Robert S. Maier

Simulation of flow through bead packs using the lattice Boltzmann method

The lattice Boltzmann method (LBM) is used to simulate viscous fluid flow through a column of glass beads. The results suggest that the normalized velocity distribution is sensitive to the spatial resolution but not the details of the packing. With increasing spatial resolution, simulation results converge to a velocity distribution with a sharp peak near zero. A simple argument is presented to explain this result. Changes in the shape of the distribution as a function of flow rate are determined for low Reynolds numbers, and the large-velocity tail of the distribution is shown to depend on the packing geometry. The effect of a finite Reynolds number on the apparent permeability is demonstrated and discussed in relation to previous results in the literature. Comparison with velocity distributions from NMR (nuclear magnetic resonance) spectroscopy finds qualitative agreement after adjusting for diffusion effects in the NMR distributions.

Pore-scale simulation of dispersion

Tracer dispersion has been simulated in three-dimensional models of regular and random sphere packings for a range of Peclet numbers. A random-walk particle-tracking (PT) method was used to simulate tracer movement within pore-scale flow fields computed with the lattice-Boltzmann (LB) method. The simulation results illustrate the time evolution of dispersion, and they corroborate a number of theoretical and empirical results for the scaling of asymptotic longitudinal and transverse dispersion with Peclet number. Comparisons with nuclear magnetic resonance (NMR) spectroscopy experiments show agreement on transient, as well as asymptotic, dispersion rates. These results support both NMR findings that longitudinal dispersion rates are significantly lower than reported in earlier experimental literature, as well as the fact that asymptotic rates are observed in relatively short times by techniques that employ a uniform initial distribution of tracers, like NMR.

Hydrodynamic dispersion in confined packed beds

Pore-scale simulations of monodisperse sphere packing and fluid flow in cylinders have reproduced heterogeneities in packing density and velocity previously observed in experiment. Simulations of tracer dispersion demonstrate that these heterogeneities enhance hydrodynamic dispersion, and that the degree of enhancement is related to the cylinder radius, R. The time scale for asymptotic dispersion in a packed cylinder is proportional to R2/DT, where DT represents an average rate of spreading transverse to the direction of flow. A generalization of the Taylor–Aris model of dispersion in a tube provides qualitative predictions of the long-time dispersion behavior in packed cylinders.

Enhanced dispersion in cylindrical packed beds

The effective longitudinal dispersion constant, DLeff, in cylindrical packed beds is larger than in the bulk due to the existence of radial inhomogeneities induced by the cylinder walls. For dense random packed beds, DLeff can be several times larger than the bulk value, even for arbitrarily large cylinder radius, R. The time-scale for attaining asymptotic dispersion rates in a cylindrical geometry is neither the convective nor the diffusive time-scale, but rather DT/R2, where DT is the bulk transverse dispersion rate. Similar effects are predicted for packed beds confined in ducts of any cross-sectional geometry. The case of a rectangular duct, compared with an infinite slit, provides an intuitive model for the influence of walls in the limit as R goes to infinity.

Pore-Scale Flow and Dispersion

Pore-scale simulations of fluid flow and mass transport offer a direct means to reproduce and verify laboratory measurements in porous media. We have compared lattice-Boltzmann (LB) flow simulations with the results of NMR spectroscopy from several published flow experiments. Although there is qualitative agreement, the differences highlight numerical and experimental issues, including the rate of spatial convergence, and the effect of signal attenuation near solid surfaces. For the range of Reynolds numbers relevant to groundwater investigations, the normalized distribution of fluid velocities in random sphere packings collapse onto a single curve, when scaled with the mean velocity. Random-walk particle simulations in the LB flow fields have also been performed to study the dispersion of an ideal tracer. These simulations show an encouraging degree of quantitative agreement with published NMR measurements of hydrodynamic and molecular dispersion, and the simulated dispersivities scale in accordance with published experimental and theoretical results for the Peclet number rangek 1 ≤Pe≤1500. Experience with the random-walk method indicates that the mean properties of conservative transport, such as the first and second moments of the particle displacement distribution, can be estimated with a number of particles comparable to the spatial discretization of the velocity field. However, the accurate approximation of local concentrations, at a resolution comparable to that of the velocity field, requires significantly more particles. This requirement presents a significant computational burden and hence a numerical challenge to the simulation of non-conservative transport processes.

A simple computational model for bubble plumes

Abstract A simple approximation is proposed for the buoyant force in a bubble plume. Assuming a uniform radius and slip velocity for the entire bubble column, an expression is derived for the vertical acceleration of liquid in the column, which is directly proportional to the injected gas flow-rate and inversely proportional to depth and velocity. This bubble-induced acceleration has been implemented with a k – ϵ turbulence model in a three-dimensional, single-phase computational fluid dynamics (CFD) code, whose numerical predictions indicate that the velocity outside the plume is relatively insensitive to the column radius and the bubble slip velocity. Using a median observed value of 25 cm/s for the bubble slip velocity, and a column radius given by an empirical formula based on the work of Cedarwall and Ditmars, the model renders predictions for velocity that compare favorably with experimental data taken outside single and double plumes in water. Predicted velocity increases in less-than-linear fashion with the gas flow-rate, and the flow-rate exponent approaches 1/2 in the lower limit, and 1/3 in the upper limit. In the range of flow-rates (200–22,000 cm 3 /s) for which the model is validated herein, the exponent is roughly 2/5.

Application of the lattice‐Boltzmann method to study flow and dispersion in channels with and without expansion and contraction geometry

The lattice-Boltzmann (LB) method, derived from lattice gas automata, is a relatively new technique for studying transport problems. The LB method is investigated for its accuracy to study fluid dynamics and dispersion problems. Two problems of relevance to flow and dispersion in porous media are addressed: (i) Poiseuille flow between parallel plates (which is analogous to flow in pore throats in two-dimensional porous networks), and (ii) flow through an expansion-contraction geometry (which is analogous to flow in pore bodies in two-dimensional porous networks). The results obtained from the LB simulations are compared with analytical solutions when available, and with solutions obtained from a finite element code (FIDAP) when analytical results are not available

A parallel QR algorithm for the nonsymmetric eigenvalue problem

Abstract This paper describes a prototype parallel algorithm for approximating eigenvalues of a dense nonsymmetric matrix on a linear, synchronous processor array. The algorithm is parallel implementation of the explicitly-shifted QR, employing n distributed-memory processors to deliver all eigenvalues in O ( n 2 ) time. The algorithm uses Givens rotations to generate a series of unitary similarity transformations. The rotations are passed between neibouring processors and applied, in pipeline fashion, to columns of the matrix. The rotations are also accumulated in a unitary transformation matrix, enabling the solution of eigenvectors via back-substitution and back-transformation. The algorithm involves only local communication, and confronts the problems of convergence, splitting and updating the shift in a pipelined scheme. The algorithm is implemented on a hypercube, using a ring of processors to simulate a systolic array. Experimental results on the NCUBE/seven hypercube show O ( n ) speedup over competing sequential codes, despite the overhead of interprocessor communication. Speedup and efficiency are estimated by comparing with EISPACK performance.

Parallel solution of large-scale, block-angular linear

Many important large-scale optimization problems can be formulated as linear programs with a block-angular structure. This structure lends itself naturally to parallel solutions and is used to great advantage in the solution method described. To demonstrate the efficiency of the method, it has been implemented and computationally tested on both a shared-memory vector multiprocessor (CRAY-2) and a local-memory hypercube (NCUBE/seven) with 64 processors. Computational results for problems with as many as 24,000 rows and 74,000 columns (1,024 blocks and 1.4 million nonzero elements) are presented. A problem of this size was solved on the NCUBE in less than four minutes and the CRAY-2 in 37 seconds.

Large–scale minimization on the CM-200

A finite-element formulation of the minimal-surface problem is described and solved with I the limited-memory BFGS (LM-BFGS) minimization algorithm on a 32K-processor Connection Machine (CM-200). Implementation of the problem is described with an emphasis on the data structures necessary for matching the finite-element code with the LM-BFGS minimization code. Performance of the complete code is evaluated and compared with a vectorized implementation on the CRAY-2, for problems ranging in size from n= 104 to n> 106> knowns. The results of the comparison indicate the effectiveness of the CM-200 for very large-scale problems. The implementation is representative of a large class of variational problems and suggests useful techniques for the software interface between minimization codes and nonlinear finite-element problems.