Nonlinear correction to Darcyʹs law for a flow through periodic arrays of elliptic cylinders

Nonlinear correction to Darcyʹs law for a two-dimensional flow through periodic arrays of elliptic cylinders is investigated using the lattice Boltzmann equation method. For small Reynolds number, we find that the leading correction for the anisotropic system is a third-degree homogeneous function of the average fluid velocity, and possible terms of the function are determined by symmetry. The dimensionless coefficients of the terms, which depend only on the geometry of the system, are determined for various values of the volume fraction and eccentricity of the cylinders. Also, it is found that small deformation of the shape of the cylinder does not significantly change the coefficients.