# Optimum Coastal Slopes Exposed to Waves: Experimental and Numerical Study

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}and the temperature of the Earth. The European Union has pledged to reduce greenhouse gas emissions by 20% compared to 1990 by 2020 while improving energy efficiency by 20% and increasing the share of renewable energy by 20%. In October 2014, EU Energy agreed to reduce greenhouse gas emissions by at least 40% below 1990 levels by 2030 [1]. Renewable energies are divided into several different categories, one of which is the energy of sea waves. The energy of sea waves is also divided into several different categories, one of which is a point absorber. The point absorber transfers the energy of sea waves to linear generators, linear converters, mechanical rotors, or hydraulic pumps using a float that is much smaller than the wavelength [2].

## 2. Experimental Wave Tank

#### 2.1. Wave Generation

#### 2.2. Wave Energy Dissipation

#### 2.3. Wave Gauges

## 3. Simulation Method

#### 3.1. Governing Equations

#### 3.2. Numerical Method

#### Dynamic Mesh

Algorithm 1 Computer programming codes provided for flap in Fluent software. |

#include “udf.h” DEFINE_CG_MOTION (Case1, dt, vel, omega, time, dtime) { if (time < 14.0801) omega [2] = −0.3722 ∗ cos(5.7120 ∗ time−2 ∗ atan(1.0)); else if (time < 66) omega [2] = −0.3722 ∗ cos(5.7120 ∗ (time + 0.2199)−2 ∗ atan(1.0)); else omega [2] = 0; |

## 4. Simulation Results and Discussions

#### 4.1. Mesh Independence

#### 4.2. Model Validation

#### 4.3. Wave Generation Results

_{s}represents the wave height in millimeters, which is given to the device software, and H is the depth of water from the beginning of the flap. These cases are chosen to investigate the effect of different wave heights (H

_{s}) and frequencies of the waves on the experimental and numerical simulation results.

_{s}of the wave, in this case, is reduced more than in the previous cases to 20 mm. Figure 20 illustrates that the discrepancy between the numerically simulated results and the experimental ones has increased compared with cases 1 and 2. From Figure 19 and Figure 20, it can be concluded that the accuracy of the simulation wave is reduced by decreasing the height of the wave.

_{s}or it is like case 3 but with higher frequency. A comparison of Figure 22 with previous results shows that there is a high discrepancy between the numerical and experimental results of this case, and especially at a distance of 5 m, this difference becomes very large. The reason as can be seen in the next part is the coast slope of 1:3. When the wave is not completely dampened, the reflected waves fall into the numerical wave tank and cause a disturbance in the waves. This disturbance is lower at a distance of 3 m from the flap because the reflected waves need more time to reach the distance of 3 m from the flap, while it is large at a distance of 5 m from the flap (Figure 22).

#### 4.4. Study of the Beach Slope

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Amanatidis, G. European Policies on Climate and Energy towards 2020, 2030 and 2050. Policy Department for Economic, Scientific and Quality of Life Policies. European Parliament. 2019. PE 631.047. Available online: https://policycommons.net/artifacts/1335288/european-policies-on-climate-and-energy-towards-2020-2030-and-2050/1941726/ (accessed on 29 December 2022).
- Pecher, A.; Kofoed, J.P. Handbook of Ocean Wave Energy, 1st ed.; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Lal, A.; Elangovan, M. CFD Simulation and Validation of Flap Type Wave-Maker. Int. J. Math. Comput. Sci.
**2008**, 2, 708–714. [Google Scholar] - Anbarsooz, M.; Passandideh-Fard, M.; Moghiman, M. Fully nonlinear viscous wave generation in numerical wave tanks. Ocean Eng.
**2013**, 59, 73–85. [Google Scholar] [CrossRef] - Han, Z.; Liu, Z.; Shi, H. Numerical study on overtopping performance of a multi-level breakwater for wave energy conversion. Ocean Eng.
**2008**, 150, 94–101. [Google Scholar] [CrossRef] - Collins, K.M.; Stripling, S.; Simmonds, D.J.; Greaves, D.M. Quantitative metrics for evaluation of wave fields in basins. Ocean Eng.
**2018**, 169, 300–314. [Google Scholar] [CrossRef][Green Version] - Alamian, R.; Shafaghat, R.; Ketabdari, M.J. Wave simulation in a numerical wave tank using BEM. Am. Inst. Phys. Conf. Proc.
**1648**, 2015, 770008. [Google Scholar] - Wu, Y.T.; Hsiao, S.C. Generation of stable and accurate solitary waves in a viscous numerical wave tank. Ocean Eng.
**2018**, 167, 102–113. [Google Scholar] [CrossRef] - Dao, M.H.; Chew, L.W.; Zhang, Y. Modeling physical wave tank with flap paddle and porous beach in OpenFOAM. Ocean Eng.
**2018**, 154, 204–215. [Google Scholar] [CrossRef] - Houtani, H.; Waseda, T.; Fujimoto, W.; Kiyomatsu, K.; Tanizawa, K. Generation of a spatially periodic directional wave field in a rectangular wave basin based on higher-order spectral simulation. Ocean Eng.
**2018**, 169, 428–441. [Google Scholar] [CrossRef] - Li, Z.; Deng, G.; Queutey, P.; Bouscasse, B.; Ducrozet, G.; Gentaz, L.; Ferrant, P. Comparison of wave modeling methods in CFD solvers for ocean engineering applications. Ocean Eng.
**2019**, 188, 106237. [Google Scholar] [CrossRef] - Hu, Z.; Zhang, X.; Li, Y.; Li, X.; Qin, H. Numerical simulations of super rogue waves in a numerical wave tank. Ocean Eng.
**2021**, 229, 108929. [Google Scholar] [CrossRef] - Lv, C.; Zhao, X.; Li, M.; Xie, Y. An improved wavemaker velocity boundary condition for generating realistic waves in the numerical wave tank. Ocean Eng.
**2022**, 261, 112188. [Google Scholar] [CrossRef] - Westphalen, J.; Greaves, D.M.; Williams, C.J.K.; Hunt-Raby, A.C.; Zang, J. Focused waves and wave-structure interaction in a numerical wave tank. Ocean Eng.
**2012**, 45, 9–21. [Google Scholar] [CrossRef] - Hu, Z.Z.; Greaves, D.; Raby, A. Numerical wave tank study of extreme waves and wave-structure interaction using OpenFoam
^{®}. Ocean Eng.**2016**, 126, 329–342. [Google Scholar] [CrossRef][Green Version] - Kim, S.Y.; Kim, K.M.; Park, J.C.; Jeon, G.M.; Chun, H.H. Numerical simulation of wave and current interaction with a fixed offshore substructure. Int. J. Nav. Archit. Ocean Eng.
**2016**, 8, 188–197. [Google Scholar] [CrossRef][Green Version] - Bruinsma, N.; Paulsen, B.T.; Jacobsen, N.G. Validation and application of a fully nonlinear numerical wave tank for simulating floating offshore wind turbines. Ocean Eng.
**2018**, 147, 647–658. [Google Scholar] [CrossRef] - Tian, X.; Wang, Q.; Liu, G.; Deng, W.; Gao, Z. Numerical and experimental studies on a three-dimensional numerical wave tank. IEEE Access
**2018**, 6, 6585–6593. [Google Scholar] [CrossRef] - Martínez-Ferrer, P.J.; Qian, L.; Ma, Z.; Causon, D.M.; Mingham, C.G. Improved numerical wave generation for modeling ocean and coastal engineering problems. Ocean Eng.
**2018**, 152, 257–272. [Google Scholar] [CrossRef][Green Version] - Anbarsooz, M.; Rashki, H.; Ghasemi, A. Numerical investigation of front-wall inclination effects on the hydrodynamic performance of a fixed oscillation water column wave energy converter. Proc. Inst. Mech. Eng. Part A J. Power Energy
**2018**, 233, 262–271. [Google Scholar] [CrossRef] - Kim, S.J.; Kim, M. The nonlinear wave and current effects on fixed and floating bodies by a three-dimensional fully-nonlinear numerical wave tank. Ocean Eng.
**2022**, 245, 110458. [Google Scholar] [CrossRef] - Finnegan, W.; Goggins, J. Numerical simulation of linear water waves and wave-structure interaction. Ocean Eng.
**2012**, 43, 23–31. [Google Scholar] [CrossRef][Green Version] - Zabihi, M.; Mazaheri, S.; Mazyak, A.R. Wave Generation in a Numerical Wave Tank. Int. J. Ocean Coast. Eng.
**2015**, 5, 33–43. [Google Scholar] - Fathi-Moghadam, M.; Davoudi, L.; Motamedi-Nezhad, A. Modeling of solitary breaking wave force absorption by coastal trees. Ocean Eng.
**2018**, 169, 87–98. [Google Scholar] [CrossRef] - Jiang, C.; Liu, X.; Yao, Y.; Deng, B. Numerical investigation of solitary wave interaction with a row of vertical slotted piles on a sloping beach. Int. J. Nav. Archit. Ocean Eng.
**2019**, 11, 530–541. [Google Scholar] [CrossRef] - Casella, F.; Aristodemo, F.; Filianoti, P. A comprehensive analysis of solitary wave run-up at sloping beaches using an extended experimental dataset. Appl Ocean Res.
**2022**, 126, 103283. [Google Scholar] [CrossRef] - Lee, W.D.; Yeom, G.S.; Kim, J.; Lee, S.; Kim, T. Runup characteristics of a tsunami-like wave on a slope beach. Ocean Eng.
**2022**, 259, 111897. [Google Scholar] [CrossRef] - Machado, F.M.M.; Lopes, A.M.G.; Ferreira, A.D. Numerical simulation of regular waves: Optimization of a numerical wave tank. Ocean Eng.
**2018**, 170, 89–99. [Google Scholar] [CrossRef] - Prasad, D.D.; Ahmed, M.R.; Lee, Y.-H. Studies on the performance of Savonius rotors in a numerical wave tank. Ocean Eng.
**2018**, 158, 29–37. [Google Scholar] [CrossRef] - Dean, R.G.; Dalrymple, R.A. Water Wave Mechanics for Engineers and Scientists; Advances Series on Ocean Engineering: Volume 2; World Scientific Publishing Company: Singapore, 1991. [Google Scholar]
- Ursell, F.; Dean, R.G.; Yu, Y.S. Forced small-amplitude water waves: A comparison of theory and experiment. J. Fluid Mech.
**1960**, 7, 33–52. [Google Scholar] [CrossRef]

**Figure 1.**An overview of the experimental wavemaker tank: (

**a**) a real picture and (

**b**) schematic illustration.

**Figure 23.**Wave height of case 3 for different beach slopes at a distance of (

**a**) 3 m and (

**b**) 5 m from the flap.

**Figure 24.**Wave height of case 5 for different beach slopes at a distance of (

**a**) 3 m and (

**b**) 5 m from the flap.

**Figure 25.**Results of the wave reflection coefficient for different beach slopes in different cases.

Cell height (m) | 0.008 | 0.005 | 0.004 | 0.003 | 0.002 | 0.001 |

Mesh number produced | 25,213 | 37,686 | 54,355 | 64,659 | 86,555 | 210,883 |

Case Number | Period (s) | Stroke (cm) | Still Water Depth (m) | H/S | 2πd/L |
---|---|---|---|---|---|

Small wave steepness | |||||

1 | 2.1 | 8.45 | 0.3 | 0.25 | 0.5481 |

2 | 1.4 | 8.45 | 0.3 | 0.43 | 0.8746 |

3 | 1.05 | 3.15 | 0.45 | 0.88 | 1.7449 |

4 | 1.5 | 2.4 | 0.8 | 1.25 | 2.9351 |

5 | 1.4 | 8.57 | 0.5 | 0.65 | 1.2212 |

High wave steepness | |||||

6 | 0.84 | 8.47 | 0.4 | 1.003 | 2.3242 |

7 | 1.05 | 8.45 | 0.3 | 0.67 | 1.2783 |

8 | 0.84 | 8.45 | 0.3 | 0.84 | 1.8051 |

9 | 1.05 | 6.3 | 0.9 | 1.3 | 3.2925 |

10 | 1.05 | 4.69 | 0.67 | 1.13 | 2.4789 |

Case | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

H_{s} (mm) | 90 | 60 | 40 | 60 | 40 |

H (m) | 0.62 | 0.62 | 0.62 | 0.62 | 0.62 |

Frequency (Hz) | 1.1 | 1.1 | 1.1 | 1.2 | 1.2 |

Wave period (s) | 0.9 | 0.9 | 0.9 | 0.83 | 0.83 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zandi, R.; Lari, K.; Najafzadeh, M. Optimum Coastal Slopes Exposed to Waves: Experimental and Numerical Study. *Water* **2023**, *15*, 366.
https://doi.org/10.3390/w15020366

**AMA Style**

Zandi R, Lari K, Najafzadeh M. Optimum Coastal Slopes Exposed to Waves: Experimental and Numerical Study. *Water*. 2023; 15(2):366.
https://doi.org/10.3390/w15020366

**Chicago/Turabian Style**

Zandi, Reza, Khosro Lari, and Mohammad Najafzadeh. 2023. "Optimum Coastal Slopes Exposed to Waves: Experimental and Numerical Study" *Water* 15, no. 2: 366.
https://doi.org/10.3390/w15020366