## 2 Methods
### Simulation Method  
In this paper, the CFD simulation software CONVERGE v3.0 was used and the VOF-Spray One-Way Coupling method was used for simulation calculation. The methodology entailed initially simulating the flow within the injector utilizing the VOF approach without accounting for the cavitation impact. Subsequently, the spray characteristics at the spray orifice outlet were ascertained, and these data were then utilized as the spray boundary conditions for the spray simulation calculation within the spray field.  
The injector used in the simulation calculation process of this paper was a six-hole nozzle with a hole diameter of 0.3 mm and a hole length of 1 mm, and the geometric model is shown in Figure 3. To simplify the calculation, the 1/6 model was used in the VOF calculation, and the specific structural parameters of the nozzle holes are shown in Table 2. A 4 mm base grid size was used for VOF calculations, with a 3-level scale at the nozzle, and an adaptive mesh refinement (AMR) depending on the velocity gradient and the temperature gradient. The turbulence model adopted was the LES model, and the mesh is shown in Figure 4, with the XZ plane as the long-axis plane of the elliptic orifice, and the YZ plane as the short-axis plane of the elliptic orifice.  
\begin{table}
\begin{tabular}{c c} \hline Parameters & Value \\ \hline Density (kg/m\({}^{3}\)) & 848 \\ Kinetic viscosity (mm\({}^{2}\)/s) & 3.6 \\ Surface tension (N/m) & 0.028 \\ \hline \end{tabular}
\end{table}
Table 1: Physical parameters of diesel fuel.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Nozzles** & **Major Axis [a]** & **Minor Axis [b]** & **Cross-Sectional** & **Office Length** & **Aspect Ratio** \\  & **(mm)** & **(mm)** & **Area (mm\({}^{2}\))** & **(mm)** & **Aspect Ratio** \\ \hline Circular & 0.3 & 0.3 & 0.071 & 1 & 1 \\ Elliptical-1.25 & 0.335 & 0.268 & 0.071 & 1 & 1.25 \\ Elliptical-1.5 & 0.367 & 0.245 & 0.071 & 1 & 1.5 \\ \hline \end{tabular}
\end{table}
Table 2: The parameters of different nozzles.  
Figure 2: Constant Volume Chamber.  
In order to select the most suitable base grid size, three base grid sizes of 1.6 mm, 1.8 mm, and 2.0 mm were used for calculation, with STP selected for validation. The boundary conditions were consistent with the experiments and the results are shown in Figure 7. The results showed that the maximum difference between the STP of base grid size 2.0 mm and that of base grid size 1.8 mm was 12.03%, and there was no significant difference between the STP of base grid size 1.8 mm and that of base grid size 1.6 mm, from which it can be seen that further reduction in the base grid size had less effect on the results, so a 1.8 mm base grid size was selected for the calculation. Furthermore, a 3-level scale should be employed for the spray concentration area in the vicinity of the nozzle axis. The mesh is illustrated in Figure 8.  
To verify the reasonableness of the model selection, the simulation value of the STP was compared with the experimental value, and the experimental value of spray tip penetration was obtained from the ensemble average of 6 tests. The results are shown in Figure 9. The average error between the simulation value and the experimental value was 4.24%, which was less than 5%, and it could be concluded that the model selection was reasonable. Simulation calculation working conditions are shown in Table 4.  
Figure 5: One-dimensional simulation model of injector.  
Figure 6: Inlet pressure curve.  
Figure 8: Mesh of spray calculation.  
Figure 7: Spray tip penetration (STP) of different base grid sizes.  
Figure 9: STP of experiment and simulation.