# Numerical Investigation of Wave Run-Up and Load on Fixed Truncated Cylinder Subjected to Regular Waves Using OpenFOAM

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model and Its Validation

#### 2.1. Governing Equation

#### 2.2. Model Validation

#### 2.2.1. Mesh Convergence Validation

#### 2.2.2. Validation of the Established Model

#### 2.3. Model Setup

## 3. Results Analysis

#### 3.1. Analysis of Maximum Wave Run-Up Height

#### 3.1.1. Maximum Run-Up Height Change

#### 3.1.2. Wave Surface Time Series Curve and Fourier Analysis

#### 3.2. Spatial Distribution of Wave Run-Up

#### 3.2.1. Circumferential Run-Up Height Distribution

#### 3.2.2. Radial Run-Up Height Distribution

#### 3.3. The Variation of Horizontal Wave Force with Scattering Parameters

## 4. Conclusions

- (1)
- The fixed cylinder draft has little effect on the maximum wave run-up height, but has a significant effect on the horizontal wave force. The maximum wave run-up height of a fixed truncated cylinder can be predicted theoretically by a fixed vertical cylinder run-up height estimation formula. The estimation formula proposed by Vos et al. [11] has good prediction at low wave steepness, but it will be underestimated with the increase of wave steepness, while the estimation formula proposed by Bonakdar et al. [12] can predict the wave run-up height well.
- (2)
- At the same wave steepness, the radial dimensionless run-up height increases with the increase of scattering parameters. The radial run-up height distribution gradually decreases in the upstream along the radial direction of the cylinder, and increases in the downstream, indicating that before the wave propagates to the cylinder, there is a large wave surface uplift in the upstream due to the scattering of the cylinder. At the same time, the increase of wave steepness will also lead to the increase of run-up height in the upstream region of the cylinder, and the decrease of run-up height in the corresponding downstream region.
- (3)
- Under the same wave number, the average amplitude of horizontal wave force increases linearly with the increase of incident wave radiation. Under the wave parameters and structural parameters simulated in this paper, the dimensionless horizontal wave force increases nonlinearly with the increase of scattering parameters, and the wave steepness parameter is not the significant influence parameter. When the scattering parameter is small, the theoretical value is consistent with the simulation results. With the increase of scattering parameters, the linear diffraction theory will overestimate the horizontal wave force.

## 5. Future Work

- (1)
- The effects of larger wave steepness on the wave run-up and wave load could be considered.
- (2)
- The wave conditions in practical engineering applications are irregular, such as solitary waves, extreme waves, etc., and wave breaking occurs when waves interact with structures. When the extreme waves break, water and air will react violently, and there will be an obvious turbulence effect, which will have a strong impact on structures, and are different from the effects caused by unbroken waves [34]. Therefore, the effects of extreme waves on the wave run-up and wave load could be considered.
- (3)
- There are many factors that affect the wave run-up, such as the scattering parameters, the wave steepness, the wavelength to diameter ratio, the water depth, etc. This paper mainly focuses on the effects of scattering parameters and wave steepness on the wave run-up and wave load; the effects of other parameters could be studied in further investigations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Time series of wave surface elevation of different mesh refinement schemes at $x=9\mathrm{m}$.

**Figure 4.**Comparison of instantaneous wave surface shape and theoretical wave surface shape at $t=15\mathrm{s}$.

**Figure 5.**Schematic of experimental model (Vos et al. [6]).

**Figure 7.**Comparisons of time series results of wave elevation between numerical simulation and experiments at different measuring points.

**Figure 8.**Circumferential and radial wave probes distribution: (

**a**) distribution of circumferential wave probes; (

**b**) distribution of radial wave probes.

**Figure 11.**Comparison of numerical simulation results and estimation formulas by Vos et al. [11].

**Figure 12.**Comparison of numerical simulation results and estimation formulas by Bonakdar et al. [12].

**Figure 13.**Time series of wave surface elevation and frequency spectrum at different wave steepness parameters ($T=2.5\mathrm{s}$): (

**a**) 0° time series of wave surface elevation; (

**b**) 0° frequency spectrum; (

**c**) 157.5° time series of wave surface elevation; (

**d**) 157.5° frequency spectrum; (

**e**) 180° time series of wave surface elevation; (

**f**) 180° frequency spectrum; (

**g**) 225° time series of wave surface elevation; (

**h**) 225° frequency spectrum.

**Figure 14.**The process of wave field change around a cylinder during a wave period(plane): (

**a**) ${t}_{0}$; (

**b**) ${t}_{0}+T/4$; (

**c**) ${t}_{0}+2T/4$; (

**d**) ${t}_{0}+3T/4$.

**Figure 15.**Comparison of dimensionless maximum run-up height distribution of different scatter parameters under the same wave steepness parameter: (

**a**) $ka=0.33$; (

**b**) $ka=0.22$; (

**c**) $ka=0.1157$; (

**d**) $ka=0.12$.

**Figure 16.**Comparison of dimensionless maximum run-up height distribution of different scatter parameters under the same wave steepness parameter: (

**a**) $ka=0.05$; (

**b**) $ka=0.1$; (

**c**) $ka=0.15$.

**Figure 17.**Distributions off dimensionless maximum run-up height along rows A, B, C, and D ($kA=0.05$): (

**a**) $T=1.4\mathrm{s},H=0.049\mathrm{m}$; (

**b**) $T=1.73\mathrm{s},H=0.073\mathrm{m}$; (

**c**) $T=2.1\mathrm{s},H=0.102\mathrm{m}$; (

**d**) $T=2.5\mathrm{s},H=0.133\mathrm{m}$.

**Figure 18.**Distributions off dimensionless maximum run-up height along rows A, B, C and D ($kA=0.15$): (

**a**) $T=1.4\mathrm{s},H=0.146\mathrm{m}$; (

**b**) $T=1.73\mathrm{s},H=0.218\mathrm{m}$; (

**c**) $T=2.1\mathrm{s},H=0.305\mathrm{m}$; (

**d**) $T=2.5\mathrm{s},H=0.398\mathrm{m}$.

**Figure 19.**Time series of horizontal wave force on fixed cylinder under various wave conditions: (

**a**) $T=1.4\mathrm{s}$; (

**b**) $T=1.73\mathrm{s}$; (

**c**) $T=2.1\mathrm{s}$; (

**d**) $T=2.5\mathrm{s}$.

**Figure 21.**Wave force crest mean, through mean, total value, and mean changes with different periods ($kA=0.15$).

Mesh Type | 1 | 2 | 3 | 4 |
---|---|---|---|---|

H/Δy | 5 | 10 | 15 | 20 |

λ/Δx | 40 | 80 | 120 | 160 |

Δy (m) | 0.0142 | 0.0071 | 0.0047 | 0.0036 |

Δx (m) | 0.1108 | 0.0554 | 0.0369 | 0.0277 |

Δx/Δy | 7.8 | 7.8 | 7.8 | 7.8 |

Mesh Type | 1 | 2 | 3 | 4 |
---|---|---|---|---|

height(m) | 0.0690 | 0.0706 | 0.0706 | 0.0708 |

relative error | −2.86% | −0.89% | −0.61% | −0.28% |

Name of Wave Probes | $\mathit{\alpha}\left(0\xb0\right)$ | $\mathit{r}$ |
---|---|---|

A1–A4 | 0 | a,1.2a,1.6a,2a |

B1–B4 | 45 | a,1.2a,1.6a,2a |

C1–C4 | 90 | a,1.2a,1.6a,2a |

D1–D4 | 135 | a,1.2a,1.6a,2a |

Number | $\mathbf{Period}\mathit{T}\left(\mathbf{s}\right)$ | $\mathit{k}\mathit{A}=0.05\mathit{H}\left(\mathbf{m}\right)$ | $\mathit{k}\mathit{A}=0.1\mathit{H}\left(\mathbf{m}\right)$ | $\mathit{k}\mathit{A}=0.15\mathit{H}\left(\mathbf{m}\right)$ | $\mathit{k}\mathit{a}$ |
---|---|---|---|---|---|

F1 | 1.3 | 0.042 | 0.084 | 0.126 | 0.3815 |

F2 | 1.4 | 0.049 | 0.098 | 0.146 | 0.3295 |

F3 | 1.5 | 0.056 | 0.112 | 0.167 | 0.2879 |

F4 | 1.6 | 0.063 | 0.126 | 0.189 | 0.2543 |

F5 | 1.7 | 0.073 | 0.146 | 0.218 | 0.2197 |

F6 | 1.8 | 0.078 | 0.156 | 0.235 | 0.2044 |

F7 | 1.9 | 0.087 | 0.174 | 0.258 | 0.1857 |

F8 | 2.0 | 0.094 | 0.188 | 0.282 | 0.1701 |

F9 | 2.1 | 0.102 | 0.204 | 0.305 | 0.1570 |

F10 | 2.3 | 0.118 | 0.236 | 0.353 | 0.1361 |

F11 | 2.5 | 0.133 | 0.266 | 0.398 | 0.1204 |

Estimation Formula | $\mathit{B}\mathit{i}\mathit{a}\mathit{s}\left(\mathbf{m}\right)$ | ${\mathit{I}}_{\mathit{a}}$ | ${\mathit{R}}^{2}$ | $\mathit{S}\mathit{I}\left(\%\right)$ | Estimation Formula |
---|---|---|---|---|---|

Vos | 0.0188 | 0.9474 | 0.9771 | 30.13 | Vos |

Bonakdar | 0.0031 | 0.9941 | 0.979 | 9.76 | Bonakdar |

Wave Probes | 0° | 157.5° | 180° | 225° | |||||
---|---|---|---|---|---|---|---|---|---|

Second Order | Third Order | Second Order | Third Order | Second Order | Third Order | Second Order | Third Order | ||

T = 1.4 s | kA = 0.05 | 0.107 | — | 0.048 | — | 0.071 | — | 0.028 | — |

kA = 0.1 | 0.310 | — | 0.118 | — | 0.189 | — | 0.077 | — | |

kA = 0.15 | 0.320 | — | 0.142 | — | 0.265 | — | 0.210 | — | |

T = 1.73 s | kA = 0.05 | — | — | — | — | — | — | — | — |

kA = 0.1 | 0.144 | 0.027 | 0.052 | — | 0.077 | — | 0.029 | 0.047 | |

kA = 0.15 | 0.211 | 0.029 | 0.075 | 0.014 | 0.112 | 0.02 | 0.072 | 0.089 | |

T = 2.1 s | kA = 0.05 | — | — | — | — | — | — | — | — |

kA = 0.1 | 0.178 | 0.06 | 0.097 | — | 0.105 | 0.031 | 0.07 | 0.075 | |

kA = 0.15 | 0.252 | 0.112 | 0.111 | — | 0.12 | 0.035 | 0.124 | 0.121 | |

T = 2.5 s | kA = 0.05 | 0.104 | — | 0.074 | — | 0.08 | — | 0.053 | — |

kA = 0.1 | 0.176 | 0.074 | 0.116 | — | 0.127 | — | 0.057 | 0.058 | |

ka = 0.15 | 0.264 | 0.121 | 0.153 | 0.048 | 0.169 | 0.046 | 0.052 | 0.063 |

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**MDPI and ACS Style**

Wang, B.; Li, Y.; Wu, F.; Gao, S.; Yan, J.
Numerical Investigation of Wave Run-Up and Load on Fixed Truncated Cylinder Subjected to Regular Waves Using OpenFOAM. *Water* **2022**, *14*, 2830.
https://doi.org/10.3390/w14182830

**AMA Style**

Wang B, Li Y, Wu F, Gao S, Yan J.
Numerical Investigation of Wave Run-Up and Load on Fixed Truncated Cylinder Subjected to Regular Waves Using OpenFOAM. *Water*. 2022; 14(18):2830.
https://doi.org/10.3390/w14182830

**Chicago/Turabian Style**

Wang, Bin, Yu Li, Fei Wu, Shan Gao, and Jun Yan.
2022. "Numerical Investigation of Wave Run-Up and Load on Fixed Truncated Cylinder Subjected to Regular Waves Using OpenFOAM" *Water* 14, no. 18: 2830.
https://doi.org/10.3390/w14182830