**2. Structure Model**

The waves are generally generated by wind blowing across the surface of the ocean. The motion of waves is regular and periodic on the water surface, with the amplitude decreasing exponentially with depth. When the depth to be greater than half the wavelength, the wave-induced motions is only approximately 4% of those at the surface and thus could be considered to be insignificant [10]. Therefore, this depth range is defined as the hydrostatic layer, in which the motion of waves is hardly perceived as the depth increases. Based on the above characteristics of wave motions, a novel heaving wave energy point absorber is designed, which can be used as a power module for the low-power unmanned ocean device [7]. Figure 1 shows the working scene of this point absorber. First, when the low-power unmanned ocean device is working on the ocean surface, the value of the battery energy is decreasing from the maximum to the lowest safe. Secondly, utilizing the release mechanism of the low-power unmanned ocean device, the novel heaving point absorber is released. The Underwater PTO subsystem of this absorber converts the captured wave energy into electricity. The value of the battery energy reaches the highest energy. Finally, utilizing the recovery mechanism of the ocean device, the absorber is recovered. The above process cycles back and forth, of which ensures the energy supply of unmanned ocean device for long-range and long-term work.

**Figure 1.** The working scene of the novel heaving wave energy point absorber.

The system configuration and working principle of the novel heaving point absorber are illustrated in Figure 2. The absorber mainly includes two parts, a floating body and an Underwater PTO, which are connected with each other by a steel cable. The floating body floats on the ocean surface and the Underwater PTO suspends the hydrostatic layer at a depth of about 40 m. The Underwater PTO mainly consists of a power generator, upper impeller, lower impeller, steady blade, and transmission shaft and planet-gear increaser. The impellers are mainly composed of blades, connecting rods, locking devices, center wheels, and external fixation rings. The center wheel and the external fixation ring are connected by eight radials arranged in connecting rods. Eight fan-shaped blades are fixed on corresponding connecting rods with locking devices and are arranged in a centrally symmetric circumferential array. In addition, the steel cable not only acts as dragging the Underwater PTO, but also transmissions electricity and control signals.

**Figure 2.** The system configuration and working principle of the novel heaving point absorber. (**a**) Rising state; and, (**b**) Sinking state.

The working principle of the novel heaving point absorber is shown in Figure 2. (1) During the movement of the floating body from the wave trough to the wave crest, the Underwater PTO is pulled up by the steel cable. The upper surface of the blade is impacted by the water flow and the blade adaptively swings downward, as shown in Figure 2a. Due to the limitation of the locking devices, the blade then stops swinging and is in a slanted state after reaching the maximum angle of inclination. The water flow continues to impact the slanted blades and propel the blade forward. The circumferential array of the blades enables the impeller to be subjected to circumferential thrust. Since the blades of the upper and lower impellers are arranged in opposite directions, the upper impeller is clockwise rotated by the water flow and the lower impeller is anticlockwise rotated by the water flow, and they are relatively reversed. The upper and lower impellers are fixedly connected with the stator and rotor of the generator, respectively, and then drive the generator to generate electricity. (2) During the movement of the floating body from the wave crest to the wave trough, the Underwater PTO sinks under the influence of gravity. The lower surface of the blade is impacted by the water flow and the blade adaptively swings upward, as shown in Figure 2b. Due to the limitation of the locking devices, the blade then stops swinging and it is in a slanted state after reaching the maximum angle of inclination. Since the direction of the water flow impinging on the blade does not change, the water flow continues to impact the slanted blades and propel the blade forward. The upper impeller is continuously clockwise rotated by the water flow and the lower impeller is continuously anticlockwise rotated by the water flow. Therefore, the direction of rotation of the generator does not change and it continues to generate electricity. According to the different impact directions of water flow, the impellers' blades adaptively adjust the blade deflection. The upper and lower impellers act as components that interact directly with the water flow and they provide continuous rotational motion to the generator during rising and sinking of the Underwater PTO.

#### **3. Mathematical Model**

The floating body is an important part of the novel heaving wave energy point absorber and is in direct contact with the waves on the ocean surface. In order to increase the energy harvesting of floating body, it is necessary to increase the heave amplitude, heave velocity, and wave force in the vertical direction. It can increase the rising and sinking amplitude of the Underwater PTO and the relative rotational speed of the upper and lower impellers. Moreover, the power production of the novel heaving point absorber can also be increased. This section analysis of the force of the floating body, establishes its motion equation, and solves its wave excitation force and the Underwater PTO damping force. Also, this section derives the model of energy conversion efficiency between the floating body and the wave energy, and calculates its energy conversion efficiency.

The hydrodynamics model and energy efficiency model of the floating body is established to calculate the wave force and the conversion efficiency: Section 3.1 considers nonlinear Froude-Krylov force in the vertical direction and Underwater PTO damping force for the force analysis of the floating body in micro-wave amplitude. Section 3.2 presents the capture width ratio of the floating body for the energy efficiency analysis.

#### *3.1. Hydrodynamics Model of the Floating Body*

We consider that the floating body of the novel heaving point absorber floats freely in water of uniform depth. Under the action of linear regular waves, the floating body does micro amplitude heave motion. The fluid is assumed inviscid and the incident flow irrotational and incompressible. Figure 3 is the coordinates and force analysis of the floating body. The right-handed inertial reference frame is centered at the hydrostatic equilibrium position of the body. *X* is the vertical distance from the reference fluid surfaces to the body waterline (positive downwards). *Y* is the vertical distance from the reference fluid surface to the fluid surface (positive downwards). *Y* = *r* cos *ϕt*, where *r* is the wave amplitude and *ϕ* is the wave angular frequency. *Z* is the vertical distance from the reference fluid surface to any point in the fluid (positive downwards). *D* is the waterline depth of the floating body. Newton's second law can be used to describe the system dynamics, as follows:

$$m\ddot{\mathbf{X}}(t) = \mathbf{G} - \iint\limits\_{S(t)} P(t)\mathbf{n}d\mathbf{S} - \mathbf{Q} - I + \mathbf{F}\_{\text{PTO}}(t) \tag{1}$$

where, *m* is the mass of the novel heaving point absorber, *X* is the heaving displacement of the body from its hydrostatic equilibrium position, .. *X* is the heaving acceleration of the body, *G* the gravity force, *S* the submerged surface, *P* the pressure, **n** a vector normal to the surface, *Q* the viscous damping force, *I* the inertia force of attached mass effect, and **F**PTO the Underwater PTO damping force.

The pressure *P* can be derived from the incident flow applying Bernoulli's equation:

$$P(t) = -\rho \lg z(t) - \rho \frac{\partial \phi(t)}{\partial t} - \rho \frac{|\nabla \phi(t)|^2}{2} \tag{2}$$

where, *ρ* the water density, *g* the acceleration of gravity, *P*st = −*ρgz*, hydrostatic pressure and *φ* the potential flow, which can be decomposed as the sum of the undisturbed incident flow potential *φ*I, the diffraction potential *φ*D, and the radiation potential *φ*R:

$$
\phi = \phi\_\mathrm{I} + \phi\_\mathrm{D} + \phi\_\mathrm{R} \tag{3}
$$

The Airy's wave theory assumes that the motion of the floating body is a small amplitude. The solution of Equation (1) is solved around the equilibrium position of the buoy. Under the linear assumption, the wetted surface is constant. However, the nonlinear of Froude-Krylov forces to be considered in the actual calculation process. Thus, the wetted surface is exactly instantaneous, namely integrating the fluid pressure over the actual submerged portion of the buoy, as it moves

through the water. Froude-Krylov forces include the static and dynamic forces. They depend on the instantaneous wetted surface, which depends both on the incident wave elevation and the displacement of the buoy. Froude-Frylov force can be written as:

$$\mathbf{F}\_{FK} = \mathbf{F}\_{FK\_{\text{st}}} + \mathbf{F}\_{FK\_{\text{dy}}} = \iint\limits\_{S(t)} \left( P\_{\text{sl}}(t) + P\_{\text{dy}}(t) \right) \mathbf{n} dS \tag{4}$$

where, **F***FK*st is the static Froude-Krylov force, given as the balance between the gravity force and the Archimedes force, and **F***FK*dy is the dynamic Froude-Krylov force.

$$\mathbf{F}\_{FK\_{st}} = \iint\limits\_{S(t)} P\_{st}(t)\mathbf{n}dS = -\iint\limits\_{S(t)} \rho \mathbf{g} \mathbf{z}dS \tag{5}$$

$$\mathbf{F}\_{FK\_{dy}} = \iint\limits\_{S(t)} P\_{dy}(t)\mathbf{n}dS \tag{6}$$

where, *<sup>P</sup>*dy <sup>=</sup> <sup>−</sup>*<sup>ρ</sup> ∂φ<sup>I</sup> <sup>∂</sup><sup>t</sup>* <sup>−</sup> *<sup>ρ</sup>* |∇*φI*<sup>|</sup> 2 <sup>2</sup> is the dynamic pressure.

**Figure 3.** The coordinates and force analysis of floating body.

The time-dependence annotation will be omitted for brevity hereafter. Equation (1) can be rewritten as [42]:

$$m\ddot{\mathbf{X}} = \mathbf{G} + \mathbf{F}\_{FK} + \mathbf{F}\_D + \mathbf{F}\_R + \mathbf{F}\_{\text{PTO}} \tag{7}$$

where, **F***<sup>D</sup>* is the diffraction force and **F***<sup>R</sup>* is the radiation force.

Note that, since the fluid is assumed to be inviscid, irrotational, and incompressible, no viscous force and inertia force appears in Equation (7). In addition, due to Froude-Krylov force already includes the inertia force that is caused by the attached mass effect, the inertia force also does not appear in this Equation.

Wave excitation force is the force of wind blows the waves to disturb the motion of the floating body on the ocean surface. This force can be written as:

$$\mathbf{F}\_{\rm ex} = \mathbf{C} \mathbf{F}\_{\rm FK\_{dy}} + \mathbf{F}\_{D} \tag{8}$$

where, *C* is the diffraction correction coefficient.

The linear approach assumes that radiation and diffraction forces are linear. Therefore, the radiation and diffraction potential is negligible when the floating body dimension is considerably smaller than the wave length [43–48]. When combining Equations (4)–(8), Equation (1) can be rewritten as:

$$m\ddot{\mathbf{X}} = \mathbb{C} \iiint\limits\_{S(t)} P\_{dy} \mathbf{n} dS + \mathbf{F}\_{\text{PTO}} \tag{9}$$

The algebraic calculation of the integral in Equation (9) requires the explicit definition of the dynamic pressure *P*dy, the infinitesimal surface element **n***dS*, and the limits of integration. Under the Airy's wave theory for deep water waves, the dynamic pressure at any point on the floating body in the local coordinate system can be written as:

$$P\_{\rm dy} = \frac{\rho gH}{2} \frac{chk\overline{z}}{chkd} \cos(k\overline{x} - \omega t) \tag{10}$$

where, *H* is wave height, *x* is the direction of wave propagation in the local coordinate system, *z* is the vertical displacement of the floating body in the local coordinate system (positive upwards), and *d* is water depth and *ω* is the wave circular frequency. Wave Number *k* is defined by the dispersion equation *k*tanh(*kd*) = *ω*2/*g*.

During the motion of the novel heaving point absorber on the ocean surface, the coordinates at any point on the floating body in the global coordinate system can be written as:

$$
\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} X\_0 \\ Y\_0 \\ Z\_0 \end{bmatrix} + [T] \begin{bmatrix} \overline{x} \\ \overline{y} \\ \overline{z} \end{bmatrix} \tag{11}
$$

where, (*X*0,*Y*0, *Z*0) is the coordinates of the center of gravity on the floating body in the global coordinate system and (*x*, *y*, *z*) is the coordinates at any point on the floating body in the local coordinate system. [*T*] is coordinate transformation matrix between the local coordinate system and global coordinate system.

Therefore, the dynamic pressure at any point on the floating body in the global coordinate system can be written as:

$$P\_{\rm dy} = \frac{\rho gH}{2} \frac{chk(d-z)}{chkd} \cos(k(x+\overline{x}) - \omega t) \tag{12}$$

where, *x* is the direction of wave propagation in the global coordinate system and *z* is the vertical displacement of the floating body in the global coordinate system (positive upwards).

Furthermore, the force components of the wave excitation force in the vertical direction can be written as the following:

$$F\_{\rm ex}^{\rm x} = \mathbb{C}\_{V} \iint\limits\_{S(t)} P\_{dy} n\_{z} dS \tag{13}$$

where, *nz* is the projection of the normal for the wetted surface of the floating body in the vertical direction and *CV* is the diffraction correction coefficient in the vertical direction.

Under the movement of the novel heaving point absorber on the ocean surface, the heave motion of the floating body does work by overcoming the damping of the water. The Underwater PTO subsystem of this absorber converts the captured wave energy of the body into electricity. In the motion model of this absorber under the Airy's wave theory, the linear damping model is used to analyze the damping force of the Underwater PTO system. Thus, the Underwater PTO damping force can be written as: .

$$\mathbf{F\_{PTO}} = \mathbf{B\_{PTO}}X \tag{14}$$

where, BPTO is the Underwater PTO damping coefficient and . *X* is the heaving velocity of the floating body.

#### *3.2. Energy Efficiency Model of the Floating Body*

In the researches of the WEC, the performance of power production of the device is mainly concentrated on the energy harvesting and energy conversion. Using the capture width ratio [49,50] of the floating body as an evaluation index to measure the characteristic of energy harvesting and conversion efficiency for the novel heaving point absorber. The capture width ratio is that the ratio of the average power of the energy harvesting to the wave energy of the incident wave within the width of the body, namely the efficiency of energy harvesting in the WEC. It can be written as the following:

$$
\eta = \frac{P\_{\text{WEC}}}{P\_0} \tag{15}
$$

where, *PWEC* is the average power of the floating body to harvests wave energy and *P*<sup>0</sup> is the wave energy of the incident wave within the width of the body.

The average power of the energy harvesting is that the instantaneous power is integrated in a period and it solves the average value. It can be written as:

$$P\_{\rm WEC} = \frac{1}{T} \int\_0^T \mathbf{F}\_{\rm uavve} \cdot \mathbf{V}\_{\rm WEC} dt \tag{16}$$

where, **F**wave is the wave force, **V***WEC* is the velocity of the floating body, and *T* is the wave period.

The average power of the energy harvesting in the vertical direction of the floating body can be written as:

$$P\_{\rm WEC}^{\rm z} = \frac{1}{T} \int\_0^T \left( F\_{\rm ex}^z + F\_{\rm PTO}^z \right) X dt = \frac{1}{T} \int\_0^T \left( (\mathbb{C}\_V \iint\limits\_{S(t)} P\_{\rm dy} \eta\_z dS + B\_{\rm PTO}^z X) X \right) dt \tag{17}$$

where, *F<sup>z</sup>* PTO is the Underwater PTO damping force in the vertical direction and *<sup>B</sup><sup>z</sup>* PTO is the Underwater PTO damping coefficient in the vertical direction.

The wave energy of the incident wave within the width of the floating body can be written as the following:

$$P\_0 = \rho g H^2 \frac{\omega}{16k} \left[ 1 + \frac{2kd}{sh(2kd)} \right] \cdot B \tag{18}$$

where, *B* is the heading wave width of the floating body.

#### **4. Algebraic Solution**

The characteristic parameters, such as profile parameters, immersion depth, and wetted surface distribution, play an important role in the performance analysis, structural design, manufacture, and employment of the floating body. These parameters affect the motion amplitude, motion velocity, wave force, and wave energy harvesting of the floating body. Therefore, selects the profile parameters and solves the algebraic solution of the wave force is critical. Geometry shapes of floating body are two types of axisymmetric and unaxisymmetric. The axisymmetric body is a curved surface body formed by a generatrix rotating around a fixed vertical axis. The shape of the axisymmetric body is dependent on the shape of generatrix and relative position of the generatrix and the fixed axis. The axisymmetric bodies include cylinder, cone, sphere, and so on. The unaxisymmetric body is formed by a non-rotating curved surface. This body consists of rectangular, trapezoid, polyprism,

and so on. In this paper, three kinds of shapes are selected for the floating body, such as rectangular, cylinder, and sphere. This section mainly solves the algebraic solution of the wave force of the above shape. Table 1 show that the algebraic solutions of the wave excitation force in the vertical direction of multi-type floating bodies. The geometric parameters of the axisymmetric body are described in cylindrical coordinates [42]. The axisymmetric bodies are described in rectangular coordinates.

**Table 1.** The algebraic solutions of the wave excitation force in the vertical direction of multi-type floating bodies.


#### **5. Numerical Results**

Based on the above analysis of the hydrodynamic performance and energy conversion characteristic for the multi-type floating bodies of the novel heaving wave energy point absorber, the wave force, heaving velocity, heaving displacement, and capture width ratio of the multi-type floating bodies were numerically simulated and comparatively analyzed.

During the numerical simulation, the parameters of the multi-type floating bodies are as follows: the length of horizontal rectangular *a* is 0.3 m, the length along the wave direction of horizontal rectangular *b* is 0.3 m, the height of horizontal rectangular *c* is 0.5 m, the radius of vertical cylinder *R* is 0.3 m, the height of vertical cylinder *l* is 0.5 m, the radius of sphere *R* is 0.3 m, the mean immersion depth is 0.15 m, the Underwater PTO damping coefficient in the vertical direction *B<sup>z</sup>* PTO is 20 KNs/m, and the mass of the novel heaving point absorber *m* is 10 kg. According to the model test results of previous research [51–53], the diffraction correction coefficient in the vertical direction *CV* of horizontal rectangular, vertical cylinder, and sphere can be obtained, respectively.

In this study, three reference sea states are used for evaluation the energy efficiency of the novel heaving wave energy point absorber. Sea state 1: the significant wave height *HS* is 0.1 m, the peak wave period *TP* is 2 s; Sea state 2: the significant wave height *HS* is 0.3 m, the peak wave period *TP* is 3.5 s; and, Sea state 3: the significant wave height *HS* is 0.5 m, the peak wave period *TP* is 5 s. The first sea state covers *HS* values from 0.0 to 0.1 m, the second sea state covers *HS* values from 0.1 to 0.3 m, and the third sea state covers *HS* values from 0.3 to 0.5 m. The combination of *HS* and *TP* is representative for the same wave tank, the wave climate date come from the simulation test. The model scales of the novel heaving point absorber, as follows: the Underwater PTO's diameter is 410 mm, the

distance between the upper and lower impeller is 400 mm, and the overall height of the Underwater PTO is about 560 mm. The scale between the dimensions of the tested heaving point absorber and the full scale device is 1:5. Another, the significant wave height *HS* is about 2.5 m in the real wave climate with the full scale device. The considered reference water depth *d* is 50 m. In addition, the incident wave is the linear regular wave, the angle between the direction of the linear regular wave and the direction of the novel heaving point absorber is 0◦. According to the above parameters of the incident wave, the working scene of the novel heaving point absorber is the deep-water waves. Therefore, the other parameters of the incident wave are obtained, including wavelength, wave velocity, and velocity of water particles. These parameters are as follows: the wavelength is 6.2/19/39 m, the wave velocity is 3.1/5.5/7.8 m/s, and the velocity of water particles is 0.16/0.27/0.31 m/s.

The results of numerical simulation are shown in Figures 4–11. The wave force, heaving velocity, heaving displacement, and capture width ratio of the multi-type floating bodies in the second sea state is shown in Figures 4–7, respectively. Figures 8–11 show that the above parameters for comparison in the three reference sea states and the values obtained from the numerical simulation.

**Figure 4.** The wave force of the multi-type floating bodies (*H* = 0.3 m, *T* = 3.5 s).

**Figure 5.** The heaving velocity of the multi-type floating bodies (*H* = 0.3 m, *T* = 3.5 s).

**Figure 6.** The heaving displacement of the multi-type floating bodies (*H* = 0.3 m, *T* = 3.5 s).

**Figure 7.** The capture width ratio of the multi-type floating bodies (*H* = 0.3 m, *T* = 3.5 s).

**Figure 8.** The wave forces of the multi-type floating bodies in the reference sea state.

**Figure 9.** The heaving velocity of the multi-type floating bodies in the reference sea state.

**Figure 10.** The heaving displacement of the multi-type floating bodies in the reference sea state.

**Figure 11.** The capture width ratio of the multi-type floating bodies in the reference sea state.

#### **6. Discussion**

As shown in Figures 4–6, the vibration frequency curve of the wave force, heaving velocity, and heaving displacement are less of the same for the vertical cylinder buoy and the sphere buoy. However, as to the former, the curve of the wave force and the heaving displacement is steeper, the heaving velocity is faster, wave follower is better, and the generated energy is higher. Nevertheless, the wave force and heaving displacement of the horizontal rectangular floating body are smaller than those two types of floating bodies. Therefore, the generated energy and wave follower is weaker by this one less than those two types of floater bodies. In Figure 7, the capture width ratio of the vertical cylinder floating body is higher and more stable than the sphere floating body. While, the horizontal rectangular floating body is much more less than those two types of floating bodies.

In addition, as shown in Figures 8–11, the wave force, heaving velocity, heaving displacement, and capture width ratio of the floating bodying is affected by the peak wave period and the wave

height at a certain sea state. The above parameters are become higher when the sea state increases. Another, the vertical cylinder floating body outperforms the other floating body in the three reference sea states. As shown in Figure 11, the curve of the capture width ratio is steeper and the value of this parameter is higher when the sea state big changes.

In summary, with the linear regular wave, the cylindrical floater vertically placed in the wave surface is the first optional shape for the novel heaving point absorber and follower is the sphere floater, which can increase the quality of power extracting and the efficient of the WEC system design. At last, the horizontal rectangular floating body is carefully selected.

## **7. Conclusions**

This paper presents a small novel heaving point absorber of energy supply for low-power unmanned ocean devices that are based on the counter-rotating self-adaptive mechanism, with the advantages of small space device, stability, and reliable energy conversion process. For improving the efficiency of this absorber's power production, the wave force and energy efficiency are analyzed by the Froude-Krylov method and the optimal floating body is selected, the following conclusions are drawn:


The study that was carried out in this paper focuses on the energy generation efficiency of the novel heaving point absorber for the supply power of the low-power unmanned ocean devices. Moreover, the multi-type floating bodies are optimized base on the Froude-Krylov method. Although a series of numerical analyses above has been conducted in this paper, research on the novel absorber and floating body is still not thorough enough. Therefore, a further step toward is that the verification experiments of wave tank will be done in the power production region for the novel absorber. In addition, the geometric parameters of the vertical cylinder floater will be optimized for increasing the conversion efficiency of the floating body.

**Author Contributions:** D.C., J.S. and Z.L. conceived the study design; D.C. and C.S. performed the numerical simulations; D.C. analyzed the data and wrote the manuscript; W.W. helped with editing the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China (51475465).

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