# Airplane Vortices Evolution Near Ground
## 4 Results and Discussion
### Velocity Distribution  
The potential flow equations presented in Section 2 give the value of the physical variables at every point; thus the flow behavior over the entire domain due to the vortices' presence can be obtained. In order to see a detailed representation of the flow parameters, the figures only picture the results of the right-hand side of the domain, and then the flow values at the left-hand side are symmetric. Figure 5, which was generated from Equations (14) and (15), represents the velocity field over the selected domain. Due to the circulation sign (positive in this case because the airplane's wing generates lift), the velocity vectors point downwards near the vertical axis, where the distance between the two real vortices, the one shown in Figure 5 and its symmetric, is at a minimum. At the outer part of the vortical structure, the velocity vector of the right-hand side vortex points upwards, but as the velocity vector induced from the symmetric vortex points downwards, the absolute value of the velocity magnitude is lower than at a location near the vertical coordinates' axis. As expected, and according to the definition of a free vortex, the fluid velocity magnitude increases near the center. Nevertheless, the core of the vortex is represented with constant velocity in Figures 5 and 6. This has been done to better visualize the variables in the graphs. Figure 5: Velocity distribution over the domain for the four cases defined in Table 5 and when ambient crosswind is null. The velocity at every point of the domain has been calculated to later create the vector field. The dimension of the vortices' central core was retrieved from the definition introduced in Equation (37), which is accepted as a general norm to avoid the asymptote that appears at the vortex center; this equation was used to obtain a better visualization of the variables' change in Figures 5 and 6. To calculate the central core dimension, one needs to know the initial value of the vortices' position in the wingspan, which is presented by \(B_{A}\); such a position is usually given as \(B_{A}=\frac{\pi}{4}B\). The parameter B stands for the wingspan and was initially defined in Tables 1 and 3. \[r_{c}=0.052B_{A}=0.052\cdot\frac{\pi}{4}B \tag{37}\]  
Figure 5 has been obtained by performing the domain's discretization (where each cell has a size of \(\Delta x=\Delta y=0.1\) m), and calculating the velocity and the pressure at each cell. Then, specifically for Figure 5, the streamline field was obtained with MATLAB's specific function. For the case of the medium airplane (Figure 5a), the extent of the perturbation generated by the vortex is smaller when compared to other cases. This is due to its lower initial circulation. On the other hand, in Figure 5b,c, there are inappreciable changes, because both circulation and wingspan are very similar. However for Figure 5d, although it has both initial conditions similar to the previous two figures, the initial height is double, and so its perturbation barely reaches the ground. Figure 5 also shows the velocity field colored with the same scale for all circulations presented. It can be observed how the zone perturbed by the vortex increases as the circulation rises; notice as well that the velocity at the vortex's central core is maximum. A particularity of the selected discretization of the domain can be clearly seen in Figure 5a,b, where despite the circulation being higher in the case of Figure 5b, it seems that the velocity at the central core of the medium airplane, shownFigure 5a, is higher. This is because the central core radius in the second case is larger due to the higher dimensions of the airplane, as stated in Equation (37). Almost identical phenomena can be seen in Figure 5c.