only candidate the far field allows, and the wall it slips past. The general Stokes solution has four constants; the condition at infinity kills two of them and fixes a third, leaving one to satisfy two conditions at the wall. Setting the stream function to zero there uses it up, and the tangential velocity that remains is exactly twice the free stream — for every radius, every speed, and every fluid.

The flow with no solution

Creeping flow past a sphere has a solution and everybody knows it. Creeping flow past a cylinder has none — not a difficult one, not one needing a clever method. The equations, the no-slip condition and the uniform stream at infinity are inconsistent, and the residual is exactly twice the free stream.

viscous · Stokes' paradox Two flows with one mean profile. The time-averaged velocity of a plain shear and of the same shear carrying a zero-mean disturbance. There is one line on this plot: the largest difference anywhere across the channel is four parts in 10¹⁷.

What a mean profile cannot tell anybody

Two flows are built here with mean velocity profiles that agree to four parts in 10¹⁷. One of them carries momentum across the shear and dissipates forty per cent more energy; the other carries nothing. Everything that distinguishes them is second order in a disturbance the mean cannot see at all.

kinematics · Averaging Five flows past one cylinder, every one of them a solution. Surface pressure round a cylinder in a stream, at five circulations. Each satisfies Laplace's equation, the tangency condition on the body and the condition at infinity, and each has a different lift. Nothing in the problem chooses between them: the domain has a hole in it, so the potential is many-valued and the circulation is a free constant.

The constant a hole leaves behind

In a region without holes, Laplace's equation and the boundary values have exactly one solution. Cut a hole and they have a one-parameter family. Nothing in the mathematics chooses between its members, which is why the Kutta condition has to exist and why it cannot be derived.

inviscid · Multiply-connected 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 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 The part of a field 4 cameras cannot see. Middle, a field with no symmetry on a 16 × 16 grid. Left, the part of it that projects to exactly nothing in every one of 4 views, shaded one way above zero and the other below: a pattern of streaks and hollows that cancels along every ray. Right, the field with that part taken away. The middle and right fields give identical pictures in all 4 views, to 2e-13 of the largest ray, and no reconstruction from those views can tell them apart. The invisible part is one combination of 177 independent patterns the views cannot see.

Four cameras and a field they cannot see

An axisymmetric flow can be rebuilt from one photograph because its symmetry supplies every other view. A flow without an axis has to be photographed from several directions, and a few directions do not merely give a noisy answer — they leave whole patterns of density that every camera records as nothing. Four views of a 16 × 16 field see 79 of its 256 independent patterns and are exactly blind to the rest.

misconceptions · Visualisation 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

Named alongside it

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

MeasurementModel limitBoundary conditionCirculationKutta conditionRegularisationDragIll-posedInstrumentInverse problemLiftMemory kernel

All concepts