An improved Eulerian-Lagrangian method to predict the droplet survival distance behind detonation waves

Owing to the simple geometry and high volumetric energy density, liquid-fueled detonation engines have attracted considerable interest. However, their numerical simulation remains challenging as multiple processes, such as spray atomization, droplet breakup, turbulent mixing, and detonation wave propagation, coexist and interact with each other. In particular, within the Eulerian-Lagrangian framework, existing breakup models fail to accurately predict the droplet survival distance behind detonation waves, which could result in an overestimated propagation velocity of detonation wave. To this end, a theoretical analysis of the droplet breakup process is firstly conducted. It is found that the ratio of the droplet breakup time to the relaxation time can approach unity because of the smaller diameters of the child droplets and the high surrounding flow velocity. This suggests that the breakup behavior of the child droplets may differ from that of the parent droplets. Based on the theoretical analysis, an improved method is proposed in which the breakup process is inhibited when the ratio of breakup time to relaxation time exceeds a critical threshold. The proposed method is then incorporated into the Pilch-Erdman and Reitz-Diwakar models within the OpenFOAM® framework. Numerical simulations of droplet array breakup under detonation waves demonstrate that the improved method has better predictive performances across different conditions. Specifically, for the Reitz-Diwakar model with an initial kerosene droplet diameter of 40.7 μm, the droplet survival distance is improved from 6.11 mm to 12.56 mm (experiment: 12.73 mm), and detonation wave propagation speed from 2153 m/s to 1948 m/s (experiment: 1897 m/s). Similar results are also found for Pilch-Erdman models. Overall, the proposed model is effective, robust, and readily integrable into existing frameworks.