Similarity of start-up flow in porous media for large pressure gradients

Abstract

We investigate the start-up flows through ordered porous media (hexagonal close-packed, face-centred cubic and body-centred sphere packs) by means of direct numerical simulations. The flows are initiated from rest and driven by a constant pressure gradient, allowing us to examine the transient development across a wide range of Hagen numbers. Dimensional analysis identifies two relevant time scales: the viscous diffusion time $τ_\mathrm{visc}$ and the inviscid time $τ_\mathrm{inv}$. While the small-time behaviour follows the viscous asymptotics of Johnson et al. [J. Fluid. Mech. 176, 379 (1987)], the subsequent emergence of nonlinear effects is universally governed by the inviscid time $τ_\mathrm{inv}$, rather than by any critical Reynolds number. At the pore scale, the transient evolution is characterised by the growth of thin vorticity layers on the sphere surfaces, their detachment into the pore space around $t \sim τ_\mathrm{inv}$, and the formation of inertial cores. Despite geometric differences, these processes occur in a remarkably similar sequence across all three packings. Vorticity magnitude exhibits laminar boundary-layer scaling with Hagen number, while in the body-centred cubic sphere pack case a transition towards turbulent-type scaling is observed. These results establish $τ_\mathrm{inv}$ as a unifying measure for the onset of nonlinearity in strongly accelerated porous media flows, with direct implications for the modelling of unsteady transport in natural and engineered systems.