A numerical analysis of the inertial correction to Darcy's law

Physica D: Nonlinear Phenomena, 2001

Nonlinear correction to Darcy's law for a two-dimensional ow through periodic arrays of elliptic cylinders is investigated using the lattice Boltzmann equation method. For small Reynolds number, we ÿnd that the leading correction for the anisotropic system is a third-degree homogeneous function of the average uid velocity, and possible terms of the function are determined by symmetry. The dimensionless coe cients 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 signiÿcantly change the coe cients.

Mathematical Methods in the Applied Sciences, 2001

The concept of structural stability is discussed and brie y reviewed within the context of uid ow in porous media. Then the equations for non-Boussinesq (penetrative) convection in a porous medium with anisotropic permeability are analysed. It is shown, by deriving truly a priori estimates, that the solution exists in the natural functional framework and depends continuously on changes in the permeability. Since under appropriate circumstances the onset of thermal convection may be via Hopf bifurcation rather than stationary convection, such a priori continuous dependence estimates are of much value.

Chemical Engineering Science, 2000

A momentum equation for Newtonian-#uid #ow in homogeneous and isotropic porous media is derived on the basis of momentum conservation over a representative elementary volume. The resultant equation can be expressed in the dimensionless form as

Transport in Porous Media, 2019

In this work, single-phase incompressible laminar flow in 2D model porous media is studied and the influence of microscopic structural disorder on the flow is thoroughly investigated. Emphasis is laid upon the onset of the deviation from Darcy's law and the identification of different inertia regimes observed before the flow becomes unsteady. For this purpose, six globally disordered pore structures were generated and the values of the critical Reynolds number at which the flow becomes unsteady corresponding to the first Hopf bifurcation were determined. Numerical simulations of steady laminar single-phase flow were then carried out to investigate the effects of the microstructures on the inertial correction to Darcy's law. Different flow regimes, namely weak inertia, strong inertia and the regime beyond strong inertia, are identified. Comparisons are made with results presented in the literature which were restricted to ordered and locally disordered structures. The critical Reynolds number decreases and inertia intensity increases as more disorder is introduced into the pore structure. Results on flow inertia widely extend some previous studies on the subject and show that it is mainly influenced by the shape of the obstacles (either circular or square), slightly affected by the inclination of the square cylinders and hardly disturbed by the size distribution of the obstacles.

Geophysical Research …, 2011

International Journal of Heat and Mass Transfer, 2014

ABSTRACT In this paper, we propose a methodology to derive a macro-scale momentum equation that is free from the turbulence model chosen for the pore-scale simulations and that is able to account for large-scale anisotropy. In this method, Navier–Stokes equations are first time-averaged to form a new set of equations involving an effective viscosity. The resulting balance equations are then up-scaled using a volume averaging methodology. This procedure gives a macro-scale generalized Darcy–Forchheimer equation to which is associated a closure problem that can be used to evaluate the apparent permeability tensor including inertia effects. This approach is validated through 2D and 3D calculations. Finally, the method is used to evaluate the tensorial macro-scale properties for a gas flow through structured packings.

Physical Review Letters, 1999

We investigate the origin of the deviations from the classical Darcy law by numerical simulation of the Navier-Stokes equations in two-dimensional disordered porous media. We apply the Forchheimer equation as a phenomenological model to correlate the variations of the friction factor for different porosities and flow conditions. At sufficiently high Reynolds numbers, when inertia becomes relevant, we observe a transition from linear to nonlinear behavior which is typical of experiments. We find that such a transition can be understood and statistically characterized in terms of the spatial distribution of kinetic energy in the system. [S0031-9007(99)09541-1]

Journal of Fluid …, 1997

Under fairly general assumptions, this paper shows that for periodic porous media, whose period is of the same order as that of the inclusion, the nonlinear correction to Darcy's law is quadratic in terms of the Reynolds number, i.e. cubic with respect to the seepage velocity. This claim is substantiated by reinspection of well-known experimental results, a mathematical proof (restricted to periodic porous media), and numerical calculations.

HAL (Le Centre pour la Communication Scientifique Directe), 2018