## 4 Conclusions  
The general laws that govern flow and concentration fields were determined through a numerical simulation study. The following conclusions can be drawn:  
The flow rate of air leakage in the ventilation duct is inversely proportional to the distance from the fan. Due to the effect of air resistance and friction with the tunnel walls, the airflow distribution in the duct gradually follows a parabolic distribution pattern with high flow velocity in the middle and low flow velocity at the edges. In the forced ventilation mode, the discharge of pollutants inside the tunnel is divided into movement and diffusion processes. This pattern is consistent with the mass transfer process of turbulent airflow. The discharge of harmful gases has both the transport effect of the main airflow and the diffusion effect of the turbulent airflow. The gas leakage from the ventilation duct has a significant dilution effect on the concentration of pollutants inside the tunnel. The dilution effect is mainly reflected in the vertical direction of the leakage hole. And the dilution effect of leaked gas on pollutants is directly proportional to the gas flow rate. The leaked airflow can be effectively utilized in engineering to eliminate pollutants. The models with different apertures and tunnel lengths have similar air leakage laws. The calculation of air leakage can be fitted with a logarithmic function. The matching degree between the simulation results of the model and the empirical equation depends on the length of the tunnel. The estimation function can be used to calculate the actual air supply of the working face during the tunnel construction. The estimation function can also be used to calculate the actual required air volume of the working face for fan selection. The mechanism of air leakage of the duct on the flow field distribution and pollutant transport in tunnel engineering is mastered through numerical simulation. Accurate estimation of the effective air volume of the tunnel working face avoids the waste of resources. And it provides a theoretical basis for practical engineering. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Leakage Size 1 (m)** & \
\begin{tabular}{c} **Average Air** \\ **Velocity (m/s)** \\ \end{tabular} & **Air Leakage Rate** & **Leakage Size 2 (m)** &
\begin{tabular}{c} **Average Air** \\ **Velocity (m/s)** \\ \end{tabular} & **Air Leakage Rate** \\ \hline
0 & 30.00 & 0.00\% & 0 & 30.00 & 0.00\% \\
0.08 \(\times\) 0.04 & 29.83 & 0.57\% & 0.1 \(\times\) 0.06 & 29.54 & 1.53\% \\
0.08 \(\times\) 0.06 & 29.75 & 0.83\% & 0.1 \(\times\) 0.08 & 29.52 & 1.60\% \\
0.08 \(\times\) 0.08 & 29.71 & 0.97\% & 0.1 \(\times\) 0.15 & 29.18 & 2.73\% \\
0.08 \(\times\) 0.10 & 29.68 & 1.07\% & 0.12 \(\times\) 0.2 & 29.12 & 2.93\% \\
0.08 \(\times\) 0.15 & 29.66 & 1.13\% & 0.16 \(\times\) 0.2 & 29.04 & 3.20\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Effect of hole size on air velocity and leakage rate. **Author Contributions:** All authors contributed to the study conception and design. Methodology and Software were performed by X.C., J.C. and J.R. Formal Analysis and Data Curation were organized by X.C. and R.Y. Writing--Original Draft and Writing--Review and Editing was prepared by all X.C. All authors commented on previous versions of the manuscript. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by [the National Natural Science Foundation of China], grant number [52209147], and [the Natural Science Basic Research Program of Shaanxi], grant number [2022]Q-426]. **Institutional Review Board Statement:** All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards. This article does not contain any studies with animals performed by any of the authors.