# Airplane Vortices Evolution Near Ground
## 3 Validation  
Before the validation, it is necessary to take into account another feature that will also be introduced to the study. This is the temporal variation of the vortex's circulation. Then due to the viscosity effect, circulation cannot be considered constant. In fact, the time dependent vortex circulation is expected to have considerable effects on the vortices' temporal behavior, for example their lateral motion will be slowed down because the circulation decrease with time. According to [6], the simplest expression that models the temporal evolution of the vortex's circulation is the single-parameter wake vortex decay, represented as Equation (29):  
\[\Gamma(t)=\Gamma_{0}\cdot exp\left(-\frac{t}{T_{e^{-1}}}\right) \tag{29}\]  
For solving Equation (29), the parameter \(T_{e^{-1}}\) must be known, which is retrieved from experimental data. [PERSON] et al. [17] stated that the dependence of the decay can be related to a time parameter \(t_{0}\), whose value is obtained for each case imposed by the initial circulation. Based on references [6,17], the mentioned value will be defined as \(T_{e^{-1}}=10t_{0}/\pi\), where \(t_{0}\) is given as:  
\[t_{0}=\frac{2\pi}{\Gamma_{0}}\left(\frac{\pi}{4}B\right)^{2} \tag{30}\]  
The \(\Gamma_{0}\) and \(B\) values are the vortex's initial circulation and the wingspan of the airplane, respectively. Instead of using this temporal variation, another relation can be used, which provides a two-phase decay of the vortices' circulation [10]. This second temporal variation is more complex and realistic when there is low ambient turbulence, but some probabilistic data must be used. In this paper, the temporal equation presented in [PERSON] et al. [6] is initially employed. Once Equation (29) is coupled to the study, by means of substituting the value retrieved from this equation at each time step to Equations (1), (14), (15) and (28), the initial validation of the mathematical procedure shall be done. To validate the in-house program with the one-phase decay theory (Equation (29)), since the temporal variation of the circulation simply depends on the initial circulation, the main focus is put on variables whose mathematical formulation is different in each method, for example the velocity domain and the position of the vortices' center throughout time. The comparison between the results obtained in the present paper and the ones presented by [PERSON] et al. [6] are introduced in Figure 2. The vortex central core's vertical and horizontal velocity distribution throughout time, the vortex's vertical position and the separation between the two vortices as a function of time are represented. The temporal variation of the vortex's central position is obtained by combining Equations (24) and (25) together with the time elapsed, and the separation between vortices is drawn from the horizontal velocity and the time elapsed. The evolution of the variables is almost the same in all figures; minor differences are observed in the vortices' positions and separation between vortices. Therefore, the results obtained from the present equations and methodology need to be seen as accurate. The comparison was done using the initial values introduced in Table 1 and extracted from reference [6], where \(\rho\) stands for the air density, the height is the vertical distance from the ground at which the vortex is formed, \(\Gamma_{0}\) is the initial value of the circulation and \(B\) is the wing span of the airplane. \begin{table}
\begin{tabular}{l} \(\rho=1.225\) kg/m\({}^{3}\) \\ Height = 50 m \\ \(\Gamma_{0}=500\) m\({}^{2}\)/s \\ \(B=63.7\) m \\ \end{tabular}
\end{table} 
Table 1: Initial values used for the validation between one-phase decay methods. Once it is seen that the mathematical model is accurate regarding other theories with one-phase decay, the next step is to perform a validation comparing to a two-phase decay model (obtained from reference [10]). The comparison was done using the initial values introduced in Table 2 and extracted from the same reference.