## 4 Effect of the Injection Rate Shape
### Mixture Properties
At low mass flow rates, the field flow will be smooth without a recirculation zone. On the other hand, when the mass flow rate increases, the vortex will increase. This intense recirculation will increase heat transfer compared to the smooth channels. Figure 22 shows the strongest TKE during the initial injection under the QIGD injection rate, while the RECT injection rate shows the strongest TKE during the injection. Subsequently, after the EOI, the GIQD injection rate shows the strongest TKE. These phenomena can be explained by the influence of TKE production, which is related to the effect of the spray structure. The simulation results of TKE distribution with different injection rates, both during injection and after the EOI, are shown in Figures 23 and 24, respectively. The turbulence phenomenon is considered in the boundary of the spray region, where the model provides sufficient mesh density. The spray images were obtained from the cut-plane of the direction of the spray. The gradient color region represents the TKE distribution, which presents the TKE data as a gradient color bar in the bottom right corner. Figure 22: Comparison of the turbulence kinetic energy (TKE) for different injection rate shapes. (**a**) TKE and (**b**) injection mass flow rate. Figure 23: Comparison of the TKE distribution during injection for different injection rate shapes. Figures 23 and 24 show the contour of the TKE with different injection rate shapes. The state of turbulence distribution affects the proportions of the length and width of the spray shape. For example, Figures 23 and 24 show the strong TKE of the QIGD injection rate case that moved to the area along the length of the spray, which resulted in long spray penetration and a lean equivalence ratio in the spray tip area, as can be seen in Figures 20 and 21. This is because a high injection rate directly affects the crossflow, resulting in significant disintegration and displacement. Figures 23 and 24 show that the TKE production of the RECT injection rate is strong in the axial area of the spray, which indicates that the TKE distribution occurs along the width of the spray. Although the TKE of the RECT injection rate  
Figure 24: Comparison of the TKE distribution after the EOI for different injection rate shapes. is the strongest during the injection period (see Figure 22), the TKE produced cannot offer penetration longer than the QIGD injection rate (Figures 23 and 24). The QIGD injection rate initially shows the strongest TKE; then, the TKE distribution moves along the length of the spray faster than the other injection rates and has the highest TKE at the spray tip area. This is due to this spray's high speed and acceleration, causing more increased turbulence levels in the downstream areas than in other injection rate shapes. This gives the QIGD injection rate the highest efficiency of spray turbulence, providing the leanest equivalence ratio at the spray tip area. The phenomenon of TKE distribution can be well illustrated by the temperature distribution in Figures 18 and 19, which show that the QIGD injection rate has a higher temperature in the spray tip area than other shapes. When considering the temperature distribution behavior after the EOI in Figure 19, the RECT injection rate shape retains the lowest temperature at the spray tip area compared to the other injection rate shapes. The highest temperature at the spray tip area of the QIGD injection rate and the lowest temperature at the spray tip area of the RECT injection rate can predict the spray turbulence behavior. A high temperature means that the area may produce high turbulence results with a lean equivalence ratio but show opposite spray behavior at low temperatures. These results are due to the high temperature influencing the droplet size and evaporation. These phenomena occur when the increased fuel injection rate causing the TKE grows, thereby leading to faster fuel spread and resulting in faster fuel and air mixing.