## 2 Phenomenological Model
### Droplet Impact Modeling  
The evaluation of impinging collection efficiency requires interaction with the macro-scale flow. Here, the amount of water hitting the airfoil is evaluated in a standard way via Lagrangian tracing of droplets within a flow field computed via CFD tools (SU2 open-source code), assuming one-way coupling (droplets do not impact on the airflow field). The impact dynamics are quite complex: a comprehensive analysis and comparison may be found in [15]. Here, we follow the model of [PERSON] et al. [13]. The droplet energy is the sum of kinetic \(E_{K}\) and surface energy \(E_{\sigma}\), which in turn is the sum of the contributions from both solid-liquid \(S_{sl}\) and liquid-air \(S_{la}\) interfaces. \[E_{\sigma}=S_{la}\sigma+S_{sl}\sigma(1-cos\theta_{s}) \tag{1}\]  
The droplet energy decreases in time due to the sum of internal viscous dissipation \(E_{\mu}\) and the capillary rim dissipation \(E_{\mu}^{*}\)[13]:  
\[\frac{dE_{K}}{dt}+\frac{dE_{\sigma}}{dt}+E_{\mu}+E_{\mu}^{*}=0 \tag{2}\]  
The surface and kinetic energy terms, as well as the internal viscous dissipation \(E_{\mu}\) (which include dissipation within the droplets and friction losses at the liquid-solid interface), are integrated approximating the droplet shape during the impact as a shallow cylinder of radius \(r\) and height \(h\), while assuming an axisymmetric velocity field in the radial \(x\) and normal coordinate \(y\):  
\[u_{x}=\frac{2\dot{r}}{rh}xy\quad u_{y}=-\frac{2\dot{r}}{rh}y^{2} \tag{3}\]  
Figure 1 shows a sketch of the approximated geometries. In Equation (3), \(\dot{r}\) is the time derivative of the base radius of the droplet, i.e., its spreading (or retracting) velocity. The rim dissipation \(E_{\mu}^{*}\) is evaluated via the following correlation [13]:  
\[E_{\mu}^{*}=4.48\pi\cdot\mu\cdot r\left(\frac{Re}{\sqrt{We}}\right)^{\frac{2 }{3}}\dot{r}^{2} \tag{4}\]The local droplet Weber and Reynolds numbers are defined as  
\[We =\frac{\rho u_{0}^{2}r_{0}}{\sigma} \tag{5}\] \[Re =\frac{\rho u_{0}r_{0}}{\mu} \tag{6}\]  
where \(r_{0}\) and \(u_{0}\) are the incoming droplet radius and velocity, while \(\sigma\), \(\mu\) and \(\rho\) represent water surface tension, viscosity and density. Substituting in (1) we obtain an ordinary differential equation that can be solved in time. As initial conditions, we assume that both kinetic and surface energies are still equal to the incoming spherical droplet values: such conditions enable the evaluation of both the initial drop diameter and its advancing velocity. Figure 2 shows the time history of the droplet base radius \(r\) (normalized with the incoming droplet radius \(r_{0}\)): agreement with experimental data as in [13] is reasonably good, in particular in terms of maximum spread, while the radius oscillation time scale is over-predicted. Time-dependent simulations at increasing \(We\) may be used to define a rebounding condition, which occurs when the droplet returns to its initial impact shape with residual kinetic energy. The identification of the transition to splashing is a complex problem. Different regimes (aerodynamic or hydrodynamic dominated) lead to different functional dependencies [15]. Here, for the sake of consistency, we follow the same approach used for the rebound limit: a further increase in \(We\) leads to a second threshold, above which the droplet radius oscillations appear overdamped and it no longer retracts down to the initial shape. This second level is assumed as the onset of corona splashing. Figure 1: Droplet evolution sketch, following [13], and coordinate system. (**a**) impinging droplet, (**b**) shallow cylinder approximating the droplet shape during the spreading and retreat phase.