## 3 Results and Discussions
### Sensitivity Study and Model Validation  
In this section, the classical Stefan problem [42; 51; 52] is simulated for model validation. Here the test cases with different mesh numbers (1600, 6400, and 25,600) and time steps (0.0005 s, 0.001 s, and 0.002 s) are designed to study the numerical convergence of the proposed method. The boundary conditions for model validation are listed in Table 1.  
Initial Conditions  
The fluid zone of the computational domain is fully filled with liquid water with a temperature of 283.15 K (10 \({}^{\circ}\)C). The properties of water and ice are listed in Table 2. The numerical study in this paper is carried out at atmospheric pressure within a temperature range from \(-20\)\({}^{\circ}\)C to \(10\)\({}^{\circ}\)C. It is assumed that the temperature variation would not induce the change of the thermo-physical properties of water and ice. The pressure variation during the simulation is also insignificant compared with atmospheric pressure. Therefore, the thermo-physical properties are set as constants. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Boundary** & \multicolumn{3}{c}{**Conditions**} \\ \hline Left & \(u_{l}=0,\;v_{l}=0,T_{l}=253.15,u_{s}=0,v_{s}=0,T_{s}=253.15,\frac{\partial u_{ s}}{\partial n}=0\) \\ Right & \(u_{l}=0,\;v_{l}=0,\frac{\partial T_{l}}{\partial n}=0,u_{s}=0,v_{s}=0,\frac{ \partial T_{l}}{\partial n}=0,\frac{\partial u_{s}}{\partial n}=0\) \\ Upper & \(u_{l}=0,\;v_{l}=0,\frac{\partial T_{l}}{\partial n}=0,u_{s}=0,v_{s}=0,\frac{ \partial T_{l}}{\partial n}=0\) \\ Lower & \(u_{l}=0,\;v_{l}=0,\frac{\partial T_{l}}{\partial n}=0,u_{s}=0,v_{s}=0,\frac{ \partial T_{l}}{\partial n}=0,\frac{\partial u_{s}}{\partial n}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The settings of the boundary conditions (model validation). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Materials** & \(\rho\) **(kg m\({}^{\text{\text{\text{23}}}}\))** & \(C_{p}\) **(J kg\({}^{\text{\text{\text{-1}}}}\) K\({}^{-\text{\text{1}}}\))** & \(K\) **(W m\({}^{\text{\text{\text{-1}}}}\) K\({}^{-\text{\text{1}}}\))** & \(\mu\) \\ \hline Water & 996.32 & 4318 & 0.5263 & 0.00176 \\ Ice & 916.2 & 2050 & 2.22 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermo-physical properties of water (0.5 \({}^{\circ}\)C, 0.1 MPa) and ice (0 \({}^{\circ}\)C) [53]. In the numerical simulation, the volume fraction change of the ice phase indicates the formation and grow of the ice layer. There is no ice in the domain at the initial state, which suggests the volume fraction of ice phase would equal to zero in the whole domain. The ice would gradually form, leading to the volume fraction of the ice phase increase along with the time goes on. As the cell plotted in Figure 3 where the ice forms and develops, is adopted to demonstrate the ice layer. The ice-water interface is tracked by the changing of the volume fraction (ice phase) in the model. Then the location of the water-ice interface is determined as the contour of ice phase with the average value of volume fraction as 0.5. On this basis, the sensitivity test about the mesh number and time step for the conduction-controlled water-ice phase change problem is conducted and the result is plotted in Figure 4. Therein, the effect of mesh numbers (mesh schemes 1600, 6400, and 25,600) on the interface location evolution is given in Figure 4a. While, results for sensitivity tests of the time steps versus 0.0005 s, 0.001 s, and 0.002 s are plotted in Figure 4b, respectively. It is seen from Figure 4a that along with the mesh number increasing, the curves of the interfacial position evolution tend to coincide, indicating the convergence of the present method. In consideration of the computational accuracy and cost, the mesh number of 6400 (grid size 0.125 mm) and the time step of 0.001 s are suitable for the numerical simulation of freezing process in this work.