## 2 Numerical Study
### Numerical Simulation Results of Pipeline Plugging Process
Boundary layer separation and vortex have occurred at the edge of the PIPR's tail, which can form a clear reflux zone with a certain fluctuation in the flow direction. Due to severe energy loss in the reflux zone, the flow velocity significantly decreases. The flow velocity distribution after the plugging process is shown in Figure 6b. The fluid inside the entire pipeline is almost stagnant, indicating that the upstream and downstream of the PIPR are isolated. At this time, the plugging operation is completed. Figure 7 shows the velocity distribution of fluid particles on the central surface of the pipeline. It is clearly shown that the highest velocity of in-pipe fluid particles exists in the gap between the PIPR and pipeline wall. Comparing Figure 7a,b shows that with the increase in plugging progress, the velocity of fluid particles in the gap becomes higher. The velocity value after the plugging process is much higher than that before the plugging process. As the plugging operation progresses, the pressure difference before and after the PIPR continues to increase, and the fluid passing through the slit also gains more kinetic energy. Figure 5: The fluid pressure distribution on the center surface of the pipeline. (**a**) Before the plugging process; (**b**) After the plugging process. The force acting on the PIPR is particularly important for the structural design and control of the PIPR, which directly determines the movement status of the PIPR. Therefore, the relationship between the force of the PIPR, plugging process, and pipeline operating parameters is analyzed, as shown in Figure 8. Figure 8a shows the relationship between the force of the PIPR and the plugging process and inlet pressure. From the initial state to the plugging state of 80%, the in-pipe upstream pressure increases, which can increase the force on the PIPR continuously. Until the plugging process reaches 80%, the force of PIPR begins to decrease due to the in-pipe pressure distribution. The pressure difference between the upstream and downstream is relatively large, and negative pressure may occur in some areas. Therefore, the pressure generated on the PIPR can be partially offset. Figure 8b shows the relationship between inlet velocity and the plugging process, as well  
Figure 6: The flow velocity distribution on the center surface of the pipeline. (**a**) Before the plugging process; (**b**) After the plugging process. Figure 7: The velocity distribution of fluid particles on the center surface of the pipeline. (**a**) Before the plugging process; (**b**) After the plugging process. as the force of the PIPR. It can be seen that the plugging process and inlet velocity have a significant impact on the force fluctuation of the PIPR. The smaller the inlet velocity, the smaller the impact force on the PIPR. When the plugging process reaches 50%, the flow velocity directly affects the in-pipe pressure distribution. In summary, all three variables involved have a significant influence on the force of the PIPR, and the effects of the plugging process and inlet velocity are more pronounced. The PIPR with the original stepped end face bears a high level of stress during the later stage of the plugging process. In addition, significant pressure fluctuations and vortex-induced vibration occur, mainly caused by the increased turbulent flow field excitation, and the plugging operation is not stable. Therefore, the PIPR's end face with a certain degree of radian is proposed to reduce the disturbance and pressure gradient of the in-pipe flow field.