Error analysis and correction for Lattice Boltzmann simulated flow conductance in capillaries of different shapes and alignments

We study the relative error in conductance calculations, for simulated flow of a single component single phase fluid through a capillary in three dimensions, by the Lattice Boltzmann (LB) method with bounce-back boundary conditions. The relative error with respect to analytical results for capillary cross-sections of circular, triangular and square shapes are calculated as a function of the cross-section diameter, a, and for different alignment of the cross-section relative to the underlying lattice grid. It is shown, when the shapes are not aligned perfectly to the lattice, that the relative error decreases systematically with the size, a, as ~1/a when a is evaluated by mapping the computed cross-sectional area, in terms of the enclosed number of grid points, to the respective geometrical shapes concerned. For perfectly aligned geometries, viz. the square capillary aligned to the LB lattice grid or rotated with its side along the diagonal of the LB grid, the relative error decreases as ~1/a^2. A simple method is suggested to locate the boundary wall depending on its orientation relative to the grid, such that the exact conductance of the new shape matches the LB computed conductance.