## 1 Introduction
This suggests that a higher injection pressure will often lead to a better spraying process, that this is one of the most effective ways to meet the efficiency requirements, and that this process has a potential benefit in diesel engine performance [13; 14; 15]. [PERSON] et al. [16] found that increasing the injection pressure reduces the number and mass of particles and increases the diesel spray velocity, which improves the atomization and evaporation process. In addition, [PERSON] et al. [17] applied a numerical simulation to examine the effects of injection time and injection rate shape on the performance and exhaust emissions of compression ignition engines. The authors found that CO, UHC, and soot emissions can be reduced by using rectangular-type and boot-type rate shapes instead of other types. Based on these previous studies, the injection rate is clearly an important parameter that requires more attention. Injection rate parameters, such as injection velocity, injection mass quantity, and injection duration, have a significant influence on the fuel mixing and combustion process [18]. Many of these investigations considered how the diesel spray mixing process behavior can increase mixing efficiency by studying the effect of the injection rate shape. Attempts to increase fuel mixing efficiency by determining the injection rate shape remain unsatisfactory. Although there have been extensive studies in the past on the influence of injection rates, most of these studies were interested in investigating the influence of injection rates on combustion efficiency and engine emissions [9; 10; 11; 12; 17]. Few studies have investigated the influence of injection rates on spray mixing behavior. Based on our previous study [19], we examined the spray mixing characteristics under different injection rate shapes using a modified one-dimensional spray model. This one-dimensional spray model can analyze the spray penetration, entrainment rate, and velocity over a cross-sectional area, as well as the equivalence ratio distributed along with the spray's axial distance. This model does not consider breakup and evaporation. Instead, it reveals general information on both the liquid and the vapor. The fuel and air are assumed to be immediately mixed uniformly, so turbulent mixing cannot be analyzed in detail. A 3D-CFD model is better for researching a spray that includes two-phase flow characteristics. Creating a spray model using CFD has become an effective way to study diesel spray to analyze the mechanism of fuel-air mixing and atomization. The results of many previous studies on creating a spray model using CFD show that the various parameters of fuel injection can be efficiently monitored and predicted [20; 21; 22]. Nevertheless, there are few studies on the injection rate shape's effects on spray mixing. Unlike a quasi-steady-state spray, it is difficult to obtain clear spray images that include liquid and vapor information under varying injection rates, so there is a lack of experimental data to validate the 3D-CFD spray model with varying injection rates. Consequently, most numerical studies focus on the effects of injection rates on combustion processes and do not offer an in-depth understanding of the spray behaviors under different injection rates. The objective of the current study is to analyze the effect of different injection rate shapes on the diesel mixing process using a numerical modeling method. In this work, the \"CONVERGE\" CFD code was adopted for a constant-volume combustion chamber with a single hole injector, particularly to study the spray breakup and spray mixing behavior. Validation of the model results involved a comparison of the spray shape and spray penetration with the experimental data from previous researchers of the Sandia National Laboratory, taken from the ECN website [23]. We found that the modified CFD spray model can predict spray behavior.