## 2 Numerical Simulation of Submerged Cavitating Jet
### Numerical Model
The turbulent kinetic energy \(k\) equation is as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial\left(\rho ku_{j}\right)}{ \partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{l}} {\sigma_{k}}\right)\frac{\partial k}{\partial x_{j}}\right]+G_{k}+G_{b}-\rho \varepsilon-Y_{M}+S_{k} \tag{4}\]  
The dissipation rate \(\varepsilon\) equation is given below:  
\[\frac{\partial(\rho\varepsilon)}{\partial t}+\frac{\partial\left(\rho eu_{j} \right)}{\partial x_{j}}=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac {\mu_{l}}{\sigma_{\varepsilon}}\right)\frac{\partial\varepsilon}{\partial x_{ j}}\right]+C_{1\varepsilon}\frac{\varepsilon}{k}C_{3\varepsilon}G_{b}+\rho C_{1}S \varepsilon-C_{2}\rho\frac{\varepsilon^{2}}{k+\sqrt{v\varepsilon}}+S_{\varepsilon} \tag{5}\]  
Figure 2 shows the numerical model of a submerged cavitating jet. The simulation area included the internal flow field of the nozzle and the external water environment domain. Since the nozzle itself is a rotational structure, a two-dimensional axisymmetric model was used to save the calculation time under the premise of ensuring the accuracy of the simulation. The pre-processing software Mesh 19.0 was used to mesh the model. The pressure-inlet was selected, and the gauge total pressure was established at 30 MPa. By utilizing the Bernoulli equation (Equation (6)), the Reynolds number equation (Equation (7)), and the turbulence intensity calculation equation (Equation (8)), the turbulence intensity at the inlet can be determined as 5%. \[P_{G}=P_{S}+\frac{1}{2}\rho LI^{2} \tag{6}\]  
\[R_{\varepsilon}=\frac{\rho LI}{\mu} \tag{7}\]  
\[I\cong 0.16(R_{\varepsilon})^{-0.125} \tag{8}\]  
In the equation, \(P_{G}\) represents the total pressure at the inlet, \(P_{S}\) represents the static pressure, \(\rho\) represents the density of water, \(U\) represents the velocity at the inlet, \(L\) represents the diameter of the inlet, and \(\mu\) is the dynamic viscosity coefficient. The outlet was assigned to the pressure-outlet, with the gauge pressure at 0.1 MPa. The pressure difference was measured according to the principles of pressure gauge measurement, implying that the actual pressure is the atmospheric pressure exerted on the gauge pressure. Consequently, the actual pressure at the outlet corresponds to the pressure when the water depth is ten meters. To expedite convergence, the pressure correction method employed the SIMPLEC algorithm. The discrete format adopted the second-orderupwind scheme, hybrid initialization was adopted, the number of iterations was set to 500, and residual accuracy was maintained at \(10^{-5}\). To ensure that the grid quality does not affect the simulation results, a grid dependence validation is necessary. Three different grid element sizes were selected, as shown in Table 2. The pressure drop was calculated based on the average total pressure at the inlet and outlet. By comparing the pressure drops obtained from the second and third grid sizes with that from the first grid size, it was observed that when the grid size reached 0.1 mm, further reducing the grid size resulted in an error of less than 1%. This indicates that decreasing the grid size further has a minimal impact on the simulation results. Therefore, a grid size of 0.1 mm can be used for meshing the model.