The Navier-Stokes equations for incompressible fluids are solved in the pore space of a hexagonal close pack of perfect spheres having a particle diameter dp of 50 microns. In the Equation 1 below u is the interstitial fluid velocity, p denotes the pressure field, and ρ and μ are the density and dynamic viscosity of the fluid, respectively.
The single domain employed represents the primitive cell for the interstitial space (see Figure 2). The applied boundary conditions are for periodic flow with a given pressure differential ΔP between the inlet (highlighted in red) and the outlet of the cell, and symmetry everywhere else.
For small ΔP values the flow is laminar and the corresponding fluid velocity changes linearly with ΔP. As the fluid velocity values increase though, ΔP apparently begins to follow a quadratic dependence on |u|, even if still within the laminar flow regime. The red curve Figure 3 exemplifies the departure from linearity for ΔP(Re), where Re is the unitless particle Reynolds number defined as ρ|u|dp/μ. The black curve represents in fact the numerical solution for the Stokes equations, i.e. the variant of the Navier-Stokes equations above with the inertial term (highlighted in red) neglected. The black curve overlaps perfectly with the red one for Re values much smaller than unity, when the viscous term dominates. The results in Figure 3 thus demonstrate the empirical Darcy’s law and Forchheimer equations for macroscopic flow, solely by solving for the Navier-Stokes equations at the microscopic scale of the pore space.
Figures 4 shows the fluid velocity streamlines for three different values of the Reynolds number: Re = 1, 100, and 2000. At Re = 1 the inertial and viscous terms are equal, with the flow being well within the laminar regime, while at Re = 2000 inertial flow largely dominates and turbulence is expected to occur even in the case of flow through straight pipes. Secondary flow in the form of vortices emerges in this model at Re ~ 40, while still in the laminar regime. Chaotic behavior, indicative of turbulence, onsets at Re ~ 500.