# Airplane Vortices Evolution Near Ground
## 3 Validation
The main conclusion from Figure 3 is that the accuracy of the results is better when using experimental circulation decays than when using the circulation decay from Equation (29). Therefore, in the remaining part of the study, experimental circulations will be used to generate the graphs. In the following comparison case, the focus again is being put on the position of the vortices' center. The figures generated to perform the comparison characterize the nondimensional horizontal and vertical vortex position variation throughout time. A polynomial regression of grade 6 has been performed to obtain the circulation temporary decay based on experimental data. The data used for the validation were extracted from reference [11], and involve two cases without crosswind, where the vortices are generated at two different heights. The nondimensionalization was done with the criteria established in Equation (31), according to [PERSON] [11]. Additionally, the comparison was performed using the initial values introduced in Table 3 and extracted from the same reference. With these initial conditions, the temporal variation of \(\Gamma\) is given by the following formulas, obtained by performing a polynomial regression of the 6 th grade of the experimental circulations obtained in [11]. Equation (33) corresponds to the case with an initial nondimensional height of \(b_{0}\) and Equation (34) corresponds to the case with an initial nondimensional height of \(2b_{0}\). \[\Gamma_{\xi_{0}-b_{0}}^{*}=4.86\cdot 10^{-6}r^{4}+6.78\cdot 10^{-5}r^{5}-3.21 \cdot 10^{-3}r^{4}+3.26\cdot 10^{-2}r^{4}-0.13r^{2}+9.64\cdot 10^{-3}r^{4}+1 \tag{33}\]  
\[\Gamma_{\xi_{0}-2b_{0}}^{*}=8.61\cdot 10^{-5}r^{4}-2.20\cdot 10^{-3}r^{5}+2.11 \cdot 10^{-2}r^{4}-8.96\cdot 10^{-2}r^{3}+0.15r^{2}-0.16r^{4}+1 \tag{34}\]  
The comparison between the results obtained from the present paper and the ones presented in [PERSON] [11] is introduced in Figure 4. Looking at the figures of the vortex's center position, the agreement is very good when considering the horizontal temporal position, especially for higher initial altitudes. When considering the vertical position, the deviation of the results obtained by the in-house program is low at early times, but as the time increases, the deviation also increases. This behavior is given by the fact that as the principal vortices come close to the ground, the vortex structure is more complex, and thus the temporal variation estimated by the potential flow theory is less precise. The vortex behavior could be better modeled if the secondary vortices, which appear due to boundary layer detachment near the ground, were also included in the in-house code. In fact, in page 223 of [PERSON] and [PERSON] [10], there is a possible parameterization to include those vortices. Some of the data needed to model them are exposed in the mentioned reference. Where, the generation height of the secondary vortexes and its relation with the height of the main ones, and the relation between their circulation and that of the main ones, are given. This aspect could greatly improve the temporary results of the in-house program, as the vortex rebound effect could be simulated. \begin{table}
\begin{tabular}{c} \hline
**A340-300** \\ \hline \(\Gamma_{0}=458\) m\({}^{2}\)/s \\ \(B=60.3\) m \\ \(b_{0}=47.4\) m \\ \(\rho\)= 1.17 kg/ m\({}^{3}\) \\ \(U_{\infty}=0\) m/s \\ \(y^{*}=b_{0}\), \(y^{*}=2b_{0}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Initial values used for the validation of the experimental circulation’s temporal variation, comparing with experimental values of the two-phase wake vortex decay theory of [PERSON] [11]. Overall, it can be concluded that as the height decreases, the temporal variation of the mathematical model is less precise. In particular, the \(y^{*}\) figure of the in-house program has a correct behavior, in both cases, until the vortex rebound phenomena takes place.