## 3 Simulation Results and Discussion
### Effect of Nozzle Hole Number and Injection Direction on the In-Cylinder Mixture Formation
On the other hand, the wall-attached jet 'b' does not form vortex structures, so it gradually dilutes over time. Figure 10: The mass fraction of the mixture with different equivalent ratios in the cylinder at \(-112\,^{\circ}\)CA, \(-92\,^{\circ}\)CA, and \(-72\,^{\circ}\)CA for the four schemes. Figure 9: Formation process of the mixture for the four schemes at \(-140\,^{\circ}\)CA of hydrogen injection timing. The mixture distribution in the circumferential section under the four schemes is shown in Figure 11. The circumferential section passes through the spark plug electrodes and is parallel to the bottom of the cylinder head. It can be observed that there is little change in the mixture distribution from \(-5\)\({}^{\circ}\)CA to \(5\)\({}^{\circ}\)CA. The mass fraction curves of the mixture with different equivalence ratios at the three different timings in Figure 12 also support this point: the mass fraction of the mixture with different equivalence ratios follows a normal distribution. There is no significant change in the geometry of the curves from \(-5\)\({}^{\circ}\)CA to \(5\)\({}^{\circ}\)CA, and the peak values gradually increase over time, indicating a more uniform mixture. Therefore, the mixture distribution at the TDC can be used as a reference for ignition timing within \(\pm 5\)\({}^{\circ}\)CA of the TDC. By analyzing the circumferential section at TDC, it can be seen that all schemes have rich mixture regions, which can lead to higher local combustion temperatures and increased NOx emissions, and cannot reflect the advantages of lean hydrogen combustion. For schemes P2 and P4, the rich mixture region is near the exhaust valve, making surface ignition easier. For schemes P1 and P4, the mixture equivalence ratio near the spark plug (as shown in Figure 5B) is relatively low, making misfire easier. Then, through the analysis of the radial and circumferential sections, it can be seen that scheme P2 is more uniform than schemes P1 and P4, except for one prominent rich region. Regarding reliable ignition, schemes P2 and P3 are preferable; furthermore, under scheme P3, the mixture distribution in both radial and circumferential sections is superior to the other three schemes. To quantify the uniformity of the in-cylinder mixture, the definition of mixture non-uniformity by [PERSON] was used as a reference [36]. The statistical average equivalence ratio inside the cylinder is calculated using the volume of each grid, as shown in Equation (1). The standard deviation (SD) of the equivalence ratio inside the cylinder is calculated as shown in Equation (2). Finally, the normalized standard deviation (NSD) is obtained, as shown in Equation (3). NSD, as a non-uniformity coefficient, is used to evaluate the non-uniformity of the mixture distribution inside the cylinder. The larger the non-uniformity, the poorer the uniformity of the mixture. \[\overline{\phi}=\frac{\sum\phi_{i}V_{i}}{V} \tag{1}\]  
\[SD=\sqrt{\frac{\sum\left(\phi_{i}-\overline{\phi}\right)^{2}V_{i}}{V}} \tag{2}\]  
Figure 11: Mixture distribution on the circumferential section near the TDC for the four schemes. \[NSD=\frac{SD}{\overline{\phi}} \tag{3}\]  
In the equations, \(\phi_{\rm i}\) and \(\rm V_{\rm i}\) represent the equivalence ratio and volume of grid \(\rm i\), respectively. \(\rm V\) is the total volume of the cylinder region. Figure 13 shows the equivalence ratio distribution and NSD of the mixture inside the cylinder at TDC for the four schemes. It can be seen that the equivalence ratio distribution ranges of P1, P3, and P4 are wider, consistent with the variation in NSD. The wider the range of equivalence ratio distribution, the higher the non-uniformity, and the worse the uniformity of the mixture inside the cylinder. Scheme P2 has the best uniformity, followed by scheme P3. To achieve better homogenization of the mixture inside the cylinder, further research is conducted based on scheme P2.