Rayleigh-Bénard Convection

Thermal Instability & Roll Formation — Onset at Ra_c = 1708
Rayleigh number (Ra)2000
Prandtl number (Pr)1.0
Aspect ratio4.0
Ra = 2000 Ra_c = 1708 Status: — Nu ≈ — k_c = π/√2 Roll count: —
Rayleigh-Bénard convection: fluid heated from below, cooled from top. Below Ra_c=1708 (Rayleigh 1916), conduction dominates. Above Ra_c, buoyancy overcomes viscous and thermal diffusion → convection rolls form at critical wavenumber k_c=π/√2 ≈ 2.22. Nusselt number Nu = heat flux / conductive flux. Marginal stability curve: Ra(k) = (π²+k²)³/k². Minimum at k_c gives Ra_c=27π⁴/4≈1708.