## 2 Temperature Field Coupling Theory (Airflow to Lining/Heat-Insulating Layer to Surrounding Rock)
### Theory of Airflow Temperature Field in Tunnel Based on Turbulence Equation  
The energy conservation equation employed in analyzing the temperature fields of airflows in a tunnel is expressed by:  
\[\frac{\partial(\rho\mathcal{C}_{\mathrm{P}}T)}{\partial t}+\
abla\cdot\left[- \lambda_{g}\
abla T+\rho C_{\mathrm{P}}\mathbf{u}\right]=S_{h}, \tag{1}\]  
where \(\rho\) is the air density; \(\mathcal{C}_{\mathrm{P}}\) is the specific constant pressure heat of air; \(T\) is the air temperature; \(t\) is the time; \(S_{h}\) is the internal heat; \(\lambda_{g}\) is the thermal conductivity of air; and \(\mathbf{u}\) is the airflow velocity. The mass conservation equation used to analyze the temperature fields of airflows inside a tunnel is given by:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\mathbf{u}\right)=0, \tag{2}\]  
In general, the airflow inside a tunnel is usually considered as a three-dimensional unsteady turbulent flow. Theoretically, the distribution of the turbulent airflow field can be obtained straightforwardly by solving the Navier-Stokes equations. However, the computational costs for large-scale numerical problems would be unacceptable. Therefore, the turbulent flow in this study is simulated using a simplified numerical model based on \(k\)-\(\epsilon\) equations, which are given by:  
\[\left\{ \begin{array}{c}\rho\frac{\partial k}{\partial t}+\
abla\cdot\left( \rho k\mathbf{u}\right)=\
abla\cdot\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}} \right)\
abla\cdot k\right]+G_{k}+G_{b}-\rho\epsilon-Y_{M}+S_{k}\\ \rho\frac{\partial\epsilon}{\partial t}+\
abla\cdot\left(\rho\epsilon\mathbf{ u}\right)=\
abla\cdot\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right) \
abla\cdot\epsilon\right]+\frac{\mathcal{C}_{\mathrm{G}}\epsilon}{k}+\left[G _{k}+C_{\mathrm{\epsilon}3}G_{b}\right]-\frac{\mathcal{C}_{\mathrm{G}}\rho \epsilon^{2}}{k}+S_{\epsilon}\end{array} \right., \tag{3}\]  
where \(\mu_{t}\) is given by:  
\[\mu_{t}=\frac{\rho C_{\mathrm{P}}k^{2}}{\epsilon}, \tag{4}\]  
where \(k\) is the turbulent kinetic energy; \(G_{k}\) is the turbulent kinetic energy generated by the mean velocity gradient; \(G_{b}\) is the turbulent kinetic energy generated by buoyancy; \(Y_{M}\) is the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate; \(S_{k}\) and \(S_{\epsilon}\), are user-defined sources; \(\varepsilon\) is the dissipation rate of turbulent kinetic energy; \(\mu\) is the dynamic viscosity of air; \(\mu_{t}\) is the turbulent (or eddy) viscosity; and \(C_{\varepsilon 1}\), \(C_{\varepsilon 2}\), \(C_{\varepsilon 3}\), \(C_{\mu}\), \(\sigma_{k}\), and \(\sigma_{\varepsilon}\) are constants. The convective heat transfer coefficient between the fluid domain and the solid domain can be obtained by using the field measurement method, the empirical method, or the temperature wall function method. The field measurement method is the most straightforward in the three methods, but it highly depends on field conditions, the accuracy of measurement instrumentations, and measurement techniques. Due to this reason, the measurement of the convective heat transfer coefficient may not be accurate and may not always be reliable. Regarding the empirical method, so far, it is very difficult to find a proper mathematical model, which is recognized as the most reliable one by all researchers, although there are many existing mathematical models. For the wall function method, the convective heat transfer coefficient between the fluid domain and the solid domain can be determined based on the turbulent flow model. This means that once the parameters of the turbulent flow model are confirmed, the heat transfer coefficient can be calculated accordingly. It was found that the mesh resolution of the numerical model may affect the accuracy of the convective heat transfer coefficient, but this influence can be largely reduced after using a finer meshing scheme and doing the re-mesh procedure.