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

Eulerian and Lagrangian — where it appears

The two ways of describing a flow: by the field at fixed points, or by following the parcels. They give different averages of the same motion, and a quantity that is carried — a pollutant, a float, a marked parcel — moves with the second.

Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.

The acceleration field of a steady flow. How hard the fluid is being accelerated at each point of a steady flow past a cylinder. The flow does not change with time anywhere in this picture, and yet almost nowhere in it is a parcel travelling at constant velocity — the pattern stands still while the fluid running through it is thrown about.

Steady does not mean nothing is happening

Photograph the flow past a cylinder twice and the two pictures are identical. Every parcel of air in them is being thrown about — braked to a dead stop, hauled round the shoulder at nearly twice the free-stream speed, braked again. Both statements are exactly true.

kinematics · Acceleration 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 The fluid a moving cylinder carries with it. Kinetic energy density around a cylinder moving through fluid that is at rest far away. The fluid is not dragged along in a lump: it is pushed aside in front and closes in behind, and the energy in that motion is what has to be supplied to change the body's speed.

The force of getting going

The exact theory says a body moving steadily through an ideal fluid feels no force at all. It does not say the fluid is free. Accelerating the body has to accelerate the fluid too, and the bill for that is exactly the mass of fluid the body displaces.

inviscid · Dalembert Wound up, and worth exactly what it started with. A material loop in a steady cellular flow — an exact solution of Euler's equations — drawn at four times. Each streamline in the cell has its own period, so the loop is stretched steadily into a spiral: by the last frame its perimeter is 5.7 times what it started as. The circulation round it is 0.903741 at the start and 0.903666 at the end. Nothing about the curve survives except the number.

What survives being wound up

Draw a loop of marked fluid particles and let the flow carry it. It will be stretched, folded and wound into a spiral until nothing about its shape is recognisable, and the circulation round it will not have moved at all — provided three conditions hold, each of which can be broken on purpose.

inviscid · Kelvin Four turns of a wave, and the parcel is not back. Two parcels traced through four periods of a linear deep-water wave of steepness 0.1, by integrating the exact velocity field. Each orbit is very nearly a closed circle and misses closing by a little, every time, in the same direction — that miss is the whole of the Stokes drift. The near-surface parcel advances 0.0357 of a wavelength over the four cycles and the one a tenth of a wavelength down advances 0.0122, a third as far — because the drift falls off twice as fast with depth as the orbit's own size does.

The drift in a wave that has none

The velocity at any fixed point under a passing wave averages to exactly zero, and every parcel of water in it moves steadily forward anyway. The orbits do not close, they miss by the same amount every time, and the missing amount is the square of the steepness times the wave speed.

kinematics · Stokes drift A material region, and the dye that stays inside it. The same fluid at four times, carried by an unsteady straining flow whose strain rate oscillates. The outline is a circle of the fluid at the first instant, tracked by integrating the velocity field; the shading is a blob of passive dye. The region is stretched to nearly seven to one and its area is unchanged to fifteen decimal places, because the flow is incompressible. The amount of dye inside it is unchanged to thirteen, because the dye is carried by the same fluid.

A rate of change that will not hold still

Three boxes drawn in one flow at one instant give three different answers to how fast the dye inside them is changing — one falling, one falling twice as fast, one rising. All three reconcile with a single material rate, and that rate is zero.

kinematics · Transport theorem A patch of dye, folded. A circle of marker particles carried by the flow, at four times. It is stretched into a filament and folded through itself, and its area does not change at any point of that — which is not visible in the picture, and is the whole difficulty. A scheme that lost eight per cent an orbit would produce a picture indistinguishable from this one.

The area that must not move

A patch of dye in an incompressible two-dimensional flow keeps exactly the area it started with, for ever. Two respectable integrators are put on the same flow: one respects that identically at any step size, the other does not, and the pictures they draw are the same picture.

kinematics · Advection And the net drift, which is where they disagree. The Stokes drift plus each return flow. All three drift forward at the surface, because the Stokes drift there swamps any return current of the right size. Below that they part company completely: the uniform current has most of the column moving upstream, and the two that satisfy no slip have almost none of it. The reversal depths span 77 per cent of the water column.

The drift a closed box will not allow

A wave in a wave tank carries mass forward, and the tank has nowhere to put it. So a return current appears carrying exactly the opposite transport — exactly, from mass conservation and nothing else. Which fixes a total and leaves the answer anybody wants entirely open.

kinematics · Stokes drift 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 Six orbits that do not close. One parcel's path under a linear deep-water wave of steepness 0.05, at a fifth of a wavelength down, released at the phase that centres the orbit on its release depth. Each loop returns almost to where it began and not quite.

A drift made of two things that average to zero

Stokes drift is usually explained as a parcel spending longer in the forward half of its orbit. That is true and it is not a formula. The formula is a correlation between a displacement and a gradient, each of which averages to exactly nothing, and it splits into two halves that are equal to twelve figures.

kinematics · Stokes drift A float under swell on a rotating planet goes round instead of away. The track of a float at the surface over 1 inertial periods (16.9 hours) after a 8 s swell of amplitude 1 m arrives at latitude 45°, in kilometres, the waves travelling to the right. Without friction the float runs round a circle of radius Uₛ/f = 0.479 km and comes back to where it started every 16.92 hours. With a drag on the Eulerian current it spirals out into a steady drift veered to the right: 24.3 per cent of the drift at 76.0° for a drag of 0.25 f; 70.7 per cent of the drift at 45.0° for a drag of 1 f; 94.9 per cent of the drift at 18.4° for a drag of 3 f. In a non-rotating ocean the same float would have gone 3.0 km straight on.

The drift a rotating planet takes back

In a wave tank the Stokes drift is cancelled by a return current because the tank has walls. The open ocean has none, and the drift is cancelled anyway: the Coriolis force acts on the water's real motion, drives an Eulerian current that answers it, and leaves the depth-integrated transport exactly zero at every viscosity. A float under steady swell with nothing to stop it goes round a circle instead of away.

kinematics · Stokes drift The kept transport spirals into nothing as the sea deepens. The net Lagrangian transport as a vector, scaled on the Stokes transport, traced as the water depth increases from a quarter of an Ekman depth to eight, for an 8-second swell with an eddy viscosity of 0.01 m²/s. Shallow water keeps the whole transport pointing with the waves, at the right-hand end. As the sea deepens the vector shortens and swings to the right, crosses the across-wave axis near two Ekman depths, and winds into the origin, which is the open ocean's exact cancellation.

The floor that gives the drift back

In the open ocean the Coriolis force drives a current that cancels a swell's Stokes transport exactly. Over a continental shelf the sea floor holds a stress, and whatever it holds is transport the rotation does not take back. How much survives depends almost only on the depth in Ekman depths; which way it points depends on the wave.

kinematics · Stokes drift

Named alongside it

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

ConservationTransportMass conservationMeasurementStokes driftWavesAveragingBoundary conditionControl volumeMaterial derivativeModel limitModel validity

All concepts