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Home/Thermodynamics/Bénard Convection (Rayleigh)

Bénard Convection (Rayleigh)

Heated-from-below layer: Ra vs Ra_c ~1708; schematic hex/roll pattern.

Rayleigh–Bénard toy

2500
1×

Measured values

Ra_c (rigid–rigid ~)1708
Supercriticality (toy)46%

About this model

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.