Direct Numerical Simulations of
Freely Moving Spheres: A Dynamic Drag Correlation

Y. (Yali) Tang^a*,
M.A. (Martin) van der Hoef^b, E.A.J.F.
(Frank) Peters^a, J.A.M. (Hans) Kuipers^a

^a
Department of
Chemical Engineering and Chemistry, Eindhoven University of Technology,
Eindhoven, The Netherlands

^b
Faculty of
Science and Technology, University of Twente, Enschede, The Netherlands

(^*
Corresponding E-Mail: y.tang2@tue.nl )

Keywords: direct numerical simulation,
immersed boundary method, particle mobility, drag correlation

Abstract

Accurate description of
fluid-particle interactions is one of the biggest challenges in the community
of multiphase flow, which however is essential for the prediction of the flow
behavior of gas-solid flows at the engineering scale by means of computational
fluid dynamics (CFD).  Direct numerical simulations (DNS) has become a
powerful tool for developing the associated closures that are required at the
predictive (coarse-grained) levels of CFD modeling. For momentum transfer,
several drag correlations have been proposed based on DNS studies of
particulate systems.  However, most of these DNS studies have been limited
to static particle arrays. Only recently, few studies^1-3have
considered the dynamic particles, which show a need to explore the effect of
the particle dynamics on the fluid-particle hydrodynamics especially at
intermediate Reynolds (Re) numbers.

In the present work, we carry out
a numerical investigation of freely moving spheres in the intermediate-Re flow
range. This is a further study of our previous work on DNS of dense gas-solid
flows: (1) a methodology⁴ was developed and validated for
highly accurate results using an immersed boundary method (IBM) at relatively
low grid resolutions, and (2) a new drag correlation⁵ was
obtained based on the accurate results from IBM simulations of static sphere
arrays over a wide range of Re and solids volume fraction (ϕ). The
methodology corrects for the grid-size effects in IBM models, and is also
applied to the present simulations of moving particles. By comparing the
results between simulations of static spheres and dynamic spheres, we obtain
the effect of the particle mobility, which can be formulated and included in
the drag correlation.

The main characteristics of our
immersed boundary method are: the governing equations that describe the fluid
flow are solved on a fixed and structured Eulerian
grid with the grid spacing much smaller than the particle size; the particles
are represented by sets of Lagrangian marker points,
and the dynamics of the particles are governed by the Newtonian equations of
motion; the coupling between gas and solid phases is implemented by enforcing
no-slip boundary conditions at the surface of the particles, via a forcing term
that is calculated at the position of each marker point and subjected to the
transport equations. Since the flow is resolved on the entire domain including
the volume occupied by the solids, the inertia of the artificial fluid inside a
particle is subtracted from the force density.

We consider fully periodic,
three-dimensional box of length L containing incompressible Newtonian fluid
(density ρ_f
= 1.0 kg/m³, dynamic viscosity μ_f = 1×10^-5 kg/(m·s))
and monodisperse solid spherical particles with
diameter d_p
= 0.0016 m and density ρ_p= 2526 kg/m³. The overall (volume averaged) solids volume
fraction ϕ is a constant, which we have considered from 0.1 to 0.6.
The number of particles N_p is chosen such that L = (N_pπ/6ϕ)^1/3·d_p
≈10d_p. Figure 1 shows an example of a particle
configuration and flow field at ϕ=0.4. In the IBM simulations, the
fluid and the particles are initially at rest. The flow is driven by specifying
a constant pressure gradient along with an axial direction. Subsequently, each
particle freely moves with an acceleration that arises from hydrodynamic forces
and interactions with other particles. As a result, the volumetric mean
velocity evolves and the steady-state solution implies a Reynolds number (based
on the mean superficial velocity), which is in the range of [10, 1000]. The
particles are assumed to be perfectly elastic, and particle-particle
interactions are treated using a hard-sphere collision model.

Figure 1. Particle configuration and
flow velocity profile. The particles are scaled by 0.5 for visualization.

Figure 2. F_d as
a function of mean flow Reynolds number: moving spheres vs. fixed spheres.

Figure 2 compares the
dimensionless drag force between stationary arrays and dynamic systems at
 ϕ=0.5 as a function of Re.
A clear increase of the drag force is observed due to the effect of particles
mobility. In homogeneous suspensions, fluctuations in the hydrodynamic force
experienced by particles affect the evolution of particle velocity fluctuation,
which can in turn affect the mean and variance of the hydrodynamic force. This
effect of the particle mobility (corresponding to particle velocity fluctuation
or granular temperature) on the gas-solid drag force is investigated for
different combinations of Re, ϕ and density ratio ρ_p/ρ_f. A drag correlation including such mobility effect
can provide a more accurate description of the fluid-particle hydrodynamics,
and subsequently improves the accuracy of the predictive-level CFD simulations.

References

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Tang Y, et al. A methodology for highly accurate results of
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