Pii: S0301-9322(98)00048-2

A new Lagrange-multiplier based ®ctitious-domain method is presented for the direct numerical simulation of viscous incompressible ̄ow with suspended solid particles. The method uses a ®niteelement discretization in space and an operator-splitting technique for discretization in time. The linearly constrained quadratic minimization problems which arise from this splitting are solved using conjugategradient algorithms. A key feature of the method is that the ̄uid±particle motion is treated implicitly via a combined weak formulation in which the mutual forces cancelÐexplicit calculation of the hydrodynamic forces and torques on the particles is not required. The ̄uid ̄ow equations are enforced inside, as well as outside, the particle boundaries. The ̄ow inside, and on, each particle boundary is constrained to be a rigid-body motion using a distributed Lagrange multiplier. This multiplier represents the additional body force per unit volume needed to maintain the rigid-body motion inside the particle boundary, and is analogous to the pressure in incompressible ̄uid ̄ow, whose gradient is the force required to maintain the constraint of incompressibility. The method is validated using the sedimentation of two circular particles in a two-dimensional channel as the test problem, and is then applied to the sedimentation of 504 circular particles in a closed two-dimensional box. The resulting suspension is fairly dense, and the computation could not be carried out without an e€ective strategy for preventing particles from penetrating each other or the solid outer walls; in the method described herein, this is achieved by activating a repelling force on close approach, such as might occur as a consequence of roughness elements on the particle. The development of physically based mathematical methods for avoiding particle±particle and particle±wall penetration is a new problem posed by the direct simulation of ̄uidized suspensions. The simulation starts with the particles packed densely at the top of the sedimentation column. In the course of their fall to the bottom of the box, a ®ngering motion of the particles, which are heavier than International Journal of Multiphase Flow 25 (1999) 755±794 0301-9322/99/$ see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0301-9322(98)00048-2 PERGAMON www.elsevier.com/locate/ijmulflow * Corresponding author. Tel.: 713 743 3473; Fax: 713 743 3505. the surrounding ̄uid, develops in a way reminiscent of the familiar dynamics associated with the Rayleigh±Taylor instability of heavy ̄uid above light. We also present here the results of a three-dimensional simulation of the sedimentation of two spherical particles. The simulation reproduces the familiar dynamics of drafting, kissing and tumbling to side-by-side motion with the line between centers across the ̄ow at Reynolds numbers in the hundreds. # 1999 Elsevier Science Ltd. All rights reserved.