The start-of-injection (SOI) and end-of-injection (EOI) transients are important for sprays with multiple-injection strategy. Owing to the change of needle position and sac pressure, the spray behaviors in the transient processes are significantly different from those in the quasi-steady stage. In this study, considering the sac pressurization processes during the SOI transients and effects of ''entrainment wave'' after the EOI, a theoretical zero-dimensional model for the entire development processes of the spray tip penetration (S_tip) is summarized. Then, a high-speed CMOS camera and constant-volume chamber are employed to study the spray characteristics and verify the model. The model and experimental results clarify four key time points in the entire development processes of S_tip: the sac pressurization time (t_p), breakup time (t_b), injection duration (t_i) and two injection durations (2t_i). Five stages can be divided according to these four time points: the acceleration stage (S_tip proportion to t^1.5), transition stage 1 (t¹ or t^0.75), quasi-steady stage (t^0.5), transition stage 2 (t^0.5) and decelerating stage ((t-t_i) ^0.25). Moreover, the newly developed model is compared with the Wakuri model, the Hiroyasu & Arai model, and the Naber & Siebers model. The results show that all the models can well predict S_tip during the quasi-steady stage and transition stage 2 with the optimized model constants. While the other models overpredict S_tip in the acceleration and decelerating stages, the newly developed model still works very well.

喷射开始 (SOI) 和喷射结束 (EOI) 瞬变对于采用多次喷射策略的喷雾很重要。由于针位置和囊压力的变化，瞬态过程中的喷雾行为与准稳态阶段的喷雾行为明显不同。在这项研究中，考虑到 SOI 瞬态过程中的囊加压过程和 EOI 后“夹带波”的影响，总结了喷雾尖端穿透 ( S _tip )整个发展过程的理论零维模型。然后，采用高速 CMOS 相机和定容室来研究喷雾特性并验证模型。_尖端：囊加压时间（吨_p），破裂时间（吨_b），喷射持续时间（吨_我）和两个喷射持续时间（2吨_我）。五个阶段可以根据这四个时间点可分为：加速阶段（ š_尖端比例吨^1.5），过渡阶段1（吨¹或吨^0.75），准稳态阶段（吨^0.5），过渡阶段2（吨^0.5 ) 和减速阶段 (( t - t _i ) ^0.25）。此外，将新开发的模型与 Wakuri 模型、Hiroyasu & Arai 模型和 Naber & Siebers 模型进行了比较。结果表明，所有的模型能够很好地预测小号_尖端在准稳态阶段和过渡阶段2与优化模型常数。虽然其他模型在加速和减速阶段高估了S_尖峰，但新开发的模型仍然运行良好。