## 2 Numerical Study
### Numerical Simulation Results of Pipeline Plugging Process
The component of the self-weight of the PIPR in the velocity direction is:  
\[F_{g}=mg\sin\gamma \tag{6}\]  
Among them, \(m\) is the mass of PIPR, \(g\) is the gravity acceleration, \(\gamma\) is the angle between the pipeline and the horizontal line. The friction force \(F_{f}\) is caused by the interaction between the cup and the pipeline inner wall, which is an elastic deformation force. The friction force \(F_{f}\) of the PIPR is obtained according to the matching relationship between the cup and pipeline inner wall. \[\left\{ \begin{array}{l}F_{f}=2\pi r_{p}L_{pw}\mu_{f}N_{f}\\ N_{f}=\frac{E_{i}\delta_{p}}{r_{p}(1-v_{i})}\end{array} \right. \tag{7}\]  
Among them, \(r_{p}\) is the radius of PIPR, \(L_{pw}\) is the contact length between the PIPR and pipeline wall, \(\mu_{f}\) is the friction coefficient between the PIPR and pipeline wall, \(N_{f}\) is the pressure of the contact surface, \(E_{i}\) is the elastic modulus of the cupola material, \(\delta_{p}\) is the change value of the cupola in the direction of the radius, and \(\varepsilon_{i}\) is the Poisson ratio of the cupola material. By substituting Equations (6) and (7) into Equation (5), the force model of the PIPR during the plugging process can be expressed as:  
\[m\frac{dv}{dt}=(P_{1}-P_{2})A-mg\sin\gamma-2\pi r_{p}L_{pw}\mu_{f}\frac{E_{i} \delta_{p}}{r_{p}(1-\varepsilon_{i})} \tag{8}\]  
Without considering the energy input outside the PIPR, the axial motion model of the PIPR can be expressed as follows:  
\[\rho_{p}\frac{\partial v_{z}}{\partial t}+\rho_{p}v_{z}\frac{\partial v_{z}}{ \partial z}+\rho_{p}v_{r}\frac{\partial v_{z}}{\partial r}=f_{pz}+\frac{ \partial\sigma_{z}}{\partial z}+\frac{1}{r}\frac{\partial(r\tau_{rz})}{ \partial r} \tag{9}\]  
The radial motion model of the PIPR is as follows:  
\[\rho_{p}\frac{\partial v_{r}}{\partial t}+\rho_{p}v_{z}\frac{\partial v_{z}}{ \partial z}+\rho_{p}v_{r}\frac{\partial v_{r}}{\partial r}=f_{pr}+\frac{ \partial\tau_{rz}}{\partial z}+\frac{1}{r}\frac{\partial(r\sigma_{r})}{ \partial r}-\frac{\sigma_{\theta}}{r} \tag{10}\]  
Among them, \(\rho_{p}\) is the density of the PIPR, \(v_{z}\) is the axial velocity of the PIPR, \(v_{r}\) is the radial velocity of the PIPR, \(f_{prz}\) is the axial component of the volume force of the PIPR, \(f_{pr}\) is the radial component of the volume force of the PIPR, \(\sigma_{r}\) is the radial stress of the PIPR, \(\sigma_{\theta}\) is the circumferential stress of the PIPR, and \(\tau_{rz}\) is the shear stress of the PIPR. The numerical simulation of the traditional PIPR with the footstep-shaped end face is performed. In order to observe the simulation results of the in-pipe obstructed flow field more intuitively, the center surface of the pipeline is taken as the observation surface. Figure 4(a) shows the fluid pressure on the center surface of the pipeline before the plugging operation. It can be seen that the pressure downstream of the flow field shows a clear regional distribution and a decreasing trend along the flow direction. This pressure distribution is mainly caused by the external structure of the PIPR. Figure 4(b) shows the distribution of fluid pressure on the center surface of the pipeline when the plugging process reaches99%. It can be seen that a significant pressure difference has occurred between the upstream and downstream. The continuous accumulation of high-pressure fluid in the upstream leads to an increase in in-pipe pressure, which can cause large shock vibrations on the PIPR. The velocity distribution of the flow field around the PIPR is shown in Figure 6. Through the plugging operation, the velocity values of in-pipe fluid particles vary significantly. As shown in Figure 6a, before the plugging process starts, there is a local area on the front-end face of the PIPR with a flow velocity close to zero, called the stagnation zone. The blockage of the PIPR can lead to the dissipation of fluid kinetic energy. At the same time, the fluid also causes a certain impact on the PIPR.