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

Lyapunov — where it appears

The exponential rate at which two nearby trajectories of a flow separate, defined as a long-time average along one of them. It sets the predictability horizon, it is positive for any chaotic flow, and over finite windows it is a broad distribution rather than a number.

Named by 2 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

Named alongside it

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

AdvectionChaosMeasurementToleranceConservationConvergenceDiscretisationEnsembleEulerian and LagrangianIncompressibleJacobianMass conservation

All concepts