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.

Acoustic pulse at \(t=0.000625s\)


Density profiles in the centerline at \(t=0.000625s\)


\(L_2\) 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.



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


res=( 8 16 32 64 )
for i in "${res[@]}"
    rm -rf "${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