# Train-Induced Unsteady Airflow in a Metro Tunnel with a Ventilation Shaft
## 2 Theoretical Analysis
### PW-Wys  
According to the Bernoulli equation for unsteady flow, a theoretical model was developed to analyze the piston effect in a metro tunnel with a ventilation shaft [4; 14; 31]. The Bernoulli equation applied in this study is articulated as Equation (1):  
\[\frac{\rho v_{1}^{2}}{2}+P_{1}+\rho gz_{1}=\frac{\rho v_{2}^{2}}{2}+P_{2}+\rho gz _{2}+h_{f}+\rho\varepsilon\int_{S}\frac{\partial v}{\partial t}ds \tag{1}\]  
where \(\rho\) is the ambient air density (kg/m\({}^{3}\)), \(v_{1}\) (m/s) and \(v_{2}\) (m/s) are the velocity at cross-section 1 and cross-section 2,\(P_{1}\) (Pa) and \(P_{2}\) (Pa) are the pressure at cross-section 1 and cross-section 2, \(z_{1}\) (m) and \(z_{2}\) (m) are the height of cross-section 1 and cross-section 2, \(g\) (m/s\({}^{2}\)) is the acceleration of gravity, \(h_{f}\) (Pa) represents resistance loss, \(\rho\varepsilon\int_{S}\frac{\partial v}{\partial t}ds\) (Pa) is the inertia head representing the amount of kinetic energy of fluid per unit mass changing over time, and \(\varepsilon\) is a correction factor of the uneven flow velocity in the tunnel, respectively. Since the cross-sectional area of the tunnel is constant, \(\varepsilon\approx 1\) is obtained in this study [6; 25]. When the metro train travels inside the tunnel, according to the continuity equation [32], Equations (2) and (3) can be obtained:  
\[A_{T}V_{t}dt=A_{L}vdt+(A_{L}-A_{T})wdt \tag{2}\]  
\[v_{h}=w+V_{t}=\frac{A_{T}V_{T}-A_{L}v}{A_{L}-A_{T}}+V_{t}=\frac{V_{t}-v}{1-\alpha} \tag{3}\]  
where \(\alpha=A_{T}/A_{L}\), \(w\) is the velocity of airflow relative to the tunnel wall in the annular space and \(v_{h}\) is the velocity of airflow relative to the train in the annular space. In accordance with the aforementioned Bernoulli and continuity equations, the dynamics of unsteady airflow in PW-WVS mode are meticulously analyzed. The section extending from the tunnel entrance to the shaft position is designated as the tunnel's upstream, while the segment from the shaft position to the tunnel exit is identified as its downstream. As shown in Figure 2, the variation in the piston wind movement covers six \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Physical Meaning** \\ \hline \(v_{s}\) & Velocity of airflow in ventilation shaft (m/s) \\ \(w\) & Velocity of airflow relative to the tunnel walls in the annular space (m/s) \\ \(v_{h}\) & Velocity of airflow relative to the train in the annular space (m/s) \\ \(v_{h2}\) & Velocity of airflow relative to the train in the annular space downstream (m/s) \\ \(\alpha\) & Blockage ratio of the train to the tunnel \\ \(\beta\) & Ratio of shaft cross-sectional area to tunnel cross-sectional \\ \(\lambda_{L}\) & Frictional resistance coefficient of airflow along the tunnel walls \\ \(\lambda_{h}\) & Frictional resistance coefficient of airflow along the annular space \\ \(\lambda_{s}\) & Frictional resistance coefficient of airflow along the ventilation shaft walls \\ \(D_{L}\) & Hydraulic diameter of the tunnel (m) \\ \(D_{h}\) & Hydraulic diameter of the annular space (m) \\ \(D_{s}\) & Hydraulic diameter of the ventilation shaft (m) \\ \(L\) & Length of the tunnel (m) \\ \(L_{T}\) & Length of the train (m) \\ \(L_{u}\) & Length of the tunnel upstream (m) \\ \(L_{d}\) & Length of the tunnel downstream (m) \\ \(l_{ij}\) & Distance between cross-sections \(i\) and \(j\) (m) \\ \(\xi_{in}\) & Local drag coefficient at the inlet of the tunnel \\ \(\xi_{out}\) & Local drag coefficient at the outlet of the tunnel \\ \(\xi_{hin}\) & Local drag coefficient at the inlet of the annular space \\ \(\xi_{baut}\) & Local drag coefficient at the outlet of the annular space \\ \(\xi_{sin}\) & Local drag coefficient at the inlet of the ventilation shaft \\ \(\xi_{sout}\) & Local drag coefficient at the outlet of the ventilation shaft \\ \(\xi_{i-j}\) & Local drag coefficient of the airflow from cross-section \(i\) to cross-section \(j\) \\ \hline \hline \end{tabular}
\end{table}