## 2 Numerical Study
### Numerical Simulation Results of Pipeline Plugging Process  
During the plugging process, the motion of the in-pipe medium belongs to the problems of high Reynolds number. Assuming that the fluid in the pipeline is incompressible and no heat transfer occurs, the mass conservation equation and momentum conservation equation of the PIPR can be expressed as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho V_{x})}{\partial x}+\frac{ \partial(\rho V_{y})}{\partial y}+\frac{\partial(\rho V_{z})}{\partial z}=0 \tag{1}\]  
\[\left\{ \begin{array}{l}\frac{\partial(\rho V_{x})}{\partial t}+\frac{ \partial(\rho V_{x}V_{z})}{\partial x}+\frac{\partial(\rho V_{x}V_{y})}{ \partial y}+\frac{\partial(\rho V_{x}V_{z})}{\partial z}\\ =\frac{\partial x}{\partial x}(\mu\frac{\partial V_{x}}{\partial x})+\frac{ \partial y}{\partial y}(\mu\frac{\partial V_{y}}{\partial y})+\frac{\partial (\rho V_{x}V_{z})}{\partial z}+[-\frac{\partial(\rho V_{x}^{\prime 2})}{\partial x}-\frac{ \partial(\rho V_{x}^{\prime}V_{y}^{\prime})}{\partial y}-\frac{\partial(\rho V _{x}^{\prime}V_{z}^{\prime})}{\partial z}]-\frac{\partial p}{\partial x}\\ \frac{\partial(\rho V_{x})}{\partial t}+\frac{\partial(\rho V_{y}V_{z})}{ \partial x}+\frac{\partial(\rho V_{y}V_{z})}{\partial y}+\frac{\partial(\rho V _{y}V_{z})}{\partial z}\\ =\frac{\partial y}{\partial x}(\mu\frac{\partial V_{y}}{\partial x})+\frac{ \partial y}{\partial y}(\mu\frac{\partial V_{y}}{\partial y})+\frac{\partial (\rho V_{y}V_{z})}{\partial z}+[-\frac{\partial(\rho V_{x}^{\prime}V_{y}^{ \prime})}{\partial x}-\frac{\partial(\rho V_{y}^{\prime 2})}{\partial y}-\frac{ \partial(\rho V_{y}^{\prime 2})}{\partial z}]-\frac{\partial p}{\partial y}\\ \frac{\partial(\rho V_{z})}{\partial t}+\frac{\partial(\rho V_{z}V_{z})}{ \partial x}+\frac{\partial(\rho V_{z}V_{y})}{\partial z}+\frac{\partial(\rho V _{z}V_{z})}{\partial z}+\frac{\partial(\rho V_{z}V_{z})}{\partial z}\\ =\frac{\partial x}{\partial x}(\mu\frac{\partial V_{z}}{\partial x})+\frac{ \partial}{\partial y}(\mu\frac{\partial V_{y}}{\partial y})+\frac{\partial (\rho V_{z})}{\partial z}+[-\frac{\partial(\rho V_{y}^{\prime}V_{z}^{\prime}) }{\partial x}-\frac{\partial(\rho V_{y}^{\prime}V_{z}^{\prime})}{\partial y}- \frac{\partial(\rho V_{z}^{\prime 2})}{\partial z}]-\frac{\partial p}{\partial z}\end{array} \right. \tag{2}\]  
Therefore, the simple algorithm was chosen for calculation. At the same time, due to the high Reynolds number in the flow field, large vorticity vibration is generated during the plugging process, so the model is set as the standard \(k\)-\(\varepsilon\) model, whose turbulent kinetic energy (\(k\)) equation (shown in Equation (3)) and turbulent dissipation rate (\(\varepsilon\)) equation (shown in Equation (4)) can be expressed as follows:  
\[\begin{array}{l}\frac{\partial(\mu k)}{\partial t}+\frac{\partial(\rho kV_{x} )}{\partial x}+\frac{\partial(\rho kV_{y})}{\partial y}+\frac{\partial(\rho kV _{z})}{\partial z}\\ =\frac{\partial}{\partial x}\left[\left(\mu+\frac{\mu_{t}}{\alpha_{t}}\right) \frac{\partial k}{\partial x}\right]+\frac{\partial}{\partial y}\left[\left( \mu+\frac{\mu_{t}}{\alpha_{t}}\right)\frac{\partial k}{\partial y}\right]+ \frac{\partial}{\partial z}\left[\left(\mu+\frac{\mu_{t}}{\alpha_{t}}\right) \frac{\partial k}{\partial z}\right]+G_{k}+G_{b}-\rho\varepsilon\\ \frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial(\rho V_{x})}{ \partial x}+\frac{\partial(\rho V_{y})}{\partial y}+\frac{\Because the PIPR is operated under the action of in-pipe pressure difference, its motion law conforms to [PERSON]'s second law.  
\[m\frac{dv}{dt}=(P_{1}-P_{2})A_{d}-F_{g}-F_{f} \tag{5}\]  
Among them, \(v\) is the motion speed of PIPR, \(P_{1}\) is the upstream pressure of PIPR, \(P_{2}\) is the downstream pressure of PIPR, \(A_{d}\) is the equivalent cross-sectional area of PIPR, \(F_{g}\) is the component of the self-weight of the PIPR in the velocity direction, \(F_{f}\) is the friction force between the PIPR and the pipeline inner wall.