Rayleigh–Bénard Convection

DEFINITION

∂u/∂t + (u·∇)u = −∇p + ν∇²u + gα(T−T₀)ẑ,   ∇·u = 0
∂T/∂t + (u·∇)T = κ∇²T

ω = ∇×u,  ∇²ψ = −ω:
∂ω/∂t + (u·∇)ω = ν∇²ω − B ∂T/∂x

Ra = B d³/νκ,   Ra(k) = (k²+π²)³/k²

NOTES

Heat a layer of fluid from below and nothing happens. The fluid at the bottom is lighter than the fluid above it, which is unstable in the way a pencil balanced on its point is unstable, and yet the layer just sits there conducting heat upward through its own stillness — because to move at all it must push the cold fluid aside, which viscosity resists, and it must carry its warmth with it faster than diffusion can level it out. Rayleigh put a number on that competition in 1916. Buoyancy over the product of the two dissipations, cubed by the depth: below a critical value of it the layer conducts, above it the layer rolls, and the transition is sharp. The dial here is that number in units of the critical value this box computes for itself, so 1 is where the pencil falls. What Rayleigh minimised over — the wavelength the layer chooses — a box cannot choose, and this one is as wide as it is deep, so its own critical value is about four and a half times the 657.5 of a free layer, computed on the plate from its own discretisation and confirmed against its own growth rate. Just above 1 the growth is slow, and that slowness is the bifurcation rather than a defect of the plate: the closer to critical, the longer any disturbance takes to decide whether to grow. Bénard is here for the experiment of 1900 and for the cells that carry his name, with one correction the century made: his own layers were thin films with a free surface, and the hexagons he photographed are now understood to have been driven mostly by surface tension rather than by buoyancy. The instability this plate integrates is the one Rayleigh analysed when he set out to explain them. What you are looking at is the temperature field itself and not a dye added to reveal it: the hot wall along the bottom, the cold one along the top, and between them either a smooth gradient or the plumes carrying heat across in bulk. The layer is re-cooled every so often so that the breaking can be watched rather than merely arrived at, and it breaks somewhere else each time.

PROVENANCE

Origin (the experiment)

H. Bénard, "Étude expérimentale des courants de convection dans une nappe liquide. — Régime permanent: tourbillons cellulaires", Journal de Physique Théorique et Appliquée 9, 1900, 513–524 (doi:10.1051/jphystap:019000090051300); the observation methods follow in the same journal, 10, 1901, 254–266 (doi:10.1051/jphystap:0190100100025400), and the thesis is Annales de Chimie et de Physique (7) 23, 1901, 62–144

Origin (the mathematics)

Lord Rayleigh, "LIX. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side", The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, series 6, 32(192), 1916, 529–546. The linear stability analysis, the dimensionless group and the free-boundary result Ra_c = 27π⁴/4 = 657.51 at k·d = π/√2 are all in that paper

The correction, on the record

Rayleigh wrote to explain Bénard cells and the pairing has been standard ever since, but the cells Bénard photographed were largely not this instability. His layers were half a millimetre deep with a free upper surface, where the surface tension gradient across a warm spot dominates: M. J. Block, Nature 178, 1956, 650–651 (doi:10.1038/178650a0), and the analysis is J. R. A. Pearson, "On convection cells induced by surface tension", Journal of Fluid Mechanics 4(5), 1958, 489–500 (doi:10.1017/S0022112058000616). The buoyancy instability is Rayleigh. Both names belong on the plate and the twist belongs with them

Boussinesq

The approximation that density varies only in the buoyancy term and the fluid is otherwise incompressible: J. Boussinesq, Théorie analytique de la chaleur, volume 2, Gauthier-Villars, 1903, 172–176. Pre-DOI

Standing

Public domain — a linear stability analysis of the nineteenth-century equations of fluid motion, and an approximation of 1903. Implemented here from the mathematics, never from any convection code

What is solved, honestly

The two-dimensional Boussinesq system in the streamfunction and vorticity form, which is what makes it fit: velocity comes from one scalar and is divergence-free by construction, so there is no pressure to solve for and no projection to leave a residual. Advection is semi-Lagrangian after Stam, diffusion is an explicit blend toward the neighbour average, and the streamfunction is recovered from the vorticity by Jacobi relaxation, warm-started from the previous cycle. The same three-mode truncation of these equations is what Saltzman integrated in 1962 (doi:10.1175/1520-0469(1962)019<0329:FAFCAA>2.0.CO;2) and Lorenz reduced to the attractor at PL. 07 the year after — the ρ dial there is this ρ dial, a Rayleigh number in units of critical

The boundary is a decision

The lattice wraps in both directions by construction, which would make this a torus with no top and no bottom, so the vertical wrap is broken deliberately: the top and bottom rows are walls that are never advanced, holding a fixed temperature and zero vorticity and streamfunction. Zero vorticity at a flat wall is the stress-free condition exactly, which is the boundary Rayleigh solved for and the reason his 657.5 is a closed form. Sideways the wrap is kept, because a periodic layer is what the analysis assumes

The onset, measured

The critical value is derived on the plate from the lattice own operators rather than quoted, and then checked against the plate itself: the growth rate of the streamfunction crosses zero at 1.0024 of it, measured through this draw function at the published lattice, and at 1.0000 in a double-precision reference at every Prandtl number and every sweep count tried. Above it the Nusselt number the plate prints at each re-cooling runs 1.55 at one and a half times critical, 2.43 at two, 4.18 at five and 6.25 at twelve; below it nothing happens at all, for as long as anyone watches

Where the slider stops, and why

At twelve times critical, with a measured margin of four. Past about fifty times, this box stops behaving like a layer and starts behaving like a two-dimensional one: the flow condenses into a single box-filling vortex and the heat transport falls rather than rises — mean Nusselt 13.1 at fifty times critical and 1.8 at sixty, which survives both a finer lattice and a slower solve, so it is the inverse cascade of two-dimensional turbulence against stress-free walls and not the solver giving up. It is real and it is a different plate from this one

Constants

The Rayleigh dial is in units of the critical value this box computes for itself, so 1 is onset wherever the other dials sit. Prandtl is the ratio of the two diffusivities and changes the character of the flow above onset but not the onset itself. The sweep count is the quality of the streamfunction solve and is the one dial whose effect on the picture is nil — the settled Nusselt number is 4.18 at every value of it, and what moves is the cost. The lattice is not a detail dial either: the growth rate of the instability falls as the square of it, measured at frames 76, 136, 215, 309, 421 and 550 across the declared range, so twice the cells is four times the wait for the same picture resolved more finely

Source

doi:10.1080/14786441608635602