## 1 Introduction  
Most of the numerical models adopted in in-flight icing simulations are essentially based on the Messinger model [1] and its extension to three-dimensional arbitrary surfaces [2], combined with the prediction of impinging mass flow following either a Lagrangian or Eulerian [3] droplet flow field analysis. The runback water layer is implicitly assumed to behave as a continuous film whose velocity is mainly defined by the shear stress. In the most sophisticated form, this approach allows for full integration into a three-dimensional computational fluid dynamics (CFD) model, and offered efficient and accurate results in a number of highly complex applications, up to the analysis of a whole aircraft. However, it inherently skips most of the micro-scale effects related to the wettability and adhesion properties of the substrate, as well as the details of the transitions between moving droplets, rivulets, and full film flow regimes; actually, the smaller scale of these phenomena would require unpractically refined grids. An exception is given by roughness models developed on the basis of a droplet distribution analysis [4]. On the other hand, the advances in the development of micro- and nano-structured surfaces provide a number of highly specialized surfaces, designed either for hydrophobicity or low ice adhesion properties, that are promising as passive anti-icing tools [5; 6; 7; 8]. Although ice phobicity and hydrophobicity are not synonyms, and their correlation is a matter of basic research [9; 10], it is clear that some typical features of super-hydrophobic surfaces, such as the higher probability of impinging droplet rebound or the reduction of liquid-solid contact (and heat transfer) areas will help with an anti-icing strategy. Laboratory tests can define such micro-scale properties, usually via the analysis of a single droplet,but the actual performance under real-world in-flight icing conditions is highly dependent on their interaction with the outer, macroscale flow field. Thus, a kind of bridge is required to transfer the relevant micro-scale information to the macroscale analysis around a whole airfoil or a whole airplane. In particular, considering a super-hydrophobic surface, we have a few mechanisms that affect the ice accretion process. First, above a threshold impinging velocity (i.e., Weber number) and low wettability, impinging droplets may rebound, effectively reducing the collection efficiency, or (at even higher Weber) may break down into smaller droplets [11], affecting the local ice beads' related roughness and coalescence processes [12; 13]. For an extremely water repellent surface, the rebound may occur at relatively low _We_[14; 15], and it is worth noting that the physical phenomena behind the rebound are quite complex and might be strongly affected by the topological details of the surface roughness and its different scales [15; 16]. Furthermore, the jumping droplet may stick to the surface after a few rebounds at lower energies. As a second relevant effect, low hysteresis greatly enhances the runback water drainage, since most droplets are quickly removed, ideally preventing the formation of a continuous film. A population of droplets, either still or moving, will obviously have different active heat transfer surfaces with respect to a continuous film [17; 18; 19], as well as different equivalent runback water velocities, thus strongly affecting both the ice accretion and the evaporation process. In particular, solid-liquid and air-liquid heat transfer will occur through the droplet base circle (wet area fraction) and spherical cap surfaces, respectively, while the velocity will be defined by the balance of forces acting on the droplets. However, if the liquid water content is high enough, the moving droplets may merge, forming individual rivulets; the rivulets, then, may coalesce forming a continuous film [20]. Even these transitions are strongly affected by the wettability properties, which in turn control the stability of the contact lines.