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 14 essays across 2 fields — 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 Every gradient in one plane, and where each one goes. The two invariants of a trace-free velocity gradient, R across and Q up. Under the restricted Euler equation the combination 27R²/4 + Q³ never changes, so every trajectory is one of these curves, and R never decreases, so every curve is travelled from left to right. Vieillefosse's line, where the combination is zero, has two branches: the left one runs into the origin and is the only way to reach it, and the right one is where every other trajectory ends, at infinity, in finite time.

A gradient left to itself

Follow a parcel's velocity gradient with nothing acting on it but its own square and the part of the pressure a single parcel can know about. Every starting gradient but a set of measure zero reaches infinity in a finite time, and it gets there as a sheet with its spin lying in the sheet.

kinematics · Deformation Every particle leaves a flow that is a vortex at every instant. Eight fluid particles starting on a small circle in Haller's rotating-saddle flow, followed for 2.4 time units. At every instant the velocity gradient has complex eigenvalues and a positive Q — the flow is a vortex by every criterion that reads a snapshot — and every particle spirals outward, its distance growing as e to the time. The fluid is not held; it is flung.

Every snapshot says vortex, and every particle leaves

There is a flow in which the velocity gradient has complex eigenvalues at every point and every instant — a vortex by every criterion that reads a snapshot, in the frame the flow is measured in — and in which every fluid particle is flung away exponentially. The snapshot is not wrong about the gradient. It is wrong about the fluid, and it is wrong exactly when the strain's axes turn.

kinematics · Frames Remembered pressure turns the runaway into a teardrop. The velocity gradient's two invariants, Q against R, both scaled by the mean enstrophy, for an ensemble of parcels whose pressure remembers one Kolmogorov time of deformation, at a memory of a tenth of a large-eddy time. The restricted Euler equation sends every parcel off to infinity along the right-hand branch of Vieillefosse's curve, dashed. With the remembered pressure the parcels stay: they crowd along that branch in the strain-dominated quadrant and above the axis on the left where vortices are being stretched — the teardrop measured in turbulence.

The pressure a parcel remembers keeps it finite

The restricted Euler equation follows a parcel's velocity gradient with the pressure's shape thrown away, and every gradient it follows blows up. Give the parcel back a pressure that remembers how its neighbourhood was deformed over the last Kolmogorov time — nothing more — and no gradient blows up at all. The ensemble settles into the teardrop measured in turbulence, its vorticity lines up with the middle strain axis, and its intermittency grows as the memory shortens. What the memory cannot do is keep the one identity homogeneity demands, and that miss says where the rest of the pressure lives.

kinematics · Deformation Moving the saddles never lowers the quarter; fast large wobbles raise it. The smallest Stokes number at which a cloud of particles in the oscillating lattice folds within sixty time units, against the oscillation's amplitude ε, at frequencies of 1, 3 and 10 times the saddles' strain rate. With ε = 0 it is the steady lattice's 0.251. Moving the lattice never takes it below a quarter; at ε = 1 it is 0.283, 0.311 and 0.324 at the three frequencies.

Wobbling saddles keep the quarter, and raise it

Heavy particles in a steady lattice of vortices fold onto the cell walls only above a Stokes number of a quarter, the converging saddles' own threshold. Random flows fold at every Stokes number, so it was natural to suspect the quarter of belonging to steady saddles. Set the lattice wobbling and the particles cross from cell to cell, seventy per cent of them in forty time units, and still none folds below a quarter. The threshold belongs to the strongest strain the flow has; moving the saddles only moves particles out of reach of it, and raises the threshold.

kinematics · Acceleration A stronger strain folds particles below a quarter only if it lasts. The Stokes number above which particles in a lattice with pulsing strain 1 + a sin ωt fold onto its walls, against the pulse's frequency, for amplitudes of 0.15, 0.3 and 0.5, with the steady lattice's quarter dashed. A slow pulse folds them at the peak strain's quarter, 1/(4(1 + a)): 0.194 at a = 0.3 and ω = 0.05, against 0.192. A fast one folds them only at the mean's: 0.249 at ω = 5.

A stronger strain folds a cloud only if it lasts

Heavy particles in a lattice of vortices fold onto the walls above a Stokes number of a quarter of the saddles' strain. Let the strain pulse and the threshold moves — down to the quarter of the peak strain when the pulse is slow, back up to the quarter of the mean when it is fast — because a particle responds to the strain it has been in for about four strain times, not to the strain at the instant. In a random strain the same rule makes the folds arrive in bursts, at the excursions that are strong and long at once.

kinematics · Acceleration Saddles that share one signal fold a cloud in a single step; independent saddles fold it steadily, and less. The share of a cloud seeded over the whole patch that has folded, against time, at a Stokes number of 0.16, in four seeded fields: with every saddle's strain the same signal, and with each saddle its own. After 400 time units the shared signal has folded 0.89, 0.89, 0.89, 0.89; the independent saddles 0.77, 0.85, 0.66, 0.69. A ninth of each cloud starts at a vortex centre, where the shared lattice holds it still.

A cloud folds where a saddle is strong, not when

When every saddle of a lattice strengthens and weakens together, a cloud of heavy particles folds in one step, at the strain's first strong excursion. Give each saddle its own random strength and the folding spreads out into a steady rate — and falls, because a particle now has to be at a saddle while that saddle is strong. The rate follows the Gaussian tail of the strain, at about a tenth of the time the strain spends past its fold condition, and a longer-lived strain helps only up to a few strain times.

kinematics · Acceleration Noise in every frame pulls in the frame interval that stops buying, by the same share whatever the window sees. The longest frame interval at which a run still collects nine-tenths of continuous filming's independent values, over its value with no noise, against the noise's share of the window mean's variance, for a velocity window one integral scale wide, one a twentieth as wide, and a gradient in the small window; dashed, a single exponential memory in closed form. velocity, window ℓ: 0.782 at ν = 0.1, 0.199 at ν = 1; velocity, window ℓ/20: 0.804 at ν = 0.1, 0.243 at ν = 1; gradient, window ℓ/20: 0.819 at ν = 0.1, 0.242 at ν = 1.

Noise in every frame is a memory of zero, and the run cannot measure it at its own pace

A camera that adds an independent error to every frame adds a part of the signal with no memory at all. Infinitely many frames would still average it away, so the run's information still approaches continuous filming's — but the frame interval at which a faster camera stops buying is pulled in by the same share for a wide window, a zoomed one and a gradient, about half at a noise of three-tenths of the variance and a fifth at a noise equal to it. And the usual way to learn the noise, from the run's own correlation extrapolated to zero lag, works only at frames much faster than the run's information needs.

turbulence · Spectrum

Named alongside it

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

Strain rateModel limitVorticitySaddleParticleQ-criterionCritical pointInertiaInvariantMeasurementObjectivityStagnation point

All concepts