## 3 Numerical Simulation Results and Discussions
### Construction and Validation of Circulating Air Mixing Ratio Model  
It can be learned from the analysis above that: (1) \(\varphi_{\rm c}\) has no direct relation with the air velocity value but is related to the ratio of air velocity at the inlet and the outlet--\(u_{\rm in}/u_{\rm out}\); (2) with the increase of \(d\), \(\varphi_{\rm c}\) decreases nearly exponentially; (3) when \(\Delta l\leq 0\), \(\varphi_{\rm c}\) increases linearly with the increase of \(\Delta l/D\); when \(\Delta l\)>0, however, \(\varphi_{\rm c}\) will first increase and then decrease with the increase of \(\Delta l/D\); the bigger the \(d\) is, the bigger the \(\Delta l/D\) will be when \(\varphi_{\rm c}\) is the maximum. Therefore, the functional relationship between \(\varphi_{\rm c}\) and \(u_{\rm in}/u_{\rm out}\), \(d\), \(\Delta l\), and \(D\) can be described in the following forms:  
\[\varphi_{c}=f(u_{\rm in}/u_{\rm out},d,\Delta l/dD) \tag{4}\]  
Based on the CFD simulation results, setting \(\Delta l=0\) as the demarcation point, the fitted piecewise expressions of \(\varphi_{\rm c}\) can be obtained as follows via the software Matlab with the gradient descent method:  
\[\varphi_{c}=\left\{ \begin{array}{ll}0.27{\rm e}^{-0.4d}\left(\frac{u_{\rm in }}{u_{\rm out}}\right)^{1.87}\left(0.82\Delta l/Dd+0.26\right),\ \Delta l\leq 0\\ {\rm e}^{-2.06d^{2.11}}\left(\frac{u_{\rm in}}{u_{\rm out}}\right)^{0.94} \left(2.36{\rm e}^{-0.03\Delta l/Dd}-2.44{\rm e}^{-1.29\Delta l/Dd}\right),\ \Delta l>0\end{array} \right. \tag{5}\]  
Figure 13: Impacts of \(\Delta l/D\) on circulating air mixing ratio \(\varphi_{\rm c}\). A comparison between calculated results of Equation (5) and simulated values of \(\varphi_{\rm c}\) is given in Figure 14. The goodness of fit \(R_{\rm new}\) is 0.99. The circulating air mixing ratio model (Equation (5)) was validated by the measured data of two actual tunnels with small spacing. Both of these tunnels are one-way and double-lane tunnels with a horseshoe-shaped cross section. The structural parameters of the two tunnels are shown in Table 1.  
The field measurement was conducted on a windless day. Because of the toxicity of CO, this test replaced CO with CO\({}_{2}\), which is less harmful, cheaper, and easier to obtain and measure. The pollutant-releasing device of the model test mainly included a CO\({}_{2}\) cylinder, a CO\({}_{2}\) two-stage pressure-reducing valve (better pressure stabilizing effect), and a vortex flow meter. In the experiment, CO\({}_{2}\) gas was released at about 100 m away from the outlet into the outflow tunnel so as to ensure that CO\({}_{2}\) was well mixed at the outlet section. The locations of measuring points in #1 tunnel are shown in Figure 15.  
The jet velocity of the jet fan was regulated in the tunnel in order to change inlet and outlet air velocity. The air velocity of the inlet and outlet tunnel was obtained by a Testo425 thermal anemometer (Testo AG, Lenzkirch, Germany). In fact, the air velocity of a single point on the cross section of the tunnel cannot represent the average air velocity of the entire cross section. Thus, several air \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **Length L/m** & **Cross-Sectional Area \(A\)/m\({}^{2}\)D/m** & \(d\)/m** & \(\Delta l\)/m** & **Front Slope of Tunnel Portal\({}^{o}\)** \\ \hline \#1 & 3875 & 54.3 & 7.6 & 8 & 43 & 15\({}^{\circ}\) \\ \#2 & 3580 & 67.3 & 8.4 & 6 & 0 & 20\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structural parameters of the tunnels. Figure 14: Comparison of calculated circulating air mixing ratio \(\varphi_{\rm c}\) with simulated results. Figure 15: The locations of measuring points. velocity measuring points were set on the cross section according to fluid's characteristics of velocity distribution on the cross section of the tunnel, with the distribution of the serial numbers and the effective areas being shown in Figure 15(a).