## 4 Description of the Application Problem and Employed Multi-Fidelity Models
### High Fidelity: 3D CFD Condensation Model  
This work employs a 3D CFD model of the three-way junction embedded with a condensation subroutine as a high-fidelity model. The approach used is the one described by [PERSON] et al. [68] and has been validated in several ways using a series of different experimental techniques: secondary flows measured with laser particle image velocimetry [68], qualitative condensation distributions observed using planar laser-induced visualization [68], temperature fields [69] and condensation mass fluxes [69]. Indeed, this numerical tool has been able to predict which junction configuration provides the least amount of condensation mass flow rate, which in turn provides the lowest impeller wear in agreement with the level observed in experimental durability tests [70].  
The geometry of the study is an LP-EGR thee-way junction where fresh inlet air and EGR are mixed before the compressor. As shown in Figure 4, there are two inlet branches (a 50 mm diameter duct for air and a 22.4 mm diameter pipe for EGR) and one outlet duct, whose diameter is 34 mm. The nozzle section after the EGR insertion represents the tapered inlet of the compressor. To reduce the impact of boundary locations on the region of interest, both inlet ducts are extruded three additional diameters upstream, and the outlet pipe is extended two diameters downstream.  
The mesh depicted by Figure 4 comprises polyhedral cells with a base size of 1.5 mm and eight prism layers next to the walls to improve the boundary layer resolution. A mesh independence study has selected this grid of 3M cells. Employing a coarser grid of 300 thousand cells results in a difference of 10.4% in condensation at the outlet section. Refining the mesh instead until it features 11M cells provides a difference in condensation of only 3%. Considering the computational effort of the numerical campaign (more than 180 simulations), the mesh of 3M cells was therefore selected, which achieves \(y^{+}\) to be below 1.5 for 99% of the wall cells.  
Figure 4: LP-EGR three-way junction with a cross-section of the mesh employed.  
Figure 3: Scheme of the input parameters on the perfect mixing model.  
The segregated solver of Simcenter STAR-CCM+ is selected for the simulations. Regarding the turbulence formulation, Reynolds-Average Navier-Stokes (RANS) was selected with the \(SST\)\(k\)-\(\omega\) turbulence model [71]. The cases are initially run steadily, and their condensation mass flow rate is monitored. For those working points with an oscillation in this variable above 2%, the temporal treatment was switched to unsteady RANS (URANS), employing a second-order implicit transient solver with a time-step size of \(\Delta t=10^{-4}\) s. As boundary condition types, the mass flow rate, specific humidity, stagnation temperature, and turbulence intensity are specified at both inlets, whereas the static pressure is set at the outlet. The particular values for the numerical campaign are indicated in Section 5.1.1, which correspond to representative conditions for a C-segment passenger car engine.  
A condensation subroutine [42] is embedded into the 3D CFD model. This condensation submodel reads the psychrometric state of each cell and compares it to the corresponding saturated equilibrium state. When supersaturation is detected, the model instantaneously condenses the required amount of water, interacting with the transport equations solved by the 3D CFD code using the corresponding source terms. For more details, please refer to [PERSON] et al. [42].