## 2 Materials and Methods
### Numerical Stimulation Methods  
In numerical simulations, the Volume of Fluid (VOF) method is commonly utilized to define the interface between gas and liquid phases within a grid. In the VOF method, the \(F\) function is often used to represent the phase function, and the volume fraction \(\beta\) of the lattice liquid occupying the lattice space is calculated by solving the phase function. The phase function is used to characterize the distribution of the gas-liquid phases in the flow field. The definition of the \(F\)-function is shown in Equation (5):  
\[F\left(\overset{\rightarrow}{r},t\right)= \begin{cases}1,&\text{in the liquid region}\\ 0<F<1,&\text{at the inter face}\\ 0,&\text{in the gas region}\end{cases}  \tag{5}\]  
where \(\overset{\rightarrow}{r}\) is any point in the computational domain and \(t\) is time. As water in numerical simulations is treated as an incompressible fluid, the primary governing equation for the VOF method is the continuity equation for incompressible fluids, ensuring effective conservation of physical quantities. The mass conservation equation for the \(F\) function can be expressed as:  
\[\frac{\partial F}{\partial t}+\left(\overset{\rightarrow}{u}\cdot\
abla \right)F=0 \tag{6}\]  
where \(\overset{\rightarrow}{u}\) is the velocity of the grid. (CLSVOF) method proposed by [PERSON] [34] are used to establish the numerical modelling framework according to the characteristics of both the VOF and Level-set method. The CLSVOF method is based on the VOF model and is modified by introducing the level set function \(\varphi\left(\overset{\rightarrow}{r},t\right)\). The CLSVOF method can be applied to the dynamic interface of liquid droplet flow problems, such as wetting, which is related to the attachment of droplets on inclined surfaces with interfacial motion. Firstly, the method will solve the continuity equation and the Navier-Stokes equations in the computational domain [35; 36]:  
\[\
abla\cdot\overset{\rightarrow}{v}=0 \tag{10}\]  
\[\frac{\partial}{\partial t}[\rho(\varphi)\overset{\rightarrow}{v}]+\
abla \cdot[\rho(\varphi)\overset{\rightarrow}{v}\overset{\rightarrow}{v}]=- \
abla p+\
abla\cdot\left[\mu(\varphi)\left(\
abla\overset{\rightarrow}{v}+ \
abla\overset{\rightarrow}{v}^{T}\right)\right]+\rho(\varphi)\overset{ \rightarrow}{S}-\overset{\rightarrow}{F} \tag{11}\]  
where \(\overset{\rightarrow}{v}\) is the velocity vector, \(\overset{\rightarrow}{S}\) is the gravity, and \(p\) is the pressure. \(\mu(\varphi)\) and \(\rho(\varphi)\) are the dynamic viscosity and density of the droplet, respectively. \(\varphi\) is a function of the level set:  
\[\varphi(\overset{\rightarrow}{r},t)=\left\{ \begin{array}{ll}-d,&\text{in the gas region}\\ 0,&\text{at the interface}\\ d,&\text{in the liquid region}\end{array} \right. \tag{12}\]  
where \(d\) is the shortest distance between the interface and the point \(\overset{\rightarrow}{r}\). The inclusion of the level set function \(\varphi\left(\overset{\rightarrow}{r},t\right)\) allows the calculation of both interface curvature and physical properties to be based on the distance function. The continuity of the level set functions makes the calculation results more accurate. In addition to this, the CLSVOF method will use the method proposed by [PERSON] [37] via the CSF (continuous surface force) model, which is the source term of the surface tension imposed by the phase interaction condition solved. This can be expressed smoothly according to the Heaviside function:  
\[\overset{\rightarrow}{F}=\sigma\kappa(\varphi)\
abla H(\varphi) \tag{13}\]  
\[H(\varphi)= \begin{cases}0&\varphi<-a\\ \frac{1}{2}+\frac{\varphi}{2a}+\frac{1}{2\pi}sin(\frac{\pi\varphi}{a})&|\varphi |\leqslant a\\ 1&\varphi>a\end{cases}  \tag{14}\]  
where \(\kappa(\varphi)\) is the curvature of the liquid surface and \(\sigma\) is the surface tension coefficient. \(H(\varphi)\) is the Heaviside function, which can be used to calculate the physical properties of fluids within the interface. \(a\) is the thickness of the transition zone at the phase interface.