## 3 Results  
### Comparison of Predicted and Simulated Blast Effects  
Of particular interest is the comparison of numerical results with the classical Kingery-Blumash type empirical curves. The fits to these data are most conveniently given by [PERSON] (1998) in the following form:  
\[\exp\Big{(}A+B\ln K+C(\ln K)^{2}+D(\ln K)^{3}+E(\ln K)^{4}+F(\ln K)^{5}+G(\ln K)^ {6}\Big{)} \tag{7}\]  
Here, \(K\) is the \(K\) factor given by Equation (1). The curve parameters \(A-G\) for Equation (7) fitting peak incident overpressure and positive impulse are given in Tables 4 and 5, respectively.  
The results for blast overpressure are also compared with predictions from the Taylor-von Neumann-Sedov result. It is shown in ([PERSON], 2017) that from this solution, the blast radius and corresponding peak pressure are given as a function of time as:  
\[R(t) =\beta\bigg{(}\frac{Et^{2}}{\rho_{0}}\bigg{)}^{1/5} \tag{8}\] \[p(t) =\frac{2}{\gamma+1}\rho_{0}\bigg{(}\frac{2}{5}\frac{R}{t}\bigg{)} ^{2} \tag{9}\]  
Here, \(E\) is the energy of the explosion, \(\rho_{0}\) the initial density of the air. \(\gamma\) is the same parameter appearing in Equation (3), and \(\beta\) is a corresponding parameter which has a value of 1.033 for air. Solving (8) for \(t\) and substituting into (9) yields an equation for pressure as a function of blast radius:  
\[p(R)=\frac{8}{25(\gamma+1)}ER^{-3}\beta^{5} \tag{10}\]  
This result is valid for a point source explosion in a zero-pressure medium expanding spherically from the origin. In order to compare with our hemispherical results, we compare to a blast having twice the energy of 1 kg TNT; this corresponds with the fact that the symmetry conditions on the floor of our free-field simulations make them numerically equivalent to spherical blasts of the same radius, i.e., twice the weight.  
Figure 10 shows the comparisons for peak incident blast overpressure of the free field hemispherical and detonation cord simulations with Equations (7) and (10). Figure 11 shows the corresponding positive impulses calculated from the pressure tracers by numerically integrating the positive portions of the pressure tracers from the hemispherical simulations compared to Equation (7).  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline \(K\)**Values** & \(A\) & \(B\) & \(C\) & \(D\) & \(E\) & \(F\) & \(G\) \\ \hline
0.2–2.9 & 7.2106 & \(-2.1069\) & \(-0.3229\) & \(0.1117\) & \(0.0685\) & \(0.0\) & \(0.0\) \\
2.9–23.8 & 7.5938 & \(-3.0523\) & \(0.40977\) & \(0.0261\) & \(-0.01267\) & \(0.0\) & \(0.0\) \\
23.8–198.5 & 6.0536 & \(-1.4066\) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Parameters for Equation (7) for peak incident overpressure (from Swisdak).  
\begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline \(K\)**Values** & \(A\) & \(B\) & \(C\) & \(D\) & \(E\) & \(F\) & \(G\) \\ \hline
0.2–0.96 & 5.522 & \(1.117\) & \(0.6\) & \(-0.292\) & \(-0.087\) & \(0.0\) & \(0.0\) \\
0.96–2.38 & 5.465 & \(-0.308\) & \(-1.464\) & \(1.362\) & \(-0.432\) & \(0.0\) & \(0.0\) \\
2.38–33.7 & 5.2749 & \(-0.4677\) & \(-0.2499\) & \(0.0588\) & \(-0.00554\) & \(0.0\) & \(0.0\) \\
33.7–158.7 & 5.9825 & \(-1.062\) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) & \(0.0\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Parameters for Equation (7) for incident impulse (from Swisdak).