## Modeling and numerical simulation of particulate flows by
the Eulerian-Lagrangian approach

N. A. Patankar

Department of Mechanical Engineering,

Northwestern University, Evanston, IL 60208.

D. D. Joseph

Department of Aerospace Engineering and Mechanics,

University of Minnesota, Minneapolis, MN 55455

**Abstract**

In this paper we present an Eulerian-Lagrangian numerical simulation (LNS)
scheme for particulate flows. The overall algorithm in the present approach is
a variation of the scheme presented earlier by N. Patankar and Joseph (1999).
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
particles include drag from the fluid, body force and the interparticle force
that prevents the particle volume fraction from exceeding the close-packing
limit. There is freedom to use different models for these forces and to
introduce other forces. In this paper we have used two types of interparticle
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. The fluid and particle momentum equations
are coupled in the solution procedure unlike the earlier approach of N.
Patankar and Joseph (1999). A finite volume method is used to solve these
equations on an Eulerian grid. Particle positions are updated explicitly. This
numerical scheme can handle a range of particle loadings and particle types. We
solve the fluid phase continuity and momentum equations using a Chorin-type
fractional-step method. The numerical scheme is verified by comparing results
with test cases and experiments.

**Key Words**: Two-phase flow, Eulerian-Lagrangian numerical
simulation (LNS), multiphase particle-in-cell (MP-PIC) method, particulate
flows, Chorin scheme, fractional-step method, non-staggered grid, bimodal
sedimentation, inclined sedimentation, gas-solid and liquid-solid fluidization.

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