# Numerical Model of Constrained Wave Energy Hyperbaric Converter under Full-Scale Sea Wave Conditions

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## Abstract

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## 1. Introduction

## 2. Numerical Model

**In the first step**, DualSPHysics solves the following continuity and Navier–Stokes equations in SPH form [31] with the density diffusion term [32]:

**In the second step**, the linear $d\overrightarrow{V}/dt$ and angular $d\overrightarrow{\Omega}/dt$ acceleration to be applied in the centre of mass of a rigid body b are transferred to the Project Chrono library. During this time step, this library updates the motion, considering the given mechanical constraints and using the multibody dynamic model. The configuration of a rigid multibody system is described by using generalised coordinates $\overrightarrow{q}={[{\overrightarrow{R}}^{\mathrm{T}}+{\overrightarrow{\Theta}}^{\mathrm{T}}]}^{\mathrm{T}}$, where $\overrightarrow{R}$ is the translational and $\overrightarrow{\Theta}$ is the rotational coordinates that define each body in the system frame [16].The dynamics of rigid bodies are characterised by a system of two differential algebraic equations that relate the time derivative of generalised coordinates and velocity through a linear transformation and the equation of the force balance that ties the inertial forces to the applied and constraint forces in the following form [18]:

**In the third step**, position, linear and angular velocity of the rigid body are transferred back to the DualSPHysics to update the particles that form the rigid body with the information transferred from the Project Chrono library (i.e., $\overrightarrow{V}$, $\overrightarrow{\Omega}$ and ${\overrightarrow{R}}_{0}$). The velocity of each particle of a rigid body is given by:

## 3. Numerical Model Validation

## 4. Results and Discussion

#### 4.1. Simulation Set-Up

#### 4.2. Power Absorption under Regular Sea State

#### 4.3. Power Absorption under Irregular Sea State

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic view of WEHC with main components (floater, lever arm, support platform and hydraulic PTO system), including the mechanical constraints among the three different sub-systems (universal joint between lever arm and floater, revolute joints between lever arm and support platform and hydraulic pump).

**Figure 2.**Schematic (side) view of the numerical set-up: (

**a**) channel with the Wavestar model (not to scale); (

**b**) dimensions (in m) of the Wavestar model.

**Figure 3.**Time series of position (top) and velocity (bottom) of the PTO system for Test 1 with (

**a**) $D=0$, (

**b**) $D=50$ Nms, (

**c**) $D=100$ Nms and (

**d**) $D=200$ Nms [43].

**Figure 4.**Time series of position (top) and velocity (bottom) of the PTO system for Test 2 with (

**a**) $D=0$ and (

**b**) $D=200$ Nms [43].

**Figure 5.**Time series of position (top) and velocity (bottom) of the PTO system with $D=200$ Nms for (

**a**) Test 3 and (

**b**) Test 4 [43].

**Figure 6.**Schematic (side view) of the numerical set-up: (

**a**) channel with the full-scale WEHC (not to scale); (

**b**) dimensions (in m) of the WEHC model.

**Figure 7.**Influence of PTO damping on the CWR for $T=7$, 9 and 11 s and (

**a**) $H=2$ m and (

**b**) $H=2.5$ m.

**Figure 8.**Snapshots of the velocity field in the vicinity of the WEHC device for $H=2$ m and $T=7$ s with impermeable and straight breakwater at (

**a**) $t=100$ s, (

**b**) $t=102$ s, (

**c**) $t=104$ s and (

**d**) $t=107$ s.

**Figure 9.**Influence of breakwater porosity and geometry and PTO damping on the CWR for $T=7$, 9 and 11 s and (

**a**) $H=2$ m and (

**b**) $H=2.5$ m.

**Figure 10.**Snapshots of different instants of the velocity field in the vicinity of the WEHC device for $H=2$ m and $T=7$ s with porous breakwater at (

**a**) t = 100 s, (

**b**) t = 102 s, (

**c**) t = 104 s and (

**d**) t = 107 s.

**Figure 11.**Power spectrum of (

**a**) free-surface elevation measured at $x=220$ m and of (

**b**) horizontal position of hydraulic PTO system for ${H}_{s}=2.5$ m and ${T}_{p}=$ 7, 9 and 11 s.

**Figure 12.**Detail of surface elevation and CWR for (

**a**,

**b**) ${H}_{s}=2.5$ m and ${T}_{p}=7$ s, (

**c**,

**d**) ${H}_{s}=2.5$ m and ${T}_{p}=9$ s, (

**e**,

**f**) ${H}_{s}=2.5$ m and ${T}_{p}=11$ s. In the power capture curve, the horizontal dashed red line indicates the mean power capture.

**Figure 13.**Power spectra of (

**a**) free-surface elevation measured at $x=220$ m and of (

**b**) the horizontal position of the hydraulic PTO system for ${H}_{s}=2$ m and ${T}_{p}=$ 7, 9 and 11 s.

**Figure 14.**Detail of surface elevation and CWR for (

**a**,

**b**) ${H}_{s}=2$ m and $T=7$ s, (

**c**,

**d**) ${H}_{s}=2$ m and $T=9$ s, (

**e**,

**f**) ${H}_{s}=2$ m and $T=11$ s. In the power capture curve, the horizontal dashed red line indicates the mean power capture.

Test 1 | Test 2 | Test 3 | Test 4 | |
---|---|---|---|---|

H (m) | 0.1 | 0.15 | 0.25 | 0.25 |

T (s) | 1.4 | 1.4 | 1.4 | 2.8 |

$kA$ (-) | 0.103 | 0.154 | 0.257 | 0.07 |

Test 1 | Test 2 | Test 3 | Test 4 | |
---|---|---|---|---|

z (m) | 10% | 8% | 7% | 9% |

$\dot{z}$ (m/s) | 11% | 6% | 5% | 13% |

State 1 | State 2 | State 3 | State 4 | State 5 | State 6 | |
---|---|---|---|---|---|---|

H (m) | 2 | 2 | 2 | 2.5 | 2.5 | 2.5 |

T (s) | 7 | 9 | 11 | 7 | 9 | 11 |

$kA$ (-) | 0.083 | 0.054 | 0.040 | 0.104 | 0.067 | 0.050 |

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## Share and Cite

**MDPI and ACS Style**

Brito, M.; Bernardo, F.; Neves, M.G.; Neves, D.R.C.B.; Crespo, A.J.C.; Domínguez, J.M.
Numerical Model of Constrained Wave Energy Hyperbaric Converter under Full-Scale Sea Wave Conditions. *J. Mar. Sci. Eng.* **2022**, *10*, 1489.
https://doi.org/10.3390/jmse10101489

**AMA Style**

Brito M, Bernardo F, Neves MG, Neves DRCB, Crespo AJC, Domínguez JM.
Numerical Model of Constrained Wave Energy Hyperbaric Converter under Full-Scale Sea Wave Conditions. *Journal of Marine Science and Engineering*. 2022; 10(10):1489.
https://doi.org/10.3390/jmse10101489

**Chicago/Turabian Style**

Brito, Moisés, Francisco Bernardo, Maria G. Neves, Diogo R. C. B. Neves, Alejandro J. C. Crespo, and José M. Domínguez.
2022. "Numerical Model of Constrained Wave Energy Hyperbaric Converter under Full-Scale Sea Wave Conditions" *Journal of Marine Science and Engineering* 10, no. 10: 1489.
https://doi.org/10.3390/jmse10101489