# Inundation Characteristics of Solitary Waves According to Revetment Type

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Overview of Physical Modeling Test

#### 2.1. Physical Modeling Facility

#### 2.2. Free-Surface Tracking

#### 2.3. Incident Wave Conditions

## 3. Experimental Results

#### 3.1. Water Surface Elevation Distribution

#### 3.2. Water Surface Elevation Distribution in Land Area

#### 3.3. Maximum Inundation Height

## 4. Numerical Methodology

#### 4.1. Numerical Model

#### 4.1.1. Governing Equations

#### 4.1.2. Solution Techniques

#### 4.1.3. Fluid Resistance

#### 4.1.4. Turbulence Model

#### 4.1.5. Boundary Conditions

#### 4.1.6. Stability Conditions

#### 4.2. NWT Setup

#### 4.3. Wave Generation and Verification

#### 4.3.1. Wave Generation Conditions

#### 4.3.2. Grid Resolution Test

#### 4.3.3. NWT Verification

^{2}), mean square error (MSE), root mean square error (RMSE), and normalized root mean square error (NRMSE), which are shown in Figure 14. The values of R

^{2}, MSE, RMSE, and NRMSE are calculated as follows:

^{2}, MSE, RMSE, and NRMSE. Nevertheless, the accuracy of inundation analysis for the VR was considered to be at a reliable level.

## 5. Numerical Analysis Results

#### 5.1. Wave Field and Flow Field

#### 5.1.1. Overtopping and Inundation Process

#### 5.1.2. Flow Field Distribution

#### 5.2. Vertical Flow Velocity Distribution

#### 5.3. Vortex Field

#### 5.4. Maximum Water Surface Elevation Distribution

#### 5.5. Inundation Distance

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Run | ${\mathit{A}}_{\mathbf{0}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | $\mathit{\epsilon}$$\mathbf{(}{\mathit{A}}_{\mathbf{0}}\mathbf{/}\mathit{h}\mathbf{)}$ | ${\mathit{\eta}}_{\mathbf{m}\mathbf{a}\mathbf{x}}^{\mathbf{V}\mathbf{R}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{\eta}}_{\mathbf{m}\mathbf{a}\mathbf{x}}^{\mathbf{WAR}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{\eta}}_{\mathbf{m}\mathbf{a}\mathbf{x}}^{\mathbf{V}\mathbf{R}}\mathbf{/}\mathit{h}$ | ${\mathit{\eta}}_{\mathbf{m}\mathbf{a}\mathbf{x}}^{\mathbf{WAR}}\mathbf{/}\mathit{h}$ | ${\mathit{\eta}}_{\mathbf{m}\mathbf{a}\mathbf{x}}^{\mathbf{V}\mathbf{R}}\mathbf{/}{\mathit{A}}_{\mathbf{0}}$ | ${\mathit{\eta}}_{\mathbf{m}\mathbf{a}\mathbf{x}}^{\mathbf{WAR}}\mathbf{/}{\mathit{A}}_{\mathbf{0}}$ | ${\mathit{\eta}}_{\mathbf{m}\mathbf{a}\mathbf{x}}^{\mathbf{WAR}}\mathbf{/}{\mathit{\eta}}_{\mathbf{m}\mathbf{a}\mathbf{x}}^{\mathbf{V}\mathbf{R}}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 4.4 | 0.09 | 7.73 | 7.71 | 0.16 | 0.16 | 1.76 | 1.75 | 1.00 |

2 | 5.4 | 0.12 | 9.09 | 8.80 | 0.19 | 0.19 | 1.68 | 1.63 | 0.97 |

3 | 6.3 | 0.13 | 10.33 | 9.95 | 0.22 | 0.21 | 1.65 | 1.59 | 0.96 |

4 | 7.2 | 0.15 | 11.52 | 10.98 | 0.25 | 0.23 | 1.60 | 1.52 | 0.95 |

5 | 8.2 | 0.18 | 12.72 | 11.99 | 0.27 | 0.26 | 1.55 | 1.46 | 0.94 |

6 | 10 | 0.21 | 15.08 | 14.30 | 0.32 | 0.30 | 1.51 | 1.43 | 0.95 |

7 | 12.3 | 0.26 | 17.32 | 15.37 | 0.37 | 0.33 | 1.41 | 1.25 | 0.89 |

8 | 14.5 | 0.31 | 19.40 | 16.35 | 0.41 | 0.35 | 1.34 | 1.13 | 0.84 |

9 | 16.5 | 0.35 | 21.56 | 17.63 | 0.46 | 0.38 | 1.31 | 1.07 | 0.82 |

10 | 18.7 | 0.4 | 23.63 | 18.84 | 0.50 | 0.40 | 1.26 | 1.01 | 0.80 |

11 | 21.5 | 0.46 | 25.63 | 20.02 | 0.55 | 0.43 | 1.19 | 0.93 | 0.78 |

Run | ${\mathit{A}}_{\mathbf{0}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | $\mathit{\epsilon}$$\mathbf{(}{\mathit{A}}_{\mathbf{0}}\mathbf{/}\mathit{h}\mathbf{)}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}\mathbf{/}\mathit{h}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{/}\mathit{h}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}\mathbf{/}{\mathit{A}}_{\mathbf{0}}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{/}{\mathit{A}}_{\mathbf{0}}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{/}{\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 4.4 | 0.09 | 1.34 | 1.82 | 0.03 | 0.04 | 0.31 | 0.41 | 1.36 |

2 | 5.4 | 0.12 | 1.85 | 2.58 | 0.04 | 0.05 | 0.34 | 0.48 | 1.39 |

3 | 6.3 | 0.13 | 2.43 | 3.37 | 0.05 | 0.07 | 0.39 | 0.54 | 1.38 |

4 | 7.2 | 0.15 | 3.21 | 4.25 | 0.07 | 0.09 | 0.45 | 0.59 | 1.32 |

5 | 8.2 | 0.18 | 4.18 | 5.07 | 0.09 | 0.11 | 0.51 | 0.62 | 1.21 |

6 | 10 | 0.21 | 6.35 | 7.03 | 0.14 | 0.15 | 0.64 | 0.70 | 1.11 |

7 | 12.3 | 0.26 | 8.57 | 9.33 | 0.18 | 0.20 | 0.70 | 0.76 | 1.09 |

8 | 14.5 | 0.31 | 10.67 | 10.88 | 0.23 | 0.23 | 0.74 | 0.75 | 1.02 |

9 | 16.5 | 0.35 | 13.91 | 12.85 | 0.30 | 0.27 | 0.84 | 0.78 | 0.92 |

10 | 18.7 | 0.4 | 14.48 | 14.34 | 0.31 | 0.31 | 0.77 | 0.77 | 0.99 |

11 | 21.5 | 0.46 | 16.61 | 15.33 | 0.35 | 0.33 | 0.77 | 0.71 | 0.92 |

Run | ${\mathit{A}}_{\mathbf{0}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | $\mathit{\epsilon}$$\mathbf{(}{\mathit{A}}_{\mathbf{0}}\mathbf{/}\mathit{h}\mathbf{)}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}\mathbf{/}\mathit{h}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{/}\mathit{h}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}\mathbf{/}{\mathit{A}}_{\mathbf{0}}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{/}{\mathit{A}}_{\mathbf{0}}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{/}{\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 4.4 | 0.09 | 1.27 | 2.01 | 0.03 | 0.04 | 0.29 | 0.46 | 1.59 |

2 | 5.4 | 0.12 | 1.97 | 2.10 | 0.04 | 0.04 | 0.36 | 0.39 | 1.07 |

3 | 6.3 | 0.13 | 2.42 | 2.44 | 0.05 | 0.05 | 0.39 | 0.39 | 1.01 |

4 | 7.2 | 0.15 | 2.41 | 2.95 | 0.05 | 0.06 | 0.34 | 0.41 | 1.22 |

5 | 8.2 | 0.18 | 2.55 | 3.31 | 0.05 | 0.07 | 0.31 | 0.40 | 1.30 |

6 | 10 | 0.21 | 3.20 | 4.67 | 0.07 | 0.10 | 0.32 | 0.47 | 1.46 |

7 | 12.3 | 0.26 | 3.75 | 5.49 | 0.08 | 0.12 | 0.30 | 0.45 | 1.47 |

8 | 14.5 | 0.31 | 4.47 | 6.49 | 0.10 | 0.14 | 0.31 | 0.45 | 1.45 |

9 | 16.5 | 0.35 | 5.38 | 7.89 | 0.11 | 0.17 | 0.33 | 0.48 | 1.47 |

10 | 18.7 | 0.4 | 6.21 | 9.37 | 0.13 | 0.20 | 0.33 | 0.50 | 1.51 |

11 | 21.5 | 0.46 | 7.41 | 10.46 | 0.16 | 0.22 | 0.34 | 0.49 | 1.41 |

Run | ${\mathit{A}}_{\mathbf{0}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | $\mathit{\epsilon}$$\mathbf{(}{\mathit{A}}_{\mathbf{0}}\mathbf{/}\mathit{h}\mathbf{)}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}\mathbf{/}\mathit{h}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{/}\mathit{h}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}\mathbf{/}{\mathit{A}}_{\mathbf{0}}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{/}{\mathit{A}}_{\mathbf{0}}$ | ${\mathit{D}}_{\mathit{H}}^{\mathbf{WAR}}\mathbf{/}{\mathit{D}}_{\mathit{H}}^{\mathbf{V}\mathbf{R}}$ |
---|---|---|---|---|---|---|---|---|---|

1 | 4.4 | 0.09 | 1.15 | 1.62 | 0.02 | 0.03 | 0.26 | 0.37 | 1.42 |

2 | 5.4 | 0.12 | 1.62 | 1.89 | 0.03 | 0.04 | 0.30 | 0.35 | 1.17 |

3 | 6.3 | 0.13 | 2.06 | 2.03 | 0.04 | 0.04 | 0.33 | 0.32 | 0.98 |

4 | 7.2 | 0.15 | 2.32 | 2.46 | 0.05 | 0.05 | 0.32 | 0.34 | 1.06 |

5 | 8.2 | 0.18 | 2.52 | 2.51 | 0.05 | 0.05 | 0.31 | 0.31 | 1.00 |

6 | 10 | 0.21 | 3.11 | 3.92 | 0.07 | 0.08 | 0.31 | 0.39 | 1.26 |

7 | 12.3 | 0.26 | 3.55 | 3.41 | 0.08 | 0.07 | 0.29 | 0.28 | 0.96 |

8 | 14.5 | 0.31 | 4.23 | 4.55 | 0.09 | 0.10 | 0.29 | 0.31 | 1.07 |

9 | 16.5 | 0.35 | 6.62 | 5.39 | 0.14 | 0.11 | 0.40 | 0.33 | 0.82 |

10 | 18.7 | 0.4 | 5.79 | 6.21 | 0.12 | 0.13 | 0.31 | 0.33 | 1.07 |

11 | 21.5 | 0.46 | 7.04 | 6.99 | 0.15 | 0.15 | 0.33 | 0.33 | 0.99 |

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**Figure 1.**Wave flume and test setup: (

**a**) vertical revetment (VR); (

**b**) wave absorbing revetment (WAR).

**Figure 2.**Images of solitary wave overtopping and inundation on the VR (left) and WAR (right): (

**a**) Run-1 (${A}_{0}=4.4\mathrm{cm}$, $\epsilon =0.09$); (

**b**) Run-7 (${A}_{0}=12.3\mathrm{cm}$, $\epsilon =0.26$); (

**c**) Run-11 (${A}_{0}=21.5\mathrm{cm}$, $\epsilon =0.46$).

**Figure 4.**Distributions of maximum water surface elevations due to $\epsilon $ at the front of the vertical wall—comparison between VR and WAR.

**Figure 5.**Comparisons of water surface elevations at WG3, WG4, and WG5 for the VR and WAR: (

**a**) Run-1 (${A}_{0}=4.4\mathrm{cm}$, $\epsilon =0.09$); (

**b**) Run-7 (${A}_{0}=12.3\mathrm{cm}$, $\epsilon =0.26$); (

**c**) Run-11 (${A}_{0}=21.5\mathrm{cm}$, $\epsilon =0.46$).

**Figure 12.**Spatial distributions of measured and calculated water surface elevations around the VR (left: Run-4 (${A}_{0}=8.2\mathrm{cm}$, $\epsilon =0.18$), right: Run-10 (${A}_{0}=18.7\mathrm{cm}$, $\epsilon =0.4$))—comparison between the measured and calculated results.

**Figure 13.**Spatial distributions of measured and calculated water surface elevations around the WAR (left: Run-4 (${A}_{0}=8.2\mathrm{cm}$, $\epsilon =0.18$), right: Run-10 (${A}_{0}=18.7\mathrm{cm}$, $\epsilon =0.4$))—comparison between the measured and calculated results.

**Figure 15.**Time-series of wave and flow fields for Run-7 (${A}_{0}=12.3\mathrm{cm}$, $\epsilon =0.26$) around the revetment—comparison between the values for the VR (left) and WAR (right).

**Figure 16.**Wave and flow fields around the VR when the maximum water level occurs at the front of the vertical wall ($x/h=0$).

**Figure 17.**Wave and flow fields around the WAR when the maximum water level occurs at the front of the vertical wall ($x/h=0$).

**Figure 18.**Vertical distribution of horizontal velocity at the front of the vertical wall ($x/h=0$)—comparison between VR and WAR.

**Figure 19.**Vertical distribution of vertical velocity at the front of the vertical wall ($x/h=0$)—comparison between VR and WAR.

**Figure 20.**Vertical distribution of horizontal velocity at the center of the separated region—comparison between VR and WAR; (

**a**) $\frac{x}{h}=0.04;\left(b\right)\frac{x}{h}=0.11;\left(c\right)x/h=0.15$.

**Figure 21.**Vertical distribution of vertical velocity at the center of the separated region—comparison between VR and WAR; (

**a**) $\frac{x}{h}=0.04;\left(b\right)\frac{x}{h}=0.11;\left(c\right)x/h=0.15$.

**Figure 22.**Wave and vortex fields around the VR when the maximum water level occurs at the front of the vertical wall ($x/h=0$).

**Figure 23.**Wave and vortex fields around the WAR when the maximum water level occurs at the front of the vertical wall ($x/h=0$).

Run | $\mathit{h}$ (cm) | ${\mathit{A}}_{\mathbf{0}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{L}}_{\mathit{e}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | $\mathit{\epsilon}$$\mathbf{(}{\mathit{A}}_{\mathbf{0}}\mathbf{/}\mathit{h}\mathbf{)}$ | ${\mathit{A}}_{\mathbf{0}}\mathbf{/}{\mathit{F}}_{\mathit{b}}$ | ${\mathit{A}}_{\mathbf{0}}\mathbf{/}{\mathit{L}}_{\mathit{e}}$ |
---|---|---|---|---|---|---|

1 | 47 | 4.4 | 654.31 | 0.09 | 0.88 | 0.007 |

2 | 5.4 | 587.92 | 0.12 | 1.08 | 0.009 | |

3 | 6.3 | 544.31 | 0.13 | 1.26 | 0.012 | |

4 | 7.2 | 509.15 | 0.15 | 1.44 | 0.014 | |

5 | 8.2 | 477.1 | 0.18 | 1.64 | 0.017 | |

6 | 10 | 427.77 | 0.21 | 2 | 0.023 | |

7 | 12.3 | 389.55 | 0.26 | 2.46 | 0.032 | |

8 | 14.5 | 358.78 | 0.31 | 2.9 | 0.040 | |

9 | 16.5 | 336.33 | 0.35 | 3.3 | 0.049 | |

10 | 18.7 | 315.93 | 0.4 | 3.74 | 0.059 | |

11 | 21.5 | 294.64 | 0.46 | 4.3 | 0.073 |

Test | $\mathbf{\Delta}\mathit{x}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | $\mathbf{\Delta}\mathit{z}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | ${\mathit{L}}_{\mathit{e}}\mathbf{/}\mathbf{\Delta}\mathit{x}$ | ${\mathit{A}}_{\mathbf{0}}/\mathbf{\Delta}\mathit{z}$ | $\mathbf{\Delta}\mathit{z}\mathbf{/}\mathbf{\Delta}\mathit{x}$ |
---|---|---|---|---|---|

1 | 2 | 0.5 | 327.2 | 8.8 | 1/4 |

2 | 0.6 | 0.2 | 3271.6 | 22.0 | 1/3 |

3 | 0.3 | 0.1 | 6543.1 | 44.0 | 1/3 |

$\mathit{\epsilon}$ | $\mathbf{Inundation}\mathbf{Distance}{\mathit{I}}_{\mathit{D}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | $\mathbf{\Delta}{\mathit{I}}_{\mathit{D}}\mathbf{\left(}\mathbf{cm}\mathbf{\right)}$ | $\mathbf{\Delta}{\mathit{I}}_{\mathit{D}}\mathbf{/}\mathit{h}$ | ${\mathit{I}}_{\mathit{D}}\mathbf{/}\mathit{h}$ | ${\mathit{I}}_{\mathit{D}}^{\mathbf{W}\mathbf{A}\mathbf{R}}\mathbf{/}{\mathit{I}}_{\mathit{D}}^{\mathbf{V}\mathbf{R}}$ | ||
---|---|---|---|---|---|---|---|

${\mathit{I}}_{\mathit{D}}^{\mathbf{V}\mathbf{R}}$ | ${\mathit{I}}_{\mathit{D}}^{\mathbf{W}\mathbf{A}\mathbf{R}}$ | ${\mathit{I}}_{\mathit{D}}^{\mathbf{W}\mathbf{A}\mathbf{R}}\mathbf{-}{\mathit{I}}_{\mathit{D}}^{\mathbf{V}\mathbf{R}}$ | ${\mathit{I}}_{\mathit{D}}^{\mathbf{V}\mathbf{R}}\mathbf{/}\mathit{h}$ | ${\mathit{I}}_{\mathit{D}}^{\mathbf{W}\mathbf{A}\mathbf{R}}\mathbf{/}\mathit{h}$ | |||

0.09 | 109.25 | 121.5 | 12.25 | 0.26 | 2.32 | 2.59 | 1.11 |

0.12 | 197.5 | 231.5 | 34 | 0.72 | 4.20 | 4.93 | 1.17 |

0.13 | 250.2 | 338.5 | 88.3 | 1.88 | 5.32 | 7.20 | 1.35 |

0.15 | 361.5 | 415.5 | 54 | 1.15 | 7.69 | 8.84 | 1.15 |

0.17 | 458.5 | 526.5 | 68 | 1.45 | 9.76 | 11.20 | 1.15 |

0.21 | 631.5 | 762.5 | 131 | 2.79 | 13.44 | 16.22 | 1.21 |

0.26 | 906.5 | 958.5 | 52 | 1.11 | 19.29 | 20.39 | 1.06 |

0.31 | 1038.5 | 1132.5 | 94 | 2.00 | 22.10 | 24.10 | 1.09 |

0.35 | 1183.5 | 1258.5 | 75 | 1.60 | 25.18 | 26.78 | 1.06 |

0.4 | 1284.5 | 1437.5 | 153 | 3.26 | 27.33 | 30.59 | 1.12 |

0.46 | 1378.5 | 1570.5 | 192 | 4.09 | 29.33 | 33.41 | 1.14 |

Mean value (cm) | 86.69 | 1.84 | Mean value | 1.15 |

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## Share and Cite

**MDPI and ACS Style**

Lee, W.-D.; Hwang, T.; Kim, T. Inundation Characteristics of Solitary Waves According to Revetment Type. *Water* **2022**, *14*, 3814.
https://doi.org/10.3390/w14233814

**AMA Style**

Lee W-D, Hwang T, Kim T. Inundation Characteristics of Solitary Waves According to Revetment Type. *Water*. 2022; 14(23):3814.
https://doi.org/10.3390/w14233814

**Chicago/Turabian Style**

Lee, Woo-Dong, Taegeon Hwang, and Taeyoon Kim. 2022. "Inundation Characteristics of Solitary Waves According to Revetment Type" *Water* 14, no. 23: 3814.
https://doi.org/10.3390/w14233814