## 3 Theoretical Analysis and Numerical Model  
### Theoretical Analysis of Air Flow Direction  
The air flow in the ventilation network of the double-hole tunnel follows the law of resistance, the law of energy conservation, the conservation of nodal air flow volume, and the law of loop energy balance. Figure 4 is a ventilation network diagram of a double-hole, single-cross passage, wherein the junction of the cross passage and the advance side tunnel is node 3, the junction of the cross passage and the undercut side tunnel is node 4, the air flow direction in the tunnel depends on the pressure of nodes 3 and 4, the air flows from the node with high energy level to the node with low energy potential, and when the energy potential of the two nodes is the same, the air flow is stagnant.  
Figure 3: Air flow velocity distribution at the measurement points of the double-hole tunnel. (**a**) The air flow velocity distribution at measurement points of the undercut side tunnel; (**b**) the air flow velocity distribution at measurement points of the advance side tunnel.  
Figure 2: On-site air flow velocity test chart.  
According to the law of energy balance of the loop, Equation (3) can be obtained:  
\[R_{1}{Q_{1}}^{2}+R_{3}{Q_{3}}^{2}=R_{2}{Q_{2}}^{2}+R_{4}{Q_{4}}^{2} \tag{3}\]  
The pressure energy difference between node 3 and node 4 is:  
\[E_{3-4}=R_{1}{Q_{1}}^{2}-R_{2}{Q_{2}}^{2} \tag{4}\]  
When there is no air flow in the traffic cross passage, \(E_{3-4}=0\):  
\[R_{1}{Q_{1}}^{2}=R_{2}{Q_{2}}^{2},R_{3}{Q_{3}}^{2}=R_{4}{Q_{4}}^{2} \tag{5}\]  
From (5), it can be obtained:  
\[\frac{R_{1}R_{4}}{R_{2}R_{3}}=\left(\frac{Q_{2}Q_{3}}{Q_{1}Q_{4}}\right)^{2}=1 \tag{6}\]  
When the air flow direction in the cross passage is from node \(3\rightarrow\) node 4, the pressure energy of node 3 is higher than that of node 4, \(E_{3-4}>0\), which is:  
\[R_{1}{Q_{1}}^{2}>R_{2}{Q_{2}}^{2},R_{3}{Q_{3}}^{2}>R_{4}{Q_{4}}^{2} \tag{7}\]  
From (7), it can be obtained:  
\[\frac{R_{1}R_{4}}{R_{2}R_{3}}<\left(\frac{Q_{2}Q_{3}}{Q_{1}Q_{4}}\right)^{2}<1 \tag{8}\]  
In the same way, when the air flow direction in the cross passage is changed from node 4 \(\rightarrow\) node 3, the relation is:  
\[\frac{R_{1}R_{4}}{R_{2}R_{3}}>\left(\frac{Q_{2}Q_{3}}{Q_{1}Q_{4}}\right)^{2}>1 \tag{9}\]  
Combining (6), (8), and (9) gives discriminant (10):  
\[K=\frac{R_{1}R_{4}}{R_{2}R_{3}}\left\{\begin{array}{l}>1,\text{ The wind direction in the transverse channel is from node 4 to node 3;}\\ =1,\text{ The wind flow in the transverse channel is stagnant;}\\ <1,\text{ The wind direction in the transverse channel is from node 3 to node 4.}\\ \end{array}\right. \tag{10}\]  
Through numerical simulation, the flow \(Q_{\text{m}}\) of A-A, B-B, C-C, and D-D sections in the four branches is obtained, as well as the resistance \(h_{\text{m}}\) of each branch and the air flow resistance \(R_{\text{m}}\) of each branch, and then the air flow resistance \(R_{\text{m}}\) of each branch is substituted into Equation (10) for theoretical calculation, and the air flow direction of the double-hole tunnel can be determined by the \(K\) obtained.  
Figure 4: Ventilation network diagram of double-hole, single-cross passage.