# A Study of Cavitation Erosion in Artificial Submerged Water Jets
## 2 Numerical Simulation
### Mesh Configuration and Boundary Conditions in Flow Domain  
As shown in Figure 1a, \(d\) is the diameters of the inner nozzle; \(L_{1}\), \(L_{2}\), and \(\delta\) denote the throat length, divergent segment length, and divergent angle of the inner nozzle; \(\beta\) is the convergent angle of the outer nozzle; and \(D\) stands for the inner clearance between the inner and outer nozzle. The standoff distance is \(S\), which is the same for the inner and external nozzle [34]. To fully capture the cavitation cloud distribution pattern on the workpiece wall, the workpiece diameter \(\hat{\rho}_{w}\) should be large enough relative to the nozzle diameter. The dimensionless standoff distance \(S_{D}\) is the ratio of standoff distance to the diameter of the inner nozzle. The nozzle and workpiece dimensions are listed in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(d\) & 1 mm \\ \(L_{1}\) & 3 mm \\ \(L_{2}\) & 3 mm \\ \(\delta\) & 0, 30\({}^{\circ}\) \\ \(\beta\) & 0, 30\({}^{\circ}\) \\ \(D\) & 3 mm \\ \(S\) & 20, 30, 40, 50 mm \\ \(\hat{\rho}_{w}\) & 80 mm \\ \(S_{D}\) & 20, 30, 40, 50 \\ \hline \hline \end{tabular} \The computational domain and boundary conditions are illustrated in Figure 1b. The model created using SOLIDWORKS is imported into ANSYS ICEM to generate high-quality hexahedral structured meshes, as illustrated in Figure 1c. To ensure an accurate representation of the boundary layer near the inner wall of the nozzle, it is recommended to position the center of the first computational cell in the logarithmic law region, where the dimensionless wall distance, denoted as \(y+\), should fall within the range of \(20\leq y+\leq 200\)[35]. In the Reynolds number, \(Re=UL/v^{\bullet}\), \(U\) represents the flow velocity at the nozzle throat, \(L\) is the thickness of the first layer of meshes on the inner wall of the nozzle throat, and \(v^{\bullet}\) is the kinematic viscosity. A flow velocity of 150 m/s is selected as the flow velocity at the nozzle throat. The thickness of the first mesh layer on the inner wall of the nozzle throat is set to 0.05 mm, and the mesh size increases outward by a factor of 1.1, \(y+\) resulting in a value of approximately 30.  
The non-slip wall condition is applied to the surface of the nozzle. The artificial submerged cavitation water jet consists of two channels: a high-pressure channel and a low-pressure channel. The inlet conditions for both channels are set as pressure inlet. As the outer flow field surrounding the water jet corresponds to an air domain, the inlet of the outer flow field is specified as a pressure inlet with a gauge pressure of 0 MPa, while the inlet air volume fraction is set to 1. The outlet boundary condition of the flow field is defined as the pressure outlet with a pressure set to 0 MPa. The calculation of velocity and pressure is iteratively decoupled using the pressure-based SIMPLE (Semi-Implicit Pressure Linked Equation Method) solver. The simulation employs a physics time step of \(\Delta T=1*10^{-6}\), with 20 internal iterations conducted within each physics time step. Moreover, the momentum equation's space discretization adopts the bounded central difference scheme, while the time discretization employs the bounded second-order implicit scheme. To capture the periodicity of cavitation in an artificially submerged cavitation water jet, a total physics time exceeding 10 ms is allocated for each mesh level.  
Figure 1: (a) Schematic diagram of the geometric size of the nozzle structure; (b) layout and boundary conditions of simulation domain; (c) hexahedral-structured meshes.