# Influence of the Drag Force on the Average Absorbed Power of Heaving Wave Energy Converters Using Smoothed Particle Hydrodynamics

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## Abstract

**:**

## 1. Introduction

## 2. Smoothed Particle Hydrodynamics—DualSPHysics

#### 2.1. Governing Equations

#### 2.2. Boundary Conditions

#### 2.3. Floating Bodies

#### 2.4. Coupling Dualsphysics—Project Chrono

## 3. Methodology

#### 3.1. Determination of Hydrodynamic Coefficients Using DualSPHysics

#### 3.2. Derivation of the Equation of Motion and the Optimal Damping Coefficient Including the Effect of Drag Forces

#### 3.2.1. Linear Damping PTO System

#### 3.2.2. Coulomb Damping PTO System

## 4. Test Cases and Numerical Setup

- A heaving sphere with a diameter of 5 m, as studied in [32]. For this WEC, the added mass and hydrodynamic damping were also estimated wtih DualSPHysics.
- A cylindrical WEC with a diameter of 0.3 m and draft of 0.28 m, as studied in [34,57], hereafter referred to as ’cylinder2’. This WEC is studied in more detail by analyzing its response amplitude operator (RAO) and comparing it with the experimental results found in [57]. The WEC is also simulated with two kinds of PTO systems (a linear damping and a Coulomb damping PTO system) and the average absorbed power is compared for a range of damping coefficients.

- Simulations were carried out with two different types of boundary conditions: Dynamic Boundary Conditions (DBC) and modified Dynamic Boundary Conditions (mDBC), as described in Section 2.2. Both SPH results were compared to the theoretical force calculated with Equation (23), with the hydrodynamic coefficients from NEMOH and ${C}_{d}=0.45$ as in [32]. It is clear from Figure 4 that mDBC gave significantly better results compared to DBC. When mDBC was applied, the repulsive forces exerted by the boundary particles of the sphere were much smaller than when applying DBC, resulting in a smaller gap between the sphere and the fluid.
- Artificial viscosity was applied with an artificial viscosity coefficient $\alpha =0.01$. Artificial viscosity was introduced into SPH in [14] and was used primarily due to its simplicity [18]. It was stated in [59] that this artificial viscosity corresponds to an equivalent kinematic viscosity of $\frac{15}{112}\alpha {c}_{s0}h$ (in 2D), which is generally much higher than the real kinematic viscosity of water $\nu $ = 1 × 10${}^{-6}$ m²/s. Therefore, one way of reducing the numerical dissipation caused by artificial viscosity is by lowering $\alpha $; however, this was not preferred since the value of $\alpha =0.01$ has been proven to give the best results in the validation of wave flumes to study the wave propagation and wave loadings exerted onto coastal structures [21,42] and is also the value used when simulating the WEC in regular waves. Only in the case where the hydrodynamic coefficients ${A}_{33}$ and ${B}_{33}$ are computed is $\alpha $ set to be equal to zero, as described in Section 3.1.
- The initial speed of sound was set to ${c}_{s0}=15\sqrt{gd}$, with d being the depth of the numerical wave basin. It was found that convergence was reached with a lower resolution when the speed of sound ${c}_{s0}$ was decreased. This can be related to the influence of ${c}_{s0}$ on the viscosity: it is stated in [59] that the equivalent kinematic viscosity associated with the artificial viscous term has the form $\frac{15}{112}\alpha {c}_{s0}h$. Further decreasing ${c}_{s0}$ leads to overly large timesteps and therefore less accurate results.
- A convergence test was done by varying the interparticle distance $dp$ and studying the resulting hydrodynamic coefficients ${A}_{33}$, ${B}_{33}$ and the drag coefficient ${C}_{d}$, calculated with Equations (28), (30) and (31), respectively. The hydrodynamic coefficients were compared to results from potential flow theory obtained with NEMOH and the drag coefficient was compared to results from previous experimental or numerical tests from [32,33,34]. The results of these convergence tests are described in Section 5.
- The domain size of the basin was set to be large enough to avoid interaction with side walls (see Figure 5). Sloped sidewalls were provided as well as numerical damping layers, with the aim of reducing side wall reflection.

## 5. Results and Discussion

#### 5.1. Estimation of the Hydrodynamic Coefficients and of the Drag Coefficient

#### 5.2. Cylindrical WEC in Regular Waves

#### 5.2.1. Undamped Heaving WEC

#### 5.2.2. Linear Damping PTO System

#### 5.2.3. Coulomb Damping PTO System

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

a | Heave amplitude of the WEC (m) |

${A}_{d}$ | Cross-sectional area of the heaving WEC (m²) |

${A}_{33}$ | Added mass (kg) |

${B}_{33}$ | Hydrodynamic damping in heave (Ns/m) |

${B}_{PTO,l}$ | Linear PTO system damping coefficient (Ns/m) |

${B}_{PTO,c}$ | Coulomb damping PTO system damping coefficient (N) |

${C}_{d}$ | Drag coefficient (-) |

${c}_{s}$ | Speed of sound in DualSPHysics (m/s) |

D | WEC diameter (m) |

d | Depth of the numerical basin (m) |

$dp$ | Interparticle distance in DualSPHysics (m) |

${F}_{e}$ | Excitation force (N) |

${F}_{hs}$ | Hydrostatic force (N) |

${F}_{r}$ | Radiation force (N) |

${F}_{PTO}$ | PTO system force (N) |

${F}_{PTO,coulomb}$ | Coulomb damping PTO system force (N) |

${F}_{z}$ | Rotal vertical force acting on the heaving WEC (N) |

${f}_{k}$ | Force per unit mass acting on boundary particle k (N/kg) |

g | Gravitational acceleration (m/s²) |

H | Wave height (m) |

h | Smoothing length in DualSPHysics (m) |

${K}_{33}$ | Hydrostatic spring stiffness (N/m) |

k | Wave number (1/m) |

m | WEC’s mass (kg) |

P | Fluid pressure (Pa) |

${P}_{av}$ | Average absorbed power (W) |

${r}_{k}$ | Position of particle k in DualSPHysics (m) |

T | Wave period (s) |

t | Time (s) |

${v}_{0}$ | Vertical fluid velocity (m/s) |

v | WEC’s heave velocity (m/s) |

${v}^{\prime}$ | WEC’s relative velocity, equal to $v-{v}_{0}$, (m/s) |

W | Kernel function |

${z}_{b}$ | Half of the WEC’s draft (m) |

$\alpha $ | Artificial viscosity coefficient applied in DualSPHysics |

${\alpha}_{v}$ | Ratio of the WEC’s relative velocity amplitude to the WEC’s velocity amplitude |

$\Pi $ | Artificial viscosity term in DualSPHysics (m²/s) |

$\rho $ | Fluid density (kg/m³) |

$\Phi $ | Velocity potential |

${\varphi}_{\alpha}$ | Phaseshift between ${v}^{\prime}$ and v (rad) |

$\Omega $ | Rotational velocity (rad/s) |

$\omega $ | Angular wave frequency (rad/s) |

$\widehat{}$ | Complex amplitude |

CFD | Computational Fluid Dynamics |

DBC | Dynamic Boundary Condition |

mDBC | Modified DBC |

PTO | Power take-off |

SPH | Smoothed Particle Hydrodynamics |

WEC | Wave Energy Converter |

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**Figure 3.**Vorticity in DualSPHysics surrounding a heaving cylindrical WEC in regular waves with $H=0.16$ m, $T=1.5$ s, ${B}_{PTO,l}=1100$ Ns/m (

**left**), compared to experimental measurements of the heaving WEC, as performed in [56] (

**right**).

**Figure 4.**Vertical force ${F}_{z}$ acting on a sphere with prescribed heave motion, $a=1.5$ m $T=9$ s.

**Figure 5.**Dimensions of the numerical wave basin for hydrodynamic coefficients and drag coefficient test for a heaving sphere.

**Figure 6.**Hydrodynamic coefficients ${A}_{33}$ and ${B}_{33}$ calculated with NEMOH and with Smoothed Particle Hydrodynamics (SPH)–DualSPHysics for a sphere with a diameter of 5 m.

**Figure 8.**Dimensions of a basin for a heaving cylindrical WEC in waves with $T=1.2$ s in DualSPHysics.

**Figure 9.**Response amplitude operator (RAO) for the cylindrical WEC cylinder2 without a power take-off (PTO) system, calculated with (i) linear potential flow theory with ${C}_{d}$ = 0.0, (ii) linear potential flow theory with ${C}_{d}$ = 1.5, (iii) SPH–DualSPHysics and (iv) obtained from experiments [34], H = 0.08 m.

**Figure 10.**Average absorbed power of cylinder2 with (

**a**) a linear damping PTO system and (

**b**) a Coulomb damping PTO system for a range of PTO system damping coefficients, calculated with linear potential flow theory with (i) ${C}_{d}$ = 0.00, (ii) ${C}_{d}$ = 1.50 and (iii) with DualSPHysics—$H=0.15$ m, $T=1.2$ s.

**Figure 11.**Velocity of cylinder2 with (

**a**) a linear damping PTO system, ${B}_{PTO,l}=25$ Ns/m and (

**b**) a Coulomb damping PTO system, ${B}_{PTO,c}=10N$ calculated with linear potential flow theory with (i) ${C}_{d}$ = 0.00, (ii) ${C}_{d}$ = 1.50 and (iii) with DualSPHysics—$H=0.15$ m, $T=1.2$ s.

**Table 1.**Heave amplitude a and heave period T applied during the forced oscillation with DualSPHysics of a spherical WEC and two cylindrical WECs, as well as the obtained drag coefficient ${C}_{d}$.

Spherical WEC | Cylinder1 | Cylinder2 | |
---|---|---|---|

T [s] | 9 | 1.5 | 1.2 |

a [m] | 1.5 | 0.045 | 0.1 |

${\mathit{C}}_{\mathit{d}}$ [-] | 0.78 | 1.65 | 1.50 |

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**MDPI and ACS Style**

Quartier, N.; Ropero-Giralda, P.; M. Domínguez, J.; Stratigaki, V.; Troch, P.
Influence of the Drag Force on the Average Absorbed Power of Heaving Wave Energy Converters Using Smoothed Particle Hydrodynamics. *Water* **2021**, *13*, 384.
https://doi.org/10.3390/w13030384

**AMA Style**

Quartier N, Ropero-Giralda P, M. Domínguez J, Stratigaki V, Troch P.
Influence of the Drag Force on the Average Absorbed Power of Heaving Wave Energy Converters Using Smoothed Particle Hydrodynamics. *Water*. 2021; 13(3):384.
https://doi.org/10.3390/w13030384

**Chicago/Turabian Style**

Quartier, Nicolas, Pablo Ropero-Giralda, José M. Domínguez, Vasiliki Stratigaki, and Peter Troch.
2021. "Influence of the Drag Force on the Average Absorbed Power of Heaving Wave Energy Converters Using Smoothed Particle Hydrodynamics" *Water* 13, no. 3: 384.
https://doi.org/10.3390/w13030384