A distributed Lagrange multiplier fictitious domain method for particulateflows

A new Lagrange-multiplier based fictitious-domain method is presented for t he direct numerical simulation of viscous incompressible flow with suspende d solid particles. The method uses a finite-element discretization in space and an operator-splitting technique for discretization in time. The linear ly constrained quadratic minimization problems which arise from this splitt ing are solved using conjugate-gradient algorithms. A key feature of the method is that the fluid-particle motion is treated im plicitly via a combined weak formulation in which the mutual forces cancel- explicit calculation of the hydrodynamic forces and torques on the particle s is not required. The fluid flow equations are enforced inside, as well as outside, the particle boundaries. The flow inside, and on, each particle b oundary is constrained to be a rigid-body motion using a distributed Lagran ge multiplier. This multiplier represents the additional body force per uni t volume needed to maintain the rigid-body motion inside the particle bound ary, and is analogous to the pressure in incompressible fluid flow, whose g radient is the force required to maintain the constraint of incompressibili ty. The method is validated using the sedimentation of two circular particles i n 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. T he resulting suspension is fairly dense, and the computation could not be c arried out without an effective strategy for preventing particles from pene trating 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 fluidized suspensions. The simulation starts with the particles packed densely at the top of the s edimentation column. In the course of their fall to the bottom of the box, a fingering motion of the particles, which are heavier than the surrounding fluid, develops in a way reminiscent of the familiar dynamics associated w ith the Rayleigh-Taylor instability of heavy fluid above light. We also present here the results of a three-dimensional simulation of the s edimentation of two spherical particles. The simulation reproduces the fami liar dynamics of drafting, kissing and tumbling to side-by-side motion with the line between centers across the flow at Reynolds numbers in the hundre ds. (C) 1999 Elsevier Science Ltd. All rights reserved.