# Cellules de convection de Rayleigh–Bénard Visualisation interactive de la convection de Rayleigh–Bénard. Observez la formation de rouleaux de convection réguliers lorsque la différence de température dépasse le nombre de Rayleigh critique Ra_c ≈ 1708. > Page canonique: https://elysiatools.com/fr/visualizations/rayleigh-benard-convection - **Catégorie:** Physics ## Présentation 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