## 4 Analysis of the Influence of Water Depth on Air Layer Drag Reduction
### Effect on Air Layer Morphology  
For saturated jetting conditions at two different model speeds, \(F_{r}\) = 0.057 and \(F_{r}\) = 0.091, \(F_{r}\) is the velocity Froude number:  
\[F_{r}=\frac{V}{\sqrt{gl}} \tag{11}\]  
where \(V\) is speed, \(g\) is the gravitational acceleration, and \(l\) is the characteristic length of the object. Figures 8 and 9 show the effects of water depth change on the air layer morphology in the groove and the gas overflow at the stern. The left side of the figure is the nephogram of the gas layer thickness distribution, and the darker the color, the larger the gas layer thickness. The figure on the right is a comparison of the composition of the gas phase and the liquid phase, and the darker the color, the more the gas phase content. The results show that variations in water depth have an impact on air layer morphology in the groove and how the gas overflows from the stern. However, there is no discernible air layer fragmentation or new overflow occurrences, demonstrating that the groove keeps the air layer stable and contained until it reaches the extremely shallow water depth. The groove may still be able to sustain a stable, enclosed air layer, according to this evidence. Gas overflow reduces from both sides of the stern and increases from the mid-profile to the rear as the water depth becomes shallower. The gas generally overflows from the mid-profile to the rear when the water depth reaches the extremely shallow water depth (\(h/T\) = 1.18), and localized fragmentation takes place in the middle region of the air layer. This shows that the air layer in the bottom groove is substantially more disturbed by the exceedingly shallow water depth. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & \(h\) **(m)** & \(h\)/\(T\) & **Category** \\ \hline
1 & 1.36 & 4 & deep water \\
2 & 0.765 & 2.25 & moderately deep water \\
3 & 0.459 & 1.35 & shallow water \\
4 & 0.4 & 1.18 & extremely shallow water \\ \hline \hline \end{tabular}
\end{table} 
Table 6: List of calculation conditions for water depth variation. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**No. **} & \multirow{2}{*}{\(\mathbf{F_{r}}\)} & \multicolumn{4}{c}{**Resistance for Non-Jet (N)**} & \multicolumn{4}{c}{**Resistance for the Fully Saturated Jet (N)**} \\ \cline{3-8}  & & **Simulated** & **Experimental** & **Error** & **Simulated** & **Experimental** & **Error** \\ \hline
1 & 0.045 & 5.493 & 5.351 & 2.66\% & 3.984 & 4.001 & 0.42\% \\
2 & 0.057 & 9.293 & 9.152 & 1.54\% & 6.541 & 6.123 & \(-\)6.83\% \\
3 & 0.068 & 14.330 & 14.015 & 2.25\% & 9.842 & 9.74  
The air layer fluctuation and overflow in the groove along the longitudinal section are compared in Figure 10 from extremely deep to extremely shallow water depth conditions. Figure 10 shows that the number of air layer fluctuation cycles decreases as the water depth becomes shallower, which was caused by the increase in the flow velocity near the hull due to shallow water effects. In terms of air layer morphology and coverage, in the case of extremely deep water, the air layer thickness is considerable, and the trough points of the air layer coincide with the lowest point of the stepped bottom of the groove while the peak points remain somewhat apart from the groove bottom. Without fragmentation or flow overflow from the rear end throughout the longitudinal section, the air layer creates a well-sealed effect against the hull's incline at the back of the groove. Under the interference of the bottom of the groove, however, the air layer fluctuation pattern in the groove was changed relative to when the water depth was exceptionally shallow (\(h\)/\(T\) = 1.18). The distance between the peak points and the bottom of the groove is dramatically shortened toward the front region of the groove, where nonlinear fluctuations are significantly enhanced. Localized air layer fragmentation results from local contact between the peak points and the groove's bottom.