(142cn) Numerical Simulations of Dilute Sedimenting Suspensions At Finite-Particle Reynolds Numers

              Sedimenting suspensions can be found in various natural and man-made processes.
Examples include the sedimentation of dust in the atmosphere, the
centrifugation of proteins, and the deposit of contaminants in waste water. Attempts have been made by researchers to
understand the fundamental properties of sedimenting
suspensions using numerical simulations. However, a numerical method that can
correctly reproduce experimental observations is not yet achieved.

         We
present an alternative numerical method for simulations of suspension flows
with application to sedimenting suspensions at
finite-particle Reynolds numbers. The locally averaged conservation equations
for the fluid phase are discretized using an extended lattice-Boltzmann scheme.
A Lagrangian particle tracking
model is used to track the trajectories of individual particles. The present
method enables us to simulate large-scale suspension systems with reasonable
computational expenses.

         The
present method is able to capture the main features of sedimenting
suspensions observed in experiments, e.g., swirl structures, helical
structures, and saddle points. At low-particle Reynolds numbers, the simulated
particle velocity fluctuation amplitudes closely follow the scaling derived
from the experimental data in the literature. At finite-particle Reynolds
numbers, our simulation results suggest a transition in the fluctuation
amplitudes at a particle Reynolds number around 0.08 (Fig. 1(a)) in accordance
with the work of Yin & Koch [1] who used a surface-resolved numerical
simulation method. We numerically demonstrate that there exists a range of
domain sizes in which the fluctuation amplitudes have a strong domain size
dependence, beyond which the fluctuation amplitudes saturate (Fig. 1(b)).  This is the first time that the
dependency of the fluctuation amplitudes on the domain size is numerically
reproduced. Furthermore, we found that the magnitude and domain size dependence
of the fluctuation amplitudes at finite-particle Reynolds numbers are
well-represented by introducing new fluctuation amplitudes scaling functions
and characteristic scaling function in the correlation derived by Segre et al.
[2].

Figure
1. (a) Long-term average fluctuation amplitudes ratio as a function of the
particle Reynolds number in the direction parallel (filled symbols) and
perpendicular (empty symbols) to gravity. The solid and dotted lines are the
calculation from the empirical correlation presented in the literature and our
correlation, respectively. (b) Long-term average fluctuation amplitudes in the
direction parallel (squares) and perpendicular (triangles) to gravity versus
the domain size. Dotted line is the calculation from our correlation.

References

[1] Yin
X., Koch D. L. Velocity fluctuations and hydrodynamic diffusion in
finite-Reynolds number sedimenting suspensions. Phys.
Fluids. 2008;20:043305.

[2] Segre P. N., Herbolzheimer
E., Chaikin P. M. Long-range correlations in
sedimentation. Phys. Rev. Lett. 1997;79:2574-2577.