## 2 Phenomenological Model
### Droplet Behavior and Runback Water Modeling  
Modeling the runback water flow over a hydrophobic surface requires a simulation of the evolution of the droplet population that takes properly into account the wetting properties of the surface. These phenomena involve small-scale modeling, on the size of the individual droplets, and the information required from the larger-scale flow field is limited essentially to the amount of incoming water (i.e., the collection efficiency) and its geometrical and dynamic characteristics (diameter, velocity components normal and parallel to the wall), as well as the wall shear stress and pressure gradients. A possible choice is to follow the evolution of each individual droplet after impinging via a Lagrangian approach: since the number of involved droplets is very high, a full multiphase CFD computation would quickly become unpractical. However, it is possible to simplify the process with a phenomenological model that describes the main steps of the evolution, namely impinging density, growth or reduction via coalescence and/or phase change and aerodynamic drag-induced motion. These computations will be performed over a small portion of the surface and are meant to provide, via integration, useful information in order to offer corrections to the standard, continuous film models usually assumed for icing simulation on regular surfaces. We will assume that the droplets have a spherical cap shape, defined by their static contact angle \(\theta_{s}\), while the hysteresis between advancing and receding contact angles will be used in order to define the droplet's force and energy balances. Additional elements that need to be tracked are frozen beads and possibly rivulets. The former appears as spherical segments, covering the base of the droplet and progressively growing until the liquid phase disappears; the latter is modeled as an ensemble of interconnected droplets. The simulation is carried out on a two-dimensional domain, which may be representative of an arbitrarily shaped surface as long as we provide the proper distribution of shear stress, collection efficiency and pressure gradients from the CFD solution of the macro-scale problem. The main workflow of the procedure may be summarized in the following steps:  
1. Generation of \(N\) impinging sites at random locations, depending on local collection efficiency. 2. Coalescence check between new impinging droplets and existing ones. 3. Heat transfer balance for any single droplet. 4. Evaluation of freezing or evaporating mass flow. 5. Frozen mass will either contribute to the growth of an existing ice bead or create a new one at the freezing bead location. 6. Droplet stretching limit check, detecting possible rivulet onset. 7. Droplet and rivulet movement check, including coalescences along the moving path. 8. Next time step. The procedure described above was originally developed for a high-fidelity modeling of icing roughness [4], and it has already been validated and applied to different problems [18; 19; 24]. A population of incoming droplets is randomly generated at each time step. As an example, on a rectangular domain \(L_{x}\times L_{y}\) we define a maximum number of impinging sites at any time step as  
\[N^{0}=\frac{u_{0}LWC\cdot\Delta t\cdot L_{x}\times L_{y}}{\rho V_{d}} \tag{10}\]  
Then, given three random numbers \(a_{x},a_{y},a_{c}\): \(0<\alpha<1\) the possible impinging sites \((x,y)_{i},\ i=1,N^{0}\) are defined as \(\left(L_{x}a_{x},L_{y}a_{y}\right)_{i}\), but they get activated only if \((a_{c})_{i}<\beta\), thus statistically complying with the computed collection efficiency. Once an initial distribution of the impinging droplets is given, we check if any coalescence occurs. Coalescence takes place when the distance between the centers of two droplets is less than the sum of their radii, and it results in a larger drop whose volume is equal to the sum of the volumes of the parent droplets, located at the center of mass of the two parent droplets.