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

Trajectory — where it appears

The path a particle or a body actually takes through space, as against the streamlines of the field it moves in. It is an integral of the forces along the way, so two bodies launched identically into different histories arrive in different places.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 Five points, and where each one's fluid came from. Back-trajectories through six units of time in an unsteady double gyre. Each curve ends at the place the fluid now at the marked point started; nothing about that place can be read off the velocity at the marker.

A scalar is a record of where its fluid was

A conserved scalar has no value of its own. Its value at a point is whatever it was at the place that point's fluid started from, which makes a dye field a photograph of the past — and makes the map from now to then the only thing in the flow that carries the past at all.

kinematics · Advection Spin against distance flown, not against time. The spin of a struck golf ball as a fraction of its launch spin, against how far it has flown. The integrated flight sits exactly on an exponential in distance, because the spin-down torque is proportional to speed times spin and the speed cancels. The constant is 1,202 metres and the drive is 200, so the ball arrives with five sixths of the spin it left with.

The ball that never forgets its spin

Commentary explains a late-swerving ball by saying the spin is dying away. The spin-down torque goes as speed times spin, so spin decays over a distance rather than a time — 1,202 metres for a golf ball against a 200-metre drive — and what dies away is the speed, which makes the swerve stronger.

applied · Sport ball The line the dye actually draws. A streakline in an oscillating uniform stream: everything released from the origin over the last 3.4 units of time, drawn where it has got to at one instant. The parameter along it is release time, not distance and not time of flight, which is why it is a record rather than a curve of the field. The mean speed of 1 exceeds the amplitude of 0.6, so the fluid never reverses and the filament is single-valued in x.

The line the dye actually draws

The streakline is the curve most photographs really show, and of the three curves it is the hardest to compute, because it needs the whole history of the flow. This draws it, in a flow where all four curves have closed forms, and the third curve turns out to be a different kind of object from the other two rather than a third example of the same one.

kinematics · Flow curves

Named alongside it

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

Flow mapInitial conditionMeasurementMemory kernelMixingModel validityAdvectionAngular momentumAveragingConservationDimensional analysisDispersion

All concepts