## 2 Materials and Methods
### Free-Field Detonation of TNT  
The free-field detonation of hemispherical TNT was simulated under 2D axisymmetric conditions. Figure 1 depicts a cartoon of the setup. The \(x=0\) axis is the axis of rotational symmetry, while the \(y=0\) has symmetry boundary conditions which are used to crudely approximate the ground; however, this approximation causes the simulation to be equivalent to a spherical charge of the same radius exploding in air. The air domain extends from the origin to 40 m in the \(x\) and \(y\) directions. The outer boundaries have three different boundary conditions applied; \"pressure continuous\" provides ghost nodes external to the boundary which keeps the pressure constant on the other side, which keeps the initially pressurized gas from expanding and depressurizing as soon as the simulation starts. \"Non-reflecting\" boundary conditions dampen out any reflected incident waves to minimize boundary effects. The \"outflow\" condition allows material given outbound velocity to leave the domain. \begin{table}
\begin{tabular}{l l l} \hline \(\gamma\) & \(\rho_{0}\) (g/cm\({}^{3}\)) & \(P_{0}\) (kPa) \\ \hline
1.40 & \(1.225\times 10^{-3}\) & \(1.0135\times 10^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Gamma law parameters for air. \begin{table}
\begin{tabular}{l l l l l} \hline \(E\) (kPa) & \(k\) (kPa) & \(\
u\) & \(\rho_{0}\) (g/cm\({}^{3}\)) & \(y_{c}\) \\ \hline
2.344 \(\times 10^{6}\) & \(1.119\times 10^{5}\) & 0.4 & 1.218 & \(2.086\times 10^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Power law parameters for Lexan. Four simulations in total were performed with of charges with radius 0.0261 m, 0.051 m, 0.0643 m, and 0.081 m, yielding hemispherical charge weights of approximately 0.06123 kg (0.135 lb), 0.45359 kg (1 lb), 0.90718 kg (2 lb), and 1.81436 kg (4 lb), respectively. Note that the charge radius is around 3 orders of magnitude smaller than the domain length; even in 2D, a uniform Cartesian mesh small enough to adequately resolve the TNT would lead to an intractably-large numbers of zones. Instead, a graded mesh approach was use, coarsening with distance from the origin. Initial zone sizes range from approximately \(4.5\times 10^{-3}\) m at the center of the charge out to \(5.4\times 10^{-2}\) m at the outer edge of the domain. The simulations ultimately contained around 7.1 million zones. Figure 2 shows representative temporal snapshots of pressure in the system as the explosive wave propagates in air. The peak overpressure occurs near the wavefront but rapidly decreases to the ambient pressure and then dips below it for some time before returning. The magnitude of this peak pressure decreases as the wave propagates further from the source. Fixed (Eulerian) pressure tracers were placed every 2 m in the domain just off the \(y\)-axis in order to study the structure and evolution of the blast wave. Figure 3 shows the results of these pressure tracer time histories. Each tracer shows a pronounced positive overpressure phase followed by a negative phase where pressure dips below ambient. The effects of these negative pressure phases have been studied and are in general not negligible [29; 30]. Figure 1: A “cartoon” depiction of the setup of the free-field TNT detonation simulations with materials and boundary conditions labeled (Not to scale). Figure 4 shows the pressure waves recorded at 36.576 m (120 ft) from the source. Note that when compared to some of the larger pressure, early time pressure waves on the left-hand side of the plots in Figure 3, these waves are relatively smooth and have a shallower initial slope as they ramp up to maximum incident overpressure. This corresponds to the fact that at this distance the waves are no longer proper shock waves. In fact, the wave velocity is approximately that of the speed of sound in air. The area under the positive portion of the overpressure wave is the total incident impulse at this point.