## 2 Materials and Methods  
### Numerical Calculation Model  
Airflow under natural conditions needs to be simulated, but the computational cost of full-scale wind tunnel numerical simulation is relatively high. Therefore, the scaled-down wind tunnel is usually used to reproduce the outflow field [29]. In order to study the motion of droplets on the windshield at a certain wind speed, the 3D geometric model used for the study is a scaled down model of the inclined surface with model dimensional parameters, as shown in Figure 1.  
According to the wind tunnel design theory, the size of the computational domain used in the numerical simulation needs to be determined based on the size of the three-dimensional model to ensure the accuracy of the computational results, as well as the appropriateness of the computational cost at the mean time. The computational domain and the boundaries used in this study are shown in Figure 2, in which the length is 11 L, the width is 7 W, and the height is 5 H, and L, W, and H are the length, width, and height of the three-dimensional model, respectively. The 3D model is located on the symmetry plane of  
Figure 1: Schematic of the 3D model. the computational domain, and the distance between the trailing edge of the model and the airflow outlet boundary is 7 L. In addition, the boundary conditions of the computational domain are shown in Table 1.  
Water and air are made as the dispersed phase and continuous phase, respectively. The densities of water and air are 998.2 kg/m\({}^{3}\) and 1.225 kg/m\({}^{3}\), respectively, in the numerical simulation. The viscosities of water and air are 1.003 \(\times\) 10\({}^{-3}\) kg/m-s and 1.79 \(\times\) 10\({}^{-5}\) kg/m-s, respectively. The interfacial tension \(\sigma\) between the droplet and the air is 0.0728 N/m. The droplet adhering to the hydrophilic surface of a windshield in a stationary state is shown in Figure 3a. In actual situations, the droplets appear cap-shaped on the surface. Considering the impact of droplet size on simulation results in numerical simulations, this study investigates droplets with volumes \(V\)' of 14.14 \(\mu\)L, 22.45 \(\mu\)L, and 33.51 \(\mu\)L, respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Boundary** & **Boundary Condition Setting** \\ \hline Inlet of the computational domain & Velocity inlet \\ Outlet of the computational domain & Pressure outlet \\ Upper wall and the side wall & Symmetric boundary \\
3D model and the bottom wall & No slip \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Boundary conditions of the computational domain. Figure 3: Schematic diagrams of the adhesive droplet: (**a**) Experiment; (**b**) Numerical stimulation. Figure 2: Schematic diagram of the computational domain. Using the spherical volume formula of \(V^{\prime}=\frac{4\pi}{3}\left(\frac{D}{2}\right)^{3}\), the spherical diameters \(D\) of the droplets are determined to be 3 mm, 3.5 mm, and 4 mm, respectively. Future measurements of droplet size will be uniformly annotated based on the spherical diameter \(D\). To assess the wettability of droplets on the hydrophilic windshield surface, the static contact angle \(\theta_{e}\) of the droplets was measured. On the substrate of the hydrophilic windshield used, the water droplets studied form a moderate static contact angle \(\theta_{e}=66.8^{\circ}\). As shown in Figure 3b, to determine the initial state of droplet motion on the windshield, the initial shape of the droplet on the hydrophilic windshield is considered to be a spherical cap.