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

Topology — where it appears

The properties of an arrangement that survive continuous deformation — how many holes a region has, how many times one curve threads through another. They constrain a flow more rigidly than continuous quantities do, because changing them requires something discrete to happen.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

A double integral that comes out an integer. The Gauss linking integral evaluated on six pairs of closed curves. It is not constrained to be a whole number by anything in its own definition — it is a double integral of a smooth kernel — and it returns one to within two parts in ten thousand on two hundred points per curve, because what it is computing is a topological count.

The knot a flow cannot untie

Ideal flow conserves energy, circulation and impulse, and all three are what they look like. It conserves a fourth quantity that is not: a volume integral of the velocity dotted into the vorticity, whose value counts how many times the vortex lines are linked through one another.

kinematics · Helicity 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 4 nodes and 2 saddles, and the difference is the shape. The skin-friction pattern on a body, drawn as the streaks a film of oil would leave, looking at the nose. Every critical point of the field is marked: 4 nodes, where the streaks converge or diverge, and 2 saddles, where two streaks cross. Their indices sum to 2, which is the Euler characteristic of a sphere, and nothing about the flow enters that number — the stream is at 0 and the crossflow at 0, and neither may change it. What the flow decides is where the points are and what they look like; what the body decides is how many there are of each.

The count computed on a body

The rule for a closed surface is usually quoted and seldom solved for. This solves for one, and what the computation adds is not confirmation — it is the discovery that the count survives two events in which the number of stagnation points falls, and that both of them have closed forms.

kinematics · Topology Four kinds of critical point, and the curve that separates them. The invariants of a trace-free velocity gradient, with the discriminant curve 27R²/4 + Q³ = 0 drawn through them. Inside the two upper lobes the cubic has one real root and a complex pair, which is a spiral being stretched along its own axis on the left and squeezed on the right; below the curve all three roots are real and the point is a node with two saddle directions. Of 820 random incompressible gradients, 509 land in the spiral region and 311 in the real one. A plane flow is the vertical line R = 0 and nothing else, which is why a plane has two kinds and space has four. The marked points are the cases the calculation checks that fall inside this window; the two vortex cases it also checks sit at Q = 3.25 and |R| = 4.25, off the top corners, because a window wide enough to hold them would flatten the curve the figure is about.

Two kinds is a plane flow's privilege

A plane incompressible flow has a saddle or a centre and nothing else, and the proof is one line about a trace. The same line in three dimensions constrains three numbers instead of two, which is far less, and what it leaves is four kinds of point separated by a curve — with the one a plane cannot have being the structure the whole of turbulence is made from.

kinematics · Topology A pair of points meets on R = 0, the one line an index can change on. The paths of the ABC flow's stagnation points across the (R, Q) diagram as C rises from 0.3 towards √2 with A = B = 1. The four points of index +1 share one path on the left and the four of index −1 its mirror image on the right. At C = 0.3 they sit at R = −0.088, Q = −1.045; at C = 1 they touch the discriminant curve at R = −0.707, Q = −1.500, where the strain has a repeated rate, and turn away from it without crossing; and as C approaches √2 they run in to R = -7.5e-3, Q = −2.000. Crossing into a lobe would have changed a node into a focus, which a Beltrami flow's stagnation point cannot be; reaching R = 0 is where each meets a partner of the other index.

The sign a stagnation point carries in space

In three dimensions a stagnation point's index is the sign of one determinant, and that determinant is minus the R of the invariant diagram — so the diagram's left and right halves are the two indices. An exact Euler flow in a periodic box has eight such points, four of each sign, never a spiral among them, and they can only disappear in pairs that meet on the one line where the sign is allowed to change.

kinematics · Topology 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.

CirculationCritical pointInvariantVorticityBifurcationConserved quantityEuler characteristicModel limitPoincare hopfStrain rateVelocity gradientBeltrami flow

All concepts