Direct Simulation of the Motion of Particles in Flowing Liquids
The clusters and anisotropic microstructures observed in solid-liquid flows,
such as those shown here.
Figure 1: Flow-induced anisotropy of fluidized suspensions:
(a) Particles across the stream are induced by drafting, kissing and tumbling
in water; (b) Chain of spheres settle in a polyox solution.
These are the result of particle migrations produced by particle-particle and
particle-wall interactions. These local rearrangement mechanisms mediated by
things like hydrodynamic forces at stagnation and separation points, wake
interactions, vortex shedding, and turning couples on long bodies (and on pairs
of spherical bodies in momentary contact). Direct simulation may be the only
theoretical tool available for studying all these nonlinear and geometrically
There are striking differences in the observed microstructures between
Newtonian and viscoelastic particulate flows, which are not well understood.
Results from direct simulations bearing on cooperative effects associated with
these microstructures have as yet to be obtained for viscoelastic fluids.
Simulations to date using embedded domains indicate that these differences are
associated with the sign of the pressure due to normal stresses.
Particle pair interactions are fundamental mechanisms which enter strongly
into all practical applications of particulate flows. They are due to inertia
and normal stresses and they appear to be maximally different in Newtonian and
viscoelastic liquids. The principal interactions between neighboring spheres
can be described as drafting, kissing and tumbling in Newtonian
Figure 2: Drafting, kissing and tumbling.
and as drafting, kissing and chaining in viscoelastic liquids. The
drafting and kissing mechanisms involved are distinctly different, despite
In a Newtonian liquids, when one falling sphere enters the wake of another, it
experiences reduced drag, drafts downward toward the leading sphere,
and kisses it. The two kissing spheres momentarily form a single long
body aligned parallel to the stream. But the parallel orientation for a falling
long body is unstable: hydrodynamic turning couples tend to rotate it to the
broadside-on orientation (perpendicular to the stream). The pair of kissing
spheres therefore tumbles to a side-by-side configuration. Two
touching spheres falling side-by-side are pushed apart until a stable
separation distance between centers across the stream is established; they then
fall together without further lateral migrations .
This local rearrangement mechanism implies that globally, the only stable
configuration is one in which the most probable orientation between any pair of
neighboring spheres is across the stream. The consequence of this
microstructural property is a flow-induced anisotropy, which leads ubiquitously
to lines of spheres across the stream; these are always in evidence in
two-dimensional fluidized beds of finite size spheres. Though they are less
stable, planes of spheres in three-dimensional beds can also be found.
In viscoelastic liquids, on the other hand, two spheres falling side-by-side
will be pushed apart if their initial separation exceeds a critical value.
However, if their initial separation is small enough, they will attract
(``draft''), kiss, turn and chain. One might say that we get dispersion in the
Newtonian liquid and aggregation in the viscoelastic liquid.
This chaining of falling spheres, is not well understood. It seems to be
related to the reversal of pressure due to normal stresses; there is a tension
between chained spheres. The exact mechanism needs to be clarified.
Particle-wall interactions also produce anisotropic microstructures in
particulate flows, such as clear zones near walls, and the like. If a sphere is
launched near a vertical wall in a Newtonian liquid, it will be pushed away
from the wall to an equilibrium distance at which lateral migrations stop.
If the same sphere is launched near a vertical wall in a viscoelastic liquid,
it will be sucked all the way to the wall, and will rotate anomalously as it
falls. This is very strange since the sphere appears to touch the wall where
the friction would make it rotate in the other sense.
Here is a video animation (1.5 MB mpeg) made from
an actual dynamical simulation. It shows 6 particles falling under gravity in
an Oldroyd B fluid. Particles behave differently in viscoelastic fluids than
they do in Newtonian fluids. In Newtonian fluids, particles draft, kiss, and
tumble. In viscoelastic fluids, by contrast, particles draft, kiss, and chain.
Long chains fall faster than short chains.
Statistical analysis of simulations is yet another window in which to view
the fundamentals of solid-liquid flows. The time-averaged particle (or bubble)
dynamics in a periodic or infinite domain can be described in terms of the
number density, velocity current, and force correlations, and their Fourier
transforms. The number density correlation gives the relative arrangement and
motion of the particles; the velocity current correlation gives the propagation
velocities of the dominant modes; and the force correlation gives the form of
the forcing term driving the particle system. For a numerical solution, the
above distributions can be easily obtained by recording the particles' (or
bubbles') coordinates, velocities and forces at regular time intervals.
One of the great engineering opportunities of the present day is the use of
direct numerical simulations to construct empirical correlations, of the kind
usually generated from experiments. We can hope to construct correlations
similar to that of Richardson and Zaki for fluidized suspensions, and to the
friction factor vs. Reynolds number correlation for slurries. There are many
other possibilities. However, engineering practice would not admit such
numerically generated correlations without first verifying that they work in
benchmark cases; therefore, experiments must be considered.
In the past, solid-liquid flows were studied using continuum modeling. When
done rigorously, using spatial, temporal or ensemble averaging, this leads to
``two-fluid'' models in which one of the two fluids is the solids phase. The
equations are formally correct, but the terms of interactions must be modeled,
and models which work for one flow may not work for another. Direct simulations
can provide clues for modeling the interaction terms and a standard to judge
the performance of modeling assumptions.
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Last updated October 16, 2000