# Airplane Vortices Evolution Near Ground
## 4 Results and Discussion
### Vortex Movement and Circulation under Crosswind Conditions
The main characteristic in the relative pressure field over the ground (which is obtained by fixing the Y-axis position to 0 in Equation (1) is that as the crosswind speed increases and the minimum pressure peak moves towards the left, see Figure 18. It can also be seen that as \(U_{\infty}\) increases, the initial pressure drop keeps growing and the pressure recovery along the abscissa axis becomes smaller. In other words, the typical pressure recovery observed when there is no crosswind keeps disappearing as the crosswind effect is considered. In fact, the relative pressure in the right part of the domain remains negative when the crosswind effects are considered. This information is directly extracted from Equation (1). It is also to be noticed that at a certain distance from the vortex's central core, the pressure and flow fields are no longer influenced by the vortices' effect; the only perturbation is given by the \(U_{\infty}\), and pressure and flow fields will just depend on the crosswind. Finally, looking at the different cases presented in Figure 18, the case whose pressure drop is bigger is the one of the heavy airplane with a higher initial position; additionally, this peak is the one that has moved further to the right. Among the other airplanes, the biggest pressure drop is given by the medium airplane, because it gets closer to the ground and the pressure and velocity fields are particularly influenced by it. Figure 17: Vortices’ centers’ positions throughout time with \(U_{\infty}=2\) m/s. Figure 18: Comparison of the relative pressure variation over the ground when different crosswinds (From \(U_{\infty}=0\) m/s to \(U_{\infty}=2\) m/s) are applied to different airplane models (Table 6). Another parameter that is not affected by the crosswind is the temporal variation of the circulation; as it be seen from Equations (33)-(36), its value is independent from the crosswind velocity. This means that all of the cases with the same initial \(\Gamma_{0}\) will also have the same temporal variation. This is why Figure 8 will stay true for all crosswind cases. The temporal variation of the vortex position when the crosswind velocity effect is considered is shown in Figure 19. It can be observed that as the crosswind velocity increases the vortex is further displaced sideways, but the final vertical position remains unchanged. This is a direct effect of the vertical and horizontal velocity, because even though the vertical velocity is not dependent on the crosswind, the horizontal is. This means that as the crosswind speed increases, the eccentricity of the hyperbola will be higher. From this figure it can also be extracted that the movement along the Y-axis is a function of the airplane's dimension rather than of the circulation. As is seen in Figure 11, the vertical velocity for the two different circulations of the heavy airplane (with the same initial height) is almost the same, and therefore the final Y-axis position in these cases will be also nearly the same. Yet, due to the temporal variation of \(\Gamma\), the vortices with higher circulation will end their path in a slightly higher position in comparison with the ones with a lower circulation. This small change can be produced because the initial separation between vortices is lower for the case of lower circulation, which can generate a bigger interaction between them. Additionally, in the case that the initial height of the vortex is higher, the final position will also be slightly higher, compared to those vortices with the same initial circulation but initialized at a lower height. A lower circulation means that the vertical velocity induced by the vortices themselves is smaller, and thus they will move slower and stop the movement earlier.