Thermal convection
Here we explore thermal convection in a 2D Boussinesq model. We augment the vorticity formulation of fluid dynamics by adding a temperature field to arrive at
\[\begin{aligned} \pd{\omega}{t} &= \nu \nabla^2 \omega - \pd{\psi}{y} \pd{\omega}{x} + \pd{\psi}{x} \pd{\omega}{y}+ \pd{b}{x},\\ \varepsilon \pd{\psi}{t} &= \nabla^2 \psi + \omega\\ \pd{b}{t} &= \kappa \nabla^2 b -\left( \pd{\psi}{y} \pd{b}{x} - \pd{\psi}{x} \pd{b}{y}\right), \end{aligned}\]where $b$ is the difference of the temperature from the top boundary, $\kappa$ is a thermal conductivity constant, and heating is provided at the bottom boundary via the parameter $T_b$.
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Load the interactive boundary-driven convection model. By default, the temperature perturbation $b$ is plotted.
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The heating at the lower boundary will become unstable to high-frequency perturbations, which will grow and coalesce into larger Rayleigh–Bénard cells. You can also click to add a small region of warm air, which will then convect upwards.
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Different parameter values and initial conditions can lead to qualitatively-similar behaviour, but with different scales involved. Here is a simulation with larger initial data that undergoes the instability away from the boundary.
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Another change would be to use forcing at the top and bottom in the temperature field, and to change the horizontal conditions from periodic to no-flux conditions. This boundary simulation implements these changes. One can then more clearly see convective cells forming and merging.
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One can also play with colour schemes, blending, and add a bit of stochastic forcing into the model to achieve a cloudy simulation over Durham cathedral.
This page was suggested and written with the help of Mathew Barlow.