# A New Criterion for the Splashing of a Droplet on Dry Surface from High-Fidelity Simulations
## 3 Physical Model and Numerical Method Validation
### Numerical Method Validation  
To validate the accuracy of our numerical simulation, we compared our simulation results to the numerical and experimental results from [PERSON] et al. [20]. The simula  
Figure 1: Schematic illustration of the computational domain.  
tion settings are as follows: initial diameter of droplet \(D_{0}=2.14\) mm, initial velocity of droplet \(V_{0}=0.64\) m/s, contact angle \(CA=156^{\circ}\), density of liquid \(\rho_{l}=998\) kg/m\({}^{3}\), density of gas \(\rho_{g}=1.185\) kg/m\({}^{3}\), viscosity of liquid \(\mu_{l}=8.89\times 10^{-4}\) Pas, viscosity of gas \(\mu_{g}=1.84\times 10^{-5}\) Pas, and surface tension coefficient \(\sigma=0.0728\) N/m. No slip boundary condition was adopted for the solid bottom surface, and pressure outlet boundary conditions were adopted for other surfaces.  
The grid of the computational domain is initialized to a resolution of level 6, which means that the minimum grid size is \(1/2^{6}\) of the side length of the simulation domain. During the computation process, we dynamically adjust the grid based on the velocity and the volume fraction \(\alpha\). Grid refinement is applied to regions with high velocity gradients and significant changes in the volume fraction (near the gas-liquid interface), with a maximum refinement level set to 10. Conversely, grid resolution is reduced in other regions (far from the impact area), as shown in Figure 2. The maximum number of the grids is typically around 10 million for most cases and is around 50 million for a few cases, resulting in a huge computational cost for the present study. A higher refinement level is limited by our computational resources.  
Figure 3 depicts a comparative illustration of the different stages of droplet-wall interaction. From the figure, it is evident that the numerical simulation in the present study faithfully reproduces the behavior of the droplet impacting the wall, with the shape of the droplet in the simulation closely matching the experimental observations. At the onset of collision, the droplet exhibits an incomplete spherical shape, gradually spreading into a pancake-like morphology until reaching its maximum diameter. Subsequently, the droplet recoils and rebounds before finally detaching from the surface. Notably, during the spreading phase of the droplet, pronounced capillary waves appear on the upper half of the droplet surface due to surface tension, a detail not captured in the simulation results of [PERSON] et al. (2020), highlighting the accuracy of our simulations.  
The non-dimensional time is defined as  
\[t^{*}=\frac{Ut}{D_{0}} \tag{4}\]  
In order to quantify the simulation results, the non-dimensional parameter spreading factor \(\xi\) is defined as  
\[\xi(t)=\frac{D(t)}{D_{0}} \tag{5}\]  
where \(D(t)\) is the diameter of the wetted area during the impact process.  
Figure 2: Spatial resolutions: (**a**) t = 0 ms; (**b**) t = 0.8 ms.  
As can be seen from Figure 4, both our simulation results and those of [PERSON] et al. [20] are in good agreement with the experimental results. The final spreading factor \(\zeta\) values for our simulation, [PERSON] et al. [20], and the experiments are 1.432, 1.42, and 1.428, respectively. The relative deviation between our simulation and the experiments is only around 0.28%.