A pressure-projection pre-conditioning multi-fractional-step method for Navier-Stokes Flow in Porous Media

A new pressure-projection pre-conditioning multi-fractional-step (PPPMFS) method for incompressible Navier-Stokes flow in porous media is presented. This fractional step method is applied to decouple the pressure and the velocity; thereby, overcoming the computational costs and difficulty associated with the resolution of the nonlinear term in the Navier-Stokes equation for fine geometric models. Specifically, time evolution is decomposed into a sequence of multi-fractional solution steps. In the first step, an elliptic problem is solved for pressure (p) with a no-slip boundary condition. This gives the Stokes pressure and velocity fields. In the second step, the obtained pressure p is then projected onto the field p∗ and used to solve for velocity field (u) required for the pre-conditioning of the solution to the Navier-Stokes equation. The pressure and velocity fields are obtained from the solution of the Navier-Stokes equation in the third step. Numerical and geometric discretisation of porous samples were carried out using finite-element method. For flow in simple channel models represented by two- and three-dimensions and in systems with high conductivity, the Stokes and Navier-Stokes numerical solutions produced close pressure and velocity field approximations. For flow around a cylinder, computation time is consistently higher in the Navier-Stokes equation by a factor of 2 with a pronounced non-symmetric pressure field at high mesh refinements. This computation time is desirable given that Navier-Stokes computation without preconditioning can be orders of magnitude more expensive.