## 2 Materials and Methods
### Numerical Stimulation Methods
The physical properties of fluids \(\rho(\varphi)\) and \(\mu(\varphi)\) can be expressed as:  
\[\rho(\varphi)=\rho_{g}+\left(\rho_{l}-\rho_{g}\right)H(\varphi) \tag{15}\]  
\[\mu(\varphi)=\mu_{g}+\left(\mu_{l}-\mu_{g}\right)H(\varphi) \tag{16}\]  
where the subscripts \(g\) and \(l\) denote the gas phase and liquid phase, respectively. To consider the effect of the contact angle via the wall boundary condition, the surface normal at the cell next to the wall is expressed as:  
\[\hat{n}=\hat{n}_{w}cos\theta+\hat{\tau}_{w}sin\theta \tag{17}\]  
where \(\hat{n}_{w}\) and \(\hat{\tau}_{w}\) are the unit vectors perpendicular and tangent to the wall. Establishing a steady-state flow field with constant wind speed in the absence of droplets is important to ensure the observation of droplet motion in the presence of a stable external flow field. The droplet motion process on the windshield was then simulated by establishing transient conditions based on the steady-state flow field. Combining the values of the VOF and Level Set functions with the geometric PLIC method, an initializationprocedure is applied after each time step [38]. The calculations are performed on a fixed Eulerian structured grid using the CLSVOF method to follow the gas-liquid interface. The Mach number of the external flow field of the windshield in the realistic environment is relatively low, and the airflow separation is easily formed around the vehicle during the driving process. Therefore, it can be regarded as an incompressible viscous isothermal flow field. The turbulence model is usually used in the numerical simulation of the external flow field of the car. In this study, the standard \(k\)-\(\epsilon\) model is used with good convergence and lower residual values. The movement of droplets on the windshield surface is influenced by the droplets' physical properties and the surface wettability. The wettability is determined by interactions at the solid-gas, solid-liquid, and gas-liquid interfaces and is assessed using two parameters: the static contact angle \(\theta_{e}\) and the dynamic contact angle \(\theta_{D}\). In the airflow, imbalances in the forces acting on the droplet can lead to dynamic changes in wettability behavior. When the contact line velocity \(U_{cl}\) exceeds 0, the dynamic contact angle is utilized to evaluate the motion of the droplet's contact lines. When the contact lines begin to move, its dynamic contact angle parameters are illustrated in Figure 4. The contact angle at the onset of wetting by the three-phase contact line is defined as the advancing contact angle \(\theta_{adv}\), while the contact angle during the retreat of the three-phase contact line along the wetted surface is defined as the receding contact angle \(\theta_{rec}\)[39]. The dynamic contact angle model used in association with a flow model has been found to be an accurate method to represent the contact angle variation in fluid flow [40]. Thus, the dynamic contact angle is introduced to reproduce the transient contact angle values between the droplet and the wall more accurately. The dynamic contact angle (\(\theta_{D}\)) is defined as a function of the contact line velocity in the combination of [PERSON] and [PERSON]'s methods, which is expanded in the form of the capillary number \(Ca\), the contact line's diffusion velocity \(U_{cl}\), the static advancing contact angle \(\theta_{adv}\), the static receding contact angle \(\theta_{rec}\), and the Hoffman empirical function \(f_{H}(x)\) and its inverse \(f_{H}^{-1}(x)\). It can be expressed as:  
\[\theta_{D}= \begin{cases}f_{H}\Big{(}Ca+f_{H}^{-1}(\theta_{adv})\Big{)}&U_{cl} \geq 0\\ \Big{(}\theta_{rec}^{3}-72 Ca\Big{)}^{\frac{1}{3}}&U_{cl}<0\end{cases}  \tag{18}\]  
\[f_{H}(x)=arccos\Bigg{(}1-2 tanh\Bigg{(}5.16\bigg{(}\frac{x}{1+1.31x^{0.99}} \bigg{)}^{0.706}\Bigg{)}\Bigg{)} \tag{19}\]  
where the capillary number \(Ca\) is defined as \(Ca=\frac{\mu_{i}U_{cl}}{\sigma}\). \(\mu_{i}\) and \(\sigma\) in the definition equation are the dynamic viscosity and surface tension of the liquid, respectively.