# Rayleigh–Bénard Convection Cells Interactive Rayleigh–Bénard convection visualization. Watch regular convection rolls form when the temperature difference exceeds the critical Rayleigh number Ra_c ≈ 1708. > Canonical page: https://elysiatools.com/en/visualizations/rayleigh-benard-convection - **Category:** Physics ## Overview Interactive Rayleigh-Bénard convection visualization with dual side and top views. When a horizontal fluid layer is heated from below, buoyancy drives hot light fluid upward and cold dense fluid downward; distinct from the existing Rayleigh-Taylor instability (a vertical-density-gradient, inertia-dominated interface overturn) this case is a horizontal thin layer where thermal diffusion, viscosity and buoyancy compete under the Boussinesq approximation, and is the textbook example of a dissipative structure — order emerging far from thermodynamic equilibrium (Prigogine), in the same family as the Belousov-Zhabotinsky oscillating reaction and Turing reaction-diffusion patterns. The side-view governing equations are the dimensionless incompressible Navier-Stokes coupled to a temperature equation with a buoyancy term +Ra·Pr·T ĵ, or equivalently the vorticity-streamfunction form: a vorticity transport equation ∂ω/∂t + (u·∇)ω = Pr∇²ω + Ra·Pr·∂T/∂x, a temperature advection-diffusion equation ∂T/∂t + (u·∇)T = ∇²T, and a Poisson equation ∇²ψ = −ω for the streamfunction. The top-view planform is modeled by the Swift-Hohenberg pattern-formation equation ∂u/∂t = r·u − (1+∇²)²·u + α·u² − u³ (r = Ra/Ra_c − 1 the control parameter, α the quadratic term breaking up-down symmetry that enables hexagons), solved with the ETD1 spectral method (exact integration of the stiff biharmonic operator, φ₁ function). Linear stability analysis (Rayleigh 1916, Chandrasekhar 1961) gives the critical Rayleigh number Ra_c ≈ 1707.76 for rigid no-slip plates at critical wavenumber k_c ≈ 3.117 (preferred wavelength ≈ 2 layer depths); below Ra_c heat travels by conduction alone (Nusselt Nu=1), above it regular convection rolls appear and enhance heat transfer (Nu>1), and the hexagon-to-roll transition occurs at r* = α²/3 (Cross-Hohenberg amplitude-equation result). Four visualization panels: (1) Side-view convection cross-section rendered as a temperature colormap (blue cold → red hot) into an offscreen grid-resolution ImageData scaled to display size, with overlaid velocity-vector arrows scaled by local speed. (2) Top-view Bénard planform showing the classic hexagonal cell pattern from above, the Swift-Hohenberg order-parameter field rendered as a diverging colormap (rising fluid red, sinking fluid blue), self-organizing from three seeded 120°-spaced wavevectors plus noise. (3) Temperature profile plot showing the horizontally-averaged T(y) bending from a straight conductive line into the characteristic S-shaped convective profile, against the dashed conduction reference. (4) Nusselt-over-time plot tracking heat-transfer enhancement with the Nu=1 conduction reference line. Adjustable parameters: temperature difference ΔT (1–50 °C), fluid type (air/water/glycerin/silicone oil switching α/ν/κ), aspect ratio W/H (1–3), Prandtl number Pr (0.5–10), hexagon bias α (0–3, where α=0 forces stripes), and a velocity-arrow toggle. Four scenario presets: Stable (Ra