A modified distributed Lagrange multiplier/fictitious domain method (DLM) that allows particles to undergo collisions is developed for particulate flows. In the earlier versions of the DLM method for Newtonian and viscoelastic liquids the particle surfaces were restricted to be more than one velocity element away from each other. A repulsive body force was applied to the particles when the distance between them was smaller than this critical value. This was necessary for ensuring that conflicting rigid body motion constraints from two different particles are not imposed at the same velocity nodes.In the modified DLM method the particles are allowed to come arbitrarily close to each other and even slightly overlap each other. When conflicting rigid body motion constraints from two different particles are applicable on a velocity node, the constraint from the particle that is closer to that node is used and the other constraint is dropped. An elastic repulsive force is applied when the particles overlap each other. In our simulations, the particles are allowed to overlap as much as one hundredth of the velocity element size. The modified DLM method is implemented for both Newtonian and viscoelastic liquids. Our simulations show that when particles are dropped in a channel, and the viscoelastic Mach number (M) is less than one and the elasticity number (E) is greater than one, the particles form a chain parallel to the flow direction. As in experiments, the new method allows particles in the chain to approximately touch each other. The particles dropped in a Newtonian liquid, on the other hand, undergo characteristic drafting, kissing and tumbling. During the touching phase, as in experiments, the two particles touch each other. The modified method thus allows hydrodynamic forces to be fully resolved to within the tolerance of the mesh and thus the extra artificial force in a security zone outside the particle which are used in all other methods are not needed.
Bibliographical noteFunding Information:
This work was partially supported by National Science Foundation KDI Grand Challenge grant (NSF/CTS-98-73236), New Jersey Commission on Science and Technology through the New-Jersey Center for Micro-Flow Control under award number 01-2042-007-25, the Department of Basic Energy Science at DOE and the University of Minnesota Supercomputing Institute.
- Direct numerical simulations
- Finite element method
- Oldroyd-B fluid
- Particle collisions
- Particulate flows
- Viscoelastic fluid