## 3 Numerical Method  
### Geometry of a Sessile Drop  
Three different cases in which the droplets rested on a flat, dry surface are shown in Figure 4. The droplets were set to the liquid phase. Ideally, the shape of a stationary droplet on a flat surface is determined by the surface tension of the droplet \(\gamma\), the adhesion coefficient of the liquid-solid surface \(\delta\), the volume of the droplet V, and the contact angle of the surface \(\theta\). The solution to the shape of a droplet at rest on a plane is a conicity derived from the Young-Laplace equation. The expression of the equation is as follows:  
\[\cos\theta=\frac{\sigma}{\gamma}-1 \tag{1}\]  
Each point on the surface of the droplet can be obtained in the following form of the Young-Laplace equation:  
\[\gamma\bigg{(}\frac{1}{R_{1}}+\frac{1}{R_{2}}\bigg{)}=\Delta p \tag{2}\]  
\(\Delta p\) indicates the pressure difference between the inside and outside of the droplet. \(R_{1}\) and \(R_{2}\) are the two main radii of the curvature of the surface at that point. The Young-Laplace equation is a partial differential equation in which different boundary conditions determine the shape of the droplet. The final shape of the water droplet is the result of the balance of surface tension and gravity. At the origin coordinates, the equation in Equation (3) is satisfied. \[x(0)=z(0)=\alpha(0)=V(0)=A(0)=0 \tag{3}\]  
The governing equation for droplets standing on a plate is as follows:  
Figure 4: The shapes of droplets in three different cases. Figure 3: Outline of the initial state of the droplet. \[\left\{ \begin{array}{l}\dfrac{dx}{ds}=\cos\alpha\\ \dfrac{dz}{ds}=\sin\alpha\\ \dfrac{dx}{ds}=2b+cz-\dfrac{\sin\alpha}{x}\\ \dfrac{dV}{ds}=\pi x^{2}\sin\alpha\\ \dfrac{dA}{ds}=2\pi x\end{array} \right. \tag{4}\]  
In Equation (3), \(b\) is the curvature of the curve at the initial point, which can be used as a target parameter, \(c\) is the curvature of the curve at the point \((x_{i},z_{i})\), \(s\) is the contour curve of the droplet, \(\alpha\) is the angle between the tangent at the point \((x_{i},z_{i})\) on \(s\) and the \(x\)-axis, \(V\) is the volume of the droplet, and \(A\) represents the cross-sectional area of the droplet. When the value of \(s\) is equal to 0, the form of the third equation in the formula needs to be changed to Equation (4). \[\dfrac{d\alpha}{ds}=b \tag{5}\]  
According to this formula, using the volume of the water droplet and the contact angle of the wall surface in the initial stage, the contour of the water droplet can be calculated at the known gravitational acceleration, the surface tension of the droplet, and the droplet density. The initial conditions for the numerical simulation of static droplet freezing in this study were the droplet shapes calculated by this equation. In this paper, the time from the appearance of the growth peak at the bottom of the droplet to the time when the growth peak reaches the top of the droplet is defined as the complete freezing time \(\mathrm{T_{t}}\). The accuracy of the experiment was tested by comparing the heights of the growth peaks at different times. The complete freezing time \(\mathrm{T_{t}}\) of the droplet was used to measure the influence of different factors on the freezing rate. A physical model, as shown in Figure 5, was established based on the water droplet profile calculated by the Young-Laplace equation. Figure 6 shows the location and boundary conditions of the droplets in the computational domain on the left and the grid on the right. The left side is the rotational symmetry axis, the right and upper sides are the pressure outlet, and the lower side is the wall surface. Different cold surfaces can be obtained by setting the wall surface temperature, and the ambient temperature can be set according to the needs of the study. Water droplets and air are modeled separately, and both use structured grids.