## 3 Two-Dimensional Flyer Projection  
A general velocity profile along the width of the flyer from the previous section was observed. However, it creates another important question about the relation between the velocity profile and the width of the flyer. As addressed before, if the speed of rarefaction intrusion in the gas, or tension wave in the flyer, is a key to understanding the velocity profile [7], then a flyer with more width should be affected, because more width means more distance to travel for the rarefaction or tension wave. This should be an important clue to answer because it may provide a way to advance the conventional Gurney model into a more realistic, two-dimensional approach. Note that the conventional Gurney model assumes that the flyer/charge system is under a one-dimensional configuration, and disregards the effect from the side rarefaction. In order to address this limitation, [PERSON] proposed an approach with effective charge mass (\(\frac{M}{c_{s}}\)) based on the side rarefaction (Figure 1) [3]. A more detailed description will be followed later in this paper. In order to answer the question above, a series of hydrocode simulations was accomplished using Autodyn(tm). For more information regarding the simulation configuration, please refer to Figure 11 and Table 11.  
In this series of simulations, four different flyer widths of 12, 25, 50, and 75 (or 100 mm), under varied \(M/C\) are simulated, and the results are tabulated in Figures 7-9, and Table 1. The flyer's material is a 2 mm thick, copper flyer, and the explosives charge material is a 10 mm thick, C4 charge (see Appendix A, Table 11 for detailed simulation parameters). The simulation was run until the velocity converging stage in order to observe the entire three stages of velocity profiles. Note 1: Some gauge points near the edge of flyer were physically separated during the projection and those data points were neglected from these graphs. Figure 7: Typical velocity profile of different widths of flat metallic flyers (refer to Figure 5 for the gauge location). Note 2: Due to the plotting limitation on the number of data points in the graph, the graphs for 50 and 75 mm only show up to 0.2 ms after detonation, and the converging points are not shown in the graph (for the converging time, see Table 1). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Thickness of Charge (mm)**} & \multirow{2}{*}{\begin{tabular}{c} **Thickness of** \\ **Flyer (mm)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **M/C** \\ *** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Width of** \\ **Flyer (mm)** \\ \end{tabular} } & \multicolumn{2}{c}{**Simulation**} & \multicolumn{2}{c}{**Baum’s**} & \multirow{2}{*}{
\begin{tabular}{c} **Gurney** \\ **Model (m/s)** \\ \end{tabular} } \\ \cline{5-8}  & & & 12 & 0.063 & 501 & 666 & 834 \\
5 & 2 & 2.23 & 25 & 0.191 & 656 & 755 & 834 \\  & & & 50 & 0.396 & 743 & 795 & 834 \\  & & & 75 & 0.498 & 777 & 808 & 834 \\ \hline \multirow{8}{*}{10} & \multirow{8}{*}{2} & \multirow{8}{*}{\(1.12\)} & \(12\) & 0.023 & 674 & 858 & 1388 \\  & & & 25 & 0.035 & 1061 & 1156 & 1388 \\  & & & 50 & 0.307 & 1265 & 1277 & 1388 \\  & & & 75 & 0.560 & 1330 & 1315 & 1388 \\  & & & 100 & 0.759 & 1365 & 1334 & 1388 \\ \hline \multirow{8}{*}{15} & \multirow{8}{*}{2} & \multirow{8}{*}{0.74} & \(12\) & 0.024 & 726 & 859 & 1793 \\  & & & 25 & 0.035 & 1251 & 1370 & 1793 \\ \cline{1-1}  & & & 50 & 0.069 & 1584 & 1597 & 1793 \\ \cline{1-1}  & & & 75 & 0.379 & 1720 & 1665 & 1793 \\ \hline \hline \end{tabular}  
* Open-face sandwich configuration; ** Some as the terminal velocity of the given flyer in the simulation; *** Gurney model calculation with the effective charge mass (open-face sandwich configuration).  
\end{table} 
Table 1: Flyer velocity comparison. Figure 8: Terminal velocity (or converging velocity) comparison for various M/C.