# A New Criterion for the Splashing of a Droplet on Dry Surface from High-Fidelity Simulations
## 4 Results and Discussion
### Criterion for Splashing  
As mentioned in the Introduction, various criteria for the occurrence of splashing have been proposed. Although some of these criteria may conflict with each other, they share a common form represented by \(K=ReWe^{n}\). Therefore, considering the critical Weber number, we define a critical parameter for splashing occurrence as follows:  
\[K=\left(We-We_{c}\right)^{\frac{1}{n+1}}Re^{\frac{n}{n+1}} \tag{6}\]  
where \(We_{c}\) is the critical Weber number, and \(n\) is a parameter. Figure 6: Droplet impact at different contact angles for \(Re=6500\) and \(We=60\). By fitting to our data from 270 simulation cases, the parameter \(n\) is determined as follows:  
\[n= \begin{cases}-0.02 CA+2.49,\text{ for }CA\leq 90^{\circ}\\ 1,\text{ for }CA>90^{\circ}\end{cases}  \tag{7}\]  
From Equation (7), when the contact angle (\(CA\)) is less than or equal to \(90^{\circ}\), there exists a good linear relationship between \(n\) and \(CA\). However, when the contact angle exceeds \(90^{\circ}\), \(n\) remains constant and is equal to unity. When the \(CA\) is small, the hindering effect of surface tension on droplet spreading is relatively weak because of the geometry of the edge of the droplet, and the viscous force dominates. In the spreading stage, the kinetic energy of the droplet is primarily dissipated by viscous forces, preventing the droplet's edge from having enough kinetic energy to overcome the surface tension and form secondary droplets. As \(CA\) increases, the geometry of the edge of the droplet changes, and thus the effect of surface tension gradually increases(the surface tension and viscous forces together dissipate the droplet's kinetic energy), leading to the decrease in \(n\), which means that the relative importance of the Reynolds number to the Weber number decreases. \(n\) being 1 leads to \(K=(We-We_{c})^{\frac{1}{2}}Re^{\frac{1}{2}}\), which is consistent with the results of [PERSON] et al. (2018), who established the criterion only for the contact angle \(CA=156^{\circ}\) and did not consider the critical Weber number. Since both the powers of \(We\) and \(Re\) are positive, an increase in the impact velocity, droplet size, or droplet density, or a decrease in the droplet viscosity or surface tension, would promote the occurrence of splashing. To illustrate the effects of the contact angle on the splashing threshold \(K_{c}\), we present the relationship between \(K_{c}\) and the contact angle \(CA\) in Figure 7. From the figure, for contact angles less than \(90^{\circ}\), the value of \(K_{c}\) remains almost unchanged with variation in \(CA\). However, for contact angles greater than \(90^{\circ}\), the value of \(K_{c}\) decreases with increasing \(CA\). This observation is consistent with the experimental results of [PERSON] et al. (2018), indicating that the hydrophobic surface greatly promotes droplet splashing. Our value of \(K_{c}\) is smaller than those of [PERSON] et al. (2018) and [PERSON] et al. (2018). This may be caused by the difference in judging the occurrence of splashing. In the present study, we deem it as splashing as long as any secondary droplet appears, which might not be observed or regarded as splashing in the experiments. In addition, the wall roughness and the properties of the surrounding gas that are not considered in the criterion may differ in different studies and result in discrepancy. Figure 7: The splashing threshold \(K_{c}\) as a function of the contact angle. The blue line represents the critical curve obtained by Equation (8). The critical value of \(K\) is determined from our data in Figure 7, as follows:  
\[K_{c}= \begin{cases}333,\text{ for }CA\leq 90^{\circ}\\ -0.021 CA^{2}+2.7 CA+260,\text{ for }CA>90^{\circ}\end{cases}  \tag{8}\]  
Figure 8 displays the phase diagrams in terms of the Weber and Reynolds numbers for different contact angles.