## 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.