Rayleigh number: ratio of buoyancy to viscous and thermal diffusion. Bénard convection cells form when $Ra \gt Ra_c \approx 1708$.
$$Nu = C \cdot Ra^n$$
Nusselt number (heat-transfer ratio): measures the strength of convection. $n \approx 1/3$ corresponds to turbulent convection at high $Ra$.
$$\alpha = \frac{k}{\rho c_p}$$
Thermal diffusivity (m²/s): $k$ is the thermal conductivity, $\rho$ the density and $c_p$ the specific heat.
What Is Rayleigh–Bénard Convection?
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When I boil water, I sometimes see flow circling at the bottom of the pan. Is that a convection cell?
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Yes, exactly! Its formal name is "Rayleigh–Bénard convection". When the bottom is heated, the fluid there warms, its density falls and buoyancy makes it rise. The top is cold, so fluid cooled there sinks. This circulation forms regular "roll-shaped convection cells". The French physicist Bénard first observed it in 1900 in experiments with whale oil, and Lord Rayleigh developed the theory in 1916.
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But sometimes there is no convection if you only heat a little, right? What is the condition?
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Sharp! That is where the Rayleigh number (Ra) comes in. Ra is the ratio of "destabilisation by buoyancy" to "stabilisation by viscosity plus thermal diffusion". Convection only begins when Ra exceeds the critical value of about 1708. Below that, fluid that tries to rise cannot overcome viscosity and thermal diffusion, and heat simply escapes upwards by conduction. Choose the "Just below critical" preset (Ra < 1708) in the simulator and you should see the convection cells disappear, leaving a uniform temperature gradient.
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Why does it form a specific pattern of "cells" once Ra exceeds that? Why doesn't it just mix randomly?
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That is the fascinating part! When the instability begins, disturbances of every spatial scale try to grow a little. But there is an "optimal wavelength" that grows fastest — roughly two to three times the layer thickness — and only that one is selected and survives. That is the self-organisation of "pattern formation". As you raise Ra further, the neat initial rolls break up, motions of several scales coexist, and eventually it becomes turbulent. The same principle is at work in cumulus convection in the weather and in the Earth's mantle convection.
$g$: gravitational acceleration, $\beta$: thermal expansion coefficient, $\Delta T$: temperature difference between bottom and top, $L$: layer height, $\nu$: kinematic viscosity, $\kappa$: thermal diffusivity
Advection–diffusion equation for the temperature field (Boussinesq approximation):
$$\frac{\partial T}{\partial t}+ \vec{u}\cdot\nabla T = \kappa\,\nabla^2 T$$
Velocity field (simplified stream-function method):
$$\vec{u}\propto Ra \cdot (T - T_{\text{mean}}) \cdot \hat{y}+ \text{viscous diffusion}$$
Boundary conditions: top $T=0$ (cold), bottom $T=1$ (hot), periodic side boundaries.
Applications in Nature and Engineering
Rayleigh–Bénard convection is one of the most fundamental fluid instabilities and appears in many fields:
Weather and atmosphere:The growth of cumulus clouds (thunderclouds) and the strong updraughts in cumulonimbus are large-scale versions of this convection.
Earth's interior:Convection in the Earth's mantle (the driving force of plate tectonics) is Rayleigh–Bénard convection on a timescale of millions of years.
Electronics cooling and thermal design:Understanding Bénard convection is essential in the air cooling of heat sinks and the cooling design of data centres.
Space:The "granules" on the Sun's surface are convection cells in the solar atmosphere, forming a regular pattern about 1000 km across.
Common Misconceptions and Points to Note
There are a few points people tend to trip over when they start using this simulator, so watch out. First,it is tempting to think that "the larger the Rayleigh number, the more vigorous the convection, always", but it is not that simple. Convection does start once the critical value (about 1708) is exceeded, but as Ra rises, the neat roll pattern goes through stages of "secondary instability" in which the rolls distort or split, and finally reaches turbulence. For example, raising Ra from 10⁴ to 10⁵ shows the rolls no longer standing still but fluctuating over time. That is the entrance to "transient chaos", and raising the parameter too quickly makes you miss the transitions of the basic pattern, so the trick is to change it a little at a time.
Next,do not underestimate the aspect ratio (width/height). For example, with an aspect ratio of 2 (width twice the height), two convection rolls line up naturally, but with an in-between value such as 3.5, the number of rolls is not an integer and unstable patterns arise more easily. In nature and in real equipment the "width" of the layer is often fixed, so remember that changing the aspect ratio in the simulation actually means "changing the size of the experimental apparatus or observation area".
Finally,this simulation assumes "two dimensions" and "the Boussinesq approximation". Do not forget that. In three dimensions, hexagonal Bénard cells can appear instead of rolls, and when the temperature difference is extremely large or the compressibility of a gas cannot be ignored, this approximation does not hold. Understand it as an idealised model for learning the basic principles.
Frequently Asked Questions
When the Rayleigh number is below the critical value (about 1708), heat conduction dominates and no convection occurs. Set Ra to 2000 or more with the slider and wait a few seconds for the cells to appear. If the initial temperature disturbance is small it also takes time, so keep watching for a while.
It uses a simplified stream-function method, with buoyancy proportional to the temperature difference as the driving force, combined with viscous diffusion to approximate the velocity field. It is not a rigorous Navier–Stokes solution, but it reproduces the main behaviour — formation, growth and transition to turbulence of the convection cells — in real time.
Setting the Rayleigh number to 10⁵ or more and the aspect ratio (width/height) to the maximum of 4 makes it easier to observe several cells splitting and merging in a temporally irregular, turbulent state. When Ra exceeds 50,000, the state display changes to "Turbulent convection".
This tool is a simplified model for education and visualisation. It is useful for a qualitative understanding of real thermal convection but does not replace quantitative design or experiments. For practical use, please use dedicated CAE software that solves the full Navier–Stokes equations.
The Nusselt number Nu expresses the strength of heat transfer by convection: Nu = 1 for Ra < 1708 (pure conduction), and above the critical value it increases following a power law, Nu ≈ 0.069 Ra^{1/3}. In this simulator, raising Ra step by step shows the temperature contours turning into several cell structures and the effective heat transport increasing.
The Rayleigh number of mantle convection is very large, Ra ≈ 10⁷–10⁸, and it drives plate tectonics. Setting Ra to its maximum in this simulator produces unsteady, complex cell structures, which is close to the picture of what happens in the mantle over long timescales. In the real mantle, however, viscosity depends strongly on depth and temperature, so the similarity is only qualitative.
In theory, in a 2D container with Ra near the critical value, cells whose width and height are about equal (cells with an aspect ratio ≈ 1) are the most stable. In a container with a large aspect ratio, several cells line up, and their number is roughly the integer closest to width/height. Increasing the aspect ratio from 1 to 2 to 4 in the simulator shows 1, 2 and then 4 roll-shaped cells lining up.
Yes. Pr is the ratio of momentum diffusion to thermal diffusion (Pr = ν/κ). In liquid metals (Pr ≈ 0.01) heat diffuses quickly, so the thermal boundary layer is thicker than the velocity boundary layer, and the flow tends to become time-dependent soon after the critical value is exceeded. In silicone oil (Pr ≈ 100–1000) viscosity dominates, and the cells readily form stable, regular patterns (hexagons in 3D). This simulator assumes an ordinary fluid (Pr around 1) and has no function to change the Pr dependence directly, but Pr-dependent analysis is possible with OpenFOAM's buoyantBoussinesqPimpleFoam.
Real-World Applications
Industrial use
In epitaxial growth furnaces and CVD equipment for semiconductor manufacturing, natural convection in the gas above a heated substrate spoils the uniformity of temperature and film thickness, so the dimensions and pressure conditions of the equipment are considered using the Rayleigh number as a guide. It is also the basis for finding conditions that suppress convection cells in the heat treatment of glass substrates for LCD panels.
Research and education
In university fluid dynamics and thermal engineering courses, it is treated as a topic for visualising transitions in convection patterns as Ra changes. Watching the progression from Bénard cell formation to turbulence gives an intuitive understanding of the relationship between dimensionless numbers and flow structure. In geophysics it is sometimes considered together with analogue experiments of mantle convection.
Relation to CAE analysis and practical use
This simulator is a learning aid for grasping the conditions for the onset of convection and the rough trends of heat transfer, as a step before detailed three-dimensional CFD analysis. In design, the usual flow is to understand the trends by varying Ra and then verify them in detail with high-accuracy analysis in ANSYS Fluent, OpenFOAM and so on.
Set the strength of bottom heating with the Rayleigh number (Ra) slider (log scale, Ra = 100–100,000). Below the critical value Ra ≈ 1708 there is only heat conduction and no flow; above 1708, Bénard convection rolls develop
Changing the aspect ratio (width/height, 1–4) changes the width of the domain and therefore the number of convection rolls. At AR = 2.0 two rolls line up (this tool is a 2D model, so the cells are rolls)
Turn on the display options (temperature field, tracer particles, velocity vectors, streamline overlay) to observe the structure of the convection rolls
Worked Example
For a water layer 3 mm thick with the bottom at 30°C and the top at 20°C (ΔT = 10 K), the kinematic viscosity ν = 1.0×10⁻⁶ m²/s, thermal diffusivity α = 1.4×10⁻⁷ m²/s and volumetric expansion coefficient β = 2.1×10⁻⁴ K⁻¹ give Ra = g×β×ΔT×d³/(ν×α) ≈ 3970, so Ra > 1708 and convection occurs. Setting Ra ≈ 3981 (3.6 on the log scale) and an aspect ratio of 4.0 in this tool gives about 4 rolls side by side, with one pair of rolls (the wavelength) about 2H, close to the theoretical critical value λ ≈ 2.016H. This tool is a 2D model, so the cells are rolls; hexagonal cells are a pattern specific to 3D.
Practical Notes
In equipment that heats a fluid layer, convection begins once Ra exceeds the critical value, and the heat transport (Nusselt number) increases sharply. Where a uniform temperature is required, note in the design that the layer thickness d enters Ra to the third power
In the Earth's mantle the viscosity is extremely high but the layer is thousands of kilometres thick, so Ra reaches about 10⁷–10⁸ and the convection takes a complex three-dimensional structure
For an accurate Ra, it is important to correct the kinematic viscosity for the temperature dependence of the fluid. For water, ν = 1.01×10⁻⁶ at 20°C, but it falls to ν = 0.81×10⁻⁶ at 30°C
🎬 Watch it in motion
Bénard Convection | Cells Born from Heating Below #Shorts
