## 2 Numerical Method
### Governing Equations  
The VOF method, first proposed by [PERSON] and [PERSON] [48], introduces the volume fraction \(\alpha_{i}\) to define the ratio of the \(i\)th fluid's volume to the total volume in each grid cell, i.e.,  
\[\alpha_{i}=\frac{V_{i}}{V_{cell}}, \tag{1}\]  
Figure 1: Physical model and geometric parameters used in this study.  
Figure 2: Schematic of parameter definitions for droplet impact on a micro-pillar surface.  
where \(V_{i}\) is the volume of the \(i\)th fluid phase in the grid cell and \(V_{cell}\) is the total volume of the grid cell. Moreover,  
\[\alpha_{i}=\left(\begin{array}{c}1\\ 0\sim 1\\ 0\end{array}\right), \tag{2}\]  
where \(\alpha_{i}=1\) indicates that the grid cell is fully occupied by the \(i\)th fluid phase, \(\alpha_{i}=0\)-\(1\) indicates that the interface passes through the cell, and \(\alpha_{i}=0\) indicates that the grid cell is completely without the \(i\)th fluid phase. The evolution equation for the volume fraction \(\alpha\) in the VOF method is  
\[\frac{\partial\alpha}{\partial t}+\
abla\cdot\left(\alpha\mathbf{u}\right)\ =0. \tag{3}\]  
The Level Set method was proposed by [PERSON] and [PERSON] [49] and captures the interface by defining a symbolic distance function \(\phi(\mathbf{x},t)\):  
\[\phi(\mathbf{x},t)=\begin{cases}-d,&\text{in the gas region},\\ 0,&\text{at the interface},\\ d,&\text{in the liquid region}\end{cases}, \tag{4}\]  
where \(d\) is the shortest distance from the interface to the point \(\mathbf{x}\). The evolution equation is given by  
\[\frac{\partial\phi}{\partial t}+\mathbf{u}\cdot\
abla\phi=0, \tag{5}\]  
where \(\mathbf{u}\) is the fluid velocity vector and \(\
abla\phi\) is the gradient of the level set function, indicating the direction of the interface's normal vector.  
The VOF method ensures good conservation of physical quantities, but the volume fraction function \(\alpha\) is a discrete quantity, making it difficult to solve accurately, and it requires explicit construction of the moving interface. The Level Set method can implicitly capture interfaces, and the Level Set function remains smooth regardless of changes in the flow field. However, this method is not conservative, leading to a loss of physical quantities during computation, resulting in the smearing of sharp interfaces. To overcome the limitations of both methods, this study uses a coupled Level Set and VOF method, known as the CLSVOF method.  
For liquids, the flow governing equation is  
\[\
abla\cdot\mathbf{u}=0, \tag{6}\]  
\[\frac{\rho\partial\mathbf{u}}{\partial t}+\rho\mathbf{u}\cdot\
abla\mathbf{u}=-\
abla P+ \
abla\cdot\mu[\
abla\mathbf{u}+\left(\
abla\mathbf{u}\right)^{\mathrm{T}}]-\sigma \kappa\delta(\phi)\
abla\phi+\rho g, \tag{7}\]  
In this equation, \(\rho\), \(p\), \(\sigma\), and \(g\) represent density, pressure, surface tension coefficient, and gravitational acceleration, respectively. \(\kappa\) is the interface curvature, which can be calculated by the following equation:  
\[\kappa=\
abla\cdot\frac{\
abla\phi}{\
abla|\phi|}, \tag{8}\]  
\(\delta(\phi)\) is defined as  
\[\delta(\phi)=\begin{cases}\frac{1+\cos(\pi\phi/a)}{2a}&|\phi|<1.5\omega\\ 0&|\phi|\geq 1.5\omega\end{cases}, \tag{9}\]  
where \(a=1.5\omega\) and \(\omega\) is the minimum grid size.