## 4 Results and Discussion
### Single Circular Cylinder  
The values of the cylinder diameter, inlet flow velocity, Reynolds number, Prandtl number and Richardson number are presented in Table 1.  
Previous investigations by our group have also validated the following simulation data: core radius of \(\sigma_{V}=\sigma_{T}=0.001\) and \(M=300\) flat panels. The simulated cases for Ri = 0 and 0.168 (with buoyancy forces) were performed with 1000 time steps of magnitude \(\Delta t=0.05\). Both simulations used LES turbulence modeling and the time history of the drag and lift coefficients are shown in Figure 6. The vortex shedding mechanism can be clearly identified in the oscillations of the drag and lift coefficients. Figure 6a,b show that a numerical transient is over and the periodic steady state regime is reached (from \(t=10\) on, approximately); in consequence, the lift coefficient starts to present a variation between \(-1.25\) and \(1.25\), approximately, with a dimensionless frequency (Strouhal number, St) about twice the frequency of the drag coefficient, in accordance to the expected physics for the problem, i.e., the fluctuation of the drag coefficient has twice the frequency of the lift coefficient, since the drag curve fluctuates once for each of the upper and lower sheddings \begin{table}
\begin{tabular}{c c} \hline
**Description** & **Value** \\ \hline Diameter (D) & 1.0 \\ Inlet flow velocity (\(U_{\infty}\)) & 1.0 \\ Reynolds number (Re) & 100,000 \\ Prandtl number (Pr) & 0.71 \\ Richardson number (Ri) & 0 and 0.168 \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensionless input data necessary to simulate the single circular cylinder problem. of the vortical structures in the cylinder surface. The lift curve oscillates for each pair of counter-rotating vortical structures from the cylinder surface. The mean results for the drag and lift coefficients and the Strouhal number are shown in Table 2. The simulation indicated that when the buoyancy effects were taken into account, the drag coefficient reduced, followed by a slight reduction in the Strouhal number. These results also show good agreement with experimental data obtained by [PERSON] [43], which are \(C_{d}=1.20^{\pm 10\%}\) and \(\text{St}=0.19^{\pm 10\%}\) for \(Re=100,000\). Figure 5(b) also includes five points of interest, strategically marked as points P-T. These points are associated with time instants related to the vortex shedding mechanism and consequent pressure coefficient distribution over the cylinder surface (Figure 7). \begin{table}
\begin{tabular}{c c c c c} \hline
**Test Case** & **Ri** & \(\mathbf{C_{d}}\) & \(\mathbf{C_{I}}\) & **St** \\ \hline (A) & 0.0 & 1.23 & \(-0.027\) & 0.200 \\ (B) & 0.168 & 1.21 & \(-0.05\) & 0.199 \\ \hline \end{tabular}
\end{table} 
Table 2: Present numerical results for mean values of the drag, lift and Strouhal number. Figure 6: Temporal history of integrated aerodynamic loads: (a) Ri = 0 and (b) Ri = 0.168. Figure 7: Pressure coefficient on the cylinder surface related to points P–T. These points relate to the vortex shedding mechanism described by [PERSON] [25] too. As properly interpreted by [PERSON] et al. [42] and reproduced here, Point P refers to the time instant of the initial development stage of a clockwise vortical structure shedding on the upper side of the cylinder. Point Q indicates that the upper clockwise vortex structure continues growing and starts to attract the opposite shear layer placed on the lower side of the cylinder surface. Point R indicates that a counter-clockwise vortical structure is on its initial development stage on the lower side of the cylinder. As the next physical event, it also starts to grow, attracting the upper shear layer which is feeding the clockwise vortical structure. This event causes the detachment of the clockwise vortex structure. Point S has an analogous interpretation to that of Point Q.