## 3 Results  
For a first assessment of the capability of the procedure outlined above, the experimental setup of [29] was simulated. A small-scale NACA0012 profile is considered, with a chord of 150 mm, partially heated and coated with combinations of hydrophobic and hydrophilic sections.  
In particular, we considered a case with a freestream velocity of 40 m/s, freestream temperature of \(-5\)\({}^{\circ}\)C, LWC = 1.0 g/m\({}^{3}\), and droplet size of 30 \(\upmu\)m. The airfoil is heated for 5% of the chord length, with a heat flux of 33 W along the 400 mm spanwise length. We compare two cases: in the first, the heated region is hydrophilic (\(\theta_{s}=65^{\circ},\ \theta_{a}=105^{\circ}\), \(\theta_{r}=50^{\circ}\)) and the remaining area super-hydrophobic (\(\theta_{s}=157^{\circ},\ \theta_{a}=159^{\circ},\ \theta_{r}=154^{\circ}\)), in the second case the whole surface is super-hydrophobic.  
Figure 7 shows the collection efficiency, taking into account rebound and secondary impacts for both cases. The effect is quite dramatic, with a nearly clean leading edge region, although it is important to note that the very small thickness of the airfoil maximizes the effect of rebound. The droplets with higher _We_ will rebound around the leading edge, and due to the small thickness of the airfoil, only a small fraction of them stops on the solid wall after a few secondary impacts. However, these droplets landing somewhat downstream with respect to their first impact are responsible for the recovery at \(x=0.01\), where the collection efficiency is even higher than that in the hydrophilic region.  
The effect of the reduction in water gathering, combined with the improved drainage of the smaller, low hysteresis droplets on the super-hydrophobic surface, leads to a better performance of the fully hydrophobic airfoil. In particular, the droplet evolution was computed on a small strip near the leading edge, as shown in Figure 8. Two snapshots  
Figure 6: Droplet\(-\)rivulet transition, droplet distribution at three different time steps. From left to right: individual droplet running downstream, rivulet onset (grey area), rivulet development.  
Figure 7: Resulting collection efficiency, taking into account super-hydrophobic surface rebound.  
of the droplet population are presented: red circles refer to liquid droplets and blue ones to frozen beads. It is clear that we have better drainage in the hydrophobic setup, with smaller and quicker droplets that evaporate or run downstream before any freezing. On the other hand, the hydrophilic leading edge generates larger droplets and slower flow, and thus we still have runback water beyond the heated strips, allowing for ice accretion as observed in the experiment.  
Interactions between rebounding and incoming droplets are, on the contrary, nearly negligible: Figure 9 shows the computed collision density, i.e., the number of collisions between rebounding and incoming droplets per unit time and volume, around the airfoil profile. The maximum value is attained in the region of the secondary peak in the collection efficiency, at \(x\sim 10\) mm, and is a bit less than 140 s\({}^{-1}\cdot\)m\({}^{-2}\), i.e., it is several orders of magnitude lower than the number of the actual flying droplets at the same location.