# Hydrodynamic Modelling of Wave Overtopping over a Block-Covered Flood Defence

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## Abstract

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^{®}numerical model. Using the porousWaveFoam solver, a porous layer on the crest and landward slope is implemented, where the flow resistance of this porous layer largely depends on the resistance coefficients $\alpha $ [-] and $\beta $ [-]. The numerical model is calibrated based on resistance coefficients as introduced earlier in the literature, which showed that the resistance coefficients of $\alpha =500$ and $\beta =2.0$ performed best for the peak flow velocities and the peak pressures. The numerical model is evaluated by using these resistance coefficients in other time series of the physical tests. The evaluated model is then used to determine the hydrodynamic conditions on the landward slope, which showed that the pressure was the most influential hydrodynamic condition at the time of failure. Finally, the model showed that a porosity of $n=0.6$ and the porous layer thickness $\eta =36\text{}\mathrm{mm}$ reduced the peak pressure the most.

## 1. Introduction

## 2. Physical Model Tests

## 3. Method

#### 3.1. Numerical Model Set-Up

^{2}/s

^{2}] and $\omega $ the specific rate of dissipation of turbulent kinetic energy. The $k-\omega $ turbulence model has a stress limiter ${\lambda}_{1}$ and an effective potential flow threshold ${\lambda}_{2}$. Larsen and Fuhrman [29] suggested a value of ${\lambda}_{2}=0.05$ and for ${\lambda}_{1}$ either 0.2 and 0.875 is mentioned. Chen et al. [17] showed that ${\lambda}_{1}=0.2$ performed better in a $k-\omega $ model for wave overtopping. The stabilised $k-\omega $ model with ${\lambda}_{1}=0.2$ and ${\lambda}_{2}=0.05$ was used in this study.

#### 3.2. Calibration of the Resistance Coefficients

#### 3.3. Evaluation of the Resistance Coefficients

#### 3.4. Hydraulic Loads on Blocks

## 4. Results

#### 4.1. Validation of Wave Conditions

#### 4.2. Calibration of the Resistance Coefficients

#### 4.3. Evaluation of the Resistance Coefficients

#### 4.4. Hydraulic Loads on Blocks

## 5. Discussion

#### 5.1. Towards Improved Modelling of Porous Dike Revetments

#### 5.2. Performance and Limitations of the Model

#### 5.3. Application

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Overview of the layers on the landward slope of a block-covered dike, adapted from Bezuijen et al. [14].

**Figure 3.**(

**a**) Overview of test setup with wave paddle located at $X=0$ m, water level $h=6.9$ m and wave height meters located at $X=108.5,\text{}114.5$ and $117.5$ m. (

**b**) Cross section of the test set-up in the Deltares Delta Flume, including Wave Height Meter (WHM), Pressure Sensor (PS), and Paddle Wheel (PW), adapted from Van Steeg [13].

**Figure 5.**Overview of OceanWave3D and OpenFOAM® domain, with X [m] and Z [m] the horizontal and vertical position in the flume.

**Figure 6.**A section of generated mesh in OpenFOAM®, the locally refined mesh near the water surface is visible on the left. The grid cells are refined towards the crest and along the landward slope, which are the areas of interest.

**Figure 7.**Free surface elevation of incoming waves at X = $108.5$ m from $t=200\u2013250$ s in Test B.

**Figure 9.**(

**a**) Modelled against observed peak flow velocity u [m/s] at PW1 and PW2 per overtopping wave during $t=0$–170 s in Test B. (

**b**) Modelled against observed peak pressure P [kPa] at PS2–PS5 per overtopping wave during $t=0$–170 s in Test B.

**Figure 10.**(

**a**) Modelled against observed peak flow velocity u [m/s] per overtopping wave during $t=170$–350 s in Test B using $\alpha =500$ and $\beta =2.0$. (

**b**) Modelled against observed peak pressure P [kPa] per overtopping wave during $t=170$–350 s in Test B using $\alpha =500$ and $\beta =2.0$.

**Figure 11.**(

**a**) Relative change in porosity n [%] compared to the default $n=0.4$. The peak values are based on the time series between 290 and 310 s and compared to the default peak values using $n=0.4$. (

**b**) Relative change in porous layer thickness $\eta $ [%] compared to the default $36$ mm. The peak values are based on the time series between 290 and 310 s compared to the default peak values using $\eta =36$ mm.

**Table 1.**The wave characteristics during the two tests with the water depth h, significant wave height ${H}_{m0}$, maximum wave height ${H}_{max}$, peak wave period ${T}_{p}$, spectral wave period ${T}_{m-1,0}$, wave steepness ${s}_{m-1,0}$, and number of waves N.

Test | h | ${\mathit{H}}_{\mathit{m}0}$ | ${\mathit{H}}_{\mathit{max}}$ | ${\mathit{T}}_{\mathit{p}}$ | ${\mathit{T}}_{\mathit{m}-1,0}$ | ${\mathit{s}}_{\mathit{m}-1,0}$ | q | N |
---|---|---|---|---|---|---|---|---|

[m] | [m] | [m] | [s] | [s] | [−] | [l/s/m] | [−] | |

Test A | 6.9 | 1.48 | 2.66 | 4.89 | 4.54 | 0.046 | 9.3 | 1032 |

Test B | 6.9 | 1.63 | 2.36 | 7.23 | 6.53 | 0.024 | 34.2 | 54 |

Study | $\mathit{\alpha}$ | $\mathit{\beta}$ |
---|---|---|

Van Gent [26] | 1000 | 1.1 |

Jensen et al. [18] | 500 | 2.0 |

Losada et al. [27] | 200 | 0.8 |

# | n [-] | Relative Change | # | $\mathit{\eta}$ [m] | Relative Change |
---|---|---|---|---|---|

$n1$ | 0.2 | −50% | $\eta 1$ | 0.016 | −55.6% |

$n2$ | 0.3 | −25% | $\eta 2$ | 0.026 | −27.8% |

$n3$ | 0.4 | 0% | $\eta 3$ | 0.036 | 0% |

$n4$ | 0.5 | +25% | $\eta 4$ | 0.046 | +27.8% |

$n5$ | 0.6 | +50% | $\eta 5$ | 0.056 | +55.6% |

Test A ($\mathit{t}=0$–$500\text{}\mathbf{s}$) | ||

${H}_{m0}$ [m] | ${T}_{m-1,0}$ [s] | |

Physical | 1.282 | 4.691 |

Numerical | 1.283 | 4.662 |

Test B ($\mathit{t}=0$–$330\text{}\mathbf{s}$) | ||

${H}_{m0}$ [m] | ${T}_{m-1,0}$ [s] | |

Physical | 1.364 | 6.356 |

Numerical | 1.365 | 6.562 |

**Table 5.**Performance indicators of the flow velocity u and the pressure P for the three sets of resistance coefficients.

Resistance Coefficients | Flow Velocity u | Pressure P | ||
---|---|---|---|---|

NSE [-] | RMSE [m/s] | NSE [-] | RMSE [kPa] | |

Van Gent [26] | 0.332 | 0.874 | −0.024 | 0.452 |

Jensen et al. [18] | 0.315 | 0.885 | 0.266 | 0.382 |

Losada et al. [27] | 0.299 | 0.895 | 0.225 | 0.393 |

Sensor | NSE [-] | RMSE [m/s] |
---|---|---|

PW1 | 0.681 | 0.817 |

PW2 | 0.502 | 1.072 |

Total | 0.606 | 0.935 |

Sensor | NSE [-] | RMSE [kPa] |
---|---|---|

PS2 | −0.251 | 0.443 |

PS3 | −0.023 | 0.382 |

PS4 | −1.192 | 0.312 |

PS5 | 0.546 | 0.472 |

Total | 0.154 | 0.411 |

**Table 8.**Peak flow characteristics at location of failure in Test B with the time t, flow velocity u, flow thickness h, and pressure P.

t [s] | u [m/s] | h [m] | P [kPa] |
---|---|---|---|

43 | 1.97 | 0.31 | 0.83 |

155 | 2.87 | 0.37 | 2.67 |

220 | 3.76 | 0.25 | 1.35 |

246 | 2.36 | 0.17 | 1.22 |

300 | 3.65 | 0.41 | 5.07 |

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## Share and Cite

**MDPI and ACS Style**

Barendse, L.; van Bergeijk, V.M.; Chen, W.; Warmink, J.J.; Mughal, A.; Hill, D.; Hulscher, S.J.M.H.
Hydrodynamic Modelling of Wave Overtopping over a Block-Covered Flood Defence. *J. Mar. Sci. Eng.* **2022**, *10*, 89.
https://doi.org/10.3390/jmse10010089

**AMA Style**

Barendse L, van Bergeijk VM, Chen W, Warmink JJ, Mughal A, Hill D, Hulscher SJMH.
Hydrodynamic Modelling of Wave Overtopping over a Block-Covered Flood Defence. *Journal of Marine Science and Engineering*. 2022; 10(1):89.
https://doi.org/10.3390/jmse10010089

**Chicago/Turabian Style**

Barendse, Luuk, Vera M. van Bergeijk, Weiqiu Chen, Jord J. Warmink, Aroen Mughal, Dorian Hill, and Suzanne J. M. H. Hulscher.
2022. "Hydrodynamic Modelling of Wave Overtopping over a Block-Covered Flood Defence" *Journal of Marine Science and Engineering* 10, no. 1: 89.
https://doi.org/10.3390/jmse10010089