*4.3. VSB WEC Deformation*

Since the proposed device allows significant deformation in response to the waves, it is interesting to study the geometry variation in free motion. As shown in Figure 17, although the geometry of the device varies significantly from 0 s to 1.5 s, the shape of the device at 1.5 s indicates a trend of recovering to the original shape (at 0 s). Thus, a periodic change of the shape is expected, which is also reasonable since the gas chamber is a restoring mechanism. This periodic phenomenon can be better explained in the following figures. Figures 18 and 19 show the vertical motion of two nodes where the U=pper Node denotes the node defined originally at (40, 30, 42) m and the Lower Node denotes the node defined at (40, 30, 38). Figure 18 shows the individual motion of two nodes with respect to their original location. For a conventional fixed-shape buoy WEC (FSB WEC), the motions of these two nodes are expected to be identical since rigid body motion is assumed. Although, in this study, their motion patterns are different. The relative vertical distance between these two nodes is plotted in Figure 19. The original distance between these two nodes is 4 m, which will be kept as a constant for a FSB WEC. While the vertical distance of these two nodes for the VSB WEC changes periodically and finally reaches a steady-state deformation (approximately when *t* > 15 s). The natural period of steady-state deformation is around 1.96 s, which is almost one third of the wave peak period. The relation between the natural period of the shape deformation and the wave period can be more rigorously studied in the future by simulating the dynamic behavior of VSB WEC with varied wave conditions, which may benefit future optimization of the proposed device.

**Figure 17.** Snapshots of the deformation of VSB WEC at different time instants.

**Figure 18.** Motions of the upper node and the lower node with respect to their original location.

**Figure 19.** Relative vertical distance between the upper node and the lower node during operation.

#### *4.4. VSB WEC Motion*

The motion of VSB WEC is presented in detail in this section and it will be compared with FSB WEC motion under the same wave condition to highlight the difference in motions introduced by changing the shape. The motion difference can be later controlled and utilized to produce more wave power. Additionally, the mass and dimensions of FSB WEC are the same as VSB WEC (undeformed shape). The motions of the center of gravity of VSB WEC and FSB WEC in x, y, and z directions are shown from Figures 20–22, respectively. The motion of VSB WEC significantly differs from FSB WEC in three dimensions both in terms of phase and magnitude. The difference in vertical motion between the two devices is presented in Figure 23. The maximum difference is around 0.26 m, and the maximum z-directional motion of FSB WEC is around 0.64 m, which indicates a maximum 40.6% change in the heave motion caused by the introduction of geometry variation. This large difference represents a room for improving the performance of wave energy conversion by introducing appropriate control.

**Figure 20.** x-directional motion of center of gravity compared between FSB WEC and VSB WEC.

**Figure 21.** y-directional motion of center of gravity compared between FSB WEC and VSB WEC.

**Figure 22.** z-directional motion of center of gravity compared between FSB WEC and VSB WEC.

**Figure 23.** Difference in motions of FSB WEC and VSB WEC in heave direction.

#### **5. Discussion**

In this paper, a 3D numerical simulation architecture is introduced to simulate the fluid–structure interaction of the proposed VSB WEC. The simulation results show the proposed device has a significant shape-changing due to the highly nonlinear interaction between waves and the device. The resulting motion of VSB WEC also significantly differs from the motion of a conventional FSB WEC with the same mass and dimensions. It is noted that the design of VSB WEC can be further optimized in terms of energy extraction since this paper focuses on presenting a framework of the high-fidelity simulation of this device. The applied design is simple in terms of geometry and mechanism. For instance, a possible design presented in [16] is a VSB WEC that has a rigid body part (cylindrical shape) and a shape-changing part (truncated conical shape). A control valve is included to control the pressure oscillation in the gas chamber. As found in [16], a restoring mechanism (gas chamber in this study) is required to keep a steady-state deformation of the device in order to harvest more energy. The presented framework is applicable to simulate the performance of other designs of VSB WEC.

Hyperelastic material is applied in this study since the proposed device is required to be 'soft'. The material applied in the simulation obeys Neo-Hookean hyperelasticity with an initial shear modulus of 0.1 Mpa and an initial bulk modulus of 2000 Mpa, and the density of the material is 1000 kg·m<sup>−</sup>3. The corresponding strain–stress curve of the applied material is shown in Figure 24. Unlike vulcanized rubber [30] (initial shear modulus is 0.41 Mpa and initial bulk modulus is 414.5 Mpa), which only allows small deformation, the material used in this study is 'softer', which allows more deformation. In addition, unlike neoprene rubber (initial shear modulus is 0.027 Mpa and initial bulk modulus is 13.86 Mpa), which is too soft such that the deformation is too large to keep a stable simulation. More advanced dynamic mesh techniques need to be applied to address this challenging mesh motion. The study of the effect of different materials is interesting, though beyond the scope of this paper, therefore a material that has the hyperelastic properties between vulcanized rubber and neoprene rubber is applied such that the simulation is stable with a considerable deformation of the device.

**Figure 24.** Material properties of the applied hyperelastic material.

#### **6. Conclusions**

A high-fidelity CFD-based numerical wave tank simulation for a VSB WEC is presented in this paper. This numerical tool is demonstrated to be able to simulate a VSB spherical WEC. This highly nonlinear interaction between the device and the waves is simulated using a 2-way FSI technique. Open sea conditions are applied in this study by assuming infinite air, deep water, and multi-directional waves. The numerical results in this paper capture the shape deformation in response to the varying surface pressure from the simulated ocean waves. It was shown that the VSB spherical WEC exhibits a transient response period before it reaches a steady-state motion and deformation. It was shown that this resulting motion of the VSB WEC significantly differs from that of a FSB WEC of the same shape as that of the non-deformed VSB WEC. This difference in response characteristics (difference in motion trajectories) is key for investigating the advantage of the VSB WEC power production over the FSB WEC.

**Author Contributions:** S.Z. implementation of numerical experiment, investigation, data analyses, visualization, original draft; O.A. conceptualization of the study, investigation, funding acquisition, refining of draft. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research is funded by National Science Foundation (NSF), USA, under Grant Number 2023436.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This material is based upon work supported by the National Science Foundation (NSF), USA, under Grant Number 2023436.

**Conflicts of Interest:** The authors declare that they have no conflict of interest.

#### **References**

