Direct Simulation of the Motion of Particles in Flowing Liquids

Project Overview

The current popularity of computational fluid dynamics is rooted in the perception that information implicit in the equations of fluid motion can be extracted without approximation. A similar potential for solid-liquid flows, and multiphase flows generally, has yet to be fully exploited. To extract information implicit in the equations of motion for solid-liquid flows, it is necessary to numerically solve the coupled system of differential equations consisting of the equations of fluid motion, and the equations of rigid-body motion (governing the particle motions), together with suitable initial and boundary conditions. These equations are coupled through the no-slip boundary condition on the particle surfaces, and through the hydrodynamic forces and torques exerted by the fluid on the particles.

Developing highly efficient computational schemes for solving this coupled system of equations in two and three dimensions, both for Newtonian fluids (governed by the Navier-Stokes equations) and for the family of viscoelastic fluids (with Oldroyd~B principal parts) most frequently studied in the rheology literature, represents a grand challenge in computational mechanics.

The goal is to develop high-performance, state-of-the-art software packages called particle movers, capable of simulating the motion of 1000 particles in two-dimensional simulations in Newtonian fluids in regular and complex geometries, 100 spheres in similar circumstances in three dimensions, and 100 particles in viscoelastic fluids. Such simulations will be extremely computationally intensive. It is therefore imperative to develop the most efficient possible computational schemes, and to implement them on parallel machines, using state-of-the-art parallel algorithms.

We propose to develop two different finite element schemes to meet this challenge. The first is a generalization of the standard Galerkin finite element method in which both the fluid and particle equations of motion are incorporated into a single variational equation, in which both the fluid and particle velocities appear as primitive unknowns. The hydrodynamic forces and torques on the particles are eliminated in the formulation, so need not be computed as separate quantities. The computation is performed on an unstructured body-fitted grid, and an arbitrary Lagrangian-Eulerian moving mesh technique has been adopted to deal with the motion of the particles. This scheme is discussed further in Computational Methods .

In the second approach, an embedding method, the fluid flow is computed as if the space occupied by the particles were filled with fluid. The no-slip boundary condition on the particle boundaries is enforced as a constraint using Lagrange multipliers. This allows a fixed grid to be used, eliminating the need for remeshing, a definite advantage in parallel implementations. This scheme is also discussed further in Computational Methods .

A crucial computational issue to be addressed is the efficient solution of the various algebraic systems which arise in the schemes. These systems can be extremely large for 3-D problems, and their solution can consume up to 95 of the CPU time of the entire simulation. It is therefore imperative to use efficient iterative solution methods, with matrix-free preconditioners, and to implement them on parallel architectures.

We plan to develop a library of parallel numerical algorithms to solve these systems. This parallel library will consist of algorithms for solving nonlinear algebraic equations using variants of Newton's method, preconditioned iterative solvers for sparse symmetric indefinite and nonsymmetric linear systems, and rapid elliptic and Stokes solvers on uniform grids. This library will be used for rapid prototyping of simulation codes for the application problems referred to above. For further details, see Computational Methods .

The library will be augmented with a collection of kernels to allow it to be efficiently portable across either the massive parallelism of the Cray T3-D or its successors, or cluster-based parallelism such as that of several interconnected SGI Power-Challenge workstations. Both architectures exhibit two-level parallelism that is ideally suited for schemes such as the embedding method on a fixed grid.

The codes will also be placed in the public domain, in modules designed to encourage the widest audience of potential users. The number of potential users of public-domain software for workstations is large and continually increasing as workstations get cheaper and more powerful.

The number of potential applications for such codes is extremely large. We propose to use them to address several fundamental issues in the dynamics of solid-liquid flows, and also to study a number of problems of practical engineering interest. In the category of fundamental dynamics, we propose to reveal the local rearrangement mechanisms responsible for the clusters and anisotropic microstructures observed in particulate flows, produce statistical analyses of solid-liquid flows---mean values, fluctuation levels and spectral properties, derive engineering correlations of the kind usually obtained from experiments, and provide clues and closure data for the development of two-phase flow models and a standard against which to judge the performance of such models. These issues are discussed further in Applications .

Among the industrial applications we propose to address are sedimentation, fluidization, slurry transport of solid particles in Newtonian and viscoelastic fluids and hydraulic fracturing of hydrocarbon reservoirs. These applications are discussed further in Industrial Problems .

All simulation results will be compared with experimental data. The literature contains a large volume of experimental data. The results from simulations of actual industrial processes will be compared with actual field data whenever possible. At the moment, we are thinking of advancing our understanding of the lubricated transport of slurries, the bubbling of fluidized beds and particle placements by proppant in fractured oil reservoirs. The existing data are incomplete; we therefore plan to carry out our own experiments, under the conditions assumed in the simulations. Probably most of the experiments will be carried out in Joseph's lab using funding from other grants. The present idea is to construct equipment with controlled and continuous inputs of particles and fluids to examine slurry transport in pipes in horizontal, tilted and vertical flow.

The comparison of simulations with experiments is essential when the suspending fluid is viscoelastic because the constitutive equation for the fluid used in the experiments is never known exactly; it may be adequate for some flows and not for others. This is to be contrasted with the situation for Newtonian fluids, where a single constitutive equation applies in all the usual situations. It is therefore extremely important to develop particle movers for the viscoelastic fluids which are actually used in the fracturing industry and in other applications.

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Last updated October 16, 2000