Fluid dynamics · buoyancy instability

Rayleigh–Bénard Convection

Heat the bottom plate and cool the top of a transparent chamber. Increase the temperature difference until a linear conductive profile crosses the critical Rayleigh number and reorganizes into rising hot plumes and descending cold return flow.

Rayleigh

\mathrm{Ra}

$3.5\times10^3$

Supercriticality

\mathrm{Ra}/\mathrm{Ra}_c

$2.0$

Prandtl

\mathrm{Pr}

$35.0$

Nusselt

\mathrm{Nu}

$1.16$

Roll amplitude $0.47$

Transport regimeOrganized rolls

Under the Boussinesq approximation, rigid isothermal boundaries become unstable near $\mathrm{Ra}_c\approx1708$ . This reduced model does not directly solve fully three-dimensional turbulent convection.

Physics tutorial

Rayleigh–Bénard Convection: how a thermal gradient becomes organized motion

BackgroundBottom heating makes the lower fluid slightly lighter. At small temperature difference, thermal diffusion and viscosity suppress buoyant disturbances; past onset, rising hot fluid and descending cold return flow form closed cells.

Why it mattersAtmospheric boundary layers, mantle convection, stellar interiors, and laboratory heat transfer all share the question of when conduction loses stability.

Start with the essentials

Focus question: Why does a temperature gradient remain quiet and then suddenly select a patterned roll scale?
One-sentence intuition: Rayleigh number places buoyancy against viscous and thermal damping; rigid isothermal boundaries lose stability near $\mathrm{Ra}_c\approx1708$ .

Core mathematical model

Rayleigh number

\mathrm{Ra}=\frac{g\alpha\Delta T H^3}{\nu\kappa}

Depth enters cubed, making the onset unusually sensitive to geometry.

Prandtl number

\mathrm{Pr}=\frac{\nu}{\kappa}

Compares the diffusion rates of momentum and heat, shaping the transient roll response.

Nusselt number

\mathrm{Nu}=\frac{qH}{k\Delta T}

It equals one for pure conduction and rises when convection carries additional heat.

Common difficulties

Hot fluid does not all move upward

Typical misconceptionOnce heated from below, the entire layer rises.

Better mental modelMass conservation requires return flow: each hot updraft closes through neighboring cold descent.

The temperature image is not pointwise DNS

Typical misconceptionEvery colored pixel comes from a full three-dimensional turbulent calculation.

Better mental modelThe Lab combines the conductive base state with weakly nonlinear roll modes to expose onset and heat-transport causality.

Run the experiment

01
Inspect pure conduction
Choose Below onset with isotherms visible.
What to observe: Isotherms remain nearly horizontal, tracers do not circulate persistently, and Nusselt number stays at one.
02
Cross criticality
Switch to Stable rolls.
What to observe: Isotherms bend as warm parcels rise, cold parcels descend, and paired rolls close.
03
Amplify the depth effect
Increase fluid depth gradually.
What to observe: Rayleigh number grows rapidly through the cubic depth dependence and heat transport strengthens.