# Airplane Vortices Evolution Near Ground
## 4 Results and Discussion
### Vortex Movement and Circulation under Crosswind Conditions  
Having analyzed the case without crosswind, the next step is to evaluate the effect of the crosswind on the vortical structures. In order to validate the in-house program when crosswind is included, a comparison between the results obtained with the present mathematical model and the ones obtained by means of LIDAR measurements, taken from [PERSON] [11], is presented. The procedure to obtain the temporal variation of \(\Gamma\) is the same as the one explained in Section 3. The equations employed to characterize the circulation decay are Equations (33) and (34). As was previously explained, to generate Figure 14, the figures belonging to [PERSON] [11] were extracted from the article and then modified to overlay the circulation given by the polynomial regression done in the in-house program. Figure 12: Horizontal velocity of the vortex center when ambient crosswind is considered null. Figure 13: Separation between the vortex centers when crosswind is considered equal to 0 m/s. The behavior of the vortices is almost the same, as seen in Section 3. In the case of the horizontal evolution \(x^{*}\), the values for \(y_{0}=2b_{0}\) are very accurate. For the lower initial height \(y_{0}=b_{0}\), the accuracy of the results is acceptable, although the position of the vortex center along the y* axis is slightly overestimated. This could be due to the absence of the boundary layer in the potential flow theory, meaning that the vortex motion is not affected by it. Finally, it seems that the error is lower for higher crosswind velocities, \(U_{\infty}\). Figure 14: Comparison between the present study’s results and the experimental results obtained from reference [11] (under different crosswind conditions, at 0.5, 1 and 2 m/s). The different figures belonging to the mentioned reference were edited to later overlay the prediction obtained with the in-house program. In all cases, the experimental results from Holzapfel [11] are plotted in black color. The three upper figures characterize the results at \(z=b_{0}\) (whose results were obtained with the circulation temporal variation shown in Equation (33), and the ones at the bottom represent the case at \(z=2b_{0}\) (following Equation (34) for the temporal variation of circulation). On the other hand, when considering the vertical position evolution \(y^{*}\), the values are more accurate for the case of \(y_{0}=2b_{0}\). But in both cases, the prediction becomes loose when the vortex rebound phenomena occur. One characteristic that must be noted is that the vertical position is independent of time in the in-house program--a fact that could not be taken as valid, because as \(U_{\infty}\) increases, the vortex rebound is bigger. In order to have yet another comparison, the data from article [12] were used. There, the results of the in-house program, using a two-phase wake vortex decay circulation, are compared to a Bayesian Model Averaging (BMA) method. This method provides a slightly different circulation of decay throughout time (Figure 15c), and thus it will be useful to compare the potential flow model to another decay method. The \(\Gamma\) evolution throughout time, in this case, fits the previously stated Equation (35). In Figure 15a, the in-house program correctly predicts the behavior at early times, but after \(t^{*}=1\) it presents a deviation. Additionally, at approximately \(t^{*}=5\) it abandons the 90% envelope. To be noted is that for the potential flow model the vertical position of the vortex center is not affected. In Figure 15b, a comparison between the model presented by [PERSON] et al. [12] and the in-house program for different horizontal velocities is performed. In the figure it is observed that the experimental study has a perfect agreement with the mathematical model when considering a crosswind speed of 1.75 m/s. However, the agreement is also very good when the crosswind velocity is of \(U_{\infty}=2\) m/s.