# Implementation of Open Boundaries within a Two-Way Coupled SPH Model to Simulate Nonlinear Wave–Structure Interactions

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## Abstract

**:**

## 1. Introduction

- Accurate wave generation and wave absorption through coupling an SPH solver to an FNPF solver using the open boundary formulation by Tafuni et al. [24],
- Having an online exchange of information in two directions between the SPH solver and the FNPF solver.

## 2. Smoothed Particle Hydrodynamics

#### 2.1. SPH Fundamentals

#### 2.2. Governing Equations

^{3}in this work. The parameter $\gamma $ is the polytrophic constant, ranging between 1 and 7. The maximum limit for $\rho $ is set as $B={c}^{2}{\rho}_{0}/\gamma $. Consequently, the choice of B is of high importance, since it determines the value of c. As mentioned before, c can be artificially lowered to ensure a reasonable time step [32]. However, in DualSPHysics, c is kept at least 10 times higher than the maximum expected flow velocity $\mathbf{v}$.

#### 2.3. Delta-SPH Formulation

## 3. Boundary Conditions in DualSPHysics

#### 3.1. Open Boundary Conditions

#### 3.2. Fixed Boundary Condition with Pressure Correction

#### 3.3. Comparison to the State-of-the-Art

## 4. Coupling Methodology Using DualSPHysics and OceanWave3D

#### 4.1. Inlet Velocity Correction

#### 4.2. Outlet Velocity Correction

#### 4.3. Coupling Algorithm

## 5. 2D Coupled Model: Validation

#### 5.1. DualSPHysics Solver Options

#### 5.2. Results and Discussion

## 6. 3D Coupled Model: Proof-of-Concept

#### 6.1. Experimental Set-up

#### 6.2. Numerical Set-up

#### 6.3. Results and Discussion

#### 6.4. Computational Speed-up

#### 6.5. Visual Comparison of Overtopping

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Sketch of the implemented open boundary model, adapted from Tafuni et al. [24].

**Figure 2.**General sketch of coupling methodology between the OceanWave3D domain (

**top**) and the DualSPHysics domain (

**bottom**).

**Figure 3.**Sensitivity analysis on the number of buffer particle layers ${n}_{l}$ necessary for accurate wave propagation, represented by the ratio ${K}_{D}$ along the fluid domain.

**Figure 4.**Sensitivity analysis on distance ${d}_{WG}$ from the wave measurement locations $W{G}_{in}$ and $W{G}_{out}$ to the inlet/outlet buffer zone, necessary for accurate wave propagation, represented by the ratio ${K}_{D}$ along the fluid domain.

**Figure 5.**Coupling implementation scheme showing several options to generate and absorb waves within a two-way coupled model: wave theory solutions, the wave propagation model OceanWave3D or any external data.

**Figure 7.**Sketch of ’relaxation zones’ providing a smooth transition between the OceanWave3D domain and the DualSPHysics domain.

**Figure 8.**Numerical test set-up for simulation of the response of a floating box to a custom wave signal. The DualSPHysics domain and OceanWave3D domain are indicated. ${L}_{wav}$ is the wavelength.

**Figure 9.**Comparison of time series and error between numerical and experimental results of the three degrees of freedom of the box with heave, pitch and surge motions (top three graphs). The difference between the numerical results and the experimental results is expressed in the error values for heave err${}_{H}$, pitch err${}_{P}$ and surge err${}_{S}$ (bottom three graphs).

**Figure 10.**Comparison of the instantaneous surface elevation profile between one-way and two-way coupling results of OceanWave3D and DualSPHysics. The grey masked out zone is the region around the floating box, omitted from the coupling.

**Figure 11.**Experimental set-up for tests with a heaving disk type WEC in the large wave flume of Ghent University. The numerical DualSPHysics domain is indicated as a green zone.

**Figure 12.**Numerical set-up for 3D modelling of a heaving cylindrical WEC in a two-way coupled model.

**Figure 13.**Comparison of a one-way coupling 3D proof-of-concept with experimental data of a heaving cylinder in overtopping nonlinear waves. (

**a**) shows the surface elevation 1 m in front of the WEC; (

**b**) the surface elevation 1 m behind the WEC; (

**c**) the horizontal surge force acting on the WEC and (

**d**) the heave motion of the WEC.

**Figure 14.**Visual comparison of overtopping waves between the 3D proof-of-concept with experimental data of a heaving cylinder in overtopping nonlinear waves. The plot shows a time progression of the wave, from left to right with a time difference of 0.4 s between each frame.

**Table 1.**Imposed and extrapolated quantities for inlet and outlet buffer particles (Imp = imposed, Ext = extrapolated, Hyd = hydrostatic).

Quantity | Horizontal Velocity u | Vertical Velocity w | Surface Elevation η | Pressure p |
---|---|---|---|---|

inlet | Imp | No | Imp | Hyd |

outlet | Imp | No | Ext | Ext |

Time Integration Scheme | Verlet |
---|---|

Time Step | Variable (including CFL and viscosity) |

Kernel | Wendland |

Smoothing Length | $2.0\xb7{d}_{p}$ |

Viscosity Treatment | Artificial ($\alpha =0.01$) |

Equation of State | Tait equation |

Boundary Conditions | Open Boundary Conditions |

$\delta $-SPH | Yes ($\delta $-SPH = 0.1) |

Wave Height H [m] | 0.12 |

Wave Period T [s] | 1.2 |

Water Depth d [m] | 0.7 |

Wave Length L${}_{wav}$ [m] | 2.17 |

Particle Size d${}_{p}$ [m] | 0.01 |

Domain Length ${L}_{SPH}$ [m] | 4.34 |

Domain Width W [m] | 1.0 |

WEC Diameter D [m] | 0.5 |

WEC Draft q${}_{WEC}$ [m] | 0.113 |

Wave Theory | Stokes 5th |

Time Step Algorithm | Verlet |

Artificial Viscosity | 0.01 |

$\delta $-SPH | 0.1 |

One-Way Coupling | Full Model | Difference | |
---|---|---|---|

# Particles | 5,010,954 | 25,473,152 | 508% |

# Fluid Particles | 2,949,433 | 21,165,100 | 718% |

GPU Memory [MB] | 780 | 3941 | 505% |

Estimated Runtime [hr] | 35 | 144 | 411% |

Real Runtime [hr] | 91 | 375 | 411% |

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## Share and Cite

**MDPI and ACS Style**

Verbrugghe, T.; Stratigaki, V.; Altomare, C.; Domínguez, J.M.; Troch, P.; Kortenhaus, A. Implementation of Open Boundaries within a Two-Way Coupled SPH Model to Simulate Nonlinear Wave–Structure Interactions. *Energies* **2019**, *12*, 697.
https://doi.org/10.3390/en12040697

**AMA Style**

Verbrugghe T, Stratigaki V, Altomare C, Domínguez JM, Troch P, Kortenhaus A. Implementation of Open Boundaries within a Two-Way Coupled SPH Model to Simulate Nonlinear Wave–Structure Interactions. *Energies*. 2019; 12(4):697.
https://doi.org/10.3390/en12040697

**Chicago/Turabian Style**

Verbrugghe, Tim, Vasiliki Stratigaki, Corrado Altomare, J. M. Domínguez, Peter Troch, and Andreas Kortenhaus. 2019. "Implementation of Open Boundaries within a Two-Way Coupled SPH Model to Simulate Nonlinear Wave–Structure Interactions" *Energies* 12, no. 4: 697.
https://doi.org/10.3390/en12040697