## 3 Simulation Setup and Validation
### Spray Model
#### 3.1.1 Cavitation Sub-Model  
In this paper, a new cavitation sub-model developed by authors [23] based on the homogeneous nucleation theory is adopted. Instead of regulating the relationship between bubble nucleation and gas phase volume fraction only by empirical parameters, a theoretical analysis of the causes of bubble nucleation and nucleation scale is carried out, and the effects of turbulence, mass transfer rate, equivalent bubble radius, differences in dissolved amounts of different gases, and temperature on nucleation are considered. The homogeneous nucleation theory is applied to the bubble nucleation process, where the bubble nucleation scale within each cell is presented in terms of nucleation probability; thus, the bubble nucleation phenomenon becomes a specific process related to surface tension, temperature, and pressure. The new model also subdivides the dissolved gases into different types, and each type of gas is considered and calculated separately according to its contribution to the partial pressure inside the bubble. For each gas dissolved in the test, the overall partial pressures of dissolved gases are calculated based on Henry's law and the partial pressure law and then added to obtain them. In addition, the model also introduces the concept of equivalent bubble radius, which is calculated by combining experimental data to characterize the size of bubbles in the working mass. This method is helpful to make the numerical model close to the actual situation. Compared with the previous empirical model, the new model has better accuracy, robustness, and applicability for cavitation flow characteristics prediction.  
The internal flow simulation of the nozzle of PODE/diesel blends is carried out for the injector in order to apply the cavitation and primary crushing model to the engine combustion process. The injector used for the engine test is a CRIN-138 injector from the BOCSH electronically controlled high-pressure common rail system. It has eight injector nozzle holes with a 0.18 mm diameter and a 75\({}^{\circ}\) inclination at the center. Experiments using 0# diesel and P10, P15 are applicable to the current CFD calculation framework because solubility, saturation vapor pressure, and other characteristics of PODE/diesel blends are very similar to those of diesel. The simulation process of blended fuel is set unitedly by the physical and chemical properties measured by the test to simplify the calculation. The calculation grid, which is shown in Figure 2, is a dynamic grid that is used for the simulation from the beginning of the fuel injection to the end. The fuel injection mass of each single-hole injection is calculated based on the engine's consumption of fuel per cycle at 1400 rpm, while the needle valve movement law is set based on the injector needle valve lift law and the injection rate curve. The total mass obtained by adjusting the injection pulse width is equal to that of the single-hole injection calculated based on the mass of fuel consumed per cycle, so the transient mass flow rate at the nozzle is calculated. Additionally, the back pressure condition for the nozzle flow will be determined by the pressure in the engine cylinder during fuel injection.  
The temperature of the injected fuel rises along with the high ambient temperature surrounding the engine and the high return fuel temperature. As a result, the spray simulation for the engine differs from the cold spray simulation, the model's fuel temperature is set at 57 \({}^{\circ}\)C, and the fuel density and the saturated vapor pressure adjust in line with the temperature change. The needle valve lift is around 0.4 mm, and the common rail's rail pressure is 109 MPa. In addition, the real engine combustion process simulation is used to improve the model accuracy of the internal flow characteristics of the nozzle at thenozzle outlet, including nozzle unit position, area, gas-liquid phase velocity distribution, temperature, turbulent kinetic energy, and turbulent dissipation rate.