Direct Simulation of the Motion of Particles in Flowing Liquids


Fundamental Dynamics

Studies of Local Rearrangement Mechanisms

The clusters and anisotropic microstructures observed in solid-liquid flows, such as those shown here.

Flow-induced anisotropy

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 complicated phenomena.

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-Particle Interactions

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 liquids.

Drafting, kissing and tumbling

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 appearances.

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

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.

Direct Numerical Simulations

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

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.

Empirical Correlations

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.

Two-Fluid Modeling

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