# Experimental Study on Freezing Front Model of Alpine Tunnel under Wind Field
## 4 Conclusions  
1. With the decrease of the test air temperature, the thermonuclear region in the surrounding rock of the mountain gradually shrinks, the freezing area of the surrounding rock at the arch bottom gradually increases, and the freezing area of the surrounding rock at the arch crown gradually expands to the inside of the mountain; the influence of ventilation on the longitudinal and radial temperature distribution of the tunnel is more obvious, and the greater the wind speed is, the greater the influence is. In particular, the longitudinal temperature distribution of the tunnel is more widely affected by ventilation.
2. The inverted arch is not paved with a thermal insulation layer, and so lacks protection. The lining concrete is easily frozen by cold air, leading to the maximum distance of the freezing front in the inverted arch area. The colder air gathers at the foot of the  
Figure 14: Fitting curve of longitudinal frozen front: (**a**) Vault; (**b**) Side wall; (**c**) The foot of the wall; (**d**) Inverted arch.  
\begin{table}
\begin{tabular}{c c c} \hline
**Position** & **Governing Equation** & **Correlation Coefficient** \\ \hline Vault & \(y=9.71\times 10^{-5}x^{2}-0.054x+5.77\) & 0.9898 \\ Side wall & \(y=7.05\times 10^{-4}x^{2}-0.052x+6.16\) & 0.9762 \\ The foot of the wall & \(y=1.66\times 10^{-4}x^{2}-0.067x+6.31\) & 0.9847 \\ Inverted arch & \(y=1.49\times 10^{-4}x^{2}-0.098x+11.76\) & 0.9880 \\ \hline \end{tabular} Note: \(y\) in the table is the freezing depth; \(x\) is the tunnel depth. Unit: cm.  
\end{table}
Table 3: Governing equations of longitudinal freezing front at different positions.  
tunnel wall and the invert, and the expansion distance of the freezing front at the foot of the wall is obviously higher than that at the vault. (3) The control equations of the radial three zone freezing front and the longitudinal quadratic parabola of the tunnel model under ventilation conditions are determined. The change of the longitudinal freezing front at different locations of the tunnel is basically a parabola attenuation, and with the increase of tunnel depth the absolute value of the slope of the control curve decreases linearly, the change trend of the freezing front gradually eases and finally tends to be stable and disappear. Through this control equation, the specific distribution characteristics of the freezing front at different depths of the whole model tunnel can be predicted.  
The research limit of this paper is the size effect of the model tunnel. The smaller model tunnel may affect the accuracy of mountain temperature transmission. In the future, research will focus on the influence of ventilation environments on the frozen front of large size tunnel models in cold regions.  
Conceptualization, L.L.; validation, J.Z.; writing--original draft preparation, Z.L. ([PERSON]); supervision, Z.L. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
China University Innovation Fund (2021 DZ023); Weifang Science and Technology Development Plan (2021 GX005).  
Not applicable.  
The authors declare no conflict of interest.