## 2 Numerical Model
### Validation of Methodology  
[PERSON] et al. [38] performed experiments using three different tunnel scales, 1:10, 1:2, and a full scale, with rectangular and horseshoe cross-sections. The fire sources used were of square cross section. The aspect ratio of the tunnel cross-section dimensions, HRR, and forced longitudinal ventilation velocity was variable. From the experimental data of the 1/10 scale model, empirical formulas were exported to calculate the flame inclination, the flame height, the maximum temperature of the smoke layer, and its position. The effect of tunnel cross-section dimensions was incorporated into these models. The ratio \(\text{H}^{3/2}/\text{b}^{1/2}\), where H is the height of the tunnel model and b is the width, was found to be representative for the investigation of fire phenomena in tunnels. It was also confirmed that these empirical formulas were sufficiently applicable to predict phenomena occurring during a fire accident, close to the square fire source field, by comparing the results of experiments in 1:2 and full-scale tunnels. [PERSON] et al. [39] studied, experimentally and computationally, the maximum temperature on the tunnel roof. They first compared the FDS results with the experimental measurements, and then, with the values, they calculated with the model of [PERSON], [PERSON], [PERSON], and [PERSON] [38]. The temperature predicted by the  
Figure 3: Grid meshes and fire source. The medium grid of 0.5 m was used for the majority of the tunnel, while the fine mesh of 0.25 was used near the field of the fire. FDS was found to converge quite well with both the experimental measurements and the results of the empirical model. The [PERSON] et al. [40] study provided a theoretical analysis of the maximum temperature in the tunnel's roof, based on the axon-symmetric fire plume theory, taking into account the heat release rate, the longitudinal ventilation velocity, and the geometry of the tunnel. They conducted scale tests to investigate the maximum temperature on the tunnel roof in case of fire. [PERSON] and [PERSON] [41] analyzed the effects of different ventilation systems, ventilation speeds, heat release rates, tunnel geometries, and fire sources on the maximum temperature under the tunnel roof for large fires. They used data from many model-scale tests and large-scale tunnel fire experiments around the world. They proposed some empirical formulas for the maximum temperature on the tunnel's roof nearby the fire source for low and high ventilation speeds. [PERSON] et al. [42] merged the empirical relationships from the two studies above and defined a formula that exhibited non-linear regression and calculated temperature values as a function of distance from the fire source. They also distinguished between small and large fires because, in the event of a very large fire, the flame reaches the roof, so that the maximum temperature corresponds to the temperature of the flame, and it is difficult to estimate the mass flow rate. If a portion of the flame volume containing the combustion zone reaches the roof of the tunnel, the maximum temperature on the roof is reaching a constant value. Therefore, the maximum roof gas temperature is limited to a maximum of 1350 \({}^{\circ}\)C. The formula that calculates the temperature as a function of distance from the fire source is expressed as follows:  
\[\frac{\Delta T(x)}{\Delta T_{max}} =0.55\ \exp\left(-0.143\frac{\chi-\chi_{v}}{H}\right)+0.45\ \exp\left(-0.024\frac{\chi-\chi_{v}}{H}\right) \tag{9}\]  
where \(\chi\) is the distance from the fire source, \(H\) is the height of the tunnel, and \(\chi_{v}\) is the virtual origin, which is calculated as  
\[\chi_{v}=\left\{ \begin{matrix}L_{f}-10H,&L_{f}>10H\\ 0,&L_{f}\leq 10H\end{matrix} \right. \tag{10}\]  
where \(L_{f}\) is the flame length (m).