In this chapter we discuss numerical methods based on structured grids for direct numerical simulations of multiphase flows involving solid particles. We will focus on numerical approaches that are designed to solve the governing equations in the fluid and the interaction between the phases at their interfaces.

In methods employing structured grids, there are two distinct possibilities for handling the geometric complexities imposed by the phase boundaries. The first approach is to precisely define the phase boundaries and use a body-fitted grid in one or both phases, as necessary. The curvilinear grid that conforms to the phase boundaries greatly simplifies the specification of interaction processes that occur across the interface. Furthermore, numerical methods on curvilinear grids are well developed and a desired level of accuracy can be maintained both within the different phases and along their interfaces. The main difficulty with this approach is grid generation; for example, for the case of flow in a pipe with several hundred suspended particles, obtaining an appropriate structured body-fitted grid around all the particles is a nontrival task. There are several structured grid generation packages that are readily available. Nevertheless, the computational cost of grid generation increases rapidly as geometric complexity increases. This can be severely restrictive, particularly in cases where the phase boundaries are in time-dependent motion. The cost of grid generation can then overwhelm the cost of computing the flow.

The alternative to a body-fitted grid that is increasingly becoming popular are Cartesian grid methods. Here, irrespective of the complex nature of the internal and external boundaries between the phases, only a regular Cartesian mesh is employed and the governing equations are solved on this simple mesh.