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

We investigate start-up flow from an initially quiescent state in three ordered sphere packings (hexagonal close-packed, face-centred cubic, body-centred cubic) driven by a constant intrinsic pressure gradient using direct numerical simulation. The main finding is that the first nonlinear departure from the initial linear regime occurs at a characteristic inviscid time tau Subscript i n v Baseline equals StartRoot rho d divided by StartAbsoluteValue bold nabla left angle bracket p right angle bracket Subscript italic i Baseline EndAbsoluteValue EndRoot τ i n v = ρ d / | ∇ ⟨ p ⟩ i | $\\tau _{{inv}}=\\sqrt {\\rho d/\\vert \\boldsymbol{\\nabla }\\langle p\\rangle _{\\textit{i}}\\vert }$ (where rho ρ $\\rho$ , d d $d$ , bold nabla left angle bracket p right angle bracket Subscript italic i ∇ ⟨ p ⟩ i $\\boldsymbol{\\nabla }\\langle p\\rangle _{\\textit{i}}$ correspond to fluid density, sphere diameter, intrinsic pressure gradient), rather than at a critical instantaneous Reynolds number. At early times, the flow consists of an irrotational outer motion and thin Stokes-type boundary layers near the solid surfaces; nonlinear effects emerge when the convective term in these boundary layers becomes comparable to the other leading-order terms. At the pore scale, the transient evolution is characterised by the growth of thin vorticity layers on the sphere surfaces, their large-scale migration into the pore space after times of order tau Subscript i n v τ i n v $\\tau _{{inv}}$ 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 the governing dimensionless parameter Hagen number, whilst in the body-centred cubic sphere pack case a transition towards turbulent-type scaling is observed. These results establish tau Subscript i n v