# Effect of Fuel Mass Flow at the End of Injection on Cavitation and Gas Ingestion in the Nozzle
## 1 Introduction  
With the increasing global energy demand, the research on fuel injection technology, sprays, and combustion, especially in the transportation sector, has been steadily increasing. The effect of fuel injection and spray is inseparable from the combustion and emission performance of internal combustion engines. To produce a suitable and uniform air-fuel mixture, small orifice diameter and higher injection pressure are used in mainstream compression ignition engines. During the fuel injection, sudden changes in fuel pressure and velocity will cause fuel cavitation, then the cavitation collapses as the pressure increases [1, 2, 3, 4, 5]. The fuel flow characteristic for the fuel injector has an obvious impact on the fuel atomization and spraying [6, 7, 8, 9], thus it is of certain significance to research the internal flow of the fuel injector. Due to the extremely high fuel flow rate and small diameter of the actual fuel orifice, it is difficult to carry out actual tests. Therefore, some scholars [10, 11, 12, 13] conducted experimental studies on scaled-up transparent nozzles, which were based on the principle of similarity under the premise of ensuring the geometric similarities between the experimental and real-size nozzles. However, there were many scholars who still performed research on fuel flow on a real-sized nozzle and the effects of X-rays on a transparent real-sized nozzle [14, 15, 16]. Interestingly, [PERSON] [17] observed that there were stagnant bubbles in the nozzle before fuel injection, and this phenomenon was later confirmed to have an impact onfuel atomization [18; 19; 20] because different positions of the stagnant bubbles would cause different shapes of the fuel spray. One of the latest experimental studies on these stagnant bubbles was conducted by [PERSON] [21], who pointed out that there were three sources of these initial bubbles: in the start of fuel injection (ingestion gas from the start of injection (SOI)), ingestion gas from the end of injection (EOI) of the last injection, and fuel vapor not discharged from the EOI of the last injection (gas ingestion bubbles from the EOI of the last injection played a major role). However, due to the lack of numerical simulation, he did not explain its mechanism clearly. The fuel flow in the fuel injector is a highly transient process, limited by the current experimental technology, thus many scholars chose to utilize numerical simulation to obtain the transient information of the complex and variable flow field in the nozzle [22; 23; 24]. The numerical methods of solving fuel flow in the nozzle can be simply divided into the homogeneous equilibrium model and multi-phase flow model. Because the homogeneous equilibrium model needs less memory and has a faster calculation speed, some researchers chose the homogeneous equilibrium model to study cavitation and therefore achieved good results [25; 26]. However, compared with the multi-phase flow model, less flow field information is obtained. Thus, [PERSON] [27] used the improved volume of fluid (VOF) method to study the vapor-liquid two-phase flow, and he found that the simulation result provided a good fit to that of the experimental. Then, [PERSON] [28] verified that cavitation was caused by the decrease of fuel pressure via the two-fluid model and believed that the two-fluid model was an engineering tool suitable for large-scale processes. [PERSON] [29] established a VOF model based on Eulerian multi-phase flow to capture the two-phase interface motion, and the effect of fluid parameter on the fuel flow in the nozzle was analyzed. [PERSON] [30] proposed a large eddy simulation turbulence model by coupling multi-phase flow to investigate the fuel flow in the nozzle at the SOI and EOI, but he did not study gas ingestion at the EOI. [PERSON] [20] studied the phenomenon of fuel cavitation and atomization using an Euler/VOF/LES (large eddy simulation) model and obtained results that were consistent with their experiments.