**2. Modeling**

#### *2.1. 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:

**Figure 1.** Multi-functional Marine mapping unmanned surface vehicles.


**Table 1.** Table of the unmanned surface vehicle parameters.

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).


**Table 2.** Cases list.

<sup>1</sup> *Fr*∇: Volume Froude number, a non-dimensional parameter which presents the shape and the displacement of the ship.

The governing equations were solved by RANS (standard *k*-ε) with STAR-CCM+. Assuming that the fluid flow is viscous and incompressible, the governing equations are established as below:

$$\text{div}\!\!u = 0,\tag{1}$$

$$\frac{\partial \mu}{\partial t} + \text{div}(\mu u) = -\frac{1}{\rho} \frac{\partial p}{\partial x} + \nu \text{div}(\text{grad} u), \tag{2}$$

$$\frac{\partial \mathbf{v}}{\partial t} + \text{div}(\mathbf{v}u) = -\frac{1}{\rho} \frac{\partial p}{\partial y} + \mathbf{v} \text{div}(\mathbf{grad} \mathbf{v}),\tag{3}$$

$$\frac{\partial \mathbf{w}}{\partial t} + \text{div}(\mathbf{w}\mathbf{u}) = -\frac{1}{\rho} \frac{\partial p}{\partial z} + \text{vdiv}(\mathbf{grad}\mathbf{w}),\tag{4}$$

where *u*, *v*, *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:

$$S\_{PM}(\omega) = \frac{5}{16} (H\_S^2 \omega\_P^4) \omega^{-5} \exp\left(-\frac{5}{4} \left(\frac{\omega}{\omega\_P}\right)^{-4}\right),\tag{5}$$

where ω is the angular frequency of waves. ω*<sup>P</sup>* = (2π)/(*TP*) represents the peak frequency of the spectrum. *TP* is the peak wave period. *HS* 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:

$$
\eta = H\_{\text{s}} \cos(mx - at),
\tag{6}
$$

in which η is energy, and *m* is the number of waves. The velocity field of irregular wave is as below:

$$\begin{cases} \; \mathcal{U} = \omega \mathcal{H} \epsilon^{k\overline{z}} \cos(m\mathbf{x} - \omega t) \\ \; \mathcal{V} = 0 \\ \; \mathcal{W} = \omega \mathcal{H} \epsilon^{k\overline{z}} \sin(m\mathbf{x} - \omega t) \end{cases} \tag{7}$$

where *U* and *W* are velocity profiles on *x* and *z* direction. *H* is the wave amplitude, *k* is wave-number (2π/m). *x*-axis is the direction of wave propagation, and the *z*-axis is the direction of wave fluctuation. The wave propagation direction is parallel to the heading direction of USV.

The numerical simulation method is shown in Table 3.


Momentum Second Order Upwind Transient Formulation Second Order Implicit Mesh motion Overset motion

**Table 3.** Numerical simulation methods.

The computational domain is shown in Figure 2:

**Figure 2.** Computational Domain.

The boundary conditions are as follows: The initial pressure of the whole domain basin is zero, the inlet boundary is Velocity-Inlet, and the velocity direction is perpendicular to the flow. Pressure-Outlet is used at the outlet boundary. The initial pressure value is zero. The flow is fully developed. The velocity gradient is zero in the normal direction of the boundary. Velocity-Inlet is used at the upper and lower boundaries of the basin, and symmetrical boundary conditions are used at the side walls.

Three grid numbers were tested on case 1. The numbers are 674921,1373495,1925630. The mean coefficient of drag force *Cd* at the monitoring point of the hull surface is taken to calculate the error *E*, which is written as:

$$\overline{C}\_d = \frac{\int\_0^T C\_d(t)dt}{T} \,\tag{8}$$

where *T* is monitoring period.

As shown in Table 4, the difference between Mesh 2 and Mesh 3 is extremely small, considering with saving the calculating time and source, Mesh 2 was selected. Its three-dimensional mesh generation and grid distribution around the wall are shown in Figure 3a,b


**Figure 3.** Mesh Generation and Distribution. (**a**) Three-dimensional Mesh Generation; (**b**) Grid Distribution Around the Wall.
