## 3 Numerical Simulation and Comparison of the Results
Because the modified Birkhoff theory is a steady state condition generating a single value in a single set of LSC configuration, a detailed match to the numerical simulation results having a velocity gradient is not expected. Especially, when the jet velocity during the liner collapse exhibits a nonlinear behavior, the comparison may increase significant errors. However, the average value of the jet velocity gradient during the liner collapse in 25%, 50% and 75% is almost identical to the 50% value, it is believed that the comparison is reasonable. \begin{table}
\begin{tabular}{c c|c|c c|c c c c} \hline \multicolumn{2}{c|}{**Thickness (mm)**} & \multirow{2}{*}{**M/C**} & \multicolumn{2}{c|}{**Modified Birkhoff**} & \multicolumn{3}{c}{**Jet Tip vel. in Simulation**} & \multicolumn{2}{c}{**Shug vel.**} \\ \cline{3-10}
**Liner** & & **Explosives** & & **Jet vel.** & **Slug vel.** & **25\%** & **50\%** & **75\%** & **Avg.** & (Simulation) \\ \hline
0.5 & 1 & 2.78 & 1.92 & 0.25 & 1.52 & 1.48 & 1.43 & 0.35 \\
0.5 & 2 & 1.39 & 3.23 & 0.43 & 3.13 & 3.01 & N/A & 3.01 ** & 0.27 \\
0.5 & 3 & 0.93 & 4.17 & 0.55 & 4.5 & 4.3 & 4.2 & 4.33 & 0.5 \\
1 & 1 & 5.56 & 1.06 & 0.14 & 0.6 & & No jet development & N/A * \\
1 & 3 & 1.85 & 2.64 & 0.35 & 3.1 & 2.85 & N/A * & 2.85 ** & 0.27 \\
1 & 4 & 1.39 & 3.23 & 0.43 & 4.1 & 3.8 & 3.7 & 2.87 & 0.35 \\
1 & 5 & 1.11 & 3.74 & 0.5 & 4.9 & 4.6 & 4.4 & 4.63 & 0.46 \\
1 & 6 & 0.93 & 4.17 & 0.55 & 5.7 & 5.3 & 5.1 & 5.37 & 0.45 \\
1.5 & 2 & 4.17 & 1.37 & 0.18 & 1.5 & 1.21 & 0.9 & 1.20 & 0.29 \\
1.5 & 3 & 2.78 & 1.92 & 0.25 & 2.85 & 2.3 & N/A * & 2.30 ** & 0.17 \\
1.5 & 5 & 1.67 & 2.85 & 0.38 & 4.16 & 3.71 & N/A * & 3.71 ** & 0.26 \\
1.5 & 7 & 1.19 & 3.58 & 0.47 & 5.2 & 4.84 & 4.7 & 4.91 & 0.35 \\
1.5 & 9 & 0.93 & 4.17 & 0.58 & 5.87 & 5.76 & 5.6 & 5.74 & 0.4 \\ \hline \multicolumn{2}{c}{all velocities are in km/s. * simulation stopped due to too much deformation. ** 50\% liner collapse completion value was used.} & & & & & & \\ \end{tabular}
\end{table} 
Table 2: Modified Birkhoff and numerical simulation comparison. Figure 7: Modified Birkhoff and numerical simulation comparison. The comparison between the analytical calculation and the numerical simulation results shows a clear trend. The two values follow a similar curve in the entire range of M/C values, however, the values are close in higher M/C, and somewhat different in lower M/C value. One of the reasons of the differences in lower M/C range can be the rarefaction issues. In a higher M/C configuration with less explosives charge, the liner should be affected less by rarefaction effects from the explosives charges surface during the detonation, generating quite similar \(V_{\mathrm{p}}\) values in between the two results. In the lower M/C range, however, the liner should experience a significant effect from the rarefaction from the charge surface due to the comparatively large surface area, decreasing the effective charges weight [7]. Because the analytical model doesn't consider the rarefaction effects or effective charge weight in the calculation, we see differences in jet velocity between the model and the hydrocode predictions in the low-M/C range. In this case, the velocity of the modified Birkhoff theory would be larger than the simulation results, but it shouldn't be considered here without experimental data. Another issue for the difference is the under-estimation of the Gurney calculation in small M/Cs. In general the Gurney approach exhibits a gradually underestimated plate velocity below M/C = 1 and when it reaches below M/C = 0.3, it should be recognized that the Gurney results will be a lower bound for the actual velocity [8]. Because the jet velocity is greatly affected by the liner velocity, a slight change in the liner velocity may generate large differences in jetting. In addition, the sweeping detonation in the flat liner should be under the consideration of the Taylor projection angle.