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

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 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

- Hugues, S.A. Physical models and laboratory techniques in coastal engineering, In Particular Section 5.4: Vertical Wall; World Scientific: Singapore, 1993; pp. 216–222. [Google Scholar]
- Hattori, M.; Kawamata, R. Experiments on restoration of beaches backed by seawalls. Coast. Eng. Jpn.
**1977**, 20, 55–68. [Google Scholar] [CrossRef] - Sutherland, J.; Obhrai, C.; Whitehouse, R.J.S.; Pearce, A. Laboratory tests of scour at a seawall. In Proceedings of the 3rd International Conference on Scour and Erosion (ICSE), Amsterdam, The Netherlands, 1–3 November 2006. [Google Scholar]
- El-Bisy, M.S. Bed changes at toe of inclined seawalls. Ocean. Eng.
**2007**, 34, 510–517. [Google Scholar] [CrossRef] - Pearson, J.M. Overtopping and Toe Scour at Vertical Seawalls, Coast, Marine Structures and Breakwaters; ICE: Atlanta, GA, USA, 2010; ISBN 978-0-7277-4131-8. [Google Scholar]
- Saitoh, T.; Kobayashi, N. Wave transformation and cross-shore sediment transport on sloping beach in front of vertical wall. J. Coast. Res.
**2012**, 128, 354–359. [Google Scholar] [CrossRef] - Bang, D.P.V.; Marois, L.; Roches, M.D.; Daigle, L.F.; Letellier, P. Interactions entre un Mur de Protection Cotiere et le Transport sédimentaire: Affouillements Locaux au Pied et Abaissement Global de la Plage (Progress Report No. 3, Contract R829.1, Min. Transport; Institut National de la Recherche Scientifique: Quebec City, QC, Canada, 2020. [Google Scholar]
- Marois, L.; Stolle, J.; Bang, D.P.V. Processus d’affouillement au pied d’un mur vertical de protection cotiere. J. Natl. Génie Côtier
**2020**, 259–266. [Google Scholar] [CrossRef] - Dally, W.R.; Dean, R.G. Suspended sediment transport and beach profile evolution. J. Waterw. Port Coast. Ocean. Eng.
**1984**, 110, 15–33. [Google Scholar] [CrossRef] - Roelvink, J.A.; Stive, M.J.F. Bar-generating cross-shore flow mechanisms on a beach. J. Geophys. Res. Ocean.
**1989**, 94, 4785–4800. [Google Scholar] [CrossRef] - McDougal, W.G.; Kraus, N.C.; Ajiwibowo, H. The effects of seawalls on the beach: Part II, numerical modeling of SUPERTANK seawall tests. J. Coast. Res.
**1996**, 12, 702–713. [Google Scholar] - Gislason, K.; Fredsøe, J.; Sumer, B.M. Flow under standing waves. Part 2. Scour and deposition in front of breakwaters. Coast. Eng.
**2009**, 56, 363–370. [Google Scholar] [CrossRef] - Myrhaug, D.; Ong, M.C. Random wave-induced scour at the trunk section of a breakwater. Coast. Eng.
**2009**, 56, 688–692. [Google Scholar] [CrossRef] - Hajivalie, F.; Yeganeh-Bakhtiary, A.; Houshanghi, H.; Gotoh, H. Euler-Lagrange model for scour in front of vertical breakwater. Appl. Ocean. Res.
**2012**, 34, 96–106. [Google Scholar] [CrossRef] - Zou, Q.; Peng, Z.; Lin, P. Effects of wave breaking and beach slope on toe scour in front of a vertical seawall. Coast. Eng. Proc.
**2012**, 1, 122. [Google Scholar] [CrossRef] - Ahmad, N.; Bihs, H.; Chella, M.A.; Arntsen, A. CFD modelling of Arctic coastal erosion due to breaking waves. Int. J. Offshore Polar Eng.
**2019**, 29, 33–41. [Google Scholar] [CrossRef] [Green Version] - Ahmad, N.; Bihs, H.; Myrhaug, D.; Kamath, A.; Arntsen, A. Numerical modeling of breaking wave induced seawall scour. Coast. Eng.
**2019**, 150, 108–120. [Google Scholar] [CrossRef] - Zapata, M.U.; Zhang, W.; Bang, D.P.V.; Nguyen, K.D. A parallel second-order unstructured finite volume method for 3D free-surface flows using a σ-coordinate. Comput. Fluids
**2019**, 190, 15–29. [Google Scholar] [CrossRef] - Zhang, W.; Zapata, M.U.; Bai, X.; Pham-Van-Bang, D.; Nguyen, K.D. Three-dimensional simulation of horseshoe vortex and local scour around a vertical cylinder using an unstructured finite-volume technique. Int. J. Sediment Res.
**2020**, 35, 295–306. [Google Scholar] [CrossRef] - Phillips, N.A. A coordinate system having some special advantages for numerical forecasting. J. Meteor.
**1957**, 14, 184–185. [Google Scholar] [CrossRef] [Green Version] - Chen, C.; Liu, H. An unstructured Grid, Finite Volume, Three-Dimensional, Primitive Equations Ocean Model: Application to Coastal Ocean and Estuaries. J. Atmos. Ocean. Technol.
**2003**, 20, 159–186. [Google Scholar] [CrossRef] - Liu, X.; Mohammadian, A.; Sedano, J.A.I. Three-dimensional modeling of non-hydrostatic free-surface flows on unstructured grids. Int. J. Numer. Meth. Fluids
**2016**, 82, 130–147. [Google Scholar] [CrossRef] - van Rijn, L.C. Sediment transport, Part I: Bed load transport. J. Hydraul. Eng.
**1984**, 110, 1431–1456. [Google Scholar] [CrossRef] [Green Version] - Chorin, A.J. Numerical solution of the Navier—Stokes equations. Math. Comp.
**1968**, 22, 745–762. [Google Scholar] [CrossRef] - Temam, R. Sur l’approximation des équations de Navier-Stokes par la méthode des pas fractionnaires (II). Arch. Ration. Mech. Anal.
**1967**, 26, 367–380. [Google Scholar] [CrossRef] - Rhie, C.M.; Chow, W.L. Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J.
**1983**, 21, 1525–1532. [Google Scholar] [CrossRef] - Grasso, F.; Michallet, H.; Barthélemy, E.; Certain, R. Physical modeling of intermediate cross-shore beach morphology: Transients and equilibrium states. J. Geophys. Res. Ocean.
**2009**, 114, C9. [Google Scholar] [CrossRef] [Green Version] - Nikuradse, J. Laws of Flow in Rough Pipes; VDI Forschungsheft: Washington, DC, USA, 1933. [Google Scholar]
- Roulund, A.; Sumer, B.M.; Fredsøe, J.; Michelsen, J. Numerical and experimental investigation of flow and scour around a circular pile. J. Fluid Mech.
**2005**, 534, 351–401. [Google Scholar] [CrossRef] - Khosronejad, A.; Kang, S.; Borazjani, I.; Sotiropoulos, F. Curvilinear immersed boundary method for simulating coupled flow and bed morphodynamic interactions due to sediment transport phenomena. Adv. Water Resour.
**2011**, 34, 829–843. [Google Scholar] [CrossRef] - Khosronejad, A.; Kang, S.; Sotiropoulos, F. Experimental and computational investigation of local scour around bridge piers. Adv. Water Resour.
**2012**, 37, 73–85. [Google Scholar] [CrossRef] - Lin, P.; Li, C.W. A σ-coordinate three-dimensional numerical model for surface wave propagation. Int. J. Numer. Methods Fluid
**2002**, 38, 1045–1068. [Google Scholar] [CrossRef] - Kim, D.; Choi, H. A second-order time-accurate finite volume method for unsteady incompressible flow on hybrid unstructured grids. J. Comput. Phys.
**2000**, 162, 411–428. [Google Scholar] [CrossRef] - Barth, T.; Jespersen, D.C. The design and application of upwind schemes on unstructured meshes. In Proceedings of the 27th Aerospace Sciences Meeting, Reno, NV, USA, 9–12 January 1989; p. 366. [Google Scholar]
- Vidović, D.; Segal, A.; Wesseling, P. A superlinearly convergent Mach-uniform finite volume method for the Euler equations on staggered unstructured grids. J. Comput. Phys.
**2006**, 217, 277–294. [Google Scholar] [CrossRef] - Zapata, M.U.; Bang, D.P.V.; Nguyen, K.D. An unstructured finite volume technique for the 3D Poisson equation on arbitrary geometry using a σ-coordinate system. Int. J. Numer. Meth. Fluids
**2014**, 76, 611–631. [Google Scholar] [CrossRef] - Zhang, W.; Zapata, M.U.; Bai, X.; Bang, D.P.V.; Nguyen, K.D. An unstructured finite volume method based on the projection method combined momentum interpolation with a central scheme for three-dimensional nonhydrostatic turbulent flows. Eur. J. Mech.-B/Fluids
**2020**, 84, 164–185. [Google Scholar] [CrossRef] - Oumeraci, H.; Klammer, P.; Partenscky, H.W. Classification of breaking wave loads on vertical structures. J. Waterw. Port Coast. Ocean Eng.
**1993**, 119, 381–397. [Google Scholar] [CrossRef] - Hasselmann, K.; Munk, W.; MacDonald, G. Bispectrum of Ocean Waves; Rosenblatt, M., Ed.; Time Series Analysis; JohnWiley: New York, NY, USA, 1963. [Google Scholar]
- Bertin, X.; de Bakker, A.; van Dongeren, A.; Coco, G.; André, G.; Ardhuin, F.; Bonneton, P.; Bouchette, F.; Castelle, B.; Crawford, W.C.; et al. Infragravity waves: From driving mechanisms to impacts. Earth Sci. Rev.
**2018**, 177, 774–799. [Google Scholar] [CrossRef] [Green Version] - Lin, C.-Y.; Huang, C.-J. Decomposition of incident and reflected higher harmonic waves using four wave gauges. Coast. Eng.
**2004**, 51, 395–406. [Google Scholar] [CrossRef] - Amoudry, L.O.; Liu, P.L.-F. Two-dimensional, two-phase granular sediment transport model with applications to scouring downstream of an apron. Coast. Eng.
**2009**, 56, 693–702. [Google Scholar] [CrossRef]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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