# Airplane Vortices Evolution Near Ground
## 4 Results and Discussion
### Vortex Movement and Circulation under Crosswind Conditions
However, this last statement only applies to an airplane with the same dimensions, because as seen in Figure 19, even though the circulation of the medium airplane is smaller than the one of the heavy airplane, the final position of its vortex is shorter, approximately 10 m less when compared with the position of the vortex generated by a heavier airplane. Once the motion of the vortices is introduced, it is interesting to see how their velocity varies throughout time. First, as previously explained, the vertical velocity is independent of the crosswind, as can be seen from Equation (25), and therefore the temporal variation will be the same as in the case without crosswind (Figure 11). This is since the separation between vortices remains the same regardless of the crosswind velocity. In the case of the horizontal velocity, the only change due to the appearance of the crosswind is that this value will be added up directly to the value already calculated supposing no crosswind, as can be observed from Equation (14). It can be seen in Figure 20  
Figure 19: Comparison of the starboard vortex center’s positions throughout time, when different crosswinds (from \(U_{\infty}=0\) m/s to \(U_{\infty}=2\) m/s) are applied to different airplane models (Table 6). that as the circulation increases, the peak on the horizontal velocity becomes higher. The acceleration of the velocity on the X-axis is higher in the case of the medium airplane relative to the heavy ones. Even though at the initial time the velocity induced in this direction by the vortices themselves is almost 0, this particularly high acceleration is likely to be produced by the effect of the ground on the vortical structures. The behavior of the vortices generated by a heavy airplane at a higher altitude follows the same tendency as the case without crosswind, meaning that the peak is delayed with respect to the other models. Finally, for the medium airplane, the horizontal velocity becomes higher than the one of the heavy airplanes at approximately 110 s. This may be either due to the ground effect or due to a perturbation in the temporal variation of the circulation, which can be seen in Figure 8.  
Figure 21 represents the velocity field for different circulations and crosswind velocities. In general, when comparing Figure 5 and Figure 21, it is clearly seen how the velocity changes along the domain as the crosswind velocity increases, and the entire flow field is now more intense. Additionally, when crosswind is considered, there are zones where the velocity is null. In fact, for all of the cases studied with crosswind, the velocity is much higher below the vortex than above it; this is because below the vortex the velocity induced by the vortex and the velocity of the crosswind sum up (note the change of the light blue zone), and these two velocities flow in opposite directions above the vortices. Additionally, the area affected by the perturbed flow is wider and more irregular when crosswind is considered. Nevertheless, in both cases most of the domain remains poorly affected and the flow velocity in the nearest zone around the central core remains almost unaffected by the crosswind velocity. This is likely to be due to the large velocity field induced by the vortices, which is much higher in comparison to the induced velocity generated by the crosswind. The main difference between Figure 5c in respect to Figure 5a,b resides in the intensity of the perturbation to the flow, which is lower in the case of a lower circulation. There is another difference that happens in the zone where the velocity is near 0. This zone moves from the quasi-central upper side to the upper right-hand part of the vortex as the circulation increases. Above this zone, the particles move to the right, as the speed of the vortices in this zone is not strong enough to overcome the crosswind.