# Influence of Convex and Concave Curvatures in a Coastal Dike Line on Wave Run-up

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## Abstract

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_{d}= 210° to 270°) under perpendicular wave attack, a higher wave run-up was observed for larger opening angles at the center of curvature whereas for concavely curved dikes (α

_{d}= 90° to 150°) under perpendicular wave attack, wave run-up increases at the center of curvature as the opening angle decreases. This research aims to contribute a more precise analysis and understanding the influence of the curvature in a dike line and thus ensuring a higher level of protection in the future development of coastal structures.

## 1. Introduction

_{u2%}is used, which is exceeded by 2% of the number of incident waves (EurOtop, 2018) [1] (see Figure 1). The hypothesis is set if the wave run-up is influenced by the curvature in a dike line due to additional overlapping physical processes, i.e., refraction and diffraction. Yet there is limited information available on the influence of wave run-up on a curved dike and no detailed investigations have been done to include the factor based on curvature in the prediction formulae for wave run-up. The aim of this research is to provide an insight of wave run-up on a curved dike using numerical models validated with measurements from physical model tests. The numerical investigation is accomplished using DualSPHysics, a mesh-less model and OpenFOAM, a mesh-based model. Both of these numerical models are capable to simulate wave transformation, wave breaking and interaction with sloping structures, which made them a feasible alternative to experimental investigations to predict wave run-up numerically.

#### 1.1. Influence of a Curvature in a Dike Line

_{d}and opening radius, r

_{d}of a dike and they may influence the hydrodynamics of approaching waves. The opening angle, α

_{d}is defined as the seaward angle between the tangents of the dike flanks. The opening radius, r

_{d}is defined as the distance between the meeting point of the perpendiculars of both dike flanks and the limit of the dike curvature.

#### 1.2. Literature Review

## 2. Numerical Model

#### 2.1. DualSPHysics

#### 2.2. OpenFOAM

## 3. Physical Model

_{c}/(H

_{m}

_{0}ξ

_{m}

_{–1,0}) < 1.3), the Iribarren number (0.7 < ξ

_{m}

_{–1,0}< 1.4) and the angle of the incident waves (Table 1) were varied from test to test. For the present paper, results from regular waves only are given in Table 2 and are considered to contrast between numerical approaches.

## 4. Numerical Investigation on a Curved Dike Line

_{d}chosen for the simulation are 90°, 120°, 150°, 180°, 210°, 240° and 270°. The first three opening angles were tested for concavely curved dikes and the last three opening angles were tested for convexly curved dikes. The different angles of wave attack, β included in the simulation are 0°, 30° and 45°, respectively.

#### 4.1. Calibration Study

_{0}, as shown in Equation (1).

_{0}—deep water wave length

#### 4.2. Numerical Model Set-Up

#### 4.3. Transformation Processes on a Curved Dike

## 5. Analysis Approach

## 6. Results and Discussion

_{d}= 210° to 270°), a higher run-up was observed for larger opening angles at the center of curvature. The wave energy focuses on the corner caused by wave refraction over the slope. For large opening angles (α

_{d}= 270°) the increase of the wave run-up derived from physical model tests is larger than calculated with the numerical models. In case of concavely curved dikes (α

_{d}= 90° to 150°), wave run-up increases at the center of curvature as the opening angle decreases. The results from OpenFOAM and the physical model tests for a concavely curved dikes are almost in line, whereas the wave run-up calculated with DualSPHysics is lower for α

_{d}= 90° and 120°.

_{d}= 210° to 270°), a mild increase in wave run-up at the center of the curvature is observed for larger opening angles except for α

_{d}= 210° in DualSPHysics simulations. Nevertheless, the increase is very little and data scatter around ${\gamma}_{c}$ = 1.0. Similarly, at concave corners, a slight increase in wave run-up was noticed at α

_{d}= 90°. For α

_{d}= 120° the scatter in the wave run-up recorded from the physical model tests is high.

_{d}= 210° to 270°), a higher run-up is observed at the center of curvature for larger opening angles. Results from physical model tests cause a significantly higher wave run-up compared to the two numerical models for α

_{d}= 270°. The extremely high results are due to swash running over the convex curve. For α

_{d}= 90°, a very high run-up was observed at the center of the curvature in OpenFOAM simulations under 45° oblique wave attack. In contrast, in the physical model tests a significantly reduced wave run-up height is observed. This special case (α

_{d}= 90°) was partially biased by model effects, as the incident waves propagate over the model boundary of the slope (side of the luv slope). As the model boundaries differ in numerical and physical model test runs, a deviation is likely. In general, corresponding data points have to be evaluated with care.

## 7. Conclusions and Future Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## References

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**Figure 2.**Definition sketch of opening angle, α

_{d}and opening radius, r

_{d}of curvatures in a dike line. (

**a**) concave curvature (

**b**) convex curvature.

**Figure 3.**Physical model set-up of the impermeable 1:6 sloped dike model in the wave basin with straight front (

**a**,

**b**), convex (

**c**,

**d**) and concave curvature (

**e**,

**f**). Sensor acronyms: Overtopping unit: $Qi$; velecity probe: $ADVi$; run-up gauge: $WPAi$; wave gauge: $Ai$; camera: Ci, and $i$ the sensor number.

**Figure 4.**(

**a**) Wave run-up on a 3D straight dike with perpendicular wave attack (

**b**) Results of the wave run-up calibration on a 3D straight dike compared to Hunt (1959).

**Figure 5.**3D numerical model of curved dikes in DualSPHysics: (

**a**) 90° concave dike; (

**b**) 270° convex dike.

**Figure 6.**Wave transformation processes on a 270° convexly curved dike (Left: DualSPHysics, Middle: OpenFOAM, Right: Physical model); H = 0.10 m, T = 1.46 s, β = 0°.

**Figure 7.**Wave transformation processes on a 90° concavely curved dike (Left: DualSPHysics, Middle: OpenFOAM, Right: Physical model); H = 0.10 m, T = 1.46 s, β = 0°.

**Figure 8.**Influence factor ${\gamma}_{\beta}$ for oblique, regular wave attack from numerical simulations and the literature.

**Figure 9.**Influence of a curvature in the dike line on wave run-up for perpendicular wave attack–studied position: center of curvature.

**Figure 10.**Influence of a curvature in the dike line on wave run-up for a 30° oblique wave attack–studied position: center of curvature.

**Figure 11.**Influence of a curvature in the dike line on wave run-up for a 45° oblique wave attack–studied position: center of curvature.

Curvature | Straight | Convex | Concave |
---|---|---|---|

Opening angle α_{d} [°] | 180 | 270 | 90; 120 |

Wave direction β [°] | 0; 30; 45 | −30; −15; 0; 15; 30; 45; 60 | −30; −15; 0; 15; 30 |

Wave Height H [m] | Wave Period T [s] | Water Depth d [m] |
---|---|---|

0.07 | 1.22 | 0.55 |

0.10 | 1.46 | 0.55 |

0.10 | 1.79 | 0.55 |

**Table 3.**Influence factors for curvature ${\gamma}_{c}$ for different opening angles with different angles of wave attack.

Opening Angle α_{d.} | Influence Factors ${\mathit{\gamma}}_{\mathit{c}}$ (Position: Center of Curvature) | ||||||||
---|---|---|---|---|---|---|---|---|---|

β = 0° | β = 30° | β = 45° | |||||||

Open FOAM | Dual SPH | Phys. Model | Open FOAM | Dual SPH | Phys. Model | Open FOAM | Dual SPH | Phys. Model | |

90° | 1.20 | 1.10 | 1.25 | 1.11 | 1.04 | 0.95 | 1.30 | 1.09 | 0.61 |

120° | 1.11 | 1.03 | 1.09 | 0.96 | 0.97 | 1.08 | 1.03 | 1.02 | – |

150° | 1.03 | 1.02 | – | 0.99 | 1.04 | – | – | – | – |

180° | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |

210° | 1.05 | 1.00 | – | 1.01 | 0.97 | – | – | – | – |

240° | 1.12 | 1.17 | – | 1.07 | 1.04 | – | 1.15 | 1.15 | – |

270° | 1.09 | 1.04 | 1.32 | 1.05 | 1.00 | 1.10 | 1.12 | 1.17 | 1.62 |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Subramaniam, S.P.; Scheres, B.; Schilling, M.; Liebisch, S.; Kerpen, N.B.; Schlurmann, T.; Altomare, C.; Schüttrumpf, H. Influence of Convex and Concave Curvatures in a Coastal Dike Line on Wave Run-up. *Water* **2019**, *11*, 1333.
https://doi.org/10.3390/w11071333

**AMA Style**

Subramaniam SP, Scheres B, Schilling M, Liebisch S, Kerpen NB, Schlurmann T, Altomare C, Schüttrumpf H. Influence of Convex and Concave Curvatures in a Coastal Dike Line on Wave Run-up. *Water*. 2019; 11(7):1333.
https://doi.org/10.3390/w11071333

**Chicago/Turabian Style**

Subramaniam, Suba Periyal, Babette Scheres, Malte Schilling, Sven Liebisch, Nils B. Kerpen, Torsten Schlurmann, Corrado Altomare, and Holger Schüttrumpf. 2019. "Influence of Convex and Concave Curvatures in a Coastal Dike Line on Wave Run-up" *Water* 11, no. 7: 1333.
https://doi.org/10.3390/w11071333