## Lagrangian numerical simulation of particulate flows

#### N. A. Patankar*and D. D. Joseph

#### Abstract:

The Lagrangian numerical simulation (LNS) scheme presented in this paper is
motivated by the multiphase particle-in-cell (MP-PIC). In this numerical scheme
we solve the fluid phase continuity and momentum equations on an Eulerian grid.
The particle motion is governed by Newton's law thus following the Lagrangian
approach. Momentum exchange from the particle to fluid is modeled in the fluid
phase momentum equation. Forces acting on the particle include drag from the
fluid, body force and force due to interparticle stress. There is freedom to
use different models for these forces and to introduce other forces. The effect
of viscous stresses are included in the fluid phase equations. The volume
fraction of the particles appear in the fluid phase continuity and momentum
equations. A finite volume method is used to solve for the fluid phase
equations on an Eulerian grid. Particle positions are updated using the
Runge-Kutta scheme. This numerical scheme can handle a range of particle
loadings and particle types.

The LNS scheme is implemented using an efficient three-dimensional time
dependent finite volume algorithm. We use a Chorin-type pressure-correction
based fractional-step scheme on a non-staggered Cartesian grid. In this paper,
we consider only incompressible Newtonian suspending fluid. However, the
average velocity field of the fluid phase is not divergence-free because its
effective density is not constant. Our pressure correction based
fractional-step scheme accounts for varying properties in the fluid phase
equations. This method can also account for suspending fluids with non-constant
properties. The numerical scheme is verified by comparing results with test
cases and experiments.

**Key Words**: Approximate factorization, Two-phase flow,
Eulerian-Lagrangian numerical scheme (LNS), multiphase particle-in-cell
(MP-PIC) method, particulate flows, three-dimensional time dependent finite
volume approach, Chorin scheme, pressure-correction scheme, fractional-step
method, non-staggered grid, bimodal sedimentation, Rayleigh-Taylor instability,
flow in fracture, gravity tongue, inclined sedimentation.

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Last Modified: Monday, 10-Dec-2001 13:52:12 CST