## 4 Results and Discussion
### Aircraft Wake Vortices in Ground Effect with Crosswind  
In the attempt to simulate vortex interactions and decay in aircraft wakes, values of ground plane length, release height of the primary vortical structures, and wingspan inferred from laboratory observations were adopted [21]. In Figure 1, the ground plane length was chosen as \(\rm B=8.0\); the release height of the primary counter-rotating vortical structures was \(\rm h=2.0\); the initial center-to-center distance between primary vortical structures was \(\rm b=1.0\); the circulation strength was \(\Gamma=1.0\); and the initial radius of every primary vortical structure was \(r_{0}=0.1\). Figure 4: Diffusion of a vortex sheet—comparison of corrected core-spreading method with analytical solution: \(\rm M=30\); \(t=0.22\); \(\rm Re=7650\); \(\alpha=0.80\) and \(\rm\
u_{t}=0.0\)[36]. The method of [PERSON] [37] was adopted to predict the structure of aircraft vortex wake related to the parameter \(r_{0}\). _[PERSON]'s technique_ is simple and appropriate, although it cannot be applied in configurations of straining interactions between vortices. The experimental data presented by [PERSON] and [PERSON] [21] were utilized in a closed recirculation water tank at a fixed Reynolds number of Re = 7650, where the dissipation of primary vortical structures suffers interference from the tank wall. Another problem is that instantaneous experimental measurement of the center of the moving vortical structure undoubtedly generates some kind of error. These experimental difficulties also motivated the present research to contribute to the literature. Other adopted numerical parameters were as follows: number of vortex blobs to discretize every primary vortical structure of N = 50; number of flat panels to discretize the ground plane of M = 40; spatial refinement factor of \(\alpha\) = 0.80; initial core radius of 0.001; maximum core radius (before of partition) of 0.00125; and minimum vortex strength (before partition) of 0.0004. Table 1 shows the test cases simulated for the Reynolds number fixed at Re = 7650. The same arrangement of test cases is also simulated for the Reynolds number of Re = 75,000. All test cases were performed with 1000 time steps of magnitude \(\Delta t\) = 0.05. Figure 5 shows the instantaneous behavior of the velocity field during the interaction of the primary vortical structures with the ground plane at time \(t\) = 15 (test case 1 in Table 1). The generation of secondary vortical structures from the ground can be successfully identified from Figure 5. Those primary vortical structures will separate and rebound upon interacting with the ground plane. The formation of secondary vortical structures is the key effect responsible for the rebound. In Figure 1, the centroid of the primary vortical structure from the right is instantaneously computed during the numerical simulation in a typical Lagrangian manner. The numerical value for the centroid is compared with the experimental result measured by [PERSON] and [PERSON] [21]. Figure 6 presents comparisons of three numerical trajectories (cases 1, 4, and 7 in Table 1) with the trajectory measured by [PERSON] and [PERSON] [21]. The numerical trajectories differ among them, presenting a variation in crosswind velocities, namely U\({}_{\infty}\) = 0.0, 0.02, and 0.04. \begin{table}
\begin{tabular}{c c c} \hline
**Test Case** & **U\({}_{\infty}\)** & \(\epsilon\)/\(\Delta\)s** \\ \hline
1 & 0.0 & 0.0 \\
2 & 0.0 & 0.001 \\
3 & 0.0 & 0.007 \\
4 & 0.02 & 0.0 \\
5 & 0.02 & 0.001 \\
6 & 0.02 & 0.007 \\
7 & 0.04 & 0.0 \\
8 & 0.04 & 0.001 \\
9 & 0.04 & 0.007 \\ \hline \end{tabular}
\end{table} 
Table 1: Physical parameters used in the simulation for Re = 7650. U\({}_{\infty}\) is the crosswind velocity and \(\epsilon\)/\(\Delta\)s is the relative roughness height. Figure 5: Instantaneous velocity field distribution: \(t\) = 15; Re = 7650, U\({}_{\infty}\) = 0.0 and \(\epsilon\)/\(\Delta\)s = 0.0.