Onset of convection, roll formation, and Nusselt number scaling
—Ra
657Ra_c
—Nu (est.)
—Ra/Ra_c
—Phase
Rayleigh-Bénard instability: Fluid between hot bottom (T=1) and cold top (T=0) goes unstable when Ra = gαΔTH³/(νκ) > Ra_c ≈ 657.5 (for free-slip) or 1708 (no-slip boundaries). Critical wavenumber: k_c = π/√2 (free-slip), giving roll aspect ratio ≈ 1.4:1. Nusselt scaling: Nu ~ Raβ with β ≈ 0.28–0.31 in experiments. In ultimate regime (Ra → ∞): Nu ~ Ra1/2 (Kraichnan 1962). Visualization shows temperature field (hot=red, cold=blue) and convective rolls.