Rayleigh number

The Rayleigh number (Ra) is a dimensionless parameter in fluid dynamics that characterizes the relative strength of buoyancy-driven convection to viscous dissipation and thermal diffusion. It is defined as:

Ra = (g α ΔT h³) / (ν κ)

where g is gravitational acceleration, α is the thermal expansion coefficient, ΔT is the temperature difference across the fluid layer, h is the layer depth, ν is the kinematic viscosity, and κ is the thermal diffusivity.

The Rayleigh number determines whether a heated fluid layer will remain in a quiescent conductive state or undergo a convective instability. When Ra exceeds a critical threshold — approximately 1708 for a horizontal layer with rigid boundaries — the uniform conductive state becomes unstable and the fluid spontaneously organizes into ordered convective rolls or cells.

Beyond the Bénard problem, the Rayleigh number governs convection in the Earth's mantle, stellar interiors, atmospheric boundary layers, and industrial heat-transfer systems. It is the single most important control parameter in buoyancy-driven pattern formation.