## 4 Numerical Methodology and Simulation Setup
### Thin Film Governing Equations  
In OpenFOAM, the wall film is described by the so-called thin film approach, in which the governing equations for mass, momentum, energy and species are solved in a 2D-auxiliary mesh along the wall where spray-wall interactions take place. Considering AdBlue as an incompressible liquid and following the thin film approximation, the continuity equation can be rewritten as function of film thickness, \(\delta\), by considering single face area:  
\[\frac{\partial\delta}{\partial t}+\
abla\cdot(\delta\mathrm{U_{f}})=\mathrm{ S_{impg}}+\mathrm{S_{evap}} \tag{10}\]  
where the first term on the left side expresses the transient evolution of the film thickness while the second term accounts for the convective contribution with the film velocity, Uf. The two terms on the right side represent the source term due to spray impingement and mass transfer during the film evaporation, respectively. The term \(\mathrm{S_{impg}}\) can have both negative and positive contribution depending on whether the impact results in a deposition or impingement-induced film breakup. It can be defined as:  
\[\mathrm{S_{impg}}=\frac{4\pi\mathrm{\rho}}{3\mathrm{A}\mathrm{\rho_{f}}\mathrm{ \Delta t}}\ \mathrm{\sum_{i=1}^{N_{d}}r_{i}^{3}} \tag{11}\]  
where \(\mathrm{\rho_{f}}\) is film density, A the face area, \(\mathrm{\Delta t}\) the time step and \(\mathrm{N_{d}}\) the number of interacting droplets with radius \(\mathrm{r_{i}}\). In order to describe the transport of AdBlue wall film that contains more than one component, a species transport equation needs to be solved for \(\mathrm{Y_{k}}\) species. This yields:  
\[\frac{\partial\delta\mathrm{Y_{k}}}{\partial\mathrm{t}}+\
abla\cdot(\delta \mathrm{Y_{k}}\mathrm{U_{f}})=\mathrm{S_{impg,\ k}}+\mathrm{S_{evap,\ k}} \tag{12}\]  
where the source terms due to spray impingement and evaporation in the right side are related to individual species k. Generally, the energy balance in the film consists of heat exchange both with the carrier phase and the wall together with contribution from the impinging spray, \(\mathrm{S_{H}}\):  
\[\frac{\partial\delta\mathrm{H}}{\partial\mathrm{t}}+\
abla\cdot(\delta \mathrm{HU_{f}})=\mathrm{J_{g}}-\mathrm{J_{w}}+\mathrm{S_{H}} \tag{13}\]  
where the quantity \(\mathrm{J_{g}}\) is the heat exchange between the film and the gas phase owing to both temperature gradient and evaporation, and \(\mathrm{J_{w}}\) the heat exchange with the wall. The momentum transport in the film can be expressed as:  
\[\frac{\partial\delta\mathrm{U_{f}}}{\partial\mathrm{t}}+\
abla\cdot(\delta \mathrm{U_{f}}\mathrm{U_{f}})=\frac{1}{\mathrm{\rho_{f}}}\
abla(\delta \mathrm{P_{f}})-\mathrm{\tau_{f}}+\mathrm{\tau_{w}}+\delta\mathrm{g_{t}}+ \mathrm{S_{U}} \tag{14}\]  
where the essential contributions are represented in the right side by the source term due to the pressure gradient, the shear force along the film free surface (gas side), the shear force along the wall-film interface, the gravitational force and finally the source term due to the spray impingement on the film, respectively. The reader can find more details of this model in [35].  
The effect of the substrate wettability is not considered in the present computational model. The geometry of the total wetted spot is possibly not precise. Nevertheless, the dynamics of the liquid film in the spray impact region is considered accurately in the model. Therefore, the evolution of the film thickness in this region should be predicted correctly.