Permeable disks at low Reynolds numbers

The wake of a permeable disk can be rather exceptional: a toroidal vortex can form and remains stably at a fixed distance from the disk. The streamwise length of the vortex depends on the Reynolds and Darcy numbers. We investigate this fascinating flow for Reynolds numbers from 10 to 130 and Darcy numbers ($Da$) from $10^{-9}$ to $1$. Direct numerical simulations are performed on a 2D grid with axisymmetric boundary conditions. Three flow regimes are observed: for low $Da$ (effectively impervious), the wake is characterized by the presence of a toroidal vortex whose length is approximately equal to that of an impervious disk. For $10^{-6}