# C9. Acoustic wave in cylindrical channel

## Case description

This is the case of an acoustic wave propagating in a cylindrical channel. The geometry is a $$x$$-direction aligned circular channel with periodic boundary conditions. The acoustic wave propagates along the $$x$$ direction.

The density pulse at the center of the channel is defined as:

\begin{align}\begin{aligned}\rho'(x) = \alpha \exp(-(x/\sigma)^2)\\\rho(x) = \rho_0 + \rho'(x)\\p(x) = p_0 + \rho'(x) c_s c_s\\u(x) = c_s \rho'(x) / \rho_0\end{aligned}\end{align}

The background pressure is set to $$p_0 = 100000 erg/cm^3$$, the background density is set to $$\rho_0 = 0.0014 g/cm^3$$, $$\alpha=10^{-6} g/cm^3$$, and $$\sigma=10cm$$. The length of the channel is 100cm and the radius is 25cm. The simulations are performed for $$t=0.000625s$$. The CFL is set to 0.001 to minimize time discretization errors.

## $$L_2$$ error norm of density

The $$L_2$$ error norm for a quantity $$s$$ is defined as

$\epsilon = \sqrt{ \frac{\sum_{i=1}^{n_x} (s_i^h-s_i^*)^2 }{n_x}}$

where $$s^h$$ is the numerical solution, $$s^*$$ is the exact solution, and $$n_x$$ is the number of cells in the $$x$$-direction.

Note

The second order convergence observed here is expected for this test case as all relevant physics happen in the direction perpendicular to the EB surface.

## Running study

paren=pwd
pelec="${paren}/PeleC3d.gnu.MPI.ex" mpi_ranks=36 res=( 8 16 32 64 ) for i in "${res[@]}"
do
rm -rf "${i}" mkdir "${i}"
cd "${i}" || exit cp "${paren}/inputs_3d" .
srun -n ${mpi_ranks} "${pelec}" inputs_3d amr.n_cell="${i}${i} ${i}" > out ls -1v *plt*/Header | tee movie.visit cd "${paren}" || exit
done