# Study on Mechanism of Series-Flow of Pollutants between Consecutive Tunnels by Numerical Simulation
## 3 Results and Discussions
### Construction of Series-Flow Ratio Model  
Observing the above results of simulations, we construct a model by assuming that \(\varphi_{s}\) is mainly influenced by \(\Delta L/D\) and \(u_{\text{up}}/u_{\text{down}}\). Noting the jet characteristic that the airflow velocity distribution on the jet cross-section is mainly affected by \(\Delta L/D\) and only influenced by the negative pressure in the area very close to downstream entrance where \(u_{\text{up}}/u_{\text{down}}\) decides the degree of the influence, the functions of both \(\Delta L/D\) and \(u_{\text{up}}/u_{\text{down}}\) on the series-flow ratio \(\varphi_{s}\) are mutually independent. Therefore, the series-flow ratio model can be written as follows:  
\[\varphi_{s}=f_{1}(\Delta L/D)f_{2}\big{(}u_{\text{up}}/u_{\text{down}}\big{)} \tag{5}\]  
The main part of the flow between tunnel portals with different \(\Delta L\) can be simplified as a form primarily decided by the jet characteristic of the upstream tunnel. And according to the jet characteristic of the upstream tunnel [23], \(f_{1}(\Delta L/D)\) can be constructed in sections. When \(\Delta L/D\leq 7\), the downstream tunnel inlet is located at the potential core region of the upstream tunnel jet flow. \(\varphi_{s}\) is linearly dependent on \(\Delta L/D\) at \(u_{\text{up}}/u_{\text{down}}=1\) (Figure 7). The relation between \(\varphi_{s}\) and \(\Delta L/D\) is fitted as follows by using the simulation results of different \(\Delta L/D\) and \(u_{\text{up}}/u_{\text{down}}=1\) (R\({}^{2}\) = 0.995):  
\[\varphi_{s}=1-0.0328\Delta x/D \tag{6}\]  
When \(\Delta L/D>7\), the downstream tunnel inlet is located at the decay region of the upstream tunnel jet flow. Through surface integration of pollutant concentration and velocity near the downstream tunnel inlet, \(Q_{s}\) can be expressed as follows:  
\[Q_{s}=\iint_{S}u_{r}c_{r}\text{d}s \tag{7}\]  
where \(S\) (m\({}^{2}\)) is the cross-section area of downstream tunnel entrance and d\(s\) is the surface element at any point \(p\) on \(S\); \(r\) (m) is the distance between \(p\) and the \(x\)-axis; \(u_{r}\) (m\(\cdot\)s\({}^{-1}\)) is the velocity in the \(x\) direction at \(p\); \(c_{r}\) (ppm) is the pollutant concentration at \(p\). According to three-dimensional wall jet theory [24], expressions of \(u_{r}\) and \(c_{r}\) are as follows:  
\[u_{r}=[1-(r/\delta)^{1.5}]^{2}u_{\text{max}} \tag{8}\]  
\[c_{r}=[1-(r/\delta)^{1.5}]c_{\text{max}} \tag{9}\]  
Figure 9: Impacts of \(u_{\text{up}}/u_{\text{down}}\) on series-flow ratio \(\varphi_{s}\). In the formula, \(\delta\) (m) stands for the jet boundary radius at \(x=\Delta L\)[21], its expression is \(\delta=1.157D\) + \(0.272\Delta L\); \(c_{\max}\) (ppm) is the maximum pollutant concentration of the decay region. Expressions of \(u_{\max}\) and \(c_{\max}\) are as follows:  
\[u_{\max}=\mathrm{k}(\Delta L/D)^{-n}u_{\mathrm{up}} \tag{10}\]  
\[c_{\max}=0.73\mathrm{k}(\Delta L/D)^{-n}c_{\mathrm{up}} \tag{11}\]  
where \(c_{\mathrm{up}}\) (ppm) is the initial discharge concentration of the upstream tunnel exit; \(k\) is the velocity coefficient; \(n\) is the decay index. Simplifying the tunnel portal into a semicircle of radius \(R\) and substituting Equations (8) and (9) into Equation (7) obtain:  
\[Q_{s}=\int_{0}^{\pi}\int_{0}^{R}\big{[}1-(r/\delta)^{1.5}\big{]}^{3}u_{\max}c _{\max}r\mathrm{d}r\mathrm{d}\theta=0.5m\pi R^{2}u_{\max}c_{\max} \tag{12}\]  
In the formula, \(m=1-1.714(R/\delta)^{1.5}+1.2(R/\delta)^{3}-0.308(R/\delta)^{4.5}\). By substituting jet boundary radius \(\delta\) and hydraulic diameter \(D\) into \(m\), one can obtain \(m=1-8.944(4.254+\Delta L/D)^{-1.5}+32.68(4.254+\Delta L/D)^{-3}-43.76(4.254+ \Delta L/D)^{-4.5}\approx 1-8.944(4.254+\Delta L/D)^{-1.5}\), noting the absolute value of the omitted part is far less than the remaining part for \(\Delta L/D>7\).