Rayleigh–Bénard convection — patterns from heat

Why Rayleigh–Bénard matters

Heat the bottom of a layer of fluid, cool the top, and a sequence of flow regimes emerges as you push the heating harder. At weak heating the fluid stays still and the heat is carried purely by conduction — the temperature varies linearly from one plate to the other and nothing moves. Push the heating past a threshold and that quiet state becomes unstable: warm fluid begins to rise and cold fluid to sink, organising itself into a regular pattern of upwelling and downwelling lanes — the canonical convection cells. Push harder and those cells start to wobble, drift sideways, and shed plumes; harder still, the boundary layers next to the plates become thin and intermittent, and the interior is filled with detaching plumes swept along by a large-scale circulation. The same physics drives convection in the atmospheric boundary layer, in the ocean, in stars, and inside the Earth’s mantle. Rayleigh–Bénard convection is the canonical model system that lets all of these be studied in one set of dimensionless numbers.

It is also close to my heart: Rayleigh–Bénard convection was the topic of my PhD, and I have worked on buoyancy-driven flows ever since.

Setup

A horizontal layer of fluid is sandwiched between two plates. The bottom plate is hotter than the top by a fixed amount $\Delta T$; both plates are no-slip, so the fluid is stationary right at the plate. The side walls in this 2D demo are also no-slip and thermally insulated — heat enters and leaves only through the top and bottom. Gravity points down, so warm fluid is buoyant and tends to rise. The simulation starts from a tiny random temperature perturbation around the linear conduction profile; whether that perturbation decays or grows into circulation is the whole story.

Control parameters and response: Ra, Pr, Nu

Three dimensionless numbers do all the work.

The Rayleigh number measures how strongly buoyancy drives the flow against viscous and thermal damping:

Here $g$ is gravitational acceleration, $\alpha$ is the thermal expansion coefficient, $\Delta T$ is the imposed temperature difference between the plates, $H$ is the layer depth, $\nu$ is the kinematic viscosity, and $\kappa$ is the thermal diffusivity. Below a critical value the conduction state is stable; above it, convection cells emerge; pushing further moves the flow through the qualitative regimes described above.

The Prandtl number measures the relative strength of momentum diffusivity to thermal diffusivity:

Air has $\mathrm{Pr} \approx 0.7$, water $\mathrm{Pr} \approx 7$, oils $\mathrm{Pr} \approx 100$. Pr controls how thick the velocity boundary layer is relative to the thermal boundary layer, and so where the heat flux is concentrated.

The Nusselt number is the response: the total heat flux through the layer, normalised by what pure conduction would deliver:

Here $q$ is the vertical temperature flux through the layer (units of temperature times velocity), so the conductive reference flux is $\kappa \Delta T / H$. $\mathrm{Nu} = 1$ means convection is doing nothing — pure conduction; $\mathrm{Nu} > 1$ means convection is enhancing heat transfer. At high Ra, $\mathrm{Nu}$ scales approximately as $\mathrm{Ra}^{\beta}$ with $\beta \approx 0.28$–$0.31$ — a slow scaling that has been the subject of decades of theoretical and experimental work.

You drive the demo with Ra and Pr, and you observe Nu.

Local heat flux at the plates

The strips drawn just outside the top and bottom plates show how strongly heat crosses each boundary at every horizontal position. The plotted quantity is $\mathrm{Nu}_{\text{local}}(x) = -(H/\Delta T) \, \partial T / \partial z$, the local conductive temperature gradient made dimensionless by the pure-conduction reference gradient. Peaks mark places where plumes are actively taking heat away from a plate. In a real analysis, Nu is a space-time statistic: the local heat flux varies with horizontal position and time, so one averages over both $x$ and $t$ to obtain a reliable value. The live number in this demo is only an instantaneous diagnostic, useful for seeing the signal fluctuate as the flow evolves.

It can take a little while for the convection to get going. The demo starts from small temperature perturbations, so the buoyant motion first has to amplify before clear plumes and cells appear; some patience is required, especially at lower Ra. You can also help the simulation a bit by clicking in the domain, which injects a small warm temperature blob into the flow.

About this demo

This browser version uses the site’s NS2D solver: a two-dimensional Boussinesq Navier-Stokes model with semi-Lagrangian advection and an SOR pressure projection. It is intended to show the qualitative structure of the instability in real time. The solver is useful for seeing plumes, circulation and local heat-flux patterns develop, but the Nusselt number should be read as a live diagnostic for the demo, not as a research-grade measurement. Read the methods page →