Near threshold, weakly nonlinear theory often selects hexagonal planforms in certain symmetries; rolls are also common depending on boundaries and forcing.

Does the simulation solve the Oberbeck–Boussinesq equations?

No—it is a visualization aid tied to Ra supercriticality, not a CFD time stepper.

Bénard Convection (Rayleigh)

Rayleigh–Bénard convection arises when a horizontal fluid layer is heated from below and cooled from above: warm fluid near the bottom tends to rise (buoyancy) while viscosity and thermal diffusion oppose motion. The Rayleigh number Ra = g β ΔT d³ / (ν α) measures this balance using the gap height d, temperature drop ΔT, thermal expansion β, kinematic viscosity ν, and thermal diffusivity α. Above a critical Ra_c ≈ 1708 for rigid horizontal walls, the conductive linear profile becomes unstable and convective rolls or hexagonal cells transport heat more efficiently. The animation is a stylized pattern whose amplitude scales with (Ra − Ra_c)+, not a linear stability eigenfunction or DNS of Navier–Stokes.

Who it's for: Fluid mechanics and geophysical/astrophysical convection primers.

Key terms

Rayleigh number
Bénard cells
Buoyancy
Convective instability
Critical Ra
Navier–Stokes
Pattern formation

How it works

Bénard convection appears when a thin fluid layer is heated from below: warm fluid is buoyant. The Rayleigh number Ra compares buoyancy driving to viscous and thermal damping. Past a critical Ra_c (≈ 1708 for rigid horizontal boundaries), the conductive state goes unstable and convection rolls or hexagonal cells set the transport. This page is a qualitative animation, not a Navier–Stokes integration.

Key equations

Ra = g β ΔT d³ / (ν α) · onset ~ Ra_c = 1708 (classic)

Frequently asked questions

Why hexagons?: Near threshold, weakly nonlinear theory often selects hexagonal planforms in certain symmetries; rolls are also common depending on boundaries and forcing.
Does the simulation solve the Oberbeck–Boussinesq equations?: No—it is a visualization aid tied to Ra supercriticality, not a CFD time stepper.

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