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In March 2002 I joined the Dept. of Fossil Fuels at CIEMAT in Madrid, Spain, where I began to study fluid flows containing solid particles. Eventually, the goal is to be able to predict with better accuracy the behaviour of fluidized beds which are used in combustion facilities for such substances as coal dust, organic residues etc. However, adequate models for the various macroscopic effects of the fluid-particle interactions (e.g. formation of agglomerations) and the interaction between particles and a solid wall (e.g. increased particle concentration near walls) are still lacking. The plan is to perform 'direct simulation' of these flows in an idealized setting, i.e. for low Reynolds number and a limited number of particles. These computations should provide a valuable base for the formulation and scrutinization of closure models (e.g. as arising in the framework of two-fluid formulations).

In the framework of the present project, a numerical tool capable of simulating the flow of O(1000-10000) rigid particles suspended in a fluid and moving at particle Reynolds numbers of O(100-1000) is envisaged. For this purpose, the method of 'fictitious domains' will be used (e.g. [1][2][3][4][5]). In this approach, the solid part of the domain is treated as if it belonged to the fluid as well. The physically correct behaviour of the particles is imposed by adequate volume forces which are added to the momentum equations at the respective locations.

We have performed the following intermediate steps:

- Develop and test a two-dimensional code. This stage is useful
for verifying the implementation of the numerical method and extend
it to the desired accuracy of interpolation at the fluid-solid
interfaces.
- First, the basic solver will be assessed
*without*any immersed objects. This includes the time-accuracy of the integration scheme, the spatial accuracy and the treatment of the boundaries. A stability analysis for the discrete, linearized operators is also scheduled. - The time-dependent flow around stationary solid objects (circles, polygons, etc.) will be considered in order to get a feeling for the capabilities of the immersed boundary method and the problem of interpolation from an arbitrarily-shaped interface to a fixed cartesian grid.
- The immersed objects will then be moved by a pre-determined law (e.g. oscillation of a cylinder, etc.) such that a one-way coupling of the solid-fluid system is considered first.
- Finally, freely-moving objects (i.e. circles) will be simulated. Here, the first (numerical) issue is the treatment of the two-way coupling. From the physical point of view, the interesting topics are the formulation of a reasonable model for particle-particle interaction during contact and similar descriptions of the contact between a particle and a solid boundary.

A technical report (in pdf, 16MB) describes the details of this first step.

- First, the basic solver will be assessed
- Adapt the code to multi-processor machines, taking advantage of the MPI standard for inter-processor communication. An efficient treatment of the geometric relations (moving particles - fixed grid) is vital; an efficient data structure for dealing with a large number of particles, unevenly distributed amongst the processors and in space, needs to be found, particularly in view of the three-dimensional version of the code. The strategy employed for parallelizing the algorithm has been described in this technical report (in pdf, 5MB), covering both, aspects concerning the fluid phase and the solid-phase.
- Remedy the problems noted in stage one above, i.e. the fact
that
*direct*forcing methods produce oscillatory results when the solid objects are in motion w.r.t. the fixed grid and*indirect*(=feed-back) methods imply very small time steps. For this purpose a new force-coupling method was proposed in this report and presented at a conference. - Extend the two-dimensional method to three space dimensions. From a physical point of view, this step does not pose fundamental difficulties. However, the numerical task is considerable. We have opted to use a second-order approximate factorization procedure for the parallel solution of the Helmholtz problems (predictor step) and a multigrid solver for the Poisson problem. Both types of implicit problems are well adapted for distribution over a 3D Cartesian processor grid. As in the 2D case, we use a master-and-slave approach to the parallization of the particle-related work. Applications to the sedimentation of spherical particles have been reported at conferences (IUTAM 04, Merseburg 05) and in a full paper.

Currently we are performing extensive large-scale simulations of various problems involving dilute and dense conditions:

- "pure" sedimentation problems in tri-periodic domains at particle-diameter-based Reynolds numbers around 400;
- fully-developed turbulent flow in vertical channels with several thousand suspended heavy particles
- erosion and transport of sediment particles in horizontal channel flow;
- pattern formation by sediment particles in boundary layer flow.

markus.uhlmann AT kit.edu

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