In many industrial applications, a permeable mesh (porous screen) is used to control the unsteady (most commonly vortex) flows. Vortex flows are known to display intriguing behavior while propagating through porous screens. This numerical study aims to investigate the effects of physical properties such as porosity, Reynolds number, inlet flow dimension, and distance to the screen on the flow behavior. The simulation model includes a piston-cylinder vortex ring generator and a permeable mesh constructed by evenly arranged rods. Two methods of user-defined function and moving mesh have been applied to model the vortex ring generation. The results show the formation, evolution, and characteristics of the vortical rings under various conditions. The results for vorticity contours and the kinetic energy dissipation indicate that the physical properties alter the flow behavior in various ways while propagating through the porous screens. The numerical model, cross-validated with the experimental results, provides a better understanding of the fluid–solid interactions of vortex flows and porous screens.