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.