# A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method
## 3 Results
### Flow Past the Train
#### 3.2.2 Pressure Loads on the Train
The base pressure of the 2-D train predicted by the present separated-flow model using 200 panels is \(-0.3\). The influences of different nose shapes on the aerodynamic pressure loads of the train are investigated in this paper. The comparison of the nose lengths for Nose 1, Nose 2, and Nose 3 is presented in Figure 11, for which the values are \(0.96\;D_{tr}\), \(1.28\;D_{tr}\), and \(1.6\;D_{tr}\), respectively. It can be concluded from Figure 11 that the peak values that appear just downstream of the stagnation position decrease with the increment of the nose length. The peak values just upstream do not have the same trend with the increase of the nose length, but the differences between the two peak values shrink as the nose length increases. One of the main features of the train is that it is running on the rails and is close to the ground while cruising. The presence of the ground must be considered when investigating the flow around the train. The method of images is employed in this case to model a solid ground at z = 0 by placing the mirror image of the 2-D train at the negative value of the height, which is 0.15 from the ground plane, as shown in Figure 12, where the number 1 refers to the 2-D train and the number 2 refers to the image. The method of images involves the analysis of the flow over multiple bodies, even if the original problem is for flow over a single solid body. In addition, as the solid bodies are symmetric at about z = 0 at the ground plane, the no-penetration condition is satisfied. A total of 200 panels were used in the simulation, including the ground effect. Unlike the symmetrical simulation above, the introduction of the ground plane leads to the flow around the train to be asymmetrical. According to the integral boundary-layer prediction, the separation positions with the presence of ground is not symmetric on the upper and lower surface of the train. They are about 176\({}^{\circ}\) on the upper surface and 186\({}^{\circ}\) on the lower surface in this case, as shown in Figure 12. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Position**} & \multicolumn{2}{c}{**100 Panels**} & \multicolumn{2}{c}{**200 Panels**} & \multicolumn{2}{c}{**300 Panels**} \\  & \(\mathbf{C_{P}}\) & **Position** & \(\mathbf{C_{P}}\) & **Position** & \(\mathbf{C_{P}}\) & **Position** \\ \hline \(1\) & \(-1.12\) & \(4.6^{\circ}\) & \(-1.19\) & \(5.2^{\circ}\) & \(-1.21\) & \(5.4^{\circ}\) \\ \(2\) & \(-0.41\) & \(171.8^{\circ}\) & \(-0.7\) & \(173.0^{\circ}\) & \(-0.6\) & \(173.4^{\circ}\) \\ \(3\) & \(-0.41\) & \(185^{\circ}\) & \(-0.7\) & \(185.5^{\circ}\) & \(-0.6\) & \(185.8^{\circ}\) \\ \(4\) & \(-1.12\) & \(352.6^{\circ}\) & \(-1.19\) & \(353.5^{\circ}\) & \(-1.21\) & \(353.9^{\circ}\) \\ Base pressure & \(-0.34\) & & \(-0.3\) & & \(-0.36\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of peak values and base pressure for trains with different panel numbers. Figure 10: Results from the iterative wake length procedure of the train. The asymmetry of the separation points leads to the difference in the wake panel lengths on the upper surface and the lower surface of the train. Thus, the interpolative procedure, as elaborated above, was conducted for the upper surface and the lower surface of the train, respectively. The determined \"correct\" lengths for the upper wake panel is 4 \(D_{tr}\) and for the lower wake panel is 9 \(D_{tr}\). For the asymmetric case, the pressure distribution on the upper and lower surface is different. The pressure distribution on the upper surface is similar to the symmetric case, but with a larger minimum pressure value in the head. On the lower surface, the pressure drops in the head and then remains almost constant until near the separation, where the pressure decreases slightly and then increases to meet the base pressure.