## 3 Numerical Modeling
### CFD Modeling  
The blast overpressures were simulated using a CFD code, Viper::Blast [18], where the numerical methodology was based on [26], and the AUSMDV numerical scheme [27] was used to simulate compressible flow problems. The numerical analysis for the surface detonation of TNT was conducted in two stages, considering the numerical accuracy and computational expense. The early stage of detonation of the combustion products was simulated in the one-dimensional (1D) domain, with radial symmetry until the blast shock waves reached the reflecting boundaries. These 1D data were then remapped into a three-dimensional (3D) domain in which the tunnel was modeled [28; 29; 30]. This remapping technique is efficient in terms of numerical accuracy and computational cost because it enables the use of a finer mesh for 1D analysis and a relatively larger mesh for 3D analysis.  
The 1D model comprised a high TNT explosive and the surrounding air. Air was modeled using the ideal gas equation of state (EOS), and the TNT was modeled using the Jones-Wilkins-Lee (JWL) EOS [19; 31]:  
\[P=A\bigg{(}1-\frac{w}{R_{1}V}\bigg{)}e^{-R_{1}V}+B\bigg{(}1-\frac{w}{R_{2}V} \bigg{)}e^{-R_{2}V}+\frac{wE_{0}}{V} \tag{1}\]  
where \(P\) is the blast overpressure; \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(w\) are the JWL constants; \(V\) is the relative volume; and, \(E_{0}\) is the specific internal energy. The values of the JWL EOS parameters were obtained from the LLNL Explosives Handbook in [32], as listed in Table 1. Afterburning was not modeled because it may not be realized due to the incomplete mixing of fuel with the available oxygen and a temperature less than the ignition temperature of the fuels (<1800 K) that makes up TNT [33].  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Density, \(\rho\) (kg/m\({}^{3}\)) & 1630 \\ CJ detonation speed, \(D\) (m/s) & 6930 \\ CJ detonation pressure, \(P_{CJ}\) (GPa) & 21 \\ Constant, \(A\) (GPa) & 371.2 \\ Constant, \(B\) (GPa) & 3.231 \\ Constant, \(R_{1}\) & 4.15 \\ Constant, \(R_{2}\) & 0.95 \\ Constant, \(w\) & 0.3 \\ Specific internal energy, \(E_{0}\) (MJ/m\({}^{3}\)) & 7000 \\ \hline \hline \end{tabular}
\end{table}
Table 1: JWL EOS parameters of high explosives of TNT.  
Air was modeled in 3D to completely envelop the tunnel. The lower boundary of the air domain was the ground surface and was thus defined as a perfectly reflecting boundary. All other boundaries were defined using transmitting boundaries, such that no shock wave reflection occurred at these boundaries. The tunnel was modeled as a rigid body, and all surfaces of the model fully reflected the blast waves, which enabled complex behaviors of blast waves inside the tunnel, including repeated shock wave reflection and diffraction [19; 34; 35]. Modeling the tunnel as a rigid body ignores the interaction with structures, and there is a possibility that the blast overpressures reflected from the rigid structure may be calculated somewhat conservatively. However, the differences between the results for the rigid and deformable tunnels were considered insignificant because blast overpressures numerically calculated assuming the rigid body structures has been experimentally validated in many past studies [19; 36]. Also, the numerical results were herein validated in comparison with empirical design charts in UFC 3-340-02 [21], superseding TM 5-1300 [36; 37].