# 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 present separated flow model based on the panel method is used to predict the pressure distribution on the 2-D generic train at the Reynolds number of \(3\times 10^{5}\). The calculation procedure follows the flowchart shown in Figure 1. As explained above, the separated flow model for simulating the viscous flow has two key parameters: the separation locations at which the vorticity panels are attached, and the \"correct\" wake panels with which the wake is modelled. In the former key parameter, separation locations are predicted by the boundary-layer calculation procedure as described in Section 2.2. The latter key parameter, the \"correct\" wake panel lengths, are calculated according to the interpolating procedure based on the criterion of zero tangential velocity increment between two points on the wake panels. The 2-D generic train has a blunt nose in both the front and the rear part of it, and as the air flows around the train, it tends to separate when reaching the rear part of the train. Figure 8 shows the pressure coefficient along the 2-D train surface using the separated-flow  
Figure 7: Generic train model. model. The separation positions predicted by the integral boundary-layer method are about 176\({}^{\circ}\) on the upper surface and 184\({}^{\circ}\) on the lower surface of the train, as shown in Figure 9.  
Like the symmetrical simulation of the cylinder flow above, the generic train is geometrically symmetric and the lengths of the wake panels should be the same on the upper surface and lower surface of the train. The distributions of the tangential velocity increment on the wake panels of the train with wake lengths vary from 0.75 \(D_{tr}\) to 4 \(D_{tr}\) are investigated, as shown in Figure 8. For all wake lengths, the tangential velocity increases along the wake panels. The increments in tangential velocity for the wake panels decreases with the increasing of the wake panel lengths; in other words, longer wake panels lead to smaller tangential velocity differences. The \"correct\" wake panel lengths were determined by the interpolative procedure elaborated in Section 2.3 and validated in Section 3.1. The iteration process shown in Figure 10 starts with two assumed wake lengths, 0.75 \(D_{tr}\) and 2 \(D_{tr}\), and it is converged after six iterations to 3 \(D_{tr}\), which is used as the \"correct\" wake length in evaluating the pressure distribution around the 2-D train. To validate the grid independency of the proposed model, three different panel densities: 100 panels, 200 panels, and 400 panels were tested, as shown in Figure 9. It can be seen that the distribution and value of the pressure coefficient almost coincide for all three cases. The peak values and base pressure are compared in Table 1. The discrepancies of peak values between different panel densities are mainly caused by the locations of the control points and it can be concluded that the differences between the results of 100 panels and 200 panels are small. Thus, the results of the proposed model are independent of the panel densities. Figure 8: Distributions of tangential velocity increment on the train for various wake lengths. Figure 9: Pressure distribution on the train with different panel densities without the ground. The pressure coefficient is distributed almost symmetrically on the upper surface and the lower surface of the 2-D train. When the flow encounters the train, the stagnation point appears where the pressure coefficient is the maximum. As the flow passes the train head, the pressure decreases rapidly to the minimum, and then increases until the flow reaches the body of the train. The pressure along the train body remains constant and then drops where the body transits to the tail. When separation occurs, the pressure rises and remains constant in between two separation locations on the upper and lower surface, where the pressure is referred to as the base pressure.