Rayleigh–Bénard Convection

Compare a layer heated from below at Ra = 1500 and Ra = 2500. The first example shows conduction; the second adds steady convection rolls. Change Ra and press Show, or select an example. A new input clears the old illustration. Pause freezes only the direction markers; Reset restores the defaults.

Reference onset. Ra = g β ΔT H³ / (ν κ), with gravity g, expansion coefficient β, plate temperature difference ΔT, depth H, kinematic viscosity ν and thermal diffusivity κ. Ra_c ≈ 1708 is the rounded linear onset for the classical non-rotating Boussinesq layer between two rigid, no-slip, isothermal horizontal plates, with an unbounded horizontal extent. It is not a universal threshold for any container or boundary condition. The rounded threshold is used as this illustration’s switch: no flow at or below 1708. See Kao & Yang, 2007.

What is drawn. z runs from 0 at the hot bottom to 1 at the cold top; x is measured in layer depths. Temperature θ = (T − T_top) / ΔT has fixed plate values 1 and 0. The prescribed streamfunction is ψ = I/π · sin(πx) sin²(πz). Its derivatives u = ∂ψ/∂z and w = −∂ψ/∂x give a divergence-free field with zero velocity at both plates. θ = 1 − z + 0.2w links warm departures from conduction to upward motion and cool departures to downward motion. Horizontal continuation is periodic.

Schematic intensity. I = min(1, √max(0, Ra/1708 − 1)) is a chosen visual mapping, not a predicted velocity or heat flux. It saturates at Ra = 3416. The controls cover Ra from 0 to 5000 solely for this demonstration. No Navier–Stokes, temperature evolution, Nusselt number or turbulent transition is calculated. The temperature pattern is stationary; direction markers move along its streamlines at an illustrative speed, with no physical time unit. Reduced-motion preferences start the markers paused.

View. One to four visible pairs show different lengths of the same periodic pattern. Their count is chosen, not predicted, and does not change the reference onset. Screen proportions are illustrative. The example does not resolve rigid side walls or calculate the critical wavelength. Parameters survive switching tools; drawing starts only after Show. All computation stays in the browser.