# A New Criterion for the Splashing of a Droplet on Dry Surface from High-Fidelity Simulations
## 5 Conclusions  
Numerical simulations of a droplet impact on a surface with different wettabilities have been performed. The accuracy of our numerical simulations is validated through comparisons with experimental data from the literature. By varying the surface tension coefficient and viscosity of the droplet, we simulate the droplet-wall collisions near the splashing threshold at different Reynolds numbers (500-6500) and Weber numbers (60-250). The effects of surface wettability on the droplet-splashing threshold are investigated by changing the contact angle. Several typical modes of droplet impact are captured. A new spreading splashing mode is defined and a new criterion for splashing occurrence is established. The main findings are summarized as follows:  
1. The phenomenon where secondary droplets are generated before the droplet spreads to its maximum diameter after collision is defined as splashing. Splashing with secondary droplets generated later, relatively larger in diameter, and fewer in number is defined as spreading splashing. Splashing with secondary droplets generated earlier, smaller in diameter, and greater in number is defined as prompt splashing. Based on the symmetry of the distribution of secondary droplets, splashing can be classified into partial splashing (asymmetric distribution of secondary droplets) and splashing (symmetric distribution of secondary droplets).
2. Spreading splashing primarily occurs in cases where the contact angle is relatively small (\(CA\leq 120^{\circ}\)), while prompt splashing prevails in situations where the contact angle is larger (\(CA>120^{\circ}\)). An increase in the contact angle tends to inhibit spreading splashing, while promoting rapid splashing.
3. For contact angles greater than \(90^{\circ}\), there exists a critical Weber number, \(We_{c}\), below which splashing does not occur. From our simulations, \(We_{c}\approx 60\).
4. Based on our results, a new criterion for splashing is proposed; \(K_{c}=(We-We_{c})^{\frac{1}{n+1}}Re^{\frac{n}{n+1}}\), where \(n\) equals \(-0.02 CA+0.49\) for \(CA\leq 90^{\circ}\) and 1 for \(CA>90^{\circ}\). For \(CA\leq 90^{\circ}\), the threshold \(K_{c}=333\), and for \(CA>90^{\circ}\), \(K_{c}=-0.021 CA^{2}+2.7 CA+260\).  
Our criterion for splashing is the first one that is established based on high-fidelity numerical simulations and accounts for the effects of contact angle. Due to some possible differences, such as the judgement of the splashing occurrence, the wall roughness, and the effects of the surrounding gas, quantitative agreement between our criterion and those in the literature was not reached. Our criterion corresponds to cases where the wall is smooth, and the Reynolds and Weber numbers are varied by changing the droplet viscosity and the surface tension coefficient, while the gas viscosity and density are fixed. The effects of the gas properties, which might be measured by dimensionless quantities such as the liquid-gas viscosity and density ratios, are not investigated. This is a good subject for future study. The criteria in the literature are also inconsistent with each other. We think that our criterion is a useful complement to the existing ones.  
Conceptualization, Z.Y., H.X. and B.T.; methodology, S.J. and H.X.; formal analysis, S.J., H.X., B.T. and Z.Y.; investigation, S.J., H.X., B.T. and Z.Y.; writing--original draft preparation, S.J.; writing--review and editing, Z.Y.; supervision, Z.Y.; funding acquisition, Z.Y. and B.T. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Key Laboratory of Computational Physics, Beijing Institute of Applied Physics and Computational Mathematics, and the National Natural Science Foundation of China (Grant No. 12332015, 12072319).  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors on request.  
The authors declare no conflicts of interest.