The Euler-Lagrangian Multiphace Model

]4.2 Multiphase modeling
In this section the important parameters in the spray modeling are presented, such as framework for the multiphase simulation and droplet interaction. Since a pulsating injector is used the droplet-phase is modeled transient to solve the pulsating motion of the system. A first order implicit temporal discretization scheme, also called Euler implicit, is applied. Further information about the temporal discretization can be found in section 6.2.1.
4.2.1 The Euler-Lagrange model
In general, the droplets can either be treated as a continuum with the Eulerian approach or tracked individually with the Lagrangian approach. The studied system is defined as a dilute system with a two-way coupling between the phases and these

The Eulerian approach is therefore not suitable for treating droplets in a dilute flow. Hence, the Lagrangian particle tracking is used for modeling the droplets in the spray [22]. The exhaust gas flow is modeled as a continuum by the realizable k-ε turbulence model with an additional source term that describes the interaction between the continuous phase and the dispersed phase due to the two way-coupling. The AdBlue spray is modeled by tracking the individual droplets i.e. the Eulerian-Lagrangian approach is used for the two-phase flow. The droplets may exchange mass, momentum and energy with the exhaust gas flow. The Lagrangian approach is limited by the number of droplets in the system. Too many droplets will make the simulations too computationally expensive. Naturally, a large number of droplets are needed to represent a typical spray and because of this parcels are tracked instead of individual droplets. A parcel may be defined as a bundle of droplets which all have the same dynamic properties such as velocity and size so that the parcel may be represented by one computational droplet. This means that the local properties of the bundle of droplets can be determined by solving for the properties of the computational droplets as they move through the field [22]. The movement of the parcels is defined

The drag force is defined as
Fi,Drag = where the drag force coefficient C D for spherical and rigid droplets, which is assumed in the studied system, are calculated based on the Schiller-Naumann correlation. Further the roj is the projected area, is the density of the continuous phase and u is the velocity denoted by for the droplets and for the continuous (gas) phase.
The virtual mass force is defined as
Fi,virt = where the virtual mass coefficient, C VM , is set to 0.5 in the performed simulations which means that a volume that is equal to half of the volume of a droplet of the gas phase is accelerated with the droplet.
The operator
D = denotes the relative acceleration of the droplet in comparison with the continuous phase along the droplet