Optimal Solutions in Rayleigh-Benard Convection

This code finds exact, steady solutions that optimize vertical heat transport in 2D Rayleigh-Benard convection. Specific details on the approach and the numerics can be found in [1].

Code Overview

The flowmap algorithm is used to find steady solutions to the 2D Boussinesq equations. The Nusselt number is computed at each steady solution for a given $Ra$ and $Pr$. At each $(Ra, Pr)$ pair, steady solutions are computed in different box sizes, $L$. The size of the domain (box) in $x$ is given by $L=2\pi/\alpha$ where $\alpha$ is the wavenumber that sets the scale of the solution in $x$. The solution in box size $L$ that maximizes the Nusselt number is referred to as the optimal solution.

Input File Structure

Line #	Column 1	Column 2	Column 3
1	$Pr$	Initial $\alpha$	$\alpha$ step
2	Initial $Ra$	# of $Ra$s to do	$Ra$ increment multiplier
3	Flow map time	Time step	Leave blank
4	$y$ at bottom	$y$ at top	Leave blank
5	Nx	Ny	Nz
6	x-refinement	y-refinement	z-refinement
7	Save to VTK	Save binary restart files	Calculate actual optimal solution

Running the code

Run make to generate an executable. The basic executable will be called Ra_loop.exe. Run the executable from the command line by typing ./Ra_loop.exe. The code will automatically search for the input file input.data and read in any parameters.

Depending on how the input file is set up, the code will loop over different values of Ra. For each Ra, it will perform an optimization over the box size to find which box size maximizes the heat transport at the Ra. This optimization is done by fitting a parabola to three points. The code continues until a maximum Nu is found.

The results are written out in various formats:

• Nu.txt: Contains the Ra, Nu, optimal wavenumber, and optimal box size
• vtkdata/: Solution fields are written out in VTK format at each Ra for solutions optimizing heat transport (Note: Need to create vtkdata directory before running code)
• uy, temperature: Vertical velocity and temperature fields in binary format. Used for restarts. Note that ux can be computed from uy in 2D from the continuity equation.

Contributing to the Project

Contributions to the code are very welcome. To contribute, please do the following:

1. Create a new issue
2. Move that issue to "In Progress" on the Kanban board
3. Create a new branch
   1. To create a branch locally do git checkout -b <new_branch_name>
   2. To push this branch to the main repo do git push --set-upstream origin <new_branch_name>
      • This is only necessary the first time you create a branch.
4. Make changes to the code on your branch and push your changes when they're ready
   1. git commit locally
   2. git push to push your new changes
   3. Recommendation: Only push changes that compile
5. When you're ready, make PR to main
   • Assign a reviewer
6. After making the PR, link the issue with that PR
7. Continue making changes on that branch until they are approved to be merged to main

References

[1] Sondak, D., Smith, L.M., Waleffe, F. (2015). Optimal heat transport solutions for Rayleigh-Benard convection Journal of Fluid Mechanics, 784, 656-595.