Extended modeling of decelerating turbulent jets for diesel spray's penetration after end-of-injection

•Decrease of momentum flux is caused by injected fluid mass flow rate reducing.•Injected fluid mass flow rate over the tip cross-section was formulated.•Analytical expression of turbulent jet penetration was derived.•A new analytical model of diesel spray penetration was developed theoretically. In order to meet the growing demand for reduction of pollutant emissions and improvement in thermal efficiency, advanced combustion modes, such as low-temperature combustion (LTC), premixed charge compression ignition (PCCI), have been adopted in modern diesel engines. Compared to conventional diesel combustion, the ignition delay becomes longer in advanced combustion modes. The longer ignition delay usually results in the fuel injection ending earlier than ignition. Therefore, the spray propagation after end-of-injection (EOI) plays a significant role in diesel combustion process, and it is needed to estimate the spray evolution and mixture formation after EOI. For this purpose, a simple and analytical diesel spray model including the spray evolution after the EOI is developed in this study. To develop the model, the theoretical analysis on the decelerating process of the turbulent jet tip is performed based on the integral momentum flux and mass flow rate of the injected fluid over the cross-sectional area at the jet tip. It is observed that the decrease of mass flow rate of the injected fluid over the tip cross-sectional area causes reduction of the momentum flux over the tip cross-sectional area and the turbulent jet tip decelerating. Then the mass flow rate of injected fluid over the tip cross-sectional area is formulated, furthermore the analytical equation of the turbulent jet tip penetration during decelerating state is derived, and the correctness of the developed analytical equation has been proved theoretically. Finally, the calculation is extended to diesel spray penetration, and the calculated results are validated against the one-dimensional discrete model and the experimental data from Engine Combustion Network (ECN) respectively.