## Appendix A The Semi-Analytical Model Based on the Modified Logvinovich Model  
The derivation of the semi-analytical model reported by [PERSON] et al. [7] is shown in the following description. This model is based on the Modified Logvinovich model. The definition sketch of the impact model is illustrated in Figure 1. Note that, the meanings of symbols in this section are the same as that in Section 2. For the Modified Logvinovich model [32], when the wedge section enters the fluid at a constant velocity, the distribution of pressure in the contact region is given as follows. \[P_{r}(x,t)=\frac{1}{2}\rho V^{2}[\frac{2\dot{c}(t)}{V}\frac{c(t)}{\sqrt{c^{2} (t)-x^{2}}}-\cos^{2}\beta\frac{c^{2}(t)}{c^{2}(t)-x^{2}}-\sin^{2}\beta] \tag{1}\]  
\(c(t)\) is determined with the help of the Wagner condition, which can be written as:  
\[\int\limits_{0}^{\pi/2}f[c(t)\sin\theta]d\theta=\frac{\pi}{2}Vt \tag{2}\]  
where \(f(x)\) is the function to describe the section shape. \(\pi/2\) on the right side of the equation is the fluid surface rise coefficient and it is denoted as \(c_{v}\) in the following expressions. By comparing with results of the CFD method, \(c_{v}\) was suggested to be 1.54 in [PERSON] et al. [7]. In the case of the wedge, the Equation (2) can be expressed in the form:  
\[c(t)=\frac{c_{v}Vt}{\tan\beta} \tag{3}\]  
Influenced by \(-\cos^{2}\beta\frac{c^{2}}{c^{2}-x^{2}}\) in Equation (1), the pressure of the contact region edge tends to be negative infinity. By assuming \(P[a(t),t]=0\), the impact pressure \(P(x,t)\) in the region \(-a(t)<x<a(t)\) is positive. In order to solve the equation \(P[a(t),t]=0\), it is convenient to define the ratio \(\tau=a(t)/c(t)\) and \(\tau=\sqrt{1-X^{2}}\). Then, we obtain  
\[X=\frac{\sin(2\beta)}{2c_{v}[1+\sqrt{a constant velocity, the pressure of the contact region should always be positive [4]. The pressure distribution of the region \(-a(t)<x<a(t)\) is mapped to the region \(-c(t)<x<c(t)\) by the ratio \(\tau\) in order to correct the unreasonable negative pressure. Thus, the pressure distribution pressure in the contact region is given as:  
\[P_{r}(x,t)=\frac{1}{2}\rho V^{2}[\frac{2\dot{c}}{V}\frac{c}{\sqrt{c^{2}-\left( \tau x\right)^{2}}}-\cos^{2}\beta\frac{c^{2}}{c^{2}-\left(\tau x\right)^{2}}- \sin^{2}\beta] \tag{10}\]  
It can be seen that the pressure is equal to zero when \(x\) is equal to \(c\), and the unreasonable negative pressure is avoided. As shown in Figure 1b, a fictitious section is introduced to deal with the flow separation. On the assumption that the fluid rises along the fictitious section after flow separation, the section shape function considering the fictitious section is given as:  
\[f(x)=\left\{ \begin{array}{ll}x\tan\beta&0\leq x\leq L\cos\beta\\ b\tan\beta+(x-b)\tan\alpha&x>L\cos\beta\end{array} \right. \tag{11}\]  
where \(\alpha\) is the location angle of the fictitious section and it was suggested to be \(40^{\circ}\) for wedges with \(10^{\circ}\) and \(20^{\circ}\) deadrise angles in Yu et al. [7]. By getting the time derivative of both sides of Equation (10), we obtain  
\[\int\limits_{0}^{\pi/2}f_{x}[c(t)\sin\theta]\cdot\dot{c}(t)\cdot\sin\theta d \theta=c_{v}V \tag{12}\]  
where \(f_{x}(x)\) is given as:  
\[f_{x}(x)=\left\{ \begin{array}{ll}\tan\beta&0\leq x\leq L\cos\beta\\ \tan\alpha&x>L\cos\beta\end{array} \right. \tag{13}\]  
By introducing a parameter \(\theta_{s}\) that satisfies the equation \(c(t)\cdot\sin\theta_{s}=L\cos\beta\), Equation (12) can be transformed into the following expression:  
\[\dot{c}(t)\left\{\tan\beta\int\limits_{0}^{\theta_{s}}\sin\theta d\theta+\tan \alpha\int\limits_{\theta_{s}}^{\pi/2}\sin\theta d\theta\right\}=c_{v}V \tag{14}\]  
\[\dot{c}(t)=\frac{c_{v}V}{\tan\beta(1-\cos\theta_{s})+\tan\alpha\cos\theta_{s}} \tag{15}\]  
In general, the expression of pressure distribution before and after flow separation is the same, and the difference is the calculation of \(\dot{c}(t)\). Hence, the pressure distribution of the semi-analytical model can be expressed in the following equations.