The characteristics of flow within porous materials in the regime where non-linear inertia effects are significant have previously defied analysis. However, the combination of high speed computer facilities and modern numerical techniques provides a powerful analytical tool which is applied to the study of saturated flow through an idealised two-dimensional porous material.
The Navier-Stokes equations are solved using an iterative (squaring) technique for a number of particle configurations. The hydrodynamic characteristics obtained describe the flow within porous materials for the full laminar regime, extending from the creeping flow range into the non-linear laminar range i.e. for Reynolds numbers based on the seepage velocity, ranging from zero to about 200.
The fundamental dimensionless forms of the Navier-Stokes equations are extended to develop, with appropriate assumptions, the conventional empirical formulae used for flow through porous materials and the numerical solutions obtained are compared with these traditionally accepted relations of Darcy (l856), Porchheimer (1901), Kozeny (l927), Carman (l937) and Tek (l957).
The numerical procedures required to effect the solutions are outlined in detail, including the development of the appropriate finite-difference equations, the treatments of boundary discontinuities and the problems associated with divergence of the non-linear equations.
The numerical solutions obtained compare favourably with experimental results reported by Lord (1955) for flow through two dimensional fibres and with the analytical results of Happel and Brenner (1965) among others. These solutions are presented as piezometric head gradients in the direction of flow, as drag coefficients and as a series of contour plots. The contour plots, together with the flow characteristics at each grid point in the field give a detailed picture of the flow profile within a single pore of a porous material. Thus the development and importance of the wake behind a solid particle is illustrated as the flow increases from creeping flow to the stage where the wake bubbles virtually envelop the particles.
The study of porous media flow by the method outlined gives a detailed account of the hydrodynamic characteristics of such flows. Although the method is restricted to rather simplified boundary conditions many features of porous media flow, which previously have defied analysis, can be investigated by this approach. Some areas in which the same approach can be adopted are nominated.