# Train-Induced Unsteady Airflow in a Metro Tunnel with a Ventilation Shaft
## 2 Theoretical Analysis
### Validation  
The validity and practicality of the theoretical model employed in this study have been corroborated by numerous scholarly sources [4; 63]. Specifically, [PERSON] et al. [4] established this theoretical model for tunnels without ventilation shafts and performed numerical simulation analyses based on the experiments conducted by [PERSON] et al. [13]. Their findings indicate that the computational results \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Governing Equation**} & \multicolumn{3}{c}{\(\frac{dv}{dt}=A+Bv+Cv^{2}\)} \\ \hline Steps & PPW-TET & PPW-TCT & PPW-TCL & PPW-ATL \\ \hline Time & & \(t\in\left[0,\frac{L_{T}}{V_{T}}\right]\) & \(t\in\left[\frac{L_{T}}{V_{T}},\frac{1}{V_{T}}\right]\) & \(t\in\left[\frac{L_{T}}{V_{T}},\frac{1}{V_{T}}\right]\) & \(t\in\left[\frac{L_{T}}{V_{T}},\infty\right]\) \\ \hline \multirow{4}{*}{Equation parameters} & A & \(\frac{k_{1}V_{T}^{2}(1-a)}{2[L(1-a)+l_{12}a]}\) & \(\frac{k_{1}V_{T}^{2}(1-a)}{2[(l_{12}+l_{34})(1-a)+L_{T}]}\) & \(\frac{k_{1}V_{T}^{2}(1-a)}{2[l_{12}(1-a)+l_{3}]}\) & \(0\) \\ \cline{2-6}  & B & \(\frac{k_{1}V_{T}(1-a)}{L(1-a)+l_{12}a}\) & \(-\frac{k_{1}V_{T}(1-a)}{(l_{12}+l_{34})(1-a)+L_{T}}\) & \(-\frac{k_{2}V_{T}(1-a)}{l_{12}(1-a)+l_{23}}\) & \(0\) \\ \cline{2-6}  & C & \(\frac{k_{2}V_{T}(1-a)}{L(1-a)+l_{12}a}\) & \(\frac{(k_{2}-k_{1})(1-a)}{2[(l_{12}+l_{34})(1-a)+L_{T}]}\) & \(\frac{(k_{2}-k_{1})(1-a)}{2[l_{2}(1-a)+l_{32}]}\) & \(k_{1}\) \\ \cline{2-6}  & \(k_{1}\) & \(\xi_{out}+\lambda_{L}\frac{L-l_{12}}{D_{L}}-1\) & \(\xi_{in}+\lambda_{L}\frac{l_{12}+l_{34}}{D_{L}}+\xi_{out}\) & \(\xi_{in}+\lambda_{L}\frac{l_{12}}{D_{L}}+1\) & \(-\frac{\xi_{in}+\lambda_{L}\frac{l_{1}}{D_{L}}+1}{2L}\) \\ \cline{2-6}  & \(k_{2}\) & \(\frac{\xi_{in}+\lambda_{L}\frac{l_{12}}{D_{L}}+\xi_{out}}{(1-a)^{2}}\) & \(\frac{\xi_{in}+\lambda_{L}\frac{l_{12}}{D_{L}}+\xi_{out}}{(1-a)^{2}}\) & \(\frac{\xi_{in}+\lambda_{L}\frac{l_{12}}{D_{L}}+\xi_{out}}{(1-a)^{2}}\) & \(0\) \\ \hline \multirow{2}{*}{Remark} & \multirow{2}{*}{\(l_{12}=V_{T}t\)} & \multirow{2}{*}{\(l_{12}+l_{34}=L-L_{T}\)} & \multirow{2}{*}{\(l_{12}=V_{T}t-L_{T}\)} & \multirow{2}{*}{/} \\  & & & & \(l_{23}=L+L_{T}-V_{T}t\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Theoretical model of the piston wind without ventilation shaft. Figure 3: Schematic of the piston effect of a train traveling through a tunnel without ventilation shaft (**a**) PPW-TET, (**b**) PPW-TCT, (**c**) PPW-TCL, and (**d**) PPW-ATL. of the model are better than those of three-dimensional and two-dimensional numerical simulations. In our prior research [14], we not only developed but also validated the theoretical model concerning piston wind and examined the impact of various factors influencing the evolution of piston wind. The findings demonstrated that the theoretical results were in good agreement with the experimental results. In the current study, we employed the same methodology to construct a theoretical model of piston wind for metro tunnels both with and without a ventilation shaft. For tunnels equipped with a ventilation shaft, we further confirmed the model's precision through three-dimensional numerical simulations. The geometric model was constructed utilizing ICEM based on the dimensions of a typical tunnel, train, and shaft, as specified in Table 3. The numerical domain underwent discretization through structured meshes. To update the volume mesh, the dynamic layering method facilitated by the FLUENT code was implemented. The hexahedral grid features a length of 1 m in the x-direction, with a maximum dimension of 0.2 m in both y and z directions. Consequently, the aggregate mesh count surpasses 4.01 million. The duration of the unsteady simulation extended to 600 s, with a physical time step size set at 0.05 s. For further details on boundary conditions and solver settings used in the numerical simulation, refer to our prior studies [33; 34].