## 4 Results and Discussion
### Airplane Wingtip Vortices  
The problem of dissipation of aircraft wake vortices with crosswind is the second configuration investigated in this paper. These vortices are the result of the lift being generated by the wings. In order to perform the simulations, it is necessary to estimate some numerical parameters as presented in Table 3[11; 17]. As shown in Table 3, at the initial instant of each simulation, N = 50 vortex blobs are used to represent each of the wingtip counter-rotating vortical structures or primary vortical structures. Therefore, a pair of counter-rotating vortical structures are shed from the position (L/2 \(\pm\) b/2, h) with the circulation \(\Gamma=\pm 1\). The dimensionless parameters chosen for the simulations were defined based on operational aspects of the landing and takeoff processes at airports, in order to enable comparison with the experimental data of [PERSON] [32] for Re = 7650, which considered the absence of crosswind and buoyancy forces. Based on this, the relation h/b = 2.0 is typical for problems of landing or takeoff operations \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Runway length (L) & 8.0 \\ Release height of the primary vortical structures (h) & 2.0 \\ Center-to-center spacing of the vortical structure pair (b) & 1.0 \\ Core radius of each primary vortical structure (r) & 0.1 \\ Number of flat panels (M) & 40 \\ Number of Lamb vortex elements of each & 50 \\ primary vortical structure (N) & 0.05 \\ Time stepping (\(\Delta t\)) & 0.05 \\ Number of time steps (NSteps) & 1500 \\ Core radius of each Lamb vortex (\(\sigma_{V}\)) & 0.001 \\ Core radius of each temperature particle & 0.006 \\ \((\sigma_{T})\) & 1.5 \\ Preheating time (\(\tau\)) & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dimensionless input data necessary to simulate the airplane wingtip vortices problem [11; 17]. Figure 9: Position of the wake particles at \(t=22.5\) for case B: (**a**) wake vortices and (**b**) heat wake. between parallel runways when an aircraft develops velocity of approximately 240 km/h. Therefore, the adopted values for half the aircraft wingspan and the initial release altitude are b/2 = 0.5 and h = 2.0, respectively. The length of the airport runway, sufficient to establish the interaction between the runway boundary layer and the primary vortical structures shedding from the aircraft, is defined by L = 8b. All values were previously validated by [PERSON] et al. [11] and [PERSON] et al. [17]. The focus of the present numerical simulations is to analyze the buoyancy effects with crosswind on the trajectory of the primary vortical structures. Three values of crosswind were adopted, i.e., 0.0, 0.02 and 0.04. For each value of crosswind, simulations were performed without and with the inclusion of buoyancy forces, the latter represented by Richardson numbers of 0.0 and 1.018, respectively. In simulations involving mixed convection heat transfer, the fluid temperature far from the ground plane is fixed at \(T_{\infty}=20\)\({}^{\circ}\)C and the airport runway temperature is \(T_{W}=40\)\({}^{\circ}\)C, resulting in the corresponding Richardson number of Ri = 1.018. Table 4 shows the simulated test cases for a fixed Reynolds number of Re = 7650 and a fixed Prandtl number of Pr = 0.71. Figure 10 presents comparisons of three numerical trajectories of the primary vortical structures (test cases (1), (2) and (3) in Table 4) with experimental data from [PERSON] and [PERSON] [32]. The numerical trajectories are different from each other, as each case corresponds to a crosswind velocity. Figure 11 presents a comparison among three numerical trajectories (test cases (4), (5) and (6)) and the same experimental data. \begin{table}
\begin{tabular}{c c c} \hline
**Test Case** & \(\mathbf{U_{\infty}}\) & **Ri** \\ \hline (1) & 0.0 & 0.0 \\ (2) & 0.02 & 0.0 \\ (3) & 0.04 & 0.0 \\ (4) & 0.0 & 1.018 \\ (5) & 0.02 & 1.018 \\ (6) & 0.04 & 1.018 \\ \hline \end{tabular}
\end{table}