## 3 Numerical Modeling II
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Standoff (inches)** & **Min. Value** & **Max Value** & **Jet Width (mm)** & **Jet Width (inches)** \\ \hline
**0.75** & 9.9271 & 15.8230 & 5.8959 & 0.2321 \\ \hline
**3.00** & 6.7410 & 25.7200 & 18.9790 & 0.7472 \\ \hline
**6.00** & 2.3565 & 38.1730 & 35.8165 & 1.4101 \\ \hline
**12.00** & -6.7842 & 62.8020 & 69.5862 & 2.7396 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Jet width under different standoff distances. Figure 10: %Y-velocity distribution along the jet width in large standoff distance. Figure 9: Jet width at different standoff distance. The x-direction of the jet (cross-sectional) is split into 10 pieces each approximately 7.2 mm in width as seen in Figure 11. The x-axis in the jet runs from \(-\)36 mm to 36 mm in order to compensate for the large jet width at the 12 inches (30.48 cm) standoff distance. In order for a fair comparison for each standoff distance, this is kept throughout even though the jet width is much narrower at the shorter standoff distances. The y-axis shows the bottom 2 inches (50.8 mm) of the jet. This is also consistent with all of the models except the optimum standoff model. In this case, because the jet has not moved 2 inches (50.8 mm), the y-axis is set for the bottom 0.75 inches (19.05 mm). Note that the y-axis is not important for the analysis. It is merely to help illustrate the bottom 2 inches (50.8 cm) of the jet to put in perspective the concentration of the jet from model to model. In each section, particles in the x-axis are picked at random. The magnitude of the y-velocity is then divided by the absolute velocity (Figure 10). By dividing the magnitude of the y-velocity by the absolute velocity, the fractional component of the absolute velocity that was the y-velocity is found. Five particles are picked, the percentage of the absolute velocity that is the y-velocity is recorded, and these numbers are averaged to show an average percentage for each section. The graph shows the average percentage on the y-axis and the distance from the center of the jet on the x-axis (Figure 10). Note that the 'center' is only really the center of the jet for the 12 inches (30.48 cm) standoff distance. The axes are kept unchanged for consistency in comparing the models. Due to the fact that in all of the graphs there is at least one section that has zero particles in it, these data points are left out since it would skew the graph. This also helps illustrate the jet width as the standoff increases. As the graph shows, the shorter the standoff distance becomes the more narrow the jet becomes. This is illustrated with the optimum standoff model only having two useful data points whereas the 12 inches (30.48 cm) standoff model has 9 useful data points. Due to the jet spreading during the flight at large standoff distances, the center of the jet offsets from the true center of the LSC (or jet width). However, the graph is able to illustrate the center of the jet width by the peak point of each curve (_i.e._, the center or mid-point of the jet width in 12 inches (30.48 cm) standoff distance curve is located near \(+\)8 mm). The reason for the jet spreading during the flight in the simulation is not clearly known at this point. However, it is believed that the local unbalanced cell configurations in between the left and right wing of LSCs in the SPH model generates a miniscule time mismatch during the liner collapse, which separates the jet. In reality, this happens  
Figure 11: Cross sectional area of the jet split into 10 separate pieces for analysis. As seen in Figure 12, the LSC placed at the OSD clearly shows the maximum penetration capability. There is no significant spreading of the jet over the penetration surface and the target shows a clean cut. Figure 13 shows a reduction in penetration depth due to the 3 inches (7.72 cm) travel distance. A portion of the jet particles are skewed to the right side and a large jet entrance compared to the OSD models shows a reduction in the penetration depth.