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

Abstract

A new pressure-projection pre-conditioning multi-fractional-step (PPP-MFS) 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 $p\ast$ 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.