## 3 Results and Discussions
### Discussion about Nucleation Characteristic and Inlet Flow
#### 3.2.2 Interaction of Fluid Field and Ice Phase  
In this section, the interaction between ice growth and fluid flow (case 3, 4, 5, and 6) is studied. For the practical situation, the cooling effect of the solid wall may be generally stronger than the formation water, which possibly causes the existence of the frozen ice at the initial moment of the displacement flow process. Furthermore, depending on the different conditions, the distributions of the initial frozen ice should be different. Therefore, four test cases with different initial ice phase are designed to study the displacement flow process with the growth of ice phase. The boundary conditions for case 3, 4, 5, and 6 are listed in Table 4. The shape coefficients of nucleation \(\alpha_{\text{ey}}\) in the fluid domain, on the ice and on the wall boundary are 1.0, 0.1, and 0.0001, respectively. In the case study of the current section, the phase of cryogenic fluid is specified as n-hexanol (CH\({}_{3}\)(CH\({}_{2}\))\({}_{5}\)OH). This substance could exist as steady liquid form at \(-40\)\({}^{\circ}\)C (Melting point: \(-52\)\({}^{\circ}\)C, Boiling point: \(157\)\({}^{\circ}\)C). Moreover, its solubility in water is quite low (6 g/L, 25\({}^{\circ}\)C). Considering the test conditions (below 0\({}^{\circ}\)C most time), n-hexanol is an ideal choice for conducting the freezing-coupling immiscible displacement process with water. The thermal properties of n-hexanol is listed in Table 5.  
Similar to those of water and ice, the thermo-physical properties of n-hexanol are set as constants in simulations. To investigate the influence of the ice phase distribution on the flowing state, different distributions of the ice phase in the computational domain are specified as the initial conditions. To obtain the initial ice phase distribution data, a case of conduction-controlled water-ice phase change without any fluid inflow (case ex) is carried out with the boundary conditions listed in Table 6. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Boundary** & **Conditions** \\ \hline Left & \(u_{l}=0.01,\ v_{l}=0,T_{l}=253.15,u_{s}=0,v_{s}=0,T_{s}=253.15,\alpha_{s}=0\) \\ Right & \(\frac{\partial u_{l}}{\partial x}=0,\ \frac{\partial v_{l}}{\partial x}=0,\ \frac{\partial u_{l}}{\partial x}=0,\ \frac{\partial u_{l}}{\partial x}=0,\ \frac{\partial v_{l}}{\partial x}=0,\ \frac{\partial T_{l}}{\partial x}=0,\ \frac{\partial u_{l}}{ \partial x}=0\) \\ Upper & \(u_{l}=0,\ v_{l}=0,T_{l}=253.15,u_{s}=0,v_{s}=0,T_{s}=253.15,\frac{\partial u_{ s}}{\partial t}=0\) \\ Lower & \(u_{l}=0,\ v_{l}=0,T_{l}=253.15,u_{s}=0,v_{s}=0,T_{s}=253.15,\frac{\partial u_{ s}}{\partial t}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The settings of the boundary conditions (Case 3, 4, 5, 6). Figure 9: Comparison of temperature distribution (K) in both cases at t = 10.0 s (**a**) Case 1 and (**b**) Case 2. Initial condition in Case ex is set as: The fluid zone of the computational domain is fully filled with liquid water with a temperature of 283.15 K (10 \({}^{\circ}\)C). The main difference between case ex and case 1, 2 is that the temperature at left boundary is set as 283.15 K (10 \({}^{\circ}\)C). Therefore, the ice phase in case ex would only form and grow on the two horizontal boundaries (upper and lower). Different situations of initial ice phase distribution are obtained by importing the data of ice phase distribution at different moments in case ex into the current study cases. For this time, ice distribution at t = 50 s, 100 s, and 150 s in case ex is adopted and the distribution state in the computational domain is shown by Figure 10.  
Besides the above three initial ice phase distribution conditions, the case where there is no initial ice phase is also considered. Accordingly, four cases are included in the case study in this section. The initial ice phase distributions of the four cases are:  
(1) Case 3 (no initial ice phase),  
(2) Case 4 (Figure 10a),  
(3) Case 5 (Figure 10b), and  
(4) Case 6 (Figure 10c).