## 4 Results and Discussion
### Aircraft Wake Vortices in Ground Effect with Crosswind
The roughness effects are also able to change trajectories because of interactions between vortical structures and the rough ground plane with no crosswind at Re = 7650 (Figure 16). Figure 5:3 Instantaneous velocity field distributions: Re = 75,000, U\({}_{\infty}\) = 0.04 and \(\varepsilon/\)As = 0.007. The mutual interference between crosswind and surface roughness appears as an important control technique to investigate the potential hazard of aircraft wake vortices at high Reynolds number flows (e.g., Figure 17). Very good knowledge production by researchers is necessary in order to increase airport capacity around the world. The results of analysis of both experimental data and numerical simulations are necessary to elucidate different aspects of nonlinear hydrodynamic phenomena intrinsic to the problem of aircraft wake vortices in terms of the ground effect. Figure 5:4 Trajectories of wake vortices with roughness effects: Re = 7650 and U\({}_{\infty}\) = 0.0 [21]. Figure 5:5 Trajectories of wake vortices with roughness effects: Re = 75,000 and U\({}_{\infty}\) = 0.02 [21]. The problem is complex as it involves parameters affected by the vortex-generating aircraft, by vortex-encountering aircraft and by meteorology [25]. Of particular interest here, the motion of wake vortices (generated by a large aircraft) near the rough ground plane with crosswind was investigated to show the potentialities of a Lagrangian vortex method with LES theory, including core spreading and vortex blob splitting. The use of LES theory allows for solving small-scale lengths in the order of the vortex core radius, namely \(\sigma_{0}=0.001\). As a consequence, the final CPU time for the new scheme implemented using parallel computing (OpenMP) was about 122 min when using an Intel(r) CoreTM i7-2600 [EMAIL_ADDRESS] GHz. The use of parallel computing is necessary because of the expensive computational cost of the developed algorithm. Thus, the present technique has the following disadvantages: (a) the final CPU time of a typical numerical simulation is particularly long because of the vortex-vortex interaction (the Biot-Savart law), which is computed using Equation (10); (b) the use of the method of images contributes to an increase in the final CPU time, since vortex images have to be considered when using Equation (10); and (c) the core spreading and the vortex blob splitting contribute to an increase in the number of vortex blobs. Therefore, the development of an adaptive, parallel version of the fast multipole method (FMM) [16; 17] to the kind of resolution required to treat engineering flows is necessary. The fast multipole method is a technique allowing for the fast computation of long-range interactions between NV vortex blobs in O(NV) or O(NV log NV) steps with some prescribed error tolerance. The Biot-Savart law requires interactions between NV vortex blobs in O(NV\({}^{2}\)) steps. As a future study, we also plan to simulate buoyancy forces acting on aircraft wake vortices in the vicinity of a rough ground plane exchanging heat with the fluid flow [30]. For this, the second-order velocity structure function model of the filtered field will be incorporated into the energy equation to capture interaction between particle temperatures and vortex blobs.