# Effect of Fuel Mass Flow at the End of Injection on Cavitation and Gas Ingestion in the Nozzle
## 3 Mathematical Model
### Computational Domain and Boundary Condition  
According to the geometry shape of the experiment nozzle, a geometric model of the nozzle was gotten to guarantee the simulation results, as shown in Figure 3. The experimental conditions and geometry dimensions of the experimental nozzle were the same as those of the nozzle physical model, including sac shape, orifice shape, injection condition, back pressure, and fuel physical parameters. In this model, the inlet boundary extends to the middle of the sealing surface because the needle valve was stationary. The fuel velocity and direction changed sharply when it passed through the orifice entrance during the injection process, resulting in wear of the orifice inlet. Therefore, a small rounded corner was considered at the entrance of the orifice, and its radius was set as 0.04 mm. Considering the existence of gas ingestion and super-cavitation (super-cavitation occurs at very low pressures and/or high velocities where a large and stable vapor cavity develops; when system pressure is decreased or when flow velocity is increased a small cavity will grow, and a large single steady vapor-filled supercavity will develop [44,45]) at the outlet of the orifice during the EOI, the iterative error of the numerical calculation was reduced, and the accuracy of the calculation results was improved. A part of the external environment was considered during the modeling process, whose boundary was set as a pressure boundary. To ensure the correctness and rationality of the results, it was necessary to verify the independence of the nozzle physical model. According to the previous report [46], the size of the local encrypted mesh of the sac and sac entrance was set to 0.35 mm, the grid size of the spray area at the external boundary was set from 2 \(\upmu\)m to 20 \(\upmu\)m, and the other grid size was set to 2 \(\upmu\)m. The boundary layer, cavitation region, and non-slip condition considered, the block structure method was adopted to generate hexahedral mesh by ICEM software, and the number of meshes was about 4.53 million. Based on the above grid size and the changed basic size of the grid to verify the independence of the grid, we got the following result, as shown in Figure 4.  
When the number of cells is 4.53 million and 7.9 million, the relative difference in air volume is 0.192%. Therefore, using the above grid size is acceptable. Other thermophysical parameters of fluids are shown in Table 1.  
Figure 3: Geometry model of the nozzle. Since the fact that the needle valve opening and closing were operated periodically in the working process of the actual nozzle, it was relatively difficult to realize the motion of the needle valve with OpenFOAM. Therefore, the needle valve movement model was not considered in the numerical model of the nozzle. To control the change of the mass flux at the nozzle inlet, the corresponding fuel mass flux was calculated according to the movement law of the needle valve, and the dynamic mass flow inlet boundary was used instead of the needle valve movement. According to the experimental statistical results, the sac and 3/4 orifice were filled with liquid fuel when the model was initialized. While the mass flux at the inlet of the nozzle changed according to the change law at the EOI, the fuel cavitation flow in the nozzle was calculated. The dynamic mass flow rate is calculated as Equation (23):  
\[q=\sqrt{2}\varphi\cdot A(x)\cdot(\Delta p\cdot\rho)^{0.5} \tag{23}\]  
where \(A(x)\) is the minimum flow area; \(\varphi\) is mass flow coefficient, equal to 0.5 [47]; \(\Delta p\) is the pressure difference between the nozzle inlet and outlet, and it was equal to 30 MPa, while the environment pressure was atmospheric pressure equal to 0.1 MPa. According to experimental measurement, \(x\) denotes the needle valve lift, its value was 0.1 mm, and the movement speed of the needle valve was 0.33 m/s. In this model, an obvious interface between the diesel and diesel vapor occurred in each grid cell.