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

Index theorem — where it appears

A statement that the indices of the isolated zeros of a vector field add up to a number fixed by the topology of the space alone. In a flow it counts the stagnation and separation points that must exist, whatever the shape of the body or the details of the motion.

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

Two centres, two saddles, and a sum of nothing. A separation pattern: a uniform stream with two counter-rotating cored vortices in it, which reproduces the arrangement of critical points behind a body at a Reynolds number of a few tens. There are exactly four — a saddle where the flow divides, a centre in each recirculating cell, and a saddle where it closes — and their indices sum to 0. The winding number of a loop enclosing all of them is 0, which is what a uniform stream far away requires. A bubble costs nothing in this bookkeeping, which is why one is free to appear.

The count a pattern cannot break

A picture of a flow has stagnation points in it, and they are not free to be arranged as anybody likes. Their kinds and their number obey an integer constraint that has nothing to do with the equations of motion — and an incompressible flow in a plane is allowed only two kinds of them in the first place.

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 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

Named alongside it

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

BifurcationCritical pointSaddleStagnation pointPoincare hopfSeparationTopologyVelocity gradientDividing streamlineEuler characteristicIncompressibleInvariant

All concepts