# Unstructured Finite-Volume Model of Sediment Scouring Due to Wave Impact on Vertical Seawalls

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## Abstract

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## 1. Introduction

## 2. Laboratory Experiments

## 3. Mathematical Model

#### 3.1. Hydrodynamic Model

#### 3.2. Morphodynamic Model

#### 3.3. Sand-Slide Model

#### 3.4. Equations in the $\sigma $-Coordinate System

## 4. Numerical Method

#### 4.1. Projection Method

#### 4.2. Time Integration

#### 4.3. Unstructured Finite Volume Discretization

## 5. Numerical Results

#### 5.1. Numerical Setup

#### 5.2. Grid Convergence Study

#### 5.3. Wave Elevation

#### 5.4. Velocity Field

#### 5.5. Seawall Scour

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Water/sediment bed interface monitored by the echo-Doppler a high-resolution echo-Doppler imaging device at the Laboratory for Hydraulic and Environment (LHE) of the Institut National de la Recherche Scientifique (INRS) in Québec, Canada. (

**b**) Results of the water/sediment bed interface after some minutes.

**Figure 2.**The schematic of the wave flume used for the experimental and numerical modelling of sediment scouring due to non-breaking waves on a vertical wall. The figure is not to scale.

**Figure 3.**Physical domain and parameters used in $\sigma $-transformation with free surface and bed evolution in the Cartesian coordinate system.

**Figure 4.**Schematic plot of the control volume prism ${V}_{e}$ formed by an unstructured triangular grid in the horizontal and several layers in the vertical direction of the system $({x}^{*},{y}^{*},\sigma )$. The five faces are denoted as ${S}_{j}$ ($j=1,\cdots ,5$).

**Figure 5.**Mesh example applied in the horizontal and vertical direction of the computational domain. The bottom figures corresponds to the 2D view of the selected x-y and x-z regions of the entire domain.

**Figure 6.**Instantaneous velocity field and magnitude of the simulated wave during its impact on a vertical seawall for Test B.

**Figure 8.**Experimental and numerical results of the water elevation for Test A using Mesh 2 ($\Delta x=2.5$ cm and $\Delta z=0.2$ cm).

**Figure 9.**Experimental and numerical results of the water elevation for Test B using Mesh 2 ($\Delta x=2.5$ cm and $\Delta z=0.3$ cm).

**Figure 11.**Numerical results of the instantaneous velocity field for Test A and B. The box corresponds to the PIV measurement zone. The 4 cm and 6 cm legends mean the still water levels at the seawall.

**Figure 12.**Experimental results of the bathymetry evolution for Test A and B. The dash line refers to the division between the rigid and mobile bed area.

Name | ${\mathit{d}}_{50}$ ($\mathsf{\mu}$m) | ${\mathcal{H}}_{0}$ (cm) | T (s) | L (m) | h (cm) | ${\mathit{h}}_{\mathit{wall}}$ (cm) | ${\mathit{\beta}}_{1}$ | ${\mathit{\beta}}_{2}$ | ${\mathit{x}}_{\mathit{D}}$ (cm) |
---|---|---|---|---|---|---|---|---|---|

Test A | 216 | 1.7 | 2 | 2.209 | 13 | 4 | 1/10 | 1/25 | 100 |

Test B | 700 | 1.6 | 3 | 2.365 | 15 | 6 | 1/10 | 1/25 | 100 |

Sub-Divisions | $\mathit{\Delta}\mathit{x}$ | ${\mathit{N}}_{\mathit{z}}$ | Vertices | Cells | Prisms | |
---|---|---|---|---|---|---|

Mesh 1 | $140\times 6$ | 0.05 m | 20 | 1827 | 3360 | 62,200 |

Mesh 2 | $280\times 12$ | 0.025 m | 20 | 7013 | 13,440 | 268,800 |

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**MDPI and ACS Style**

Uh Zapata, M.; Pham Van Bang, D.; Nguyen, K.D. Unstructured Finite-Volume Model of Sediment Scouring Due to Wave Impact on Vertical Seawalls. *J. Mar. Sci. Eng.* **2021**, *9*, 1440.
https://doi.org/10.3390/jmse9121440

**AMA Style**

Uh Zapata M, Pham Van Bang D, Nguyen KD. Unstructured Finite-Volume Model of Sediment Scouring Due to Wave Impact on Vertical Seawalls. *Journal of Marine Science and Engineering*. 2021; 9(12):1440.
https://doi.org/10.3390/jmse9121440

**Chicago/Turabian Style**

Uh Zapata, Miguel, Damien Pham Van Bang, and Kim Dan Nguyen. 2021. "Unstructured Finite-Volume Model of Sediment Scouring Due to Wave Impact on Vertical Seawalls" *Journal of Marine Science and Engineering* 9, no. 12: 1440.
https://doi.org/10.3390/jmse9121440