# Airplane Vortices Evolution Near Ground
## 2 Mathematical Procedure to Obtain the Velocity and Pressure Fields  
Based on the potential flow theory, the velocity potential equation to find the velocity field in the desired domain can be obtained. Once the temporal variation of the potential is known, the pressure field can be found through the Bernoulli equation for potential flow, which reads:  
\[\rho\bigg{(}\frac{\partial\phi}{\partial t}+\frac{V^{2}}{2}\bigg{)}+P=P_{0} \tag{1}\]  
The procedure starts considering two real vortices situated at points \(Z_{0}=(X_{0},Y_{0})\) and \(Z_{1}=(-X_{0},Y_{0})\), see Figure 1. In order to simulate the presence of the ground, two mirror vortices located in the positions \(Z_{0}^{*}=(X_{0},-Y_{0})\) and \(Z_{1}^{*}=(-X_{0},-Y_{0})\) have to be considered. The complex potential function of two vortices, one real and its image, is defined using Equation (2); this equation also considers the effect of the free stream velocity. Figure 1: Coordinate system and vortices position. The properties of such functions are defined in [16]; based on this information, the final form of the complex potential function has to be the one presented in Equation (3). \[f(Z)=U_{\infty}Z+\frac{i\Gamma}{2\pi}ln\bigg{(}\frac{Z-Z_{0}^{*}}{Z-Z_{0}}\bigg{)} \tag{2}\]  
\[f(Z)=\phi+i\psi \tag{3}\]  
After substituting in Equation (2) the position of the vortices located on the right-hand side of the domain and rearranging the real and imaginary parts, we obtained:  
\[f(Z)=U_{\infty}(X+iY)+\frac{i\Gamma}{2\pi}ln\bigg{(}\frac{(X-X_{0})+i(Y+Y_{0} )}{(X-X_{0})+i(Y-Y_{0})}\bigg{)} \tag{4}\]  
Recalling the concept of a logarithm of a complex function, Equation (4) can be expressed as:  
\[ \begin{split} f(Z)=U_{\infty}(X+iY)+\frac{i\Gamma}{2\pi}\bigg{[} ln|(X-X_{0})+i(Y+Y_{0})|+i\quad\text{atan}\bigg{(}\frac{Y+Y_{0}}{X-X_{0}} \bigg{)}\bigg{]}\\ -\frac{i\Gamma}{2\pi}\bigg{[}ln|(X-X_{0})+i(Y-Y_{0})|+i\quad\text {atan}\bigg{(}\frac{Y-Y_{0}}{X-X_{0}}\bigg{)}\bigg{]}\end{split}  \tag{5}\]  
Once the logarithmic terms from Equation (5) have been multiplied by their conjugate and the real and imaginary terms separated, the resulting velocity potential and stream functions characteristic for the vortex situated at the top right zone and due to the influence of the right part of the domain, will take the form:  
\[\phi_{r}(x,y)=U_{\infty}X+\frac{\Gamma}{2\pi}\bigg{[}-\text{atan}\bigg{(}\frac {Y+Y_{0}}{X-X_{0}}\bigg{)}+\text{atan}\bigg{(}\frac{Y-Y_{0}}{X-X_{0}}\bigg{)} \bigg{]} \tag{6}\]  
\[\psi_{r}(x,y)=U_{\infty}Y+\frac{\Gamma}{4\pi}\bigg{\{}ln\Big{[}(X-X_{0})^{2}+ (Y+Y_{0})^{2}\Big{]}-ln\Big{[}(X-X_{0})^{2}+(Y-Y_{0})^{2}\Big{]}\bigg{\}} \tag{7}\]  
Taking into account that Equations (6) and (7) only consider the effect of the two right-hand side vortices, the real and the imaginary one, it is necessary to add the influence from the other two remaining vortices depicted in Figure 1. It needs to be remembered that the effect of free stream velocity needs to be considered only once; therefore when applying Equation (2) to the other two vortices, the free stream velocity does not need to be taken into account, and then the free stream velocity would be evaluated twice (It is important to note that the potential theory is linear).