# Experimental Study on Freezing Front Model of Alpine Tunnel under Wind Field
## 3 Experimental Results and Analysis
### Analysis of Freezing Front in Tunnel Model
#### 3.2.1 Radial Analysis of Freezing Front  
The study of radial freezing front has important reference value for the radial non-uniform laying of an insulation layer. In the study, the insulation layer, lining and surrounding rock are considered as a whole. When it is stable, the tunnel radial 0 \({}^{\circ}\)C line is fitted. It can be seen from Figure 12 that the change rule of the tunnel radial freezing depth is basically in the form of a quadratic parabola function (under different function types, the fitting correlation coefficient of the quadratic parabola function is the highest). The distribution of freezing depth at different sections of the tunnel is similar. Section 1 (test air temperature \(-\)10.6 \({}^{\circ}\)C, wind speed 3.6 m/s at the entrance) is taken as the fitting analysis object of freezing front. As long as the distribution law of radial freezing front at  
Figure 11: Vector Diagram of Tunnel Longitudinal Temperature Distribution at Test Air Temperature \(-\)10.6 \({}^{\circ}\)C: **(a)** 0 m/s; **(b)** 1.8 m/s. the entrance of the tunnel is obtained, the distribution characteristics of extending the range of radial freezing front at different sections can be determined along different longitudinal depths of the tunnel. Due to the different distribution forms of the radial freezing front, in order to better analyze the tunnel radial freezing front it is divided into three regions (Region I, Region II and Region III), as shown in Figure 12a. The control curve thus obtained is more consistent with the distribution form of the freezing front in each region. Figure 12b-d show the fitting curve of the freezing front in the three regions. (By comparing the correlation coefficients of other types of function fitting equations, we see that the correlation coefficients of function fitting equations in Table 2 are the highest.) \begin{table}
\begin{tabular}{c c c} \hline \hline
**Region** & **Governing Equation** & **Correlation Coefficient** \\ \hline Region I & \(y=455.55e^{-\frac{(y-4.15)}{1+4.65}}-440.01\) & 0.9413 \\ Region II & \(y=83.91e^{\left\lfloor\frac{\pi}{2}\right\rfloor}-9.19\) & 0.9128 \\ Region III & \(y=0.04x^{2}-0.002x-18.67\) & 0.9794 \\ \hline \hline \end{tabular} Note: \(y\) in the table represents the buried depth of the tunnel; \(x\) is the distance from the tunnel centerline. Unit: cm.  
\end{table} 
Table 2: Governing equations of freezing front in different regions. Figure 12: Fitting curve of frozen front in different regions: (**a**) Division area of frozen front; (**b**) Fitting curve of frozen front in Region I; (**c**) Fitting curve of frozen front in Region II; (**d**) Fitting curve of frozen front in Region III. It can be seen from the above three regional freezing front curves that the freezing front has reached a stable state, and the cold air in the tunnel makes the freezing front generate and move forward. At this time, there is no cold air to make the freezing front continue to move forward until the melting front reaches the location of the freezing front and stops. The freezing front will disappear outside the fitting curve, and a freezing thawing balance surface will appear. The freezing front distance in Region III is the largest, mainly because the inverted arch is not paved with a thermal insulation layer, which means it lacks protection, and the concrete lining can easily be frozen through by cold air; the expansion distance of the freezing front at the foot of the wall in Region II is obviously higher than that of the freezing front at the vault in Region I, mainly because the cooler air is generally concentrated at the foot of the wall and the invert of the tunnel. The specific control equation of the frozen front is shown in Table 2.  
In order to better fit the freezing front, three function types are used for fitting analysis. The correlation coefficients of the freezing front fitting functions obtained from the 3 regions are all above 0.9, indicating that the fitting effect is correct.