Dynamic similarity of vortex shedding in a superfluid flowing past a penetrable obstacle

We numerically investigate wake dynamics in a superfluid flowing past a penetrable obstacle. Unlike an impenetrable object, a penetrable obstacle does not fully deplete the density. We define an effective diameter $D_{\rm eff}$ from the Mach-1 contour of the time-averaged irrotational flow around the obstacle, which delineates the local supersonic region where quantized vortices nucleate. Using this flow-defined length scale, we construct a superfluid Reynolds number $Re_{\rm s} = (v_0 - v_c) D_{\rm eff}/ (\hbar/ m)$, where $v_0$ is the flow speed, $v_c$ is the critical velocity, and m is the particle mass. We show that $Re_{\rm s}$ organizes the wake dynamics across obstacle sizes and strengths: the transition from dipole-row emission to alternating vortex cluster shedding occurs at $Re_{\rm s}$ around 2, and both the Strouhal number and the drag coefficient collapse onto universal curves when plotted as functions of $Re_{\rm s}$. These results extend the concept of dynamic similarity in superfluid flows to penetrable obstacles and demonstrate that the dynamically relevant length scale is determined by the supersonic region rather than by the geometric obstacle size.