Flow regimes through periodic arrays of cylinders

Steady single-phase flow through porous media is well understood in the low-Reynolds number regime for which the pore-scale flow is governed by Stokes' equations and the macroscopic flow satisfies the Darcy equation. At higher Reynolds numbers, however, questions remain about how to best model flows through porous media for which pore-scale inertial effects are important. This is particularly true when the pore-scale or macro-scale flow is also unsteady. Thus motivated, we perform a broad set of 2D direct numerical simulations of incompressible single phase flow through periodic arrays of cylinders using a finite volume method with immersed boundaries. We consider flows driven by either a steady pressure gradient or a gradient that oscillates in time. For steady pressure gradients, we systematically vary the porosity between 0.25 to 0.95, and the Reynolds number from the Stokes regime to regimes characterized by pore-scale unsteady vortex shedding. In this manner, we identify up to five regimes of steady macroscopic flow, each satisfying a different macroscopic relationship between the volume averaged pressure and velocity. These are then contrasted with cases driven by oscillating pressure gradients. For these cases, we focus on identifying critical Strouhal numbers.