# A Separated-Flow Model for 2-D Viscous Flows around Bluff Bodies Using the Panel Method
## 4 Conclusions  
A 2-D separated-flow model based on the constant source/vortex panel method is proposed in this paper. The proposed model includes constant source panels to model the obstacle in the flow field and constant vortex panels to model the wake behind the body. To simulate the flow separation, the integral boundary-layer method is adapted to calculate the separation locations and an interpolative calculation is adapted to calculate the \"correct\" wake length for modelling the wake.  
The proposed method is validated by comparing the calculated pressure distribution on a 2-D circular cylinder with the experimental data. The predicted separation location agrees well with the experiment, which proves the feasibility of incorporating the integral boundary-layer approximation in the separated-flow model. Additionally, the calculated pressure loads distributed on the train agree with the experimental results. This proves that the proposed model is credible in simulating flows around bluff bodies. Moreover, simulations with different panel densities are performed and the grid independency of the proposed method is verified by the results.  
The proposed method is also applied to predict the pressure distribution along a 2-D generic train surface. According to the simulation results, the pressure changes dramatically at the front of the train. The flow then separates at the rear part of the train, with symmetrical separation locations in the absence of the ground configuration. The base pressure remains constant at the rear part of the train. Furthermore, the grid independency is validated in the 2-D train case by comparing the results of three different panel densities. In addition, the presence of the ground changes the pressure distribution pattern on the lower surface of the train: it weakens the pressure peak occurring around in the head and tail and increases the pressure on the underbody of the train. The stagnation position at the head of the train moves downward with the presence of the ground.  
As the first stage to develop the numerical model for trains in tunnels, the presented method is capable of predicting the pressure distribution along arbitrary bluff bodies such as trains with flow separations. The most outstanding feature of the proposed method is that it saves tremendous computational time and resources comparing with high fidelity CFD simulations in pressure prediction. As a meshless method, it also saves the time and energy to construct a computational mesh. However, this method still needs improvements and modifications, for example, the effect of the compressibility, in order to be practised on other train aerodynamic applications such as the modelling of trains in tunnels.  
Conceptualization, H.H.; funding acquisition, J.X. and Y.J.; investigation, R.L.; supervision, H.H.; writing--original draft, R.L.; writing--review and editing, D.S. and J.N. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant numbers 11962014 and 51965033.  
Not applicable.  
Figure 13: Comparison of pressure distributions on the trains with different ground clearance.