## 2 Modeling  
### Hydrodynamic Modeling  
Small unmanned surface vehicles are insufficient to resisting wind and waves. When sailing in constrained waters, due to the blockage effect of water boundary, the reflected wave and the generated wave of the boundary overlap or decrease each other, which results in pressure distribution difference on the hull surface, generating heave and trim motion and deducing sticking on bottom, shore touching and capsizing of the vessel and endangering the safety of navigation. Our group has developed an unmanned surface vessel with an adjustable scale shown as Figure 1 and Table 1 for marine surveying and mapping. The hull is made of two inflatable protons. The width of the equipment bracket can be adjusted and quickly disassembled/assembled in 30 minutes. The navigation status of this unmanned surface vehicle is studied with three kinds of velocity in irregular waves as an example, and then the criticality of the draft variation is estimated. The parameters of the USV are as follows:  
The ship's navigation status mainly contains three states: Heave, pitch and speed. Nine cases were set up with three speeds (5 kn, 8 kn, 11 kn) and three wave heights (0.5 m, 1 m, 1.5 m) when wave period is 2 s (shown in Table 2). The governing equations were solved by RANS (standard \(k\)-\(\varepsilon\)) with STAR-CCM+. Assuming that the fluid flow is viscous and incompressible, the governing equations are established as below:  
\[\mathrm{div}\mathbf{u}=0, \tag{1}\]  
\[\frac{\partial\mathbf{u}}{\partial t}+\mathrm{div}(\mathbf{uu})=-\frac{1}{\rho}\frac{ \partial\mathbf{p}}{\partial x}+\mathbf{v}\mathrm{div}(\mathrm{grad}\mathbf{u}), \tag{2}\]  
\[\frac{\partial\mathbf{v}}{\partial t}+\mathrm{div}(\mathbf{vu})=-\frac{1}{\rho}\frac{ \partial\mathbf{p}}{\partial y}+\mathbf{v}\mathrm{div}(\mathrm{grad}\mathbf{v}), \tag{3}\]  
\[\frac{\partial\mathbf{w}}{\partial t}+\mathrm{div}(\mathbf{wu})=-\frac{1}{\rho}\frac{ \partial\mathbf{p}}{\partial z}+\mathbf{v}\mathrm{div}(\mathrm{grad}\mathbf{w}), \tag{4}\] \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline Length & 3.2 m \\ Molded breadth & 2.2 m \\ Displacement & 92 kg (full load 140 kg) \\ Location of gravity center & (1.285, 0.000, 0.600) \\ Moment of inertia of Y axis & 400 kg/m\({}^{2}\) \\ Draft & 0.5 m \\ \hline \end{tabular}
\end{table} 
Table 1: Table of the unmanned surface vehicle parameters. Figure 1: Multi-functional Marine mapping unmanned surface vehicles. \begin{table}
\begin{tabular}{c c c} \hline
**Case Number** & **Speed (kn)** & **Wave Height (m)** & \(F_{\
u 0}\)1  \\ \hline
1 & 5 & 0.5 & 0.42 \\
2 & 5 & 1.0 & 0.42 \\
3 & 5 & 1.5 & 0.42 \\
4 & 8 & 0.5 & 0.63 \\
5 & 8 & 1.0 & 0.63 \\
6 & 8 & 1.5 & 0.63 \\
7 & 11 & 0.5 & 0.84 \\
8 & 11 & 1.0 & 0.84 \\
9 & 11 & 1.5 & 0.84 \\ \hline \end{tabular}
\end{table} 
Table 2: Cases list. where \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\) are respectively the components of velocity on the directions of \(x\), \(y\), \(z\) (see Figure 1). The waves were generated by Pierson-Moskowitz spectra, which defines the motion of irregular waves--it is written as:  
\[S_{PM}(\omega)=\frac{5}{16}\Big{(}H_{S}^{2}\omega_{P}^{4}\Big{)}\omega^{-5} \exp\biggl{(}-\frac{5}{4}\Big{(}\frac{\omega}{\omega_{P}}\Big{)}^{-4}\biggr{)}, \tag{5}\]  
where \(\omega\) is the angular frequency of waves. \(\omega_{P}=(2\pi)/(T_{P})\) represents the peak frequency of the spectrum. \(T_{P}\) is the peak wave period. \(H_{S}\) is significant wave height (m). The velocity distribution of the flexible wave generator plate is employed to generate a simulated incident, waves at the inlet boundary, the wave surface equation of irregular waves can be expressed as follows:  
\[\eta=H_{s}\mathrm{cos}(mx-\omega t), \tag{6}\]  
in which \(\eta\) is energy, and \(m\) is the number of waves. The velocity field of irregular wave is as below:  
\[\left\{ \begin{array}{l}U=\omega He^{kz}\mathrm{cos}(mx-\omega t)\\ V=0\\ W=\omega He^{kz}\mathrm{sin}(mx-\omega t)\end{array}