Rayleigh-Bénard convection of a Darcy fluid in a porous semicircle

\[\begin{split} \mathbb{S}_{\psi,c} \begin{cases} \Omega = \{(x, y)~:~0 < x^2 + y^2 < 1^2~,~y>0\} \\ \partial\Omega_{\text{upper}} = \{(x, y)~:~ x^2 + y^2 = 1^2 \} \\ \partial\Omega_{\text{lower}} = \{(x, y)~:~ y=0 \} \\ c_0(x,y)=\mathcal{N}(x,y) & \text{perturbed initial concentration} \\ c_{\text{D}}\vert_{\partial\Omega_{\text{lower}}}=1 & \text{light lower boundary} \\ c_{\text{D}}\vert_{\partial\Omega_{\text{arc}}}=0 & \text{heavy upper boundary} \\ \psi_{\text{D}}\vert_{\partial\Omega}=0 & \text{no-penetration on entire boundary} \\ \phi = 1 & \text{constant porosity} \\ \mathsf{D} = \mathsf{I} & \text{constant isotropic dispersion} \\ \mathsf{K} = \mathsf{I} & \text{constant isotropic permeability} \\ \mu = 1 & \text{constant viscosity} \\ \rho(c) = -c & \text{linear density} \\ \textbf{e}_g=-\textbf{e}_y & \text{vertically downward gravity} \\ \end{cases} \end{split}\]

from lucifex.fdm import AB2, CN
from lucifex.sim import run
from lucifex.utils.npy_utils import as_index
from lucifex.plt import plot_colormap, plot_mesh, save_figure, create_animation, display_animation
from py.C11_darcy_rayleigh_benard import darcy_rayleigh_benard_semicircle

simulation = darcy_rayleigh_benard_semicircle(
    Nradial=32,
    scaling='advective',
    Ra=300.0, 
    c_ampl=1e-3,  
    c_freq=8, 
    D_adv=AB2,
    D_diff=CN,
)

n_stop = 600
dt_init = 1e-6
n_init = 5
run(simulation, n_stop=n_stop, dt_init=dt_init, n_init=n_init)

c = simulation['c']
mesh = c.mesh

time_slice = slice(0, None, 4)
titles = [f'${c.name}(t={t:.3f})$' for t in c.time_series[time_slice]]

anim = create_animation(
    plot_colormap,
    colorbar=False,
)(c.series[time_slice], title=titles)
anim_path = save_figure(f'{c.name}(t)', return_path=True)(anim)

display_animation(anim_path)

time_indices = as_index(c.time_series, (0, 0.25, 0.5, -1), fraction=True)
for i in time_indices:
    fig, ax = plot_colormap(
        c.series[i], 
        title=f'${c.name}(t={c.time_series[i]:.2f})$',
        cartesian=False, 
    )
    if i == 0:
        plot_mesh(fig, ax, mesh, color='cyan', linewidth=0.5)
    save_figure(f'{c.name}(t={c.time_series[i]:.2f})', thumbnail=(i == -1))(fig)