Convection — the series · Fluid Flow & Aerodynamics

A threshold with a closed form

A layer of fluid heated from below sits still until buoyancy overcomes both diffusions at once, and then it convects. Unlike every other threshold in this field, that one is an eigenvalue with an exact answer, and the answer is 27π⁴/4.

part 1 · turbulence

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.

part 2 · turbulence

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.

part 3 · turbulence

Three thresholds, and none of them is one. The neutral curves for a layer heated from below, computed by taking the smallest eigenvalue of the marginal-stability operator at each horizontal wavenumber. Every curve diverges at both ends — a cell wider than the layer has to carry heat sideways for ever, a narrower one loses it to conduction — so each has a minimum, and that minimum is the critical Rayleigh number. Two free surfaces give 657.5, one rigid and one free 1100.7, two rigid walls 1707.8. Nothing but the boundary condition differs, and it carries a factor of 2.6.

The threshold the walls decide

A layer heated from below convects at a Rayleigh number of 1707.762, and nothing whatever happens at one. The free–free case has a closed form and the two that do not differ from it by a factor of 2.6 — produced by nothing but what the top and bottom surfaces are permitted to do.

part 4 · turbulence

Three modes carry heat up to a ceiling of three; the rolls they were cut from keep going. The Nusselt number — heat carried across the layer over what conduction alone would carry — against r, the Rayleigh number over its critical value, for Lorenz's three-mode truncation, 1 + 2(r − 1)/r, and for steady rolls at the same wavenumber computed with 6 and with 42 temperature modes. At r = 2, r = 5 and r = 30 Lorenz gives 2.000, 2.600 and 2.933; the 42-mode rolls give 2.143, 3.323 and 5.970. Lorenz's value can never exceed three whatever the Rayleigh number; the rolls' keeps rising. The ceiling is not in the convection. It is in the three modes, which have only one way to thin the thermal layers, and half of it is used by the time the layer is twice past onset.

The truncation that cannot carry three times the heat

Lorenz's three modes give a convecting layer's heat flux in one line — one plus twice (r − 1) over r — and it can never reach three times what conduction carries. The same rolls computed with forty-two modes agree with that line exactly at onset, carry 7.1 per cent more heat at twice the critical Rayleigh number, and twice as much at thirty times it. The ceiling is not in the convection; it is in having one sine to draw the temperature with.

part 5 · turbulence

The same wavelength drawn as rolls and as hexagons. Plan views of a convecting layer with one critical wavelength, drawn from the amplitude equations' two stable states. On the left, rolls: a single set of parallel bands, rising fluid along one set of lines and sinking along the next. On the right, hexagons: three sets of rolls at 120° to each other with equal amplitudes, whose sum has its maxima on a triangular lattice with spacing 2/√3 of the wavelength, each maximum at the centre of a hexagonal cell. At ε = 0.0250, inside the window, both are stable: rolls with amplitude 0.158 and hexagons with 0.081 in each of their three rolls. A hexagon is not a different kind of cell; it is three roll patterns that the quadratic term lets reinforce one another.

Hexagons remember how the heat was turned up

A layer heated from below convects in rolls, unless its top and bottom are not mirror images of each other. Then three sets of rolls at 120° can feed one another through a term the symmetry used to forbid, hexagonal cells appear before the layer is formally unstable, and there is a range of heating in which rolls and hexagons are both stable — so the pattern a layer shows depends on whether the heat was turned up or down to get there.

part 6 · turbulence