## 3 Structural Optimization of PIPR's End Face  
### The Improved Model of PIPR  
Based on the numerical simulation in Section 2, the PIPR's end face has a great influence on the vibration characteristics during the plugging process. Therefore, an improved model of PIPR is proposed for structural optimization of PIPR's end face. The schematic diagram of the improved model is shown in Figure 16. For practical engineering, PIPR usually works together with other functional modules [25], such as pipeline inspection gauge (PIG), anchoring module, etc. They are connected to each other by connecting rods  
Figure 14: The velocity distribution of the flow field around the PIPR with a parabolic-shaped end face. (**a**) Before the plugging process; (**b**) After the plugging process.  
Figure 15: The velocity distribution of fluid particles of PIPR with parabolic-shaped end face. (**a**) Before the plugging process; (**b**) After the plugging process.  
and ball bearings, resulting in a connection groove in the center of PIPR's end face, as shown in Figure 16. The shape parameter \(S_{p}\) is proposed for measuring the type and curvature of curved surfaces, and its value is related to the eccentricity \(\mathbf{e}\) of the conic curve. The shape of the end face is elliptical when \(0.01\leq S_{p}<0.5\). If \(S_{p}=0.5\), it is a parabolic-shaped end face. When \(0.5<S_{p}\leq 0.99\), it is an end face with the shape of a double-leaf hyperbolic.  
The parameters and boundary conditions of the flow field model remain unchanged. The pressure and velocity distribution of the original plugging state are analyzed. Figure 17 shows the cloud map of fluid pressure distribution. It can be seen that the pressure of the improved model has decreased due to the improved end face. Local high pressure has occurred at the connection groove position, and the fluid pressure changes violently in the narrow gap between the PIPR and pipeline inner wall, which causes an impact on the pipeline wall. Overall, the pressure distribution of the flow field around the PIPR from upstream to downstream is clearly localized and radially symmetrical. The low-pressure area is mainly concentrated in the flow field near the tail of the PIPR, and the generation of backflow and vortices results in a large amount of energy loss.  
Figure 18 shows the cloud map of in-pipe flow velocity distribution. It can be seen that the upstream flow field of the PIPR is stable, and the flow velocity is obviously low with the improved model. The flow velocity near the connection groove of the front end face is almost zero due to the obstruction of the connection groove on the fluid in this area, which results in a large amount of kinetic energy loss. Similar to the original structure, when the fluid passes through the narrow gap between the PIPR and pipeline inner wall, the flow velocity significantly increases due to the small area and high pressure. After passing through the narrow gap due to the opening of the flow channel, the flow velocity shows a deceleration trend. In addition, there is a clear reflux phenomenon at the tail of the PIPR, resulting in a lower velocity and a stagnation point area. And it can cause resistance to the forward movement of the PIPR. At the same time, the flow velocity contour at the tail of the PIPR is conical, which is similar to the contour of the rear-end face.  
Figure 16: The schematic diagram of the improved model of PIPR.  
Figure 17: The fluid pressure distribution of the original plugging state of the improved model.