Rayleigh-Bénard Convection Onset

Convection rolls emerge when Ra exceeds the critical value 1708 — buoyancy overcomes viscous dissipation and thermal diffusion

Parameters

500
7.0
3.0

Display

on
Ra = 500  |  Ra_c = 1708  |  Supercriticality: sub-critical  |  Nu ≈ 1.0
Rayleigh-Bénard Instability

A fluid layer heated from below: competition between buoyancy (destabilizing) and viscous + thermal diffusion (stabilizing). The dimensionless Rayleigh number Ra = gαΔTH³/(νκ).

Linear stability analysis (Chandrasekhar 1961): Ra_c = 1708 for no-slip boundaries. Above threshold, convection rolls form with wavenumber k_c = π/√2 (roll width ≈ layer height).

The Nusselt number Nu = total heat flux / conductive flux: Nu ~ Raβ with β ≈ 2/7 (Grossmann-Lohse). This simulation uses a simplified 2D finite-difference stream-function/vorticity approach.