## 2 Methodology
### Mathematical Model  
The FLACS v22.2 software consists of three main components: the first is the computer-aided scenario design (CASD), a preprocessor used to define the explosion models for the simulation. This stage involves setting up computational grids, the porosity, geometric models, and scenarios based on the input data. The second component is the FLACS simulator, which serves as the core solver module, solving the Navier-Stokes equations and enabling various model modifications. The third component is a flowchart. This is a postprocessor that visualizes the simulation results in 2D or 3D. Moreover, this component enables an intuitive interpretation of explosion outcomes such as tunnel explosion effects. The FLACS employs the finite volume method (FVM) to solve compressible conservation equations on a 3D Cartesian grid. Therefore, accurate simulation results can be guaranteed [36; 37]. The cornerstone of the FLACS is the set of fundamental governing equations for fluid dynamics. These include mass, momentum, and energy equations. As a commercial CFD code, the FLACS can be applied to model the release, dispersion, combustion, and explosion processes of gases, liquids, and vapor clouds in both unconfined and confined spaces. The computational tool includes a 3D CFD code that solves the Favre-averaged transportation equations for the mass, momentum, enthalpy, turbulent kinetic energy, rate \begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Case**} & **Ignition Position** & **Ignition Position** & **Ignition Position** \\  & **X Axis (m)** & **Y Axis (m)** & **Z Axis (m)** \\ \hline
16. & 5.1 & 5.5 & 2.1 \\
17. & 5.1 & 5.5 & 3.1 \\
18. & 5.1 & 5.5 & 4.1 \\
19. & 5.1 & 5.5 & 4.9 \\ \hline \end{tabular}
\end{table} 
Table 3: _Cont._of dissipation of turbulent kinetic energy, fuel mass fraction, and mixture fraction. These equations are solved on a structured Cartesian grid using the finite volume method [38; 39] and can be represented as follows:  
\[\frac{\partial}{\partial t}(\rho\varphi)+\frac{\partial}{\partial X_{j}}\left( \rho u_{j}\varphi-\rho\Gamma_{\varphi}\frac{\partial\varphi}{\partial x_{j}} \right)=S_{\varphi} \tag{1}\]  
Here, \(\varphi\) denotes a general variable such as mass, momentum, energy, or turbulent kinetic energy. \(\rho\) represents the gas density, \(t\) denotes time, \(u_{j}\) is the velocity component in the \(j\)-direction, \(\Gamma_{\varphi}\) is the dispersion coefficient of \(\varphi\), and \(S_{\varphi}\) refers to the source term. The FLACS utilizes the \(k-\varepsilon\) two-equation model and standard wall functions to simulate turbulence. The \(k-\varepsilon\) model consists of two additional transportation equations. The turbulent pulsation kinetic energy k is defined using the following equation. \[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho u_{ i}k)=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}} \right)\frac{\partial k}{\partial x_{j}}\right]+G_{k}+G_{b}-\rho\varepsilon- Y_{M}+S_{k} \tag{2}\]  
Additionally, the turbulent pulsation kinetic energy dissipation rate, \(\varepsilon\), is defined using the following equation. \[\frac{\partial}{\partial t}(\rho\varepsilon)+\frac{\partial}{\partial x_{i}} (\rho u_{i}\varepsilon)=\frac{\partial}{\partial x_{j}}\left[\left(\mu+\frac{ \mu_{t}}{\sigma_{\varepsilon}}\right)\frac{\partial\varepsilon}{\partial x_{ j}}\right]+C_{1\varepsilon}\frac{\varepsilon}{k}(G_{k}+C_{3\varepsilon}G_{b})-C_{2 \varepsilon}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon} \tag{3}\]  
Here, \(u\) denotes the molecular viscosity of fluids (Pa-s) and \(\mu_{t}\) represents the turbulent viscosity of fluids (Pa-s). \(\sigma_{k}\) and \(\sigma_{\varepsilon}\) correspond to the Prandtl numbers associated with k and \(\varepsilon\), which are typically considered as 1 and 1.3, respectively. \(G_{k}\) represents the generation of turbulent kinetic energy owing to the mean velocity gradient, whereas \(G_{b}\) represents the generation of turbulent kinetic energy owing to the buoyancy.