## 3 Results and Discussions
### Discussion about Nucleation Characteristic and Inlet Flow
#### 3.2.1 Influence of Homogeneous and Heterogeneous Nucleation Rates  
Here two test cases of conduction-controlled water-ice phase change without inflow of cryogenic fluid (case 1, 2) are studied. For the simplified practical process as shown in Figure 1c, the left boundaries are designed to be inlet boundary to coincide with the situation of Figure 1c, and the temperatures on the inlet and wall boundaries are designed to be the same to study the difference of homogeneous and heterogeneous nucleation due to the different kinds of boundaries. Furthermore, different shape coefficients of nucleation are designed in Case 1 and 2 to distinguish the phase-change processes between the ice generation on the wall and the ice growth on the frozen ice, which should be closer to be practical situation. The boundary condition settings of Case 1 and 2 are listed in Table 3.  
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 shape coefficients of nucleation \(\alpha_{\text{ey}}\) on the velocity-inlet boundary and wall boundary are set to 1.0 and 0.1, respectively. It reflects the difference in nucleation characteristic at wall boundary and in the fluid. Meanwhile, in order to understand the influence of the shape coefficient of nucleation on ice formation and growth, different values of \(\alpha_{\text{ice}}\) (shape coefficient on the ice) are specified with 0.1 in case 1 and 0.2 in case 2. Simulation Results  
For case 1, the distribution of ice phase at different moments are simulated and shown in Figure 7 to describe the process of ice growth. At the moment of t = 1.0 s which is shown by Figure 7a, due to the low temperature and high rate of crystallization \(J\) on the wall boundary, the ice phase firstly appears on the upper and lower boundaries. The formation of the ice at the wall boundary affects the \(J\) of water near the newly generated ice. Meanwhile, the low temperature of left boundary causes the formation of discontinuous loose ice near the velocity-inlet boundary. After a while, the form of ice phase on the three boundaries (upper, lower, and left) becomes regular and continuous at t = 5.0 s. With the time advancing, the ice phase gradually develops towards the interior of water area at t = 10.0 s. Therein, the ice layers on the upper and lower boundaries are always thicker than those on the left boundary. From Figure 7, it can be concluded that the phenomenon of ice formation and propagation can be well simulated by the numerical method. Moreover, by comparing the ice layer thickness on the left velocity-inlet boundary with that on the horizontal wall boundaries in Figure 7a, it is noted that the present method not only can consider the effect of the low temperature but also can reflect the difference between the rate of mass transfer due to homogeneous nucleation and heterogeneous nucleation for actual physical processes. \begin{table}
\begin{tabular}{c c} \hline
**Boundary** & **Conditions** \\ \hline Left & \(u_{l}=0\), \(v_{l}=0\), \(T_{l}=253.15\), \(u_{s}=0\), \(v_{s}=0\), \(T_{s}=253.15\), \(\alpha_{s}=0\) \\ Right & \(\frac{\partial u_{l}}{\partial x}=0\), \(\frac{\partial v_{l}}{\partial x}=0\), \(\frac{\partial T_{l}}{\partial x}=0\), \(\frac{\partial u_{e}}{\partial x}=0\),Figure 8 shows the distribution of ice phase at two moments (t = 1.0 s and t = 10.0 s) in case 2 which is simulated by the present method. By comparing these results to Figure 7, it can be found that the differences between case 1 (Figure 7) and case 2 (Figure 8) mainly present the freezing rate on the formed ice, which can be expressed as follows in detail. Firstly, near the left boundary, there is no continuous ice phase on the left boundary until t = 10.0 s in case 2. Secondly, during the whole process, the ice layers of case 2 on the upper and lower boundaries have less thickness than case 1.