Numerical Tests

Several numerical tests were carried out to verify our algorithm. Results from two cases, fluid flow between parallel plates and through an overlapping sphere model, are given below. For both cases we determined the fluid permeability, k, as defined by Darcy's law [1], $\langle\vec{v}\rangle=-\frac{k}{\mu}\langle\nabla P\rangle$, where $\langle\vec{v}\rangle$ is the average flow rate, $\nabla P$ is the average pressure gradient and µ is the fluid viscosity. Figure 1 shows the permeability, in units of the lattice spacing squared, as a function of the distance between parallel plates. Clearly, there is excellent agreement between the simulation and theoretical prediction. Surprisingly, very accurate results were obtained even for the case of a one node wide channel. Since permeability depends on the average flow or net flux rate of fluid, we conclude that the LB method accurately determines the net flux across a voxel surface, not the velocity at a point. Hence, resolving the actual local flow field at a point would require more nodes. We next consider the permeability of the pore space around a simple cubic array of solid spheres that are allowed to overlap for large enough radius (i.e. when the solid fraction, c, exceeds c 0.5236). In Figure 2, we compare our simulation data with that of Chapman and Higdon [12], which is based on the numerical solution of coefficients of a harmonic expansion that satisfies the Stokes equations. Note that our calculations were performed on a relatively small 64³ system. Again, agreement is very good, especially given that we used digitized spheres, while Chapman and Higdon used smooth spheres.

Figure 1: Flow through parallel plates. The permeability, k, is in units of lattice spacing squared, while the gap between plates is in units of lattice spacing.

Figure 2: Flow through spheres centered on a simple cubic lattice. The permeability is normalized by the square of the distance, d, between the sphere centers.