## A distributed Lagrange multiplier/fictitious domain method for particulate flows

## Abstract:

A new Lagrange-multiplier based fictitious-domain method is presented for the direct numerical simulation of viscous incompressible flow with suspended solid particles. The method uses a finite-element 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 conjugate-gradient algorithms.

A key feature of the method is that the fluid-particle motion is treated implicitly via a combined weak formulation in which the mutual forces cancel---explicit calculation of the hydrodynamic forces and torques on the particles is not required. The fluid flow equations are enforced inside, as well as outside, the particle boundaries. The flow 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 fluid flow, 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 effective 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 fluidized 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 fingering motion of the particles, which are heavier than the surrounding fluid, develops in a way reminiscent of the familiar dynamics associated with the Rayleigh-Taylor instability of heavy fluid above light.

We also present here the results of a three-dimensional simulation of 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 flow at Reynolds numbers in the hundreds.

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*Last updated October 16, 2000*