# Frequency Modulation and Erosion Performance of a Self-Resonating Jet
## 2 Research Method
### Numerical Simulation Setup  
The numerical simulation was also performed to verify the Gas-Spring Theory. The axisymmetric two-dimensional computation domain and boundary conditions are shown in Figure 8. The inflow velocity was adjusted appropriately until the time averaged inlet pressure agreed with the experimental inlet condition (\(P_{in}=1.5\) MPa). The constant outlet pressure was set to 1 atm, as the absolute pressure was adopted in the current simulation. It is assumed that inlet and outlet areas were in a pure liquid region and the volume fraction of water in these boundaries were set as 1 in the present study. Other boundaries were the same as the non-slip walls. Geometrical parameters were the same with the experimental nozzle. The time step was set as \(10^{-3}\) s to capture the periodical variation of the two-phase flow instead of the real transient evolution of the cavitating flow. According to the Gas-Spring Theory, the collective bubbles act as accumulators and were investigated in the present study. Although the collective bubbles consist of small bubbles, the variation of small bubbles and their interaction were not the aim of this study. Data acquisition took place at each time step. To obtain the multiple oscillating periods, the whole simulation time was 25 s.  
The standard computational grid was composed of orthogonal cells. A special form of stretching the mesh was applied in the main flow direction just after the chamber inlet, so the two-phase flow area was efficiently simulated. In the other direction, stretching was also applied close to the walls and the shear layer. Grid independence tests were performed, and the grid size was increased by 50% each time until no noticeable variance in the velocity profile was observed. In this study the investigation of the mesh's influence was performed by monitoring the axial velocity (normalized by the maximal velocity) within 10 s. The average streamwise velocity around the shear layer was also investigated, because the shear layer plays an important role in the oscillating mechanism. The monitoring lines are presented in Figure 9. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Suction Series** & **Pressure Drop** & **Exciting Frequency** \\ \hline
0.0 & 1.25 MPa & 0 Hz \\
1.0 & 1.50 MPa & 0.25 Hz \\
2.0 & 1.75 MPa & 0.50 Hz \\
2.5 & & 1.00 Hz \\
3.0 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experimental condition. Figure 8: 2D computational axisymmetric domain and boundary conditions. The results of grid convergence are presented in Figure 10. There was no obvious difference in the average axial velocity for the three mesh cases, while mesh 1 deviated from the other mesh cases in the velocity distribution around the shear layer, especially for the position near the impinging edge (lines 4-6). Due to no noticeable variance between mesh 2 and mesh 3 in the velocity profiles, mesh 2 was selected as final mesh for the present simulation. Mesh 2 contained 14,281 nodes with the minimum length of 0.066 mm streamwise and of 0.1 mm in the vertical direction. The model equations were solved by the finite-volume method. The time-dependent governing equations were discretized in both space and time domains with the SIMPLE algorithm. The first-order upwind scheme was used for spatial discretization. The unsteady first-order implicit formulation was implemented for the transient term. It was found that within 20 iterations in each time step, the RMS (root-mean-square) residuals satisfied the current convergence criteria, which was specified as at least a three orders of magnitude decline in the volume fraction of the vapor phase and a six-order decline in the mass conservation equation. In the present paper, the cavitating flow was modeled using the single fluid approach by treating the two liquid/vapor phases as a homogenous mixture.