The dynamics of dispersed phase flows is rather intricate as a result of the complex momentum and heat transfer between the dispersed phase and the surrounding fluid. In the framework of the PeliGRIFF project, our study is limited to incompressible flows and Reynolds number based on the length scale of a single particle or drop below 1000, which still leaves a large scope of investigation. Hence, the flow might  sometimes be considered as turbulent at the level of the flow domain but is often still either laminar or inertial at the level of the drop/particle.

The number of industrial and natural flows that fall into the category of dispersed phase flows is quite large, ranging from solid particles transport in pipelines or rivers to bubble columns in catalyst reactors, as illustrated below.

The key issue of dispersed phase flows is the proper modelling of the momentum exchange between the continuous and dispersed phases:

The elementary sketch above entails the following major issues:

• How to properly model each phase separately ?
• How to couple the dynamics of the two phases, i.e., how to model the momentum transfer ?
• At which scale do we look at the dynamics of the flow ?

As collectively admitted by the international multiphase flow community as a sound approch, we suggest to employ a three-level multi-scale analysis where the three levels (of scales) corresponds to:

1. the micro scale, where the characteristic length scale is the particle or the bubble/drop and the dynamics of the two phases are fully resolved,
   
2. the meso scale, often referred to in the literature as micro/macro, where the dispersed phase is modelled in a direct fashion while the continuous (fluid) phase is averagely accounted for,
3. the macro scale, where both phases are seen as continuous phases and human scale processes can be computationally examined.

As an illustration, we show below on the problem of solid particles transport what are the typical systems and their size as we apply our 3-level multi-scale approach:

DNS illustrations from Markus Uhlmann's group, KIT, Germany

At the micro scale, both phases are modelled in a direct way and the coupling is achieved via classical interface boundary conditions. In PeliGRIFF, the solution of the fluid conservation equations relies on a Finite Element or Finite Volume scheme and an operator splitting time integration scheme. Depending on the nature of the dispersed phase, we employ:

• for solid particles, the solid/solid interactions are handled by a Discrete Element method (DEM) that enables us to consider true contacts (in the sense that particles geometrically touch each other), and a Fictitious Domain method for the hydrodynamic interactions,
  

• for drops and bubbles, a non-conservative Level Set CSF (Continuum Surface Force) method.

At the meso scale (this model is considered for solid particles or non-deformable bubbles/drops only), the dispersed phase is still modelled at the micro scale, i.e., in a direct fashion, while the continuous phase (the surrounding fluid) is solved in an average way, which implies that a computational cell is much bigger than a particle/drop/bubble. Therefore, a first closure law is required to account for the hydrodynamic interactions.

Finally, at the macro scale, both phases are described as continuum and a second closure law needs to be introduced to model the dispersed phase behavior.