# Modelling the Wave Overtopping Flow over the Crest and the Landward Slope of Grass-Covered Flood Defences

^{*}

## Abstract

**:**

^{®}to simulate the overtopping flow on the grass-covered crest and slope of individual overtopping waves for a range of landward slope angles. The model provides insights on how the hydraulic forces change along the profile and how irregularities in the profile affect these forces. The effect of irregularities in the grass cover on the overtopping flow are captured in the Nikuradse roughness height calibrated in this study. The model was validated with two datasets of overtopping tests on existing grass-covered dikes in the Netherlands. The model results show good agreement with measurements of the flow velocity in the top layer of the wave, as well as the near bed velocity. The model application shows that the pressure, shear stress and normal stress are maximal at the wave front. High pressures occur at geometrical transitions such as the start and end of the dike crest and at the inner toe. The shear stress is maximal on the lower slope, and the normal stress is maximal halfway of the slope, making these locations vulnerable to cover failure due to high loads. The exact location of the maximum forces depends on the overtopping volume. Furthermore, the model shows that the maximum pressure and maximum normal stress are largely affected by the steepness of the landward slope, but the slope steepness only has a small effect on the maximum flow velocity and maximum shear stress compared to the overtopping volume. This new numerical model is a useful tool to determine the hydraulic forces along the profile to find vulnerable points for cover failure and improve the design of grass-covered flood defences.

## 1. Introduction

^{®}to simulate the flow of an individual overtopping wave over the crest and the landward slope of a grass-covered flood defence. The numerical model covers a variety of output variables including the flow velocity, layer thickness, pressure, shear stress and normal stress. This numerical study provides insights on the dominant hydrodynamic forces working on the grass cover of dikes and how these forces change along the dike profile. Moreover, the hydraulic forces are computed at every location on the dike crest and landward slope to identify locations where the hydraulic loads are high and therefore vulnerable to failure by wave overtopping. Two datasets of overtopping experiments on grass-covered dikes in the Netherlands are used to calibrate and validate the model. These datasets together with the model input variables are described in Section 2. Section 3 describes the model settings and the output variables. The model is validated for the layer thickness, the flow velocity and pressure on the crest and landward slope (Section 4). Additionally, the vertical flow structure is validated with measurements of the flow velocity near the bed and in the top layer of the flow. The pressure, shear stress and normal stress on the cover are presented to show how the hydraulic load changes along the profile for the overtopping wave. Next, the model is applied to a grass-covered dike with varying slope steepness to show how the hydraulic loads are affected by the geometry (Section 4.4). Finally, the results are discussed in Section 5 followed by the conclusions in Section 6.

## 2. Measured Overtopping Variables

#### 2.1. Data of the Field Tests

#### 2.1.1. Vechtdijk

#### 2.1.2. Millingen a/d Rijn

#### 2.2. Model Input Variables

## 3. Model Setup

#### 3.1. Model Specifications

^{®}, which solves the two-phase Reynolds-averaged Navier–Stokes equations using the finite volume method,

^{®}and used in other overtopping studies [18,20,32]—does not define these interfaces and is only applicable to flows with a Courant number $\le 0.1$, while the IsoAdvector algorithm is applicable to flows with a maximum Courant number of 1.

^{TM}i7-9700 GHz(x12) processor using only one core.

#### 3.1.1. Boundary Conditions

#### 3.1.2. Model Output Variables

#### 3.2. Calibration of the Roughness Height

^{®}and was therefore calibrated. The dataset of Millingen a/d Rijn included the widest range of overtopping volumes and the most measurement locations; thus, one overtopping wave per volume of Millingen a/d Rijn was used for calibration of the roughness height. The 11 volumes (Table 1) were modelled for 10 values of the roughness height in the range of 1–10 mm with steps of 1 mm. Smaller values of the roughness height did not lead to better results, and the maximum roughness height for calibration was set to 10 mm to stay within the application limit of the turbulence model (Section 3.1). The measured flow velocity and layer thickness at M1 were used as boundary condition, which left the flow velocity at the other seven locations (M2–M8) for calibration of the roughness height. The roughness height was calibrated on the maximum flow velocity using the dimensionless root-mean-squared error E (-) to compare the results of the various volumes:

#### 3.3. Model Validation

#### 3.3.1. Maximum Flow Velocity and Layer Thickness

#### 3.3.2. Vertical Flow Structure

#### 3.3.3. Pressure

## 4. Results

#### 4.1. Calibration of the Roughness Height

#### 4.2. Validation of the Numerical Model

#### 4.2.1. Flow Velocity and Layer Thickness

#### 4.2.2. Vertical Flow Structure

#### 4.2.3. Pressure

#### 4.3. Model Output Potential for Hydraulic Forces during Overtopping

#### 4.3.1. Flow Velocity

#### 4.3.2. Shear and Normal Stress

#### 4.3.3. Pressure on the Bed

#### 4.4. The Effect of the Slope Steepness

## 5. Discussion

#### 5.1. Model Application and Limitation

#### Individual Overtopping Volumes

#### 5.2. Roughness Height

#### 5.3. Comparison with Existing Modelling Approaches

#### 5.4. Sensitivity of the Model Settings

#### 5.4.1. The Layer Thickness

#### 5.4.2. The Near Bed Velocity

#### 5.4.3. The Effect of the Grid Size

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Effect on Slope Steepness on the Hydraulic Load

**Table A1.**Comparison of the relative increase in the hydraulic variables with increasing volume V for $cot\left(\phi \right)=3$ and with increasing slope steepness $cot\left(\phi \right)$ for $V=1$ m${}^{3}$/m for a simple grass-covered dike with a dike height of 5 m and a crest width of 2 m.

Increase from $\mathit{V}=1$ m${}^{3}$/m to $\mathit{V}=4$ m${}^{3}$/m | Increase from $cot\left(\mathit{\phi}\right)=8$ to $cot\left(\mathit{\phi}\right)=3$ | |
---|---|---|

Flow Velocity | 42% | 13% |

Pressure | 84% | 168% |

Shear Stress | 106% | 21% |

Normal Stress | 53% | 225% |

## Appendix B. Figures to Support the Sensitivity Analysis

**Figure A1.**Sensitivity of the model results to the threshold of the water fraction $\alpha $ for the Millingen a/d Rijn case: (

**a**) The modelled maximum flow velocity ${U}_{max}$ against the measured maximum flow velocity for three values of the threshold for the water fraction $\alpha $ and the one-to-one relationship. (

**b**) The modelled maximum layer thickness ${h}_{max}$ against the measured maximum layer thickness for three values of the threshold for the water fraction $\alpha $ and the one-to-one relationship.

**Figure A2.**The effect of the vertical grid size $\Delta z$ for location V3 at Vechtdijk for V = 1.0 m${}^{3}$/m and constant horizontal grid size $\Delta x=3$ cm: (

**a**) The modelled flow velocity in the top layer ${u}_{t}$ as function of the time t for three vertical grid sizes $\Delta z$ together with the measured flow velocity. (

**b**) The modelled layer thickness h as function of the time t for three vertical grid sizes $\Delta z$ together with the measured layer thickness.

**Figure A3.**The effect of the horizontal grid size $\Delta x$ for location V3 at Vechtdijk for V = 1.0 m${}^{3}$/m and constant vertical grid size $\Delta z=2$ cm: (

**a**) The modelled flow velocity in the top layer ${u}_{t}$ as function of the time t for three horizontal grid sizes $\Delta x$ together with the measured flow velocity. (

**b**) The modelled layer thickness h as function of the time t for three horizontal grid sizes $\Delta x$ together with the measured layer thickness.

**Figure A4.**The effect of the vertical grid size $\Delta z$ for location M6 at Millingen a/d Rijn for V = 2.5 m${}^{3}$/m and constant horizontal grid size $\Delta x=5$ cm: (

**a**) The modelled flow velocity in the top layer ${u}_{t}$ as function of the time t for three vertical grid sizes $\Delta z$ together with the measured flow velocity. (

**b**) The modelled layer thickness h as function of the time t for three vertical grid sizes $\Delta z$ together with the measured layer thickness.

**Figure A5.**The effect of the horizontal grid size $\Delta x$ for location M6 at Millingen for $V=$ 2.5 m${}^{3}$/m and constant vertical grid size $\Delta z=3$ cm: (

**a**) The modelled flow velocity in the top layer ${u}_{t}$ as function of the time t for three horizontal grid sizes $\Delta x$ together with the measured flow velocity. (

**b**) The modelled layer thickness h as function of the time t for three horizontal grid sizes $\Delta x$ together with the measured layer thickness.

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**Figure 1.**(

**a**) An example of wave overtopping on a grass-covered dike with a sandy core and a dike cover consisting of clay with vegetation on top. The overtopping wave is characterised by the layer thickness h, the flow velocity near the bed ${u}_{b}$ and the flow velocity in the top layer ${u}_{t}$. The wave pulls on the grass cover leading to a shear stress ${\tau}_{s}$ parallel to the dike surface and a normal stress ${\tau}_{n}$ normal to the dike surface. (

**b**) Measurements of the flow velocity $u\left(t\right)$ over time on the crest for an overtopping volume of 2 m${}^{3}$/m together with the idealised saw-tooth shape of Hughes [3] that depends on the maximum flow velocity ${U}_{max}$ and the overtopping period ${T}_{0}$.

**Figure 2.**(

**a**) The cross-dike profile of Vechtdijk with the measurement locations V1–V5determined from GPS measurements with approximately 2 m spacing. The markers indicate the variables measured at the location with the layer thickness h, the flow velocity in the top layer ${u}_{t}$, the flow velocity near the bed ${u}_{b}$ and the pressure p. (

**b**) The cross-dike profile of Millingen a/d Rijn with the measurement locations M1–M8 and P1–P2 determined from GPS measurements with approximately 1 m spacing.

**Figure 3.**The grid of the Millingen a/d Rijn case follows the dike profile with a grid size in the cross-dike direction $\Delta x$ of 5 cm and a vertical grid size $\Delta z$ of 3 cm for the bottom layer and 6 cm for the top layer.

**Figure 4.**(

**a**) The modelled maximum flow velocity ${U}_{max}$ against the measured maximum flow velocity for three values of the roughness height ${K}_{s}$ and the one-to-one relationship. (

**b**) The modelled maximum flow velocity ${U}_{max}$ against the measured maximum flow velocity for a roughness height ${K}_{s}=8$ mm where the markers indicate the overtopping volumes of the Millingen a/d Rijn case.

**Figure 5.**Comparison between the measurements (solid blue) and the calibration results for three values of the roughness height ${K}_{s}$ for an overtopping volume of 2.5 m${}^{3}$/m at Millingen a/d Rijn: (

**a**) The flow velocity in the top layer ${u}_{t}$ at M2. (

**b**) The layer thickness h at M2. (

**c**) The flow velocity in the top layer ${u}_{t}$ at M8. (

**d**) The layer thickness h at M8.

**Figure 6.**(

**a**) Comparison of the modelled and measured maximum velocity ${U}_{max}$ together with the one-to-one relationship where the markers indicate the case studies. (

**b**) Comparison of the modelled and measured maximum layer thickness ${h}_{max}$ together with the one-to-one relationship where the markers indicate the case studies.

**Figure 7.**(

**a**) The modelled flow velocity in the top layer ${u}_{t}$ as a function of time t together with the measured flow velocity for the Vechtdijk case at location V3 with $V=1$ m${}^{3}$/m. (

**b**) The modelled layer thickness h as a function of time t together with the measured layer thickness for the Vechtdijk case at location V3 with $V=1$ m${}^{3}$/m.

**Figure 8.**(

**a**) A snapshot of the modelled water fraction $\alpha $ for $V=1$ m${}^{3}$/m of Vechtdijk at $t=5.2$ s where the light, black and blue colours correspond to $\alpha =0$, $\alpha =0.5$ and $\alpha =1.0$. Water oscillations of the fragmented flow are forming in the circle and square corresponding to a peak in the layer thickness at $t=4.3$ s and $t=9.2$ s in Figure 7b. (

**b**) A photo of the wave overtopping flow on a grass dike showing the fragmented flow on the slope (photo by Vera van Bergeijk).

**Figure 9.**The modelled flow velocity $u\left(z\right)$ along the vertical z (orange circles) at V3 for the Vechtdijk case together with the theoretical relation (black line, Equation (14)) and the measured flow velocity (blue markers) at 2–3 cm from the bed and in the top layer of the three released volumes of $V=1$ m${}^{3}$/m.

**Figure 10.**Validation of the pressure for the Millingen a/d Rijn case: (

**a**) Comparison of the modelled and measured maximum pressure ${p}_{max}$ at measurement locations P1 and P2. (

**b**) Comparison of the measured and modelled pressure p over time t for an overtopping volume $V=3$ m${}^{3}$/m at P1.

**Figure 11.**Comparison of the hydraulic forces along the dike profile for four overtopping volumes. The vertical black lines indicate the start and the end of the landward slope: (

**a**) The maximum shear stress with respect to time ${\tau}_{s,max}$ for the Millingen a/d Rijn case. (

**b**) The maximum normal stress with respect to time ${\tau}_{n,max}$ for the Millingen a/d Rijn case. (

**c**) The maximum pressure with respect to time ${p}_{max}$ for the Millingen a/d Rijn case. (

**d**) The maximum flow velocity $U\left(x\right)$ with respect to time for the Millingen a/d Rijn case. (

**e**) The maximum shear stress with respect to time ${\tau}_{s,max}$ for the Vechtdijk case. (

**f**) The maximum normal stress with respect to time ${\tau}_{n,max}$ for the Vechtdijk case. (

**g**) The maximum pressure with respect to time ${p}_{max}$ for the Vechtdijk case. (

**h**) The maximum flow velocity ${U}_{max}$ with respect to time for the Vechtdijk case.

**Figure 12.**Forces for the Millingen a/d Rijn case at measurement locations M2, M4 and M8 for an overtopping volume of 2.5 m${}^{3}$/m: (

**a**) The layer thickness h as a function of time t. (

**b**) The shear stress ${\tau}_{s}$ on the dike surface as a function of time t. (

**c**) The normal stress ${\tau}_{n}$ on the dike surface as a function of time t. (

**d**) The total pressure p on the dike surface as a function of time t.

**Figure 13.**Comparison of the maximum hydraulic variables as a function of the slope steepness $cot\phi $ for four overtopping volumes. (

**a**) The maximum flow velocity U with respect to time and space. (

**b**) The maximum pressure p with respect to time and space. (

**c**) The maximum shear stress ${\tau}_{s}$ with respect to time and space. (

**d**) The maximum normal stress ${\tau}_{n}$ with respect to time and space.

**Figure 14.**Comparison of the model results for individual overtopping waves and a time series of three overtopping waves with volumes of 1 m${}^{3}$/m, 4 m${}^{3}$/m and 2 m${}^{3}$/m for the Vechtdijk case at location V5. (

**a**) The layer thickness h as a function of time t. (

**b**) The flow velocity in the top layer ${u}_{t}$ as a function of time t. (

**c**) The pressure on the dike surface p as a function of time t. (

**d**) The shear stress on the dike surface ${\tau}_{s}$ as a function of time t.

**Figure 15.**Comparison of the modelled and measured maximum velocity ${U}_{max}$ of the near bed velocity at M3, M5 and M7 of the first, second and third cell from the dike surface for the 11 overtopping volumes at Millingen a/d Rijn.

**Table 1.**Overview of the measurements with the measurement locations of the layer thickness h, the flow velocity in the top layer ${u}_{t}$ and the flow velocity near the bed ${u}_{b}$ together with the released overtopping volumes during the tests with the wave overtopping simulator (Figure 2). The volumes were released more than once resulting in the total number of waves.

Experiment | $\mathit{h}\left(\mathit{t}\right)$ | ${\mathit{u}}_{\mathit{t}}\left(\mathit{t}\right)$ | ${\mathit{u}}_{\mathit{b}}\left(\mathit{t}\right)$ | Volumes (m${}^{3}$/m) | Waves |
---|---|---|---|---|---|

Vechtdijk | V1, V2, V3, V4, V5 | V3, V5 | V3 | 0.2, 0.4, 0.6, 0.8, 1.0, 2.0, 3.0, 4.0 | 24 |

Millingen | M1, M2, M4, M6, M8 | M1, M2, M4, M6, M8 | M3, M5, M7 | 0.4, 0.6, 0.8, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 5.5 | 28 |

**Table 2.**The dimensionless root-mean-squared error E between the modelled and measured maximum velocity for the roughness height ${K}_{s}$ in the range $1\u201310$ mm with the smallest error in bold.

${K}_{s}$ (mm) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

E (-) | 0.134 | 0.144 | 0.140 | 0.123 | 0.115 | 0.113 | 0.130 | 0.106 | 0.114 | 0.113 |

**Table 3.**The root-mean squared error between the modelled and measured maximum flow velocity of Millingen a/d Rijn in m/s for an overtopping volume of 2.5 m${}^{3}$/m as a function of the roughness height ${K}_{s}$ and three values of the roughness constant ${C}_{s}$ where the bold values correspond to the lowest error per roughness constant.

${\mathit{K}}_{\mathit{s}}$ (mm) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|

${C}_{s}=0.1$ | 0.92 | 0.88 | 0.88 | 0.88 | 0.90 | 0.87 | 0.89 | 0.96 | 0.86 | 0.87 |

${C}_{s}=0.5$ | 0.38 | 0.60 | 0.68 | 0.53 | 0.44 | 0.41 | 0.43 | 0.51 | 0.60 | 0.65 |

${C}_{s}=0.9$ | 0.86 | 0.82 | 0.81 | 0.81 | 0.82 | 1.05 | 0.89 | 0.91 | 0.92 | 1.09 |

**Table 4.**The effect of the horizontal grid size $\Delta x$ and the vertical grid size $\Delta z$ on the computational time and the observed maximum flow velocity ${U}_{max,o}$ and modelled maximum flow velocity ${U}_{max,m}$ at V3 and M6 for Vechtdijk and Millingen a/d Rijn, respectively.

Profile | $\mathbf{\Delta}\mathit{x}$ (cm) | $\mathbf{\Delta}\mathit{z}$ (cm) | Time (min) | ${\mathit{U}}_{\mathit{max}}$ (m/s) |
---|---|---|---|---|

Vechtdijk 1 m${}^{3}$/m ${U}_{max,o}=5.12$ m/s | 3.0 | 2.0 | 17.4 | 5.41 |

3.0 | 1.5 | 23.3 | 5.44 | |

3.0 | 2.5 | 9.0 | 5.19 | |

2.0 | 2.0 | 32.3 | 5.50 | |

5.0 | 2.0 | 5.3 | 5.37 | |

Millingen a/d Rijn 2.5 m${}^{3}$/m ${U}_{max,0}=6.78$ m/s | 5.0 | 3.0 | 13.5 | 6.52 |

5.0 | 2.0 | 18.2 | 6.59 | |

5.0 | 4.0 | 13.0 | 6.93 | |

3.0 | 3.0 | 20.0 | 6.29 | |

7.0 | 3.0 | 5.2 | 6.64 |

© 2020 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**

van Bergeijk, V.M.; Warmink, J.J.; Hulscher, S.J.M.H. Modelling the Wave Overtopping Flow over the Crest and the Landward Slope of Grass-Covered Flood Defences. *J. Mar. Sci. Eng.* **2020**, *8*, 489.
https://doi.org/10.3390/jmse8070489

**AMA Style**

van Bergeijk VM, Warmink JJ, Hulscher SJMH. Modelling the Wave Overtopping Flow over the Crest and the Landward Slope of Grass-Covered Flood Defences. *Journal of Marine Science and Engineering*. 2020; 8(7):489.
https://doi.org/10.3390/jmse8070489

**Chicago/Turabian Style**

van Bergeijk, Vera M., Jord J. Warmink, and Suzanne J. M. H. Hulscher. 2020. "Modelling the Wave Overtopping Flow over the Crest and the Landward Slope of Grass-Covered Flood Defences" *Journal of Marine Science and Engineering* 8, no. 7: 489.
https://doi.org/10.3390/jmse8070489