Merge branch 'add_docs' of github.com:chowland/AFiD-MuRPhFi into add_…

…docs

Makefile

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 # MARENOSTRUM (Intel): fabric intel mkl impi hdf5 fftw szip
 # SUPERMUC (Intel): fftw hdf5
 # DISCOVERER:
-#	GNU: hdf5/1/1.14/latest-gcc-openmpi fftw/3/latest-gcc-openmpi lapack
+#	GNU: hdf5/1/1.14/1.14.0-gcc-openmpi fftw/3/latest-gcc-openmpi lapack
 #	Intel: hdf5/1/1.14/latest-intel-openmpi fftw/3/latest-gcc-openmpi mkl
 
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docs/examples/rbc.md

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-# Rayleigh-Bénard Convection
+# Rayleigh-Bénard Convection
+
+One of the simplest setups we can consider is the Rayleigh-Bénard problem, where fluid is contained between two stationary parallel plates which are held at fixed temperatures.
+If the lower plate is maintained at a higher temperature than the upper plate, then density differences due to temperature can drive convection in the domain.
+Stronger thermal driving, characterised by a larger Rayleigh number, drives a stronger flow.
+
+In MuRPhFi, we can take advantage of the multiple resolution grid to efficiently simulate this problem at high $Pr$.
+Since `temp` on the coarse grid and `sal` on the refined grid evolve according to the same equations (up to a different $Pr$), we can treat `sal` as $-\theta/\Delta$ for the dimensionless temperature of the system.
+We set `activeT=0` to remove the effect of `temp` on the buoyancy and run the simulation.
+
+## 2D Visualization
+
+Here, we show the results for $Ra=10^8$, $Pr=10$.
+First, we use the script `parallel_movie.py` to produce a visualisation of a vertical slice of the "temperature" field `sal`:
+
+<video width="100%" controls>
+  <source src="../../assets/RBC.mp4" type="video/mp4">
+</video>
+
+## Statistics
+
+We can then dive into the statistics output in `means.h5` to calculate the dimensionless heat flux due to the convection, the Nusselt number $Nu$, and the Reynolds number $Re$.
+The following are produced from the script `plot_time_series.py`.
+
+## Nusselt number
+
+In Rayleigh-Bénard convection, there are a number of ways to calculate the Nusselt number, all of which should be equivalent (in a time-averaged sense) in a well-resolved, statistically steady state.
+Firstly, we can measure the conductive heat flux at the top and bottom plates:
+
+$$
+{Nu}_\mathrm{pl} = \frac{H}{\Delta} \left.\frac{\partial \theta}{\partial z} \right|_{z=0}, \qquad Nu_\mathrm{pu} = \frac{H}{\Delta} \left.\frac{\partial \theta}{\partial z} \right|_{z=H}
+$$
+
+The volume average of the heat flux will give us another definition, computed as the sum of conductive and convective contributions:
+
+$$
+Nu_\mathrm{vol} = 1 + \frac{\langle w'\theta'\rangle}{\kappa \Delta/H}
+$$
+
+Substituting this into the equations for kinetic energy $\langle|\boldsymbol{u}|^2\rangle$ and temperature variance $\langle\theta^2\rangle$ provide two further definitions, linked to the volume-averaged dissipation rates:
+
+$$
+Nu_\varepsilon = 1 + \langle \partial_i u_j \partial_i u_j \rangle / (U_f/H)^2, \qquad
+Nu_\chi = \langle |\nabla \theta|^2 \rangle / (\Delta/H)^2 .
+$$
+
+![Nusselt numbers](../figures/Nusselt.svg){ width="100%" }
+
+### Reynolds number
+
+Since there is no imposed flow and no volume-averaged mean flow in Rayleigh-Bénard convection, we use the volume-averaged kinetic energy $\mathcal{K}$ to compute a Reynolds number.
+With this approach, we can also separately compute Reynolds numbers based on the vertical kinetic energy $\langle w^2\rangle$ and on the horizontal kinetic energy $\langle u^2 + v^2\rangle$.
+
+As in the Nusselt number plot above, we appear to converge to a statistically steady state for $t \geq 75 H/U_f$ after the initial transient behaviour.
+
+![Reynolds number](../figures/Reynolds.svg){ width="100%" }
+
+## 3D Visualization
+
+Finally, we can use the saved 3D snapshots to make a volume rendering of the temperature field.
+This snapshot highlights the narrow plume structures driving the large-scale circulation at high $Pr$.
+The image has been produced using ParaView.
+
+![Volume rendering](../assets/RBC_Pr10_square.jpg){ width="100%" }