The role of porosity in the transition to inertial regime in porous media flows

In this work we investigate the fundamental physical mechanism of the transition from Darcy to inertial (Darcy-Forchheimer) regime in the flow through porous media from the perspective of the vortex formation and growth. We focus on investigating the influence of porosity on the tortuosity--Reynolds number relation for periodic stochastic and simple cubic systems by numerically solving the Navier-Stokes equations in the pore-scale. We observe that the tortuosity defined by integrals over the fluid volume exhibits similar behavior in both types of systems. We also show that the tortuosity based on averaging of the streamlines length diverges from the volume-integrated one when the inertia onset takes place. We show that the discrepancy between those two tortuosities at increasing Reynolds number carries information about the geometrical confinement within the porous sample and is largely governed by the growth of the volumes occupied by vortices and their kinetic energy. Our results shed new light on the fundamental mechanisms of the transition between Darcy and the inertial regime and bring a better understanding of the role of the porous media geometry in this transition.