INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS Int. J. Numer. Meth. Fluids 2005; 47:1245–1251 Published online 27 January 2005 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/d.914 Numerical modelling of solid particle motion using a new penalty method T. N. Randrianarivelo ∗; † , G. Pianet, S. Vincent and J. P. Caltagirone TRansferts Ecoulements FLuides Energetique (TREFLE); UMR CNRS 8508; Site ENSCPB 16 Avenue Pey Berland 33607 PESSAC Cedex; France SUMMARY The present article reports on an original implicit tensorial penalty method (ITPM) for modelling solid particle motion in an incompressible ow. The basic idea is to decompose the viscous stress tensor of Navier–Stokes equation into contributions representing elongation, pure shearing and rotation. An articial viscosity is associated to each stress contribution. The penalty method is used to impose dierent stress components thanks to a generalized augmented Lagrangian method implemented by introducing four Lagrange multipliers. An iterative Uzawa algorithm is nally used to achieve the numerical solution. The classical problems of Couette’s ow between two coaxial cylinders and the settling of a particle in a tank lled with a viscous uid have been chosen to demonstrate the capability of the new method. Copyright ? 2005 John Wiley & Sons, Ltd. KEY WORDS: Navier–Stokes; penalty method; stress tensor; direct numerical simulation; multiphase ows; liquid= particle interactions 1. INTRODUCTION Numerous problems motivated by applications from environment and engineering sciences involve multiphase ows with coupled stress interactions. In this regard, modelling of solid particle motion in a surrounding uid has received a considerable research interest. The adop- tion of unstructured grid techniques for numerical modelling of such complex ow phenomena is known to be costly in terms of computational time. Numerical modelling using a xed grid mesh is an interesting alternative. However, careful handling of dierent stress components should be employed for accurate distinction between the uid and solid phases. ∗ Correspondence to: T. N. Randrianarivelo, TRansferts Ecoulements FLuides Energetique (TREFLE), UMR CNRS 8508, Site ENSCPB 16 Avenue Pey Berland 33607 PESSAC Cedex, France. † E-mail: randrian@enscpb.fr Contract=grant sponsor: Publishing Arts Research Council; contract=grant number: 98-1846389 Received 27 April 2004 Revised 2 September 2004 Copyright ? 2005 John Wiley & Sons, Ltd. Accepted 14 September 2004