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 8 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 A stagnation point can leave the flow through the wall, and the wall's point changes kind to pay for it. The heights of the interior stagnation points on the axis of a swirling stagnation flow onto a plate, against the skin friction's spreading rate σ: the lower point, of index +1, and the upper, of index −1; along the wall, the skin-friction focus, a separation of index +1 where σ < 0 and an attachment of index −1 where σ > 0. As σ falls to zero the lower point descends into the wall as z ≈ σ and the wall's focus turns from attachment to separation; as σ rises to one half the two interior points meet at a height of one and vanish. Interior plus half the wall is −½ throughout.

A stagnation point can leave the flow through the wall, and the count is kept by halves

A no-slip wall is a sheet of zeros, so the index count that holds for a flow in open space has nothing to count there. Divide the velocity by the height and the wall's zeros shrink to the skin friction's nodes, foci and saddles; reflect the result through the wall and each of them is counted once while every point in the fluid is counted twice. That is why the points on a wall are halves, and an exact swirling flow onto a plate shows the halves doing their job: a stagnation point in the fluid can disappear without a partner by sinking into the wall, where the skin friction's focus changes from attachment to separation to pay for it. A section through the flow sees none of this, because in the section every one of the points is a saddle.

kinematics · Topology A stagnation point hangs above the skin friction's saddle while the saddle is a separation, and is gone once it is not. Streamlines in the plane y = 0 of a wall flow with no axis of symmetry, at μ = 0.4: the wall's saddle of skin friction at x = -0.632, its node at 0.632. An interior stagnation point sits at height 0.15 above x = −0.5, and the saddle is a separation: the flow leaves the wall there.

A stagnation point off the axis can land only on a saddle or a node, and turns it inside out

In a swirling flow onto a plate every stagnation point lived on the axis, so one sinking to the wall always met the wall's single critical point. Without the symmetry it might seem free to land anywhere — on an ordinary stretch of skin-friction line, leaving a pair of new points behind. It cannot. Divided by the height, the velocity is continuous down to the wall, where it becomes the skin friction, so a point can only reach the wall where the skin friction already vanishes. It lands on a saddle or a node, and the count does the rest: the point it lands on turns from separation to attachment. An oil-flow picture of the wall, which shows the same pattern before and after, cannot tell which it is looking at.

kinematics · Topology

Named alongside it

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

Critical pointBifurcationCirculationInvariantModel limitPoincare hopfSeparationSkin frictionVorticityConserved quantityEuler characteristicIndex theorem

All concepts