Parameters
Ra = 2512
Temperature field
Velocity vectors
Streamline overlay
Max Temperature
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Min Temperature
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Max Flow Speed
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Convection State
Computing...
Nusselt number Nu
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Sim
Hot
Cold
Vary the Rayleigh number (Ra) and watch convection cell formation, development, and turbulent transition in real time
Rayleigh number (dimensionless parameter):
$$Ra = \frac{g\,\beta\,\Delta T\,L^3}{\nu\,\kappa}$$$g$: gravity, $\beta$: thermal expansion coefficient, $\Delta T$: temperature difference, $L$: layer height, $\nu$: kinematic viscosity, $\kappa$: thermal diffusivity
Temperature advection–diffusion equation (Boussinesq approximation):
$$\frac{\partial T}{\partial t}+ \vec{u}\cdot\nabla T = \kappa\,\nabla^2 T$$Velocity updated from buoyancy forcing (temperature anomaly drives vertical flow).
Boundary conditions: bottom $T=1$ (hot), top $T=0$ (cold), lateral: periodic.
Meteorology: Cumulus cloud formation, thunderstorm updrafts, and mesoscale convective systems are large-scale Rayleigh–Bénard convection driven by solar heating of Earth's surface.
Earth's interior: Mantle convection — the driver of plate tectonics — operates on a Rayleigh–Bénard mechanism across geological timescales, with Ra ~ 10⁷.
Electronics cooling: Natural convection in server racks and heat sink design depends directly on understanding when and how convection cells form.
The Sun: The granules visible on the solar surface are convection cells ~1000 km across in the Sun's outer convective zone, driven by the steep temperature gradient.
When you start using this simulator, there are a few common pitfalls to watch out for. First, it's easy to assume that "a higher Rayleigh number always means more vigorous convection," but it's not that simple. While convection does begin once the critical value (approximately 1708) is exceeded, increasing Ra leads through stages: from neat roll patterns, through a "secondary instability" phase where rolls distort or split, and finally to turbulence. For instance, if you increase Ra from 10^4 to 10^5, you can observe the rolls starting to fluctuate over time instead of remaining stationary. This is the gateway to "transient chaos," so the trick is to change parameters gradually; changing them too abruptly might cause you to miss the transitions between fundamental patterns.
Next, don't underestimate the importance of the aspect ratio (width/height) setting. For example, setting the aspect ratio to 2 (width is twice the height) naturally results in two convection rolls side by side. However, if you set it to an intermediate value like 3.5, the number of rolls is no longer an integer, which can easily lead to unstable patterns. In nature or real devices, the "width" of a layer is often fixed, so remember that changing the aspect ratio in a simulation is essentially equivalent to "changing the size of the experimental apparatus or observation area."
Finally, don't forget that this simulation assumes a "2D" model and uses the "Boussinesq approximation." In three dimensions, hexagonal Bénard cells can appear instead of rolls. Also, this approximation breaks down when temperature differences are extreme or when fluid compressibility cannot be ignored. Understand that this is an idealized model for learning fundamental principles.
The simulator is based on the governing equations of Convection Cells Simulator — Rayleigh–Bénard Convection. Understanding these equations is key to interpreting the results correctly.
Each parameter in the equations corresponds to a slider in the control panel. Moving a slider changes the equation's solution in real time, helping you build a direct connection between mathematical expressions and physical behavior.
For a fluid layer (silicone oil, ν=20 mm²/s, κ=0.15 mm²/s) with depth d=10 mm and ΔT=5 K: Ra = gβΔTd³/(νκ) = 9.81×0.001×5×10³/(20×0.15) ≈ 16,350. Setting slRa to 16,350 and slAR to 2.0 produces 2–3 counter-rotating hexagonal cells with maximum vertical velocity ~8 mm/s. Reducing ΔT to 2 K drops Ra to 2,548, yielding marginal oscillating rolls.