# Cavity Formation during Asymmetric Water Entry of Rigid Bodies
## 4 Symmetric Wedges with Horizontal Velocity Component--\(\gamma=0\) and \(\epsilon\
eq 0\)  
In this section, we investigate the effect of a horizontal velocity component on the pressure distribution over a rigid wedge. At first, we concentrate on rigid wedges with constant velocity and null tilt motion, since such approximation leads to a self-similar solution and the impact time can be taken out of the analysis. As a general result, the horizontal velocity component alters the fluid motion beneath the wedge up to the point a cavity is formed. Indeed, a threshold value of it exists for the onset of cavity formation. Figure 4 shows three images of the cavity predicted by the numerical solutions on a wedge with \(\beta=25^{\circ}\) and varying advancing ratios \(\epsilon\). Notably, fluid detachment is attained as \(\epsilon\) exceeds 2 and the void region increases with the advancing ratio. Indeed, pressure drops to zero in the dry region, the extension of which increases with the advancing ratio \(\epsilon\). Figure 5 details the predicted pressure distribution over a symmetric wedge with deadrise angle \(\beta=30^{\circ}\) entering the water with varying advancing ratios \(\epsilon\). These examples report wedges running at a constant speed and the tilt motion is inhibited. Each graph reports the normalized pressure over the  
Figure 4: Fluid detachment at the vertex of a \(\beta=25^{\circ}\) wedge for varying advancing ratio \(\epsilon=\frac{6}{3}\), \(\frac{7}{3}\)and \(\frac{8}{3}\) (left to right). Figure 3: Evolution of the normalized pressure along the wetted edge for variable deadrise angles. the wedge at various instants of the impact. Notably, the solution remains self-similar for all the advancing ratios considered, as the normalized pressure and void region are not influenced by the entry depth. Interestingly, the leeward wedge side remains completely dry as \(\epsilon\) exceeds 2. The self-similarity of the results allows for the effect of the advancing ratio on the pressure distribution along the body to be compared. Figure 6 reports results similar to the ones presented in Figure 5, but stacked on a single graph to ease their comparison. Results about \(\beta=20^{\circ}\) and \(\beta=30^{\circ}\) wedges with varying advancing ratios are presented. In the case of pure vertical impact, such deadrise angles experience a hydrodynamic load that is almost constant over the entire wet surface, as also predicted by the [PERSON]'s solution reported in Figure 3. However, when a horizontal velocity component is introduced, the pressure on the right side (that is, the windward side) is found to increase almost linearly moving from the vertex to the free edge, while it remains constant on the leeward side, except for the region experiencing fluid detachment. The graph reports a single solution for each case due to the self-similarity of the results. In the case of \(\beta=30^{\circ}\), fluid detachment initiates when \(\epsilon\) becomes greater than 1 and the magnitude of the void region with respect to the overall wet length increases with the advancing ratio, to eventually remain fully dry when \(\epsilon\) is greater than 2. Notably, for both the deadrise angles considered, the cavity incipient condition is met as soon as \(\beta+\arctan\epsilon\approx 1.3\), while the cavity is completely formed as soon as \(\beta+\arctan\epsilon\approx\pi/2\), a result which matches with the analytical predictions [25], validating the capability of the numerical model to correctly capture the cavity formation event. The pressure on the windward side is found to increase with the horizontal component while it decreases on the opposite side. At the higher advancing ratio considered in this study (\(\epsilon=2.5\)), the left side never makes contact with the fluid. Figure 6 also highlights, with particular reference to the \(\beta=30\) case, how the pile-up on the windward side of the wedge increases with the advancing ratio, while it decreases on the leeward side.