## 4 Results and Discussion
### Hoods at Tunnel Entrance
For all lengths, there are two peaks marked as \(w_{\rm t}\) and \(w_{\rm h}\) in the first stage of compression wave, which superpose together before they propagate to the tunnel exit. As the length of the hood increases, the time when \(w_{\rm t}\) and \(w_{\rm h}\) appear earlier with \(w_{\rm r}\) delays gradually, while the time when \(w_{\rm p}\) arrives remains unchanged. When the hood length is less than 40 m, \(w_{\rm h}\) tends to overlap with \(w_{\rm t}\) during the propagation, even with an inclination of superposing with \(w_{\rm r}\), leading to a significant increase in the pressure gradient and the MPW. When \(l_{\rm h}<50\) m, the amplitudes of the pressure gradient and the MPW do not change with the length of the hood. Therefore, a sufficient length should be ensured, but excessive length cannot provide further relief effects. In subsequent calculations, \(l_{\rm h}=80\) m is taken to eliminate the influences of hood length on the tunnel aerodynamics. The pressure gradients of the compression wave at RP point with different cross-section areas of hood are shown in Figure 8a. When \(A_{\rm t}>200\) m\({}^{2}\), the pressure gradient of \(w_{\rm p}\) is greater than \(w_{\rm h}\), which indicates that the compressibility effect caused by a train entering a tunnel plays a dominant role. With the decrease of the cross-section area, the compressibility effect of the train entering the tunnel weakens, resulting in the decrease of the pressure gradients of \(w_{\rm h}\) and \(w_{\rm r}\). However, the compressibility effects of the train entering the hood and passing through the opening increase, leading to the increase and distinction of the pressure gradients of \(w_{\rm t}\) and \(w_{\rm h}\), which means \(w_{\rm h}\) is dominant. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Type of Hoods**} & \multicolumn{3}{c}{\(\partial\varphi/\partial t\)} & \multicolumn{2}{c}{\(\Delta\varphi_{\rm m}\)} \\ \cline{2-5}  & **Maximum Value (kPa/s)** & **Mitigation Rate** & **Maximum Value (Pa)** & **Mitigation Rate** \\ \hline Without hood & 17.49 & / & 130.20 & / \\ Enlarged hood & 11.70 & 33.10\% & 88.90 & 31.72\% \\ Oblique hood & 9.93 & 43.22\% & 80.40 & 38.25\% \\ Hood with holes & 8.05 & 53.97\% & 62.76 & 51.80\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum values of the pressure gradient and MPW with different hoods. Figure 7: The effects of hood length on tunnel aerodynamics: (**a**) Pressure gradient at RP point; (**b**) MPW at RM point. The MPW at RM point with varied hood cross-section areas is shown in Figure 8b. The waves \(w_{\text{t}}\) and \(w_{\text{h}}\) have been superimposed into a single wave (denoted as \(w_{\text{h}}\)) due to the influence of inertial effects, and the superposition effect is more significant with a smaller hood area. Affected by this, the MPW caused by the train entering the tunnel will surpass other effects when a larger hood area is adopted. Therefore, there is an optimal hood area where the compressibility effects of the train passing through the holes and entering the tunnel are equivalent, thereby minimizing the amplitude of the MPW. Figure 9 shows the variation of the peak values of the pressure gradient and the MPW of the three stages. The curves show that the first peak decreases with increasing area, while the second and third peaks increase. The minimum value of the pressure gradient appears near \(A_{\text{t}}\) = 200 m\({}^{2}\), while the minimum value of the MPW appears near \(A_{\text{t}}\) = 225 m\({}^{2}\), which means that the optimal hood area is related to the tunnel length due to the inertia effect. In addition, when \(A_{\text{t}}\) > 155 m\({}^{2}\), the change rate of the first peak decreases gradually. Considering factors such as construction costs, \(A_{\text{t}}\) = 155 m\({}^{2}\) is used in subsequent calculations. Figure 8: The effects of cross-section area of entrance hood on tunnel aerodynamics: (**a**) Pressure gradient at RP point; (**b**) MPW at RM point.