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

Advection — where it appears

The carrying of a quantity by the flow itself, as distinct from its spreading by diffusion. In two dimensions and with no compressibility it is exactly area-preserving, which makes it a Hamiltonian system with the stream function as its Hamiltonian.

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

An hour for every tenfold, for ever. How long a forecast lasts, against how well the initial state is known. The relation is T = ln(tolerance/error)/lambda — exactly logarithmic — so improving the measurement by a factor of ten buys exactly the same extra time every time: ln(10)/lambda, which for this flow is 24.5 time units. It does not get harder and it does not get easier.

An hour for every tenfold

Turbulence is deterministic and unpredictable, and the exchange rate between those two is exact: measuring the initial state ten times better buys the same extra forecast time every time, for ever. A constant, and it belongs to the flow rather than to the instrument.

misconceptions · Randomness 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 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 The stretching a window of history did, drawn as a field. The largest finite-time Lyapunov exponent over eight units of time in the double gyre, darkest where two neighbouring parcels were pulled furthest apart. The bright crest is a curve across the domain, and it is a property of the eight units rather than of any instant inside them.

A boundary that only exists over a window

The curve that separates fluid going one way from fluid going another is not in any snapshot of the flow. It is the crest of a field built from a stretch of history, it moves when the stretch is changed, and reversing the direction of time gives a different curve entirely — both of them real.

kinematics · Coherent structures

Named alongside it

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

MeasurementMixingModel validityChaosFlow mapLyapunovThe Lyapunov exponentMemory kernelPredictabilityToleranceCoherent structuresConservation

All concepts