RayleighBénard

Rayleigh-Bénard convection, often written Rayleigh–Bénard convection, is a classical pattern-forming instability in a horizontal layer of fluid heated from below and cooled from above. When the imposed temperature difference is small, heat transfer is mainly conductive. Above a critical thermal driving, buoyancy from heating the lower boundary overcomes viscous and diffusive damping, producing organized convection patterns such as rolls or cell-like structures.

The key dimensionless groups are the Rayleigh number and the Prandtl number. The Rayleigh number Ra = g α ΔT d^3 / (ν κ) measures the strength of buoyancy relative to viscous and thermal diffusion, where g is gravity, α is thermal expansion, ΔT is the temperature difference, d is the layer thickness, ν is kinematic viscosity, and κ is thermal diffusivity. The system loses stability and convection sets in when Ra exceeds a critical value Ra_c, which is about 1708 for a layer with no-slip boundaries and idealized conditions. The Prandtl number Pr = ν/κ compares momentum and thermal diffusion. The Nusselt number Nu, defined as the ratio of total to conductive heat transfer, quantifies the enhancement due to convection (Nu > 1).

Governing equations arise from the Navier–Stokes equations under the Boussinesq approximation, coupled to a temperature equation. Early experiments by Henri Bénard and theoretical work related to Lord Rayleigh established this problem, which has since become a central model for studying pattern formation, nonlinear dynamics, and turbulence.

Rayleigh-Bénard convection has applications in geophysics, astrophysics, and engineering, providing insights into thermal convection in planetary mantles, stellar interiors, and cooling technologies, as well as serving as a benchmark for numerical simulations and analytical techniques.