A new and very general technique for simulating solid–fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in a companion paper (Part 2), extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite-Reynolds-number flows, are reported.

Numerical simulations of particulate suspensions via a discretized boltzmann equation: part 1. theoretical foundation

A new and very general technique for simulating solid–fluid suspensions has been described in a previous paper (Part 1); the most important feature of the new method is that the computational cost scales linearly with the number of particles. In this paper (Part 2), extensive numerical tests of the method are described; results are presented for creeping flows, both with and without Brownian motion, and at finite Reynolds numbers. Hydrodynamic interactions, transport coefficients, and the short-time dynamics of random dispersions of up to 1024 colloidal particles have been simulated.

Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results

In this paper, theoretical and experimental approaches to flow, hydrodynamic dispersion, and miscible and immiscible displacement processes in reservoir rocks are reviewed and discussed. Both macroscopically homogeneous and heterogeneous rocks are considered. The latter are characterized by large-scale spatial variations and correlations in their effective properties and include rocks that may be characterized by several distinct degrees of porosity, a well-known example of which is a fractured rock with two degrees of porosity---those of the pores and of the fractures. First, the diagenetic processes that give rise to the present reservoir rocks are discussed and a few geometrical models of such processes are described. Then, measurement and characterization of important properties, such as pore-size distribution, pore-space topology, and pore surface roughness, and morphological properties of fracture networks are discussed. It is shown that fractal and percolation concepts play important roles in the characterization of rocks, from the smallest length scale at the pore level to the largest length scales at the fracture and fault scales. Next, various structural models of homogeneous and heterogeneous rock are discussed, and theoretical and computer simulation approaches to flow, dispersion, and displacement in such systems are reviewed. Two different modeling approaches to these phenomena are compared. The first approach is based on the classical equations of transport supplemented with constitutive equations describing the transport and other important coefficients and parameters. These are called the continuum models. The second approach is based on network models of pore space and fractured rocks; it models the phenomena at the smallest scale, a pore or fracture, and then employs large-scale simulation and modern concepts of the statistical physics of disordered systems, such as scaling and universality, to obtain the macroscopic properties of the system. The fundamental roles of the interconnectivity of the rock and its wetting properties in dispersion and two-phase flows, and those of microscopic and macroscopic heterogeneities in miscible displacements are emphasized. Two important conceptual advances for modeling fractured rocks and studying flow phenomena in porous media are also discussed. The first, based on cellular automata, can in principle be used for computing macroscopic properties of flow phenomena in any porous medium, regardless of the complexity of its structure. The second, simulated annealing, borrowed from optimization processes and the statistical mechanics of spin glasses, is used for finding the optimum structure of a fractured reservoir that honors a limited amount of experimental data.

Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing

▪ Abstract The current understanding of the average flow properties of packed beds and particle suspensions, in which inertia plays a significant role on the particle length scale, is examined. The features of inertial suspensions posing challenges to theoriticians include the nonlinear and unsteady nature of the governing equations, the inability to superimpose solutions, the prevalence of hydrodynamic instabilities, and the existence of particle-particle collisions. We discuss two special cases of inertial suspensions, for which detailed kinetic theories have been developed: (a) particles in a gas, and (b) spherical, high-Reynolds number bubbles in liquid. Subsequently, we review recent applications of computational fluid dynamics to simulate the motion in particle suspensions with both inertia and vorticity in the continueous phase. The synthesis of these analytical and numerical techniques is a promising approach to address the many challenges of modelling inertial suspensions.

Inertial effects in suspension and porous-media flows

Computational fluid dynamics (CFD) is rapidly becoming a standard tool for the analysis of chemically reacting flows. For single-phase reactors, such as stirred tanks and “empty” tubes, it is already well-established. For multiphase reactors such as fixed beds, bubble columns, trickle beds and fluidized beds, its use is relatively new, and methods are still under development. The aim of this chapter is to present the application of CFD to the simulation of three-dimensional interstitial flow in packed tubes, with and without catalytic reaction. Although the use of CFD to simulate such geometrically complex flows is too expensive and impractical currently for routine design and control of fixed-bed reactors, the real contribution of CFD in this area is to provide a more fundamental understanding of the transport and reaction phenomena in such reactors. CFD can supply the detailed three-dimensional velocity, species and temperature fields that are needed to improve engineering approaches. In particular, this chapter considers the development of CFD methods for packed tube simulation by finite element or finite volume solution of the governing partial differential equations. It discusses specific implementation problems of special relevance to packed tubes, presents the validation by experiment of CFD results, and reviews recent advances in the field in transport and reaction. Extended discussion is given of two topics: heat transfer in packed tubes and the design of catalyst particles for steam reforming.

Abdelmourhite Lahbabi

Papers

Flow in periodically constricted tubes: Transition to inertial and nonsteady flows

High Reynolds number flow through cubic arrays of spheres: steady-state solution and transition to turbulence