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

Dynamical system — where it appears

A set of equations giving the rate of change of a state in terms of the state itself. Fluid flows are of this kind with infinitely many degrees of freedom, and studying small truncations of them was how the modern account of transition began.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Three equations, and the set they never leave. The Lorenz trajectory at r = 28, projected on x and z, after the transient has been discarded. It never repeats, never leaves, and never crosses itself in three dimensions. The Lyapunov exponent printed beside it is measured on this system by separating a nearby pair, so the claim of sensitive dependence is a computation.

Three numbers left of a fluid

Saltzman truncated convection to three Fourier modes and Lorenz studied what was left. The result changed science, and it stopped being a description of a fluid at about a fifth of the way to the parameter everybody quotes it at.

turbulence · Convection A millionth, doubling every three-quarters of a second. The separation of two trajectories started a millionth apart, on log axes against time. It grows as a straight line until it saturates at the size of the attractor, and the slope of that line is the largest Lyapunov exponent — measured here on the system by renormalising a nearby pair, not quoted. Sensitive dependence is what the straightness of the line means.

A millionth is enough

Two trajectories a millionth apart separate by a factor of e every three-quarters of a second, so a millionfold improvement in the measurement buys about ten seconds of extra prediction. That exchange rate, and not the size of the error, is what limits forecasting.

turbulence · Convection One streamline, sectioned, in two steady flows. Every time a single streamline crosses the plane z ≡ 0 going upwards, a point is plotted. On the left the flow is integrable and the points lie on a curve, however long the trajectory is run. On the right one coefficient of the same exact solution has been changed and the same single streamline scatters over a sixth of the plane. Both flows are steady, both are incompressible to machine precision, and both are exact solutions of the Euler equations.

Steady, three-dimensional, and mixing anyway

A steady flow that solves the Euler equations exactly, with its vorticity equal to its velocity to six parts in ten thousand million — and one of its streamlines wanders through a sixth of the box while another, started nearby, lies on a curve for ever.

kinematics · Advection Four vortices, and the end of prediction. The same three vortices as before with a fourth, weaker one added near the middle. Three point vortices have three independent invariants for three degrees of freedom and cannot be chaotic; four have the same three invariants and one more degree of freedom, and generically are. The energy and the impulses are conserved here to fourteen digits over the whole run, which is what makes the tangle a property of the system rather than of the arithmetic.

Three is the most that can be predicted

Point vortices are the simplest dynamical system fluid mechanics has — no cores, no viscosity, no approximations, four exactly conserved quantities. Three of them are integrable and cannot be chaotic. Add a fourth and the same equations, conserving the same quantities to fourteen digits, stop being predictable at all.

inviscid · Vortex dynamics The mean flux, flat across the inertial shells and equal to the dissipation. The time-averaged transfer out of the first n shells, computed as the rate at which the nonlinearity changes their energy rather than from a remembered formula. It is constant to a tenth across the middle of the ladder and equal to the dissipation, which is the cascade — and it is an average.

A flux that runs both ways

The cascade is a statement about a mean. Kolmogorov's four-fifths law fixes an average and the constant flux through the inertial range is an average, and neither says anything about what the transfer is doing at any instant — which turns out to be running backwards a substantial part of the time.

turbulence · Cascade

Named alongside it

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

Model limitPredictabilityTruncationConvectionThe Lorenz systemThe Lyapunov exponentAngular momentumAveragingBackscatterBeltrami flowBifurcationConservation

All concepts