# Airplane Vortices Evolution Near Ground
## 4 Results and Discussion  
In this section we present how the wake vortices evolve for different flow circulations, airplane dimensions and crosswind velocities. In order to have a reference, the ICAO wake turbulence categorization was used [18], where four classes of airplanes depending on their weight are defined; see Table 4 (MTOW stands for Maximum Take-Off Weight). The relation between the MTOW associated with each airplane and the circulation generated was  
Figure 4: Validation of the mathematical model. Comparison between the present study results and the experimental results obtained from [11]. At the upper part of each subfigure the results have been calculated at an initial height equal to \(y=b_{0}\), for the ones at the bottom it is equal to \(y=2b_{0}\). The in-house program’s temporal variation of the circulation has been adjusted to the experimental data retrieved from the mentioned reference. To produce this figure, the results belonging to [PERSON] [11] have been extracted from the article, and then modified to overlay the results given by the in-house program prediction. No crosswind has been supposed. retrieved from experimental data, for this particular section the information was extracted from [PERSON] et al. [17]. The dimensions and physical values of four airplane models are stated in Table 5; initially no crosswind will be considered. For all four cases, the temporal variation of \(\Gamma^{*}\) will be obtained from experimental data. Thus, from the mentioned references in Table 5, a 6 th grade polynomial regression was performed, in order to introduce the temporal variation of the circulation into the in-house program. In all cases, the measurements were made with a LIDAR sensor during the landing procedure. The values of the medium airplane correspond to an A320, which receives this categorization according to reference [7]. On the other hand, for the heavy airplanes, the second column has similar values to a Boeing B787-8 and, finally, the third and fourth columns correspond to an A340-300. The equation of the heavy airplane's circulation of the second column was extracted from LIDAR measurements that were retrieved from [PERSON] et al. [12], whose representation fits Equation (35). On the other hand, the circulation's temporal variation corresponding to the medium airplane was retrieved from reference [7], giving equation number (36). \[\Gamma^{*}_{BMA}=4.87\cdot 10^{-5}{t^{*}}^{6}-9.17\cdot 10^{-4}{t^{*}}^{5}+5.33 \cdot 10^{-3}{t^{*}}^{4}-2.77\cdot 10^{-3}{t^{*}}^{3}-0.0592{t^{*}}^{2}-0.0264{t^{*}}+1 \tag{35}\]  
\[\Gamma^{*}_{A320}=6.66\cdot 10^{-5}{t^{*}}^{6}-1.59\cdot 10^{-3}{t^{*}}^{5}+1.31 \cdot 10^{-2}{t^{*}}^{4}-3.79\cdot 10^{-2}{t^{*}}^{3}-1.74\cdot 10^{-3}{t^{*}}^{ 2}+1.06\cdot 10^{-2}{t^{*}}+1 \tag{36}\]  
The first results that are presented are the velocity and the pressure fields. A drawback associated with these results, (which were generated for the initial time step), is that the fluid viscosity has no influence on them. However, as already mentioned, we tried to simulate the viscosity effect with the temporal variation of the vortices' circulation, which is specific to each case, see Table 5. \begin{table}
\begin{tabular}{c c} \hline
**L (Light)** & MTOW of 7000 kg or less \\
**M (Medium)** & Less than 136,000 kg and more than 7000 kg \\
**H (Heavy)** & 136,000 kg or more \\
**Super Heavy** & Airbus A380-800, with a MTOW of 560,000 kg \\ \hline \end{tabular}
\end{table} 
Table 4: Types of airplanes according to ICAO categorization. MTOW: Maximum Take-Off Weight. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Medium Airplane** & **Heavy Airplane** & **Heavy Airplane** & **Heavy Airplane** \\ \hline Initial height (m) & 45 & 45 & 47.35 & 94.7 \\ \(\Gamma_{0}\) (m\({}^{2}\)/s) & 250 & 417 & 458 & 458 \\ \(B\) (m) & 34.1 & 57.3 & 60.3 & 60.3 \\ \(t_{0}\) (s) & 18.0 & 30.5 & 30.8 & 30.8 \\ Equation & (36) & (35) & (33) & (34) \\ Used reference & [7] & [12] & [11] & [11] \\ \hline \end{tabular}
\end{table} 
Table 5: Values with which the study was carried out.