About Rayleigh-Bénard Convection
This simulation shows a fluid layer heated from below and cooled from above, the classic Rayleigh-Bénard system. It solves the 2D incompressible Navier-Stokes equations under the Boussinesq approximation using the stream function-vorticity (ψ-ω) formulation on a 120×72 grid. Temperature and vorticity are advected semi-Lagrangian, diffused explicitly, and the Poisson equation ∇²ψ = -ω is relaxed by successive over-relaxation. Buoyancy enters as a source term proportional to the horizontal temperature gradient.
The Buoyancy slider scales the driving force (effectively the Rayleigh number), Viscosity ν damps momentum, and Diffusivity κ smooths temperature. Above the critical Rayleigh number Ra ≈ 1708 the still, conductive layer becomes unstable and counter-rotating convection rolls form spontaneously. The same instability governs solar granulation, atmospheric and oceanic circulation, and the slow convection of Earth's mantle, making it a cornerstone of fluid dynamics.
Frequently Asked Questions
What is Rayleigh-Bénard convection?
It is the buoyancy-driven motion that arises when a horizontal fluid layer is heated from below and cooled from above. Warm, less dense fluid near the hot lower plate rises while cool fluid sinks, organising into regular convection rolls or hexagonal Bénard cells once the heating is strong enough.
What does the Rayleigh number mean?
The Rayleigh number Ra = gβΔTH³/(νκ) compares buoyant driving against the damping effects of viscosity and thermal diffusion. When Ra exceeds the critical value of about 1708 for rigid plates, conduction alone can no longer carry the heat and convection sets in.
What do the three sliders control?
Buoyancy scales the thermal driving force and so plays the role of the Rayleigh number; raising it produces more vigorous rolls. Viscosity ν damps the velocity field, making cells sluggish or suppressing them, while Diffusivity κ controls how quickly temperature differences smooth out.
What numerical method does it use?
It uses the stream function-vorticity formulation of the 2D Navier-Stokes equations with the Boussinesq approximation. Temperature and vorticity are advected with a semi-Lagrangian scheme, diffused explicitly, and the stream function is recovered each step by solving ∇²ψ = -ω with successive over-relaxation.
Why do the rolls only appear above a threshold?
Below the critical Rayleigh number, viscosity and thermal diffusion erase any small disturbance before it can grow, so heat moves purely by conduction. Above the threshold the buoyant force wins, tiny perturbations amplify, and the layer reorganises into steady convective motion. This is a textbook example of a hydrodynamic instability and a pitchfork bifurcation.
What is the stream function-vorticity formulation?
Instead of tracking pressure and two velocity components directly, the flow is described by vorticity ω (local spin) and a stream function ψ whose gradients give the velocity. This automatically enforces incompressibility and removes pressure from the equations, which is why it is convenient for 2D simulations like this one.
What do the velocity arrows and streamlines show?
The velocity arrows mark the local direction and relative speed of the fluid, revealing the circulation within each roll. The streamline option draws contours of the stream function ψ; fluid flows along these lines, so closed loops outline the convection cells without crossing between them.
Is this simulation physically accurate?
It captures the correct qualitative physics: the conduction-to-convection transition, counter-rotating rolls, and the dependence on buoyancy, viscosity and diffusivity. However, the grid is coarse, the slider values are scaled rather than true SI units, and it is two-dimensional, so it is a teaching tool rather than a research-grade solver.
What does the Perturb button do?
Perturb injects small random temperature fluctuations into the interior of the layer. Near or above the instability threshold these seeds grow and can trigger fresh convection cells or reorganise the existing pattern, letting you watch the system select a new flow structure from noise.
Where does Rayleigh-Bénard convection appear in nature?
The same instability drives granulation on the Sun's surface, cloud streets and thunderstorm updraughts in the atmosphere, thermohaline circulation in the oceans, and the slow convection of Earth's mantle that moves tectonic plates. Henri Bénard first observed the cells in 1900 and Lord Rayleigh explained the criterion in 1916.
What boundary conditions does the model use?
The bottom plate is held hot at T = 1 and the top plate cold at T = 0, fixing the temperature drop across the layer. The plates are rigid no-slip walls, imposed through the standard vorticity wall relation, while the horizontal direction is treated as periodic so rolls can wrap around the domain.