## 2 Physical Model and Control Equation  
### Physical Model and Simulation Scheme  
The nozzle relies on the need for a high-pressure water pump when working. With the high-pressure water pump in action, water flows into the nozzle through the high-pressure hose and is then ejected from the nozzle. The jet expelled by the front nozzle provides the dredging power; the debris is discharged from the pipe through the backflow,achieving the pipe dredging function. Based on the calculation formula for water jet nozzle propulsion [24], the calculation results show that the propulsion of all schemes is positive, which meets the working requirements.  
\[F_{p}=n_{1}\times 1.5d_{1}^{2}pcos\theta_{1} \tag{1}\]  
\[F_{\rm f}=n_{2}\times 1.5d_{2}^{2}pcos\theta_{2} \tag{2}\]  
\[F=F_{\rm f}-F_{p} \tag{3}\]  
where \(F_{p}\) is rock-breaking pressure in N; \(F_{\rm f}\) is recoil in N; \(n_{1}\) is the number of front nozzle holes; \(n_{2}\) is the number of rear nozzle holes; \(d_{1}\) is front nozzle diameter in mm; \(d_{2}\) is back nozzle hole diameter in mm; \(p\) is water pressure in MPa; \(\theta_{1}\) is the inclination angle of the front nozzle; \(\theta_{2}\) is the post-spray hole inclination angle; and \(F\) is the driving force in N.  
The outer diameter of the nozzle is 30 mm, and the inner diameter is 20 mm. The nozzle is arranged as one front nozzle, and three rear nozzles are arranged around each other at 120\({}^{\circ}\) intervals. Since this paper mainly studies the flow field in the pipeline, the nozzle modeling is simplified. Compared with the traditional nozzle with only the front hole, due to the existence of the rear nozzle hole, the rear jet can provide the reverse thrust for the whole nozzle during the operation process, so that the whole nozzle can rely on the reverse thrust of the rear jet flow to advance along the path without external force, and the rear jet can improve the fluid discharge efficiency. The nozzle aperture is 2 mm, and the control group is 3.5 mm. The nozzle length is 10 mm. The schematic diagram of the nozzle is shown in Figure 1.  
In this paper, SolidWorks was utilized to model the flow field and then imported into Fluent for numerical simulation. A comparison was made between a nozzle with only a front nozzle and a nozzle with a rear nozzle, which can provide self-propelled thrust. The interaction between the front and rear nozzles can create a vortex area in the pipeline. The water vortex, with its ability to wrap gravel, influences the characteristics of the slag discharge and the injection of the nozzle jet. The water jet ejected by the front and rear nozzles generates a balanced force, enhancing the stability of the nozzle operation.  
To explore the influence of nozzles with different axis angles and apertures on the working flow field of the nozzle in the pipeline, a numerical simulation scheme was implemented. The arrangement consisted of one front nozzle and three rear nozzles, with a 120\({}^{\circ}\) interval between the rear nozzles. The study examined the flow field differences under the same initial velocity condition with varying axis angles. The research is mainly aimed at the rear nozzle inclination angle of the four schemes, 30\({}^{\circ}\), 35\({}^{\circ}\), and 40\({}^{\circ}\), for the rear nozzle axis angles, with an increased diameter of 3.5 mm as a control group for research purposes. The specific scheme design is detailed in Table 1.  
Figure 1: (a) Nozzle parameter engineering drawing; (b) Nozzle modeling 3D schematic diagram.