### Eulerian and Lagrangian

Multiphase flows are commonly simulated by treating the particles as either a continuous Eulerian phase, or as discrete Lagrangian elements. A Eulerian approach is capable of simulating a number of particles comparable to that of typical systems, but at the expense of information on individual particle trajectories. It is also worth noting that only average particle behaviour is represented, and that more complex particle interactions such as adhesion are significantly more difficult to implement in a Eulerian framework. On the other hand a Lagrangian model, sometimes implemented using Lagrangian Particle Tracking (LPT) or Discrete Element Model (DEM), accurately describes the motion of each particle. As the timescale associated with particle collisions is much smaller than that upon which the fluid phase evolves, DEM simulations can be computationally expensive. Because of this, the number of particles which can feasibly be simulated is much smaller than usually occurs in typical applications.

### Lagrangian Description

**Lagrangian description of particles**

To apply the second law of Newton on a dispersed particle, the dispersed phase is described in a so-called Lagrangian way. This is equivalent by following the dispersed particle with the substantial operator, , where the velocity of the dispersed phase is employed. Considering a few small particles in a low *Re* number uniform flow is described by the pioneering work of Basset, Boussinesq, and Oseen. The forces on particles for realistic flow conditions have been researched by many researchers, including our team. Generally speaking, the model in Multiflow

### Particle-turbulence interactions

**Particle-turbulence interactions**

Particle-laden turbulent flows can be found in many industrial and environmental processes. Examples of such processes are pneumatic transport of particles; energy conversion of fossil fuels; movement of soot particles in the atmosphere; the flow of particles in cyclones and many more. Understanding the effects of particle.. fluid interactions is of utmost importance because this will result in a more accurate implementation of these processes. Additionally, applications such as sediment transport, where the direction of gravity is perpendicular to the flow, particle-particle and particle-wall collisions become very important. Therefore, the need to understand the effects of these additional physical phenomena is of fundamental importance. Thus robust numerical simulations will therefore help the optimization and better design of industrial processes and provide a more reliable prediction of environmental processes involving particles.

There are various frameworks in which the continuous phase for gas-solid flows can be predicted, *i.e.* Direct Numerical Simulation (DNS), Large Eddy Simulation (LES) and the Reynolds Averaged Navier-Stokes (RANS) method. DNS methods offer high accuracy in resolving all scales without ad hoc modelling at the expense of large computational time. Currently, DNS can only solve flows of relatively low Reynolds (Re) numbers, which very interesting from an academic point of view, but are outside of most engineering and industrial interests.

For most industrially relevant flows, in MultiFlow we use Large Eddy Simulation (LES). LES solves the Navier-Stokes equations up to a particular length-scale due to the application of a filter. Length-scales smaller than the cut-off filter width (D) are modeled with a so-called sub-grid scale (SGS) model. The cut-off width is an indication of the smallest size eddies that are retained in the computations and eddies smaller than D, are filtered out. Due to the filtering of the Navier-Stokes equations, models are required to provide closure for the SGS stresses, which account for the effect of the unresolved scales on the convective momentum transport. Also, models are required to describe the interaction of the small scales of turbulence (smaller than the cut-off length) with the particles. This is currently a major area of development in MultiFlow.

### Fluidized Beds

**Fluidized Beds**

Forcing a fluid (typically a gas) through a bed of granular material (e.g. sand) above a critical velocity results in the gas-solid mixture behaving similarly to a fluid, for example it is possible to 'float' objects in the bed and is said to have entered a fluidised state. Fluidised beds, commonly found in industrial and power generating plant exploit this phenomenon for its mixing properties and the large gas-solid surface area resulting high rates of heat and mass transfer making them useful in for example heat exchangers or chemical reactors such as catalytic cracking operations. The ability to predict the hydro- and thermodynamic properties of these devices is vital to their successful operation, however the scaling from laboratory models to full-size plant is frequently poor and high quality modelling tools are required.

We are using Lagrangian Particle Tracking and Euler-Euler methods to gain a more detailed understanding of the governing physical processes and the application of these advanced numerical methods to industry-size applications.