INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2006; 51:531–566 Published online 20 December 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/d.1129 Direct numerical simulation of particulate ow via multigrid FEM techniques and the ctitious boundary method Decheng Wan ‡ and Stefan Turek ∗; † Institute of Applied Mathematics LS III; University of Dortmund; Vogelpothsweg 87; 44227 Dortmund; Germany SUMMARY Direct numerical simulation techniques for particulate ow by the ctitious boundary method (FBM) are presented. The ow is computed by a multigrid nite element solver and the solid particles are allowed to move freely through the computational mesh which can be chosen independently from the particles of arbitrary shape, size and number. The interaction between the uid and the particles is taken into account by the FBM in which an explicit volume based calculation for the hydrodynamic forces is integrated. A new collision model based on papers by Glowinski, Joseph, Singh and coauthors is examined to handle particle–particle and particle–wall interactions. Numerical tests show that the present method provides a very ecient approach to directly simulate particulate ows with many particles. Copyright ? 2005 John Wiley & Sons, Ltd. KEY WORDS: particulate ows; incompressible Navier–Stokes equations; multigrid; FEM; ctitious boundary method 1. INTRODUCTION Liquids containing large solid particles are common in many industrial processes, such as foods containing particles, slurry ows, mining extraction, uidization of catalyst beds, sepa- ration process using cyclones, etc. The phenomena of such particulate ows are also visible everywhere around our living environments, for instance ow around high-rise buildings, the drag force induced by driving a car accelerating in the wind, ocean current interaction with the oshore structures, sedimentation ow in estuary and sand ow in desert, etc. From the numerical point of view, particulate ows are quite hard to simulate since both the incom- pressible uid velocity and the domain in which it is dened are unknown. It can require a ∗ Correspondence to: Stefan Turek, Institute of Applied Mathematics LS III, University of Dortmund, Vogelpothsweg 87, 44227 Dortmund, Germany. † E-mail: Stefan.Turek@math.uni-dortmund.de ‡ E-mail: Wan.Decheng@math.uni-dortmund.de Received 27 April 2005 Revised 26 September 2005 Copyright ? 2005 John Wiley & Sons, Ltd. Accepted 26 September 2005