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

Velocity gradient — where it appears

The rate at which velocity changes with position, whose symmetric part is a straining and whose antisymmetric part is a rotation. Splitting it is the first thing kinematics does, and the split is what separates deformation from spin.

Named by 7 essays across one field — each of them below, with the objects they name alongside it.

A circle of fluid, 0.90 of a gradient time later. A material circle in a uniform velocity gradient, carried by the exact matrix exponential of that gradient. It becomes an ellipse — always an ellipse, for every gradient — and the axes it stretches along are the eigenvectors of the symmetric half. In an incompressible flow the area is unchanged however extreme the distortion, which is the statement that stretching in one direction is squashing in the other.

What a parcel does in the first instant

Drop a circle of dye into a flow and it becomes an ellipse. The velocity gradient that did it splits into a stretch and a spin in exactly one way, the split is not a convention, and one half of it is the reason a fluid has any stress in it at all.

kinematics · Deformation 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 One flow, two observers, two pictures. The same ideal flow past a circular cylinder, drawn in the frame of the tunnel and in the frame of the undisturbed air. The two are related by subtracting one constant velocity. On the left the flow arrives from infinity, divides at a stagnation point on the nose and closes at another on the tail. On the right the air is at rest far away, the body pushes through it, the streamlines are closed loops, and there is no stagnation point anywhere in the field. Every force, every pressure and every measurement either observer can make is identical.

The picture belongs to whoever is watching

Photograph the flow past a cylinder from the tunnel and it has two stagnation points. Photograph the same flow from a frame moving with the air and it has none at all, and its surface speed is exactly the free stream at every angle. Both pictures are correct and no measurement distinguishes them.

kinematics · Frames Two flows with the same rate of strain, doing different things to a blob. A circle of fluid carried by two flows chosen to have exactly the same rate-of-strain magnitude, drawn at four times. Pure strain pulls it into an ellipse whose axes stay put; simple shear pulls it into an ellipse whose axes rotate as fast as they stretch. Both have zero divergence, so both preserve the area. The difference between them is not the strength of the straining but what the rotation does to the direction being stretched.

Longer, with nothing pulling it

Two flows with exactly the same rate of strain. In one a line of fluid grows by a factor of 148 in five time units; in the other it grows by 10. Turn the straining axes faster than the strain rate and no line grows at all, however hard the fluid is being strained.

kinematics · Material lines 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 The one curve of the four that measures a gradient. A row of particles released along a line at one instant, seen at 4 later times, in a layer profile. Each line has been displaced by the local velocity times the elapsed time and by nothing else, so dividing the displacement back out returns the profile exactly — checked here to machine precision at every one of two hundred heights. A streamline, a pathline and a streakline each report where fluid went; this one reports how fast neighbouring fluid was going relative to its neighbour, which is the quantity a boundary layer is made of and the one the other three never show.

The curve that measures a gradient

Three of the four curves drawn through a flow answer the same question — where did the fluid go. The fourth answers a different one. A line of particles released together is displaced by the local velocity and by nothing else, so its shape is the velocity profile, and dividing the elapsed time back out returns that profile exactly rather than approximately.

kinematics · Flow curves 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

Named alongside it

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

Strain rateVorticityCritical pointModel limitStagnation pointBifurcationIncompressibleIndex theoremInvariantMeasurementObjectivityQ-criterion

All concepts