## 4 Results and Discussion
### Airplane Wingtip Vortices

Table 4: Physical parameters adopted in the simulations for Re = 7650 and Pr = 0.71. Figure 10: Crosswind interferences on the predicted trajectories of the vortical structures from wing tips (Ri = 0.0 and Re = 7650) [32]. An important analysis for Figures 10 and 11 is the position of the first loop of the primary vortical structures trajectory and last instant of the simulation (\(t\) = 75). It can be observed that crosswind of magnitude 0.04 produces the greater height for the trajectory, as expected. When analyzing only the height of the first loop, the Richardon number interference is not so significant. Nonetheless, for the last instant of the trajectory, the effect of the buoyancy forces is captured. For cases with Ri = 1.018, the trajectory of the primary vortical structure centroid from the right already exhibits the effect of the airport runway deviation. In contrast, for the simulation with Ri = 0.0, the centroid trajectory remains in an upward movement. In Figure 12, the longitudinal distance between the centroid of the left and right primary vortical structures is investigated. It can be observed that the longitudinal distance between points A and B, as well as C and D, decreases with the Richardson number interference. The observed asymmetry between the trajectories of the two structures is attributed to the diffusion scheme that employs a statistical approach through the Random-Walk Method. The use of pseudo-random numbers causes the initial symmetry of the problem to be lost over time increments. Figure 11: Crosswind interferences on the predicted trajectories of the vortical structures from wing tips (Ri = 1.018 and Re = 7650) [32]. Figure 12: Trajectory of the centroid of the left and right wingtip vortices for test case (1). Table 5 shows the separation between these points for the cases without natural convection (cases 1, 2 and 3 for Ri = 0.0) and with natural convection (cases 4, 5 and 6 for Ri = 1.018). The spacing reduction effect occurs because the simulation with buoyancy forces introduces additional momentum into the vortical structures, resulting in higher velocities compared to the case for Ri = 0.0. When analyzing the interaction between the buoyancy forces and crosswind, it can be concluded that the longitudinal distance between points C and D also decreases. Figure 13 illustrates the velocity field behavior for the final time step of the numerical simulation for all cases. In general, the cases with Ri = 1.018 demonstrate that primary vortical structures are more compact, meaning that their centroids are closer to each other. This phenomenon facilitates the interaction and approximation of vortical structures on both the right and left sides. Based on Figure 13, the most significant comparison is for Figure 13e,f. Therefore, an important analysis is to observe the vorticity distribution for these cases. Figure 14 represents the instantaneous vorticity distribution. The contour of the vorticity distribution is not so different when comparing Figure 14a,b. However, the width of the vortical structures in Figure 14a is greater than that in Figure 14b. This behavior is important, since it corroborates the interpretation given in Figure 13, in which the longitudinal distance between the centroids of the primary vortical structures is closer for cases with buoyancy forces. The present method, therefore, is able to provide good estimates for the trajectory of primary vortical structures and their interaction with a heated ground plane, and is able to predict the flow correctly in a physical sense. The main purpose of the present analysis is to investigate the primary vortical structures pair in the heated ground effect. The present results efficiently capture the rebounding of the vortices pair as it approaches the ground for isothermal cases. It can be concluded that the proximity of the primary vortical structures to the ground plane results in boundary layer separation and secondary vorticity, explaining the upward migration of the vortex pair.