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

Rayleigh number — where it appears

The ratio of buoyancy driving a heated layer to the two diffusions that oppose it, viscosity and thermal conduction. Nothing happens at a Rayleigh number of one; convection begins at an eigenvalue of the marginal-stability operator, which is 1707.76 between rigid walls.

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

Every disturbance has its own threshold; one of them is lowest. The Rayleigh number at which a disturbance of horizontal wavenumber a becomes neutral, Ra = (π² + a²)³/a². Every wavenumber has a threshold and the layer goes unstable at the lowest of them, which a golden-section search on this curve puts at a = 2.221441 and Ra = 657.5114 — the exact π/√2 and 27π⁴/4 to fourteen digits. Below the curve the layer conducts and nothing moves.

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.

turbulence · Convection 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.

turbulence · Convection 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.

turbulence · Convection

Named alongside it

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

Rayleigh–Bénard convectionConvectionConvergenceEigenvalueLinear stabilityBoundary conditionBoundary layerBuoyancyDimensionlessDimensionless numberHeat transferThe Lorenz system

All concepts