## A fictitious domain method with distributed Lagrange multipliers for numerical simulation of particulate flows

## Abstract:

The main goal of this article, which generalizes [1] considerably, is to discuss the numerical simulation} of particulate flow for mixtures of incompressible viscous fluids and rigid particles. Such flow occurs in liquid/solid fluidized beds, sedimentation}, and other applications in Science and Engineering. Assuming that the number of particles is sufficiently large, those simulations are useful to adjust parameters in the homogenized models approximating the above two-phase flow.

From a computational point of view, the methodology to be discussed in this article combines distributed Lagrange multipliers based fictitious domain methods, which allow the use of fixed structured finite element grids for the fluid flow computations, with time discretizations by operator splitting a la Marchuk-Yanenko to decouple the various computational difficulties associated to the simulation; these difficulties include collisions between particles, which are treated by penalty type methods. After validating the numerical methodology discussed here by comparison with some well documented two particle - fluid flow} interactions, we shall present the results of two and three dimensional particulate flow simulations, with the number of particles in the range 100 - 1000; these results include the simulation of a Rayleigh-Taylor instability occurring when a sufficiently large number of particles, initially at rest, are positioned regularly over a fluid of smaller density, in the presence of gravity.

The methods described in this article will be discussed with more details (of computational and physical natures) in [2]. Actually, ref. [2] will contain, also, many references to the work of several investigators, showing that the most popular methodology to simulate particulate flow has been so far the one based on ALE (Arbitrary Lagrange-Euler) techniques; these methods are clearly more complicated to implement than those described in this article (particularly on parallel platforms).

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*Last updated October 16, 2000*