*3.2. Boundary Conditions*

As shown in Figure 5, the computational domain for a steady, incompressible, two-dimensional flow of water with physically specified boundary conditions (BC) was identified. Two compounds were modeled in the flow field: Liquid water and air above the free surface of the calm sea. The AUH model was 1.0 m in diameter and 0.45 m in height. The coordinate origin was arranged at the center of gravity (CG) of the model. The computational domain was rectangular with the dimensions of 25.0 m × 3.0 m × 10.0 m, and the overset area near the AUH, where STAR-CCM+ overlapping grid technology was implemented, had the dimensions of 2.0 m × 1.0 m × 1.0 m. The velocity inlet was specified at 5.5 m upstream of the AUH bow. The pressure outlet was specified at 9.5 m downstream of the AUH. The initial height of the AUH model was specified as 0.5 m above the free surface, and the water depth was set as 6.0 m.

**Figure 5.** Computational domain for a steady, incompressible, two-phase flow of water and air with a free surface.

#### *3.3. Meshing the Computational Domain*

The CFD meshes could be divided into two categories: Structured meshes and unstructured meshes [27]. A hybrid mesh integrates structured and unstructured meshes to increase the mesh density near walls. In this paper, six hexahedron meshes were adopted by using a cutting mesh generator, and the total number of cells was 1.0 × 106. The encryption process of meshing the AUH required a minimum mesh size of 7.5 × 10−<sup>3</sup> m to ensure an accurate solution for the turbulence model that could meet y<sup>+</sup> values greater than 30 [28]. It is worth mentioning that, to clearly simulate the variable process of AUH immersing into the water, meshing encryption at the air–water interface

needs to be arranged, and connatural and miscellaneous dimensions need to be activated in the cutting mesh generator. The generated meshes of the two-phase flow and AUH are shown in Figure 6.

**Figure 6.** Mesh generations of the two-phase flow and AUH; (**a**) mesh of computational domain; (**b**) mesh partition of the AUH surface.

## **4. Simulation Results**

This chapter introduces the water inflow process of AUH. It analyzes the changes to load and velocity in all directions during the AUH water inflow, and it analyzes the maximum load of the AUH at different velocities and at different water inlet angles.

#### *4.1. Impact Force Load of the Water Entry Process*

In this paper, the *k*−<sup>ε</sup> turbulence model and the VOF method simulation technique in STAR-CCM+ software were adopted to simulate different situations of the AUH immersing into water from 0.5 m above the surface at different initial velocities. Figure 7a,b shows the two-phase flow simulation of the AUH water entry at an initial velocity of 3 m/s with different immersion angles: 30◦ and 60◦, respectively. Distribution of the volume fraction of water to air is shown in the CFD simulation results (Figure 7). A time-varying, deformable cavity formed, and free surfaces were captured while the AUH was immersed into the water with immersion angles of 30◦, 45◦, 60◦, and 90◦. The initial velocities of the AUH were set to 3–8 m/s. In summary, since the forces acting on the symmetric, disk-type, non-spinning, inclined AUH after the impact are dictated by the cavity's dynamics, they are also affected by free surface conditions in cases where the cavity forms asymmetrically. The asymmetric degree of the formed cavity is highly correlated with the water entry velocity and immersion angle of the AUH.

Figure 8a–d shows that the surge force (X-force) changes under different immersion conditions during the initial AUH free fall, where the surge impact force on the AUH was measured with the body-fixed coordinates. When the AUH immersed into the water, the surge force reached the maximum value and then slowly decreased to a stable value. At the same immersion angle, the greater the initial velocity was, the greater the impact on the AUH. At the same initial speed, with an increase in the immersion angle of the AUH, the impact force on the AUH decreased. Notably, at the highest initial velocity of 8 m/s, the impact momentum of the surge on the AUH was smaller when the immersion angle was 90◦, whereas a lower immersion angle of 30◦ caused a prominent impulsive force on the disk-type AUH.

Figure 9a–d shows the variations of the heave force (Z-force) in body-fixed coordinates versus different immersion angles over time. These results were similar to those of the surge force changes during the initial free-falling period; thus, the heave force is negligible. After the value reaches its peak, it decreases gradually over time until steady-state conditions are reached. In the case of the same immersion angle, a greater the initial velocity causes a greater impact force on the AUH. For different immersion angles with the same velocity as in Figure 9, a greater immersion angle causes a smaller impact force on the AUH. Therefore, a greater immersion angle can decrease the force of the impact on the AUH in surge and heave, which can be conducive to the floating state of the AUH when the AUH immerses into water.

**Figure 7.** Two-phase flow simulation of the AUH at an initial velocity of 3 m/s with different immersion angles of 30◦ and 60◦. (**a**) Immersion angle: 30◦, initial velocity: 3 m/s; (**b**) immersion angle: 60◦, initial velocity: 3 m/s.

**Figure 8.** The variations of surge loads versus initial velocities of 3, 5, and 8 m/s at different immersion angles of 30◦, 45◦, 60◦, and 90◦ over time. (**a**) Immersion angle: 30◦; (**b**) immersion angle: 45◦; (**c**) immersion angle: 60◦; and (**d**) immersion angle: 90◦.

**Figure 9.** The variations of heave loads versus initial velocities of 3, 5, and 8 m/s at different immersion angles of 30◦, 45◦, 60◦, and 90◦ over time. (**a**) Immersion angle: 30◦; (**b**) immersion angle: 45◦; (**c**) immersion angle: 60◦; and (**d**) immersion angle: 90◦.

#### *4.2. Variations of the AUH Water Entry Velocity*

Figure 10a–d shows the speed changes under different immersion conditions. There was an initial acceleration process for a very short period because this is the free-falling motion of the AUH. The greater the initial velocity, the shorter the free-fall duration was; the greater the immersion angle, the shorter the free-fall duration was. Subsequently, as the AUH head entered the water, the velocity of the AUH decreased over time, and the greater the initial velocity, the greater the speed reduction. For instance, when the immersion angle of the AUH was at 30◦ and the AUH hull was completely immersed in water, the velocity reduced by 7.0, 4.3, and 2.7 m/s at 8, 5, and 3 m/s, respectively. The smaller the immersion angle, the greater the maximum vertical velocity that could be achieved. The larger the immersion angle, the smaller the vertical velocity change was, and the AUH quickly stabilized.

**Figure 10.** The variations of falling velocity in heave versus initial velocity (water entry velocity: 4.34, 5.90, and 8.59m/s) and immersion angles over time. (**a**) Immersion angle: 30◦; (**b**) immersion angle: 45◦; (**c**) immersion angle: 60◦; and (**d**) immersion angle: 90◦.

#### *4.3. Maximum Load Analysis*

Figure 11 shows the peak values of the impact surge loads on the AUH at different immersion angles. At the same angle, the initial velocity was directly proportional to the surge load. When the AUH immersion angle approached 30◦ and the initial velocity was 8 m/s, the load was up to 4000 N, which would seriously impact the safety of the AUH and would even cause electronic components to fail. At the same initial velocity, the smaller immersion angle caused a greater surge load to be experienced. Therefore, the results sugges<sup>t</sup> that, to avoid a small immersion angle and a greater speed to decreasing the damage, the AUH structure should be mounted with precise sensors.

**Figure 11.** Variations of surge loads versus velocity at different immersion angles.
