
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 solidliquid flows,
and multiphase flows generally, has yet to be fully exploited. To extract
information implicit in the equations of motion for solidliquid flows, it is
necessary to numerically solve the coupled system of differential equations
consisting of the equations of fluid motion, and the equations of rigidbody
motion (governing the particle motions), together with suitable initial and
boundary conditions. These equations are coupled through the noslip 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 NavierStokes 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 highperformance, stateoftheart software packages
called particle movers, capable of simulating the motion of 1000
particles in twodimensional 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 stateoftheart 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 bodyfitted grid, and an arbitrary LagrangianEulerian 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
noslip 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 3D 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 matrixfree 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 T3D or
its successors, or clusterbased parallelism such as that of several
interconnected SGI PowerChallenge workstations. Both architectures exhibit
twolevel 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 publicdomain 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
solidliquid 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 solidliquid flowsmean 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
twophase 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
