An implicit algorithm of solving Navier-Stokes equations to simulate flows in anisotropic porous media

A coupled computational algorithm for modified Navier-Stokes equations to simulate flows in anisotropic porous media is proposed and described. The difference from the classic SIMPLE algorithm is in the completely implicit relationship between velocity and pressure owing to the implicit terms of the pressure and mass flow gradients in the continuity equation and momentum equation. One of the attractive features of this algorithm is the possibility of completely implicit discretization of off-diagonal components of the porous medium resistance tensor in the right-hand side of the momentum equation. Implicit discretization allows reducing the number of linear iterations as compared to the SIMPLE algorithm with explicit discretization of off-diagonal components. The specific features of discretization of modified equations including the discretization of boundary conditions and components of the porous medium resistance tensor are considered. The proposed algorithm is verified in comparison with the SIMPLE algorithm on a series of benchmark problems, such as the problem of a flow through a porous insert, a flow in a divided channel, and a flow through a cylindrical porous filter. The total problem runtimes and the number of iterations required for complete convergence are given to compare the two algorithms.