2.3.1. Coupled Eulerian–Lagrangian Spray Simulation

On the modeling of spray process, the most of the strategies falls into the aforesaid two basic formulations: the Eulerian–Lagrangian (EL) and the Eulerian–Eulerian (EE) methods. Compared to the EL method, the EE method is suitable for calculation of flows with higher droplet concentration [30], whereas the other approach is appropriate in the simulation of the diluted spray region. Here, the EL and EE methods are used in combination. In the light of the contributions reported in the literature, the combination of EE and EL methods is placed in the frame of two coupling approaches; these are the "ELSA model" (Euler Lagrange Spray Atomisation) [31–35] and the "ACCI server" (AVL Coupling Code Interface) [36–39].

In the ELSA methodology, the Eulerian spray is treated as a single-phase flow represented by a liquid-gas mixture. By means of additional transport equations for the liquid mass fraction and the liquid surface density, the spray atomization is modeled; if the spray is dilute enough, Lagrangian spray parcels are initiated.

In the current work, the approach based on the ACCI server is adopted, where the coupling between Eulerian and Lagrangian spray is achieved as follows. The dense spray is calculated with the Eulerian spray approach in a separate simulation client, on a highly resolved computational grid representing the near nozzle region. In the second simulation client (here, the spray-volume ambient), the Lagrangian approach is used for the dilute spray modeling. Indeed, the computational grid of the spray-volume client covers the entire simulation domain, including also the Eulerian spray region. Thus, an overlapping region is defined, in which the simulation is performed on both clients. The interactions between the gaseous and the liquid phases in the Eulerian spray simulation are transferred as source terms to the spray-volume client simulation. The data transfer between the simulation clients is managed via the ACCI server. The coupled simulation starts with the computational initialization of both simulation clients; the fluid properties are determined and the flow field of the Lagrangian client is initialized. The first exchange event taking place is the initialization, when the initial gas flow field of the Eulerian spray client is fully determined by the initial flow field of the Lagrangian client. The three velocity components—pressure, enthalpy, species mass fractions and turbulence transport quantities—are mapped and transferred via the server. The other exchange events are determined by the simulation time steps of the clients. Data exchange from the Lagrangian to the Eulerian spray client is performed to obtain the flow field boundaries of the Eulerian spray simulation; at every exchange event, the 3D flow field of the Lagrangian client is mapped onto the boundary surface of the Eulerian spray client. Data exchange in the other direction occurs due to the interaction between droplets and gas. Drag forces introduce source terms in the momentum conservation equation; mass exchange from evaporation causes sources in the continuity and species transport equations. In the overlapping domain, the gas phase flow field is calculated on both simulation clients. Thus, the source terms from the phase interactions in momentum, mass and energy conservation equations as well as the sources from the species transport equations are transferred from the Eulerian spray to the Lagrangian client. This means that, although there is no liquid phase in the overlapping domain of the Lagrangian client, the gas phase is fully encountered by the interactions with the droplet phases.

The liquid droplets leaving the Eulerian spray client initiate the spray parcels at the Lagrangian client. Conservation of liquid mass, momentum, energy and number of droplets are required criteria for the interface. The algorithm for creating new parcels by fulfilling the conservation criteria is based on the idea that a new parcel is created, if the accumulated liquid mass of a droplet phase exceeds a certain mass threshold. The preparation of new spray parcels is fully calculated by the Eulerian spray client; the interface server sends the parcel initialization data to the Lagrangian client where the new parcels are introduced. A loop over all droplet phases and over all open boundary faces is performed for every time step to check the droplet mass leaving the domain.

Since different meshes are used for the near-nozzle and spray ambient simulations (according to the scheme of Figure 4 (right)), the source terms mapping is performed in a conservative manner using weighting factors. For extensive attributes, such as mass or momentum sources, weighting factors *w fex*

are calculated from the intersection volumes between the two client domains, as shown in Equation (1). For example, the mass source term S in a control volume of the spray mesh is calculated from all values in the control volumes in near nozzle mesh, as shown in Equation (2). In the case of intensive attributes such as mass fraction, velocity, pressure or temperature, where attributes do not depend on the size of control volumes, and the weighting factors *w fin* are defined as shown in Equation(3):

$$wf\_{\varepsilon\chi} = \frac{\heartsuit V\_{NN-SV}}{\heartsuit V\_{NN}} \,\prime \tag{1}$$

$$S\_{SV} = \sum\_{s} w f\_{\text{ex}} S\_{NN\_{\prime}} \tag{2}$$

$$wf\_{\rm in} = \frac{CV\_{NN-SV}}{CV\_{SV}}.\tag{3}$$

### 2.3.2. Near Nozzle Eulerian Spray Modeling

The Eulerian spray modeling is based on a multiphase method [40]. The gas and the liquid are treated as interpenetrating continuous phases, characterized by their volume fraction in the control volume. A number *n* of phases is considered. The first (*n* = 1) is the gaseous one, and it represents the gas and the fuel vapor in the spray ambient. The other phases (*n* = 2, ... , *n*−1) represent the droplet size classes, whereas the phase *n* is the bulk liquid phase preceding the break-up process. For each phase, the set of conservation equations is solved separately. Mass—Equation (4), momentum—Equation (5) and enthalpy—Equation (6) conservation equations for the phase *k* are reported in the following. On the right hand side of each equation, there are the exchange-terms Γ*kl*, *Mkl*, *Hkl* (mass, momentum, enthalpy) between the phases *k* and *l*; these are the terms that contain the physics of the spray. The left-hand side of each equation is made of two terms, the rate of change and the convective transport of the phase flow property. The volume fraction of each phase must fulfill the compatibility condition, Equation (7):

$$\frac{\partial \mathfrak{a}\_k \rho\_k}{\partial t} + \nabla \cdot (\mathfrak{a}\_k \rho\_k \upsilon\_k) = \sum\_{l=1, \ l \neq k}^{\mathfrak{n}} \Gamma\_{kl\prime} \tag{4}$$

$$\frac{\partial \mathbf{a}\_k \rho\_k \mathbf{v}\_k}{\partial t} + \nabla \cdot (\mathbf{a}\_k \rho\_k \mathbf{v}\_k \mathbf{v}\_k) = -\mathbf{a}\_k \nabla p + \nabla \cdot \mathbf{a}\_k \left(\mathbf{\tau}\_k + \mathbf{\tau}\_k^t\right) + \mathbf{a}\_k \rho\_k f + \sum\_{l=1, l \neq k}^n \mathbf{M}\_{kl} + \mathbf{v}\_k \sum\_{l=1, l \neq k}^n \Gamma\_{kl} \tag{5}$$

$$\nabla \cdot (a\_k \rho\_k v\_k h\_k) = \nabla \cdot a\_k (q\_k + q\_k^t) + a\_k \rho\_k f \cdot v\_k + a\_k \rho\_k \theta\_k + a\_k v\_k : \nabla v\_k + a\_k \frac{d\rho}{dt} \sum\_{l=1, \ l \neq k}^n \mathbb{H}\_{kl} + h\_k \sum\_{l=1, \ l \neq k}^n \Gamma\_{kl} \tag{6}$$

$$\sum\_{k=1}^{n} \alpha\_k = 1.\tag{7}$$

The Eulerian spray model is framed in the RANS (Reynolds-averaged Navier-Stokes) approach, with *k*-epsilon closure. The Eulerian–Eulerian spray model here adopted has been extensively validated against experimental data, as reported in [39], concerning atomization and evaporation process, and in [23] concerning reactive cases.