## 4 Results and Discussion
### Aircraft Wake Vortices in Ground Effect with Crosswind
Figure 7 shows that the primary vortical structures, identified in Figure 5 at \(t=15\), evolved to the last instant of the numerical simulation. During that time interval, the primary vortical structures approach and do not leave the runway. It can be observed in Figure 6 that the centroid of the primary vortical structure from the right attained a maximum height in the order of 0.8 h during the loop, which is clearly a potential hazard to smaller aircraft operating on the same runway. The numerical result indicates that rebounding vortices may relink and even undergo additional rebound. The vortex Reynolds number defined in terms of circulation strength, \(\Gamma\), and kinematic viscosity, \(\
u\), proves suitable for capturing this type physics [21]. The literature has reported that physics associated with the rebounding effect does not occur unless the viscous boundary layer condition is imposed on the runway [22; 23; 24; 25; 26]. Figure 8 shows the phenomena above interpreted including a crosswind velocity of magnitude U\({}_{\infty}\) = 0.04 blowing the primary vortical structures. It can be concluded that there are crosswinds of sufficient magnitude to counteract the cross-runway displacement of one vortical structure and prevent it from traversing away from the ground plane. Under that condition, it is clear that the potential hazard is for aircraft approaching on the same (or a parallel) runway. Figure 6: Comparisons among vortices trajectories with no roughness effects: Re = 7650 [21]. Figure 7: Instantaneous velocity field distribution: \(t=50\); Re = 7650, U\({}_{\infty}\) = 0.0 and \(\varepsilon/\Delta\)s = 0.0. Figure 9 shows trajectories of the centroid of the primary vortical structure from right when the Reynolds number is increased to Re = 75,000. The vortical structure configuration with no crosswind attains a maximum height in the order of 0.9 h during the loop. It can also be concluded that the primary vortical structures survive the interaction with the runway and are blown by crosswinds into the path of another aircraft approaching on the same (or a parallel) runway. A comparison between trajectories for crosswind velocity of magnitude U\({}_{\infty}\) = 0.04 indicates that the vortical structure from the right tends to move faster leaving the runway at Re = 75,000 (Figures 6 and 9 can be compared). Attention should be given to smaller aircraft operating on a parallel runway during the presence of that aircraft wake in the vicinity. The sensitivity of the roughness model can be captured in Figure 10. When the relative roughness height of \(\varepsilon/\Delta\)s = 0.001 is adopted, the main result is that primary vortical structures also survive the interaction with a rough ground plane and can attain a maximum height in the order of 0.95 h during the loop (U\({}_{\infty}\) = 0.02) at Re = 7650. Interestingly, the configuration with U\({}_{\infty}\) = 0.04 does not describe the loop leaving the runway any faster (Figures 6 and 10 can be compared). Unfortunately, the experimental data [21] did not include the combined effects of crosswind and surface roughness to be compared. Figure 8: Instantaneous velocity field distribution: \(t\) = 50; Re = 7650, U\({}_{\infty}\) = 0.04 and \(\varepsilon/\Delta\)s = 0.0. Figure 9: Comparisons among vortices trajectories with no roughness effects: Re = 75,000 [21]. The results of the configuration using \(\epsilon/\Delta\)s = 0.001 and Re = 75,000 are shown in Figure 11. When compared with the numerical results in Figure 9, the looping of the primary vortical structure for two crosswind values can be identified. In Figure 11, the vortical structure configuration with a crosswind of magnitude U\({}_{\infty}\) = 0.02 attains a maximum height in the order of 1.05 h during the loop.