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

Unsteady — where it appears

Changing with time at a fixed point, so that the time derivative in the equations does not vanish. Unsteadiness separates streamlines from particle paths, adds an inertia term to every force, and makes the average of a flow a different object from the flow.

Named by 12 essays across 6 fields — each of them below, with the objects they name alongside it.

The same patch, 6 periods later, in two flows. A round patch of 848 marked particles, advanced 6 periods by the blinking flow and by a steady flow of the same strength. The steady flow has drawn the patch into a smooth ribbon along a streamline and every particle in it is still on the streamline it started on; the blinking flow has folded the patch through itself repeatedly and its particles are spread across the whole region. Neither flow has any diffusion in it and neither has lost a particle. The difference between them is that one depends on time.

No randomness, and it mixes anyway

A steady two-dimensional flow cannot mix, however fast it is stirred, because its trajectories are its streamlines. Switch two vortices on and off alternately and the same fluid, obeying an exact map with nothing random in it, folds a patch of dye through itself until neighbouring particles separate by a factor of a thousand in six periods.

kinematics · Advection α = 13.0: a plug in the middle and everything happening at the wall. The velocity profile at eight phases of one cycle, at a Womersley number of 13.0 — the human aorta's at rest. The core moves almost as a solid plug, because viscosity cannot reach it within a cycle; all the shear is in a layer of thickness √(ν/ω) = 0.77 mm against a radius of 10.0 mm. At some phases the fluid near the wall is moving backwards while the core still moves forward, which is the reversal a Poiseuille profile can never show and which is routinely measured in arteries.

Too fast for a profile

A pipe carrying a steady flow has a parabolic profile. Make the pressure oscillate and one number decides whether it still does — and above about ten the core moves as a plug, a quarter of a cycle behind the pressure, with the fastest fluid in a ring near the wall rather than on the axis.

regimes · Womersley A steady pressure field, from a flow with no steady part. The time-averaged pressure round a cylinder in a stream that oscillates as U₀cos ωt. The mean velocity is exactly zero at every point — the flow spends as long going one way as the other — and the mean pressure is not, because pressure depends on the square of the speed and a square has no sign. The mean coefficient reaches -2.00 at the shoulders and averages -1.00 over the surface, and its resultant is 6.6e-16: a real field with no force in it. The pale lines are the instantaneous streamlines, which reverse every half cycle.

The mean is not the flow

Average an unsteady flow and the result is a new object with its own properties, and it is not a solution of anything. An inviscid stream oscillating about zero has a mean velocity of exactly nothing everywhere, a mean pressure that reaches minus two dynamic pressures at the shoulders, and a missing term in its own momentum equation that can be written down in closed form.

kinematics · Averaging An oscillation with no mean, and the steady flow it drives. The steady second-order velocity through a Stokes layer, in units of U U′/ω. The first-order flow averages to zero at every height; the average of its own nonlinear term does not, and the pale curve is that forcing. Integrating it twice across the layer, with no slip at the wall and no stress at the top, gives a steady velocity that rises through the layer and settles at -0.749998 — Rayleigh's −3/4, which was not put in anywhere. Beyond about five layer thicknesses nothing more happens, which is why the number is a boundary condition for the flow outside.

An oscillation with somewhere to go

Shake a fluid back and forth over a body and it develops a steady circulation that never reverses. The driving flow has no mean at all; the mean of its own nonlinear term does, and integrating that twice across the oscillatory layer gives a slip velocity of exactly three-quarters of U dU/dx over the frequency.

viscous · Streaming Where four insects have to turn round. Wagner's function — the fraction of its eventual circulation a wing has built after a stated distance of travel — with the half-stroke of each of four insects marked on it. Every one of them reverses while the curve is still climbing, so no insect wing ever reaches the circulation a steady calculation assigns it.

A calculation with no memory in it

The bee calculation is famous for using the wrong velocity. Done with the right one it still falls short, and the reason is structural: a quasi-steady sum is a statement about a wing that has always been going, and an insect's wing travels between two and five chord lengths before it turns round.

misconceptions · Bumblebee The exposure lengths that give the true mean, and the ones that do not. How far a finite exposure of a shedding wake lands from the long-exposure mean, against how many shedding periods the shutter was open for. It is exactly zero at every whole number of periods — an average over a complete cycle is the mean, with no error at all — and between the zeros it falls as one over the exposure. Nothing about the picture tells the reader which of these they are looking at.

The shutter is part of the answer

Flow photographs are compared as though the exposure were a detail of the camera. It is a term in the measurement: an average over exactly one shedding period returns the true mean to fourteen figures, any other length carries a residue that falls only as one over the exposure, and a two-pulse velocity reading is short by exactly the sinc of the swept angle.

misconceptions · Visualisation Where a profile stops being a parabola. Two measures of how far Womersley's solution has left the quasi-steady parabola, against the Womersley number, both logarithmic. The amplitude deficit reaches a hundredth at α = 0.28 and the phase lag reaches a hundredth of a right angle at α = 0.22 — both well below α = 1, which is where the unsteady and viscous terms are equal and is the value the number is named for. By α = 1 itself the flow is already a tenth short and eight degrees late.

Where the parabola goes

A pipe carrying a steady flow has a parabolic profile, and the Womersley number is supposed to say when an oscillating one still does. For the flow rate it is very nearly honest — one per cent at α = 0.91 — and for the phase it is out by a factor of three, because a lag is second order in the number and an amplitude deficit is fourth.

regimes · Womersley A uniform scalar in a fluid at rest, under two face rules. Nothing is flowing and the scalar starts at one everywhere. The swept-volume rule leaves it at one to the last bit, at every step of the two time units. The midpoint rule moves it by two parts in ten thousand, on a mesh motion that begins and ends in the same place, and the excursion looks exactly like a physical transient.

The mesh that makes its own mass

The transport theorem holds for a region moving at any velocity, which is what makes a moving-mesh calculation possible. Discretised carelessly it is not an identity but an approximation, and a fluid at rest with a uniform density then gains density from the motion of a grid — smoothly, plausibly, and looking exactly like a physical transient.

kinematics · Transport theorem How much the peak cares about the valve at each instant. A small bump is added to the valve's opening at one moment and the peak is recomputed; the difference, divided by the bump, is plotted against the moment. It is flat and zero for the first 0.45 seconds — that part of the closure is invisible to the peak — rises through the reflection arrivals, and falls to nothing after the peak has happened, which is the one part of the shape that needs no fluid mechanics.

The part of the closure a pipe cannot see

A surge specification says how long the valve takes to shut. The line does not integrate the duration, it integrates the shape — and it is measurably blind to the first half-second of an eight-second closure, while three closures of identical length give peaks 97 per cent apart.

applied · Water hammer The one term in ground effect that really is about carried air. The added mass of a plate approaching a plane, as a multiple of its free-air value. In free air a plate borrows ρπc²/4 per unit span — exactly 0.79 for a unit chord in unit density, which is the mass of the circle its chord spans. Near a wall the fluid in the gap must leave sideways through a narrowing passage, so it moves faster than it would in the open and carries more energy: the borrowed mass grows as the inverse of the gap and diverges as the gap closes. At half a chord it is 1.21 times, at a tenth 2.06, at a twentieth 3.12. This is a cushion in the ordinary sense — fluid that has to be got out of the way and resists being — and it is the term the steady argument has none of.

The cushion that is there after all

A wing near the ground makes more lift for reasons that have no cushion in them, and that account cannot reach two cases — a wing descending, and a rotor in the hover, where there is no steady flight for the image-vortex account to be about. Those cases have a term that really is about carried air, and it grows without limit as the gap closes.

misconceptions · Ground cushion A threshold that is a curve, not a pressure. The tension a rectangular pulse has to reach to make a five-micron bubble run away, against how long the pulse lasts. At a tenth of a microsecond it takes forty bar; at a hundred microseconds it takes 1.05, which is within about one per cent of the static threshold. There is no such thing as the cavitation pressure of this bubble on its own.

A threshold that is also a duration

The cavitation number treats inception as a pressure: below it the liquid tears, above it does not. A five-micron bubble asked to grow in 0.3 microseconds needs 43 bar of tension and the same bubble given a hundred needs 1.05, because it has to make a journey and not merely respond.

applied · Cavitation A body of no volume with a finite added mass. Thin an ellipse towards a plate and the fluid it displaces goes to nothing while its broadside added mass does not move at all — it stays at πρa² to the last digit. Whatever added mass measures, it is not how much fluid a body carries with it: the ratio of the two diverges as the reciprocal of the thickness, reaching a million at a thickness ratio of a millionth.

The mass a body has to borrow

Accelerate a sphere through water and it resists as though it were half again as heavy. The half is exact, it is a rational number rather than a measurement, and almost everything a reader infers from it about carried fluid is false.

inviscid · Impulse

Named alongside it

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

Memory kernelAdded massAveragingMeasurementModel limitModel validityBoundary layerDimensionlessEulerian and LagrangianMixingMomentumNonlinearity

All concepts