# Airplane Vortices Evolution Near Ground
## 3 Validation
The meaning of the parameters presented in Table 2 are the same as the ones initially introduced in Table 1.  
The experimentally obtained circulation temporal variation from the mentioned reference has been parameterized, by means of a 6 th grade polynomial regression, to include it into the in-house program. The concept is to check how accurate are the results when the circulation temporal decay obtained experimentally is employed. The temporal variation of the circulation for this case is modeled by Equation (32). The results are represented in Figure 3, together with the ones from reference [10], and the ones obtained with Equation (29). To model the temporal circulation, nondimensional values are needed, whose criteria, according to reference [10], are those established in Equation (31). The parameter \(b_{0}\) is as well introduced and expresses the initial separation between vortices. \[x^{*}=\frac{x}{b_{0}};y^{*}=\frac{y}{b_{0}};\Gamma^{*}=\frac{\Gamma}{\Gamma_{0 }};t^{*}=\frac{t}{t_{0}};u^{*}=\frac{V_{X0}}{w_{0}};v^{*}=\frac{V_{Y0}}{w_{0}} ;w_{0}=\frac{\Gamma_{0}}{2\pi b_{0}};b_{0}=\frac{\pi}{4}B; \tag{31}\]  
Figure 2: Validation of the in-house program’s mathematical model. Comparison between the present study’s results and the ones from [PERSON] et al. [6]. Equation (29) has been used to determine the temporal variation of the circulation. Then, to validate the study, the temporal variation of the vortices’ position throughout time was plotted. The comparison between the results obtained from the in-house program presented in this paper and the ones obtained by [PERSON] and [PERSON] [10] is shown in Figure 3. From the figures of the vortex center position, it is seen that the deviation of the results obtained by the in-house program is negligible at early times, but as the time increases, the deviation increases. The major discrepancies are observed in Figure 3a, because both curves, the one based on experimental circulation decay and the one using Equation (32), abandon the \(3\sigma\) envelope for advanced times. This shows that the accuracy of the results could be better for this case, since the in-house program does not simulate the vortices' rebound. The accuracy of the results is particularly good in Figure 3b; the ones obtained with Equation (32) show an almost perfect accuracy compared to the experimental results for all the time steps. On the other hand, for Equation (29), at early time steps the accuracy is good, but after \(t^{*}=2\) the prediction becomes loose, even though it does not stray too far from the experimental results. Finally, for Figure 3c, the polynomial regression used for Equation (32) obviously fits perfectly to the experimental results, but the single-parameter temporal variation of \(\Gamma\) shows poor accuracy, meaning that it does not fit the experimental temporal variation at any point; additionally, it leaves the \(2\sigma\) envelope during \(t^{*}=1\) to \(t^{*}\approx 2.5\). \begin{table}
\begin{tabular}{c} \hline \hline
**B747-400** \\ \hline \(\Gamma_{0}=633.3\) m\({}^{2}\)/s \\ \(B=64.4\) m \\ \(\rho=1.225\) kg/m\({}^{3}\) \\ Height = 55 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Initial values used for the validation between the in-house program (by using either Equations (29) and (32) for the temporal variation of circulation), and the two-phase decay method presented by [PERSON] and [PERSON] [10]. Figure 3: Validation of the in-house program’s mathematical model. Comparison between the present study’s results and the ones from [PERSON] and [PERSON] [10]. Figures have been extracted from this reference and edited to be able to later include the in-house program results and have a clearer comparison. However, the probabilistic envelopes of \(2\sigma\) and \(3\sigma\) have been also included, to see the in-house program results’ precision. The results obtained with Equation (29) state for “Theoretical \(\Gamma\)”. The ones obtained with Equation (32) state for “Experimental \(\Gamma\)”. For this case, crosswind is considered null.