## 4 Results and Discussion
### Advection and Diffusion Convergence  
The convergence of Equation (11) is demonstrated through the problem of advection of a pair of vortex blobs. Figure 3 shows a vortex pair of equal strength \(\Gamma_{\mathrm{v}}=1\), core radius \(\sigma_{0}=0.001\), distance \(\mathrm{AB}=2.0\) apart. If these are free vortex, then they will induce velocity on each other (see Equation (10)), acting in the direction normal to the segment AB that joins them. Since there is no other contribution for the velocity computation, the vortex pair will process in a circular path (only advection motion) about the middle point of the line AB. Here, the adopted strategy is to stop the numerical simulation when the vortex blob A, starting at its original position (0.0, 1.0), approximately attains the starting position of the vortex blob B at location (0.0, \(-1.0\)).  
The relative error between the true circular drift path maintaining constant distance \(\mathrm{AB}=2.0\) and the numerical result obtained by Equation (11) is about \(4.27\%\), \(0.88\%\), and \(0.44\%\) when time increments of 0.5, 0.1, and 0.05 are adopted, respectively. Thus, the time-stepping \(\Delta\mathrm{t}=0.05\) is considered suitable to be used in the explicit Euler time marching scheme. Curiously, \(795\) time steps of magnitude \(\Delta\mathrm{t}=0.05\) were required to satisfy the numerical simulation stopping criterion. An advantage of the Lagrangian manner over Eulerian schemes to compute vorticity advection is that no explicit treatment of advective derivatives is required.  
The convergence of Equation (12) is noticed by comparing it with the analytical solution of the viscous diffusion of the finite length of a vortex sheet, say, between \(0<x<\ell\) (\(\ell\) is adopted equal to 0.3). That vortex sheet length can also be thought of as part of  
Figure 3: Convergence of the advection scheme using a pair of vortex blobs.  
the ground plane length, i.e., the surface \(S_{\rm b}\) in Figure 1. As shown by [PERSON] [36], the analytical solution is given by:  
\[\omega(y,t)=\frac{\gamma(x)/2}{\sqrt{\pi t/\rm Re}}\exp\biggl{(}-\frac{y^{2}}{4t }\rm Re\biggr{)} \tag{14}\]  
where \(\gamma(x)=2\rm U\) represents the sheet strength, \(\rm U=1\) defines the mainstream velocity, and \(4t/\rm Re\) is interpreted as vorticity penetration depth during the time t.  
The adopted values to solve Equation (12) are \(\rm M=30\) vortex blobs, every intensity \(\Gamma_{\rm v}=\gamma(x)\)\(\ell/\rm M\) and generated on \"pivotal points\" on \(y=0\); core radius \(\sigma_{0}=0.01\); time corresponding to \(t=\Delta t=0.22\); Reynolds number value of \(\rm Re=7650\); and spatial refinement factor \(\alpha=0.80\). The analytical solution obtained from Equation (14) is compared with the numerical solution given by [13]:  
\[\omega_{jk}(y,t)=\sum_{k=1}^{Z}\frac{\Gamma_{k}}{\pi\sigma_{k}^{2}}\exp \biggl{(}-\frac{r_{jk}^{2}}{\sigma_{k}^{2}}\biggr{)} \tag{15}\]  
where \(Z=120\) vortex blobs and \(\sigma_{k}=0.01\) because of the partition scheme. During the time corresponding to \(t=0.22\), the core radius attained a value of \(0.0125\), and it was split into four new vortex blobs, each with a core radius \(\sigma_{k}=0.01\).  
The test case run is illustrated in Figure 4, which shows the vorticity distribution for \(\ell/2\) (at half the length of the vortex sheet) and \(0\leq y\leq 0.04\). The comparison between numerical simulation and analytical solution was found to be very satisfactory.