Advecting cone

This is a classical test case for demonstrating the benefit of a high order discretization. It can be found in this classical reference https://epubs.siam.org/doi/book/10.1137/1.9781611970425 in chapter 11. We consider scalar advection of a cone of diameter 1 and height 1, with its center initially positioned at (1, 0). The advecting velocity is a vortex with $u=-x \pi$ and $v = y \pi$, meaning that the cone makes a full rotation in 2 seconds.

The case is run for 1 rotation and the shape of the cone is examined. Ideally it should be preserved, but in practice numerical error lead to the distortion of the shape. The idea is to see how the order of the polynomial basis affects the error, while keeping the total number of DoFs roughly the same.

Case setup

To keep the advecting velocity constant, we enable freeze for the fluid. The initial velocity distribution is set in the user file, and similarly for the scalar. We output generously: 100 data dumps during one rotation, so that one can examine the dynamics during post-processing.

To analyze the results, we use the probes simcomp to extract the values of the scalar along a line aligned with $y$ in the center of the domain. A Python script to create the probes is provided.

The mesh is generated with genmeshbox with periodic conditions in all directions, and a single element across $z$, to mimic a 2D case.

The script run.sh can be used to run the study, computing the results for several meshes and polynomial orders. We suggest inspecting the script and modifying it to your liking. Note that the change_order.py script is used to change the polynomial order in the case file.

A Python script to plot the results is also provided for convenience.

Expected results

The following curves will be obtained with the settings in the run.sh script. One can see that the results improve with higher order, even if the number of DoFs stays roughly the same.