# Cavity Formation during Asymmetric Water Entry of Rigid Bodies
## 3 Coupled FEM/SPH Numerical Model  
In this study, we assume that cavitation at the wedge vertex does not occur and flow separation depends on the flow characteristics enabling ventilation. Gravity is always neglected, while the entry velocity and tilt motion either follow the actual impact dynamics or is set to a constant value, for in the latter case the solution is self-similar in time, hence the results might be normalized by the entry depth. In the case of full fluid-structure interaction simulations (hence modelling the whole impact dynamics) the solution is not self-similar in time not only because of the role of acceleration but also because the tilt motion modifies the deadrise angle. As a result, the jet of water piling up over the counter-clock side thickens, while in the opposite side, the water is pushed by the rotating wedge and the free surface profile near the wedge deforms significantly. The 2D numerical model, presented in Figure 2, is based on the SPH solutor available within the commercial software LS-DYNA. In all the numerical solutions, the fluid is modelled by SPH particles with a constant diameter of 0.25 mm covering a region 0.8 m wide and 0.3 m deep. Non-reflecting boundaries define the bounding box. The fluid is modelled through an equation of state following the Gruneisen model and the fluid particles are solved utilizing the SPH-enhanced fluid formulation (FORM = 16). Both wedge's sides are 0.3 m long and are modelled as a rigid shell composed by 100 elements on each side, leading to an average of four SPH particles that make contact with a single element in the wet region. Such a strategy allows to smooth out the marked pressure fluctuations otherwise attained at the fluid-structure interface. Deadrise angle, initial rotation, and entry velocities are parametrically varied, while the wedge mass is 100 kg per unit width. Figure 1: Sketch of the asymmetric impact of a wedge. The numerical results will be compared against established analytical solutions. It has already been shown that the numerical scheme utilized within this work is in line with the analytical predictions provided by [PERSON]'s model for the case of pure vertical velocity [26; 27] which, in turn, was shown to provide results in line with experimental results conducted on rigid wedges [28; 29; 30]. In [PERSON]'s expanding plate model, which has been formulated for symmetric impacts only, the pressure distribution along the wedge edge is given by:  
\[\frac{p}{\rho}=\dot{V}\sqrt{r^{2}-x^{2}}+\frac{V\;r\;\dot{r}}{\sqrt{r^{2}(t)- x^{2}}}-\frac{1}{2}\frac{V^{2}\;x^{2}}{r^{2}-x^{2}} \tag{1}\]  
while the Modified Logvinovich Model (MLM), before flow separation, reads [32]  
\[\frac{p}{\rho}=\frac{1}{2}V^{2}\left[\frac{\dot{r}}{V}\frac{r(t)}{\sqrt{r^{2}( t)-x^{2}}}-\cos^{2}\beta\frac{r^{2}(t)}{cr^{2}(t)-x^{2}}-\sin^{2}\beta\right] \tag{2}\]  
where \(0<x<r\) defines the wet portion of the edge. Within this scheme, \(x\) is an abscissa oriented horizontally, i.e., following the direction of the free surface, and \(r\) is a function of the entry depth \(\xi\) and equals \(\frac{\pi}{2}\frac{\xi}{\tan\beta}\). In the case of constant entry velocity, the horizontal projection of the wet length reads \(r=\frac{\pi\;V\;t}{\tan\beta}\). MLM can be also utilized to study asymmetric impacts. However, a simplified method considering an equivalent deadrise angle equal to the average of the left- and the right-side deadrise angles is sufficiently accurate to predict the overall impact dynamics [32]. Within this numerical scheme, the distribution of the non-dimensional hydrodynamic pressure \(\bar{p}\), which is given by \(\frac{2p\tan\beta}{\rho V^{2}}\), over the horizontal abscissa \(x\) is given by  
\[\bar{p}(x)=\frac{\pi}{\sqrt{1-\left(\frac{x}{r}\right)^{2}}}-\tan\beta\frac{ \left(\frac{x}{r}\right)^{2}}{1-\left(\frac{x}{r}\right)^{2}} \tag{3}\]  
The normalized pressure distribution for varying deadrise angle is shown in Figure 3.  
Figure 2: Two-dimensional numerical model with highlighted boundary and initial conditions.