# Numerical Simulation of Scour Depth and Scour Patterns in Front of Vertical-Wall Breakwaters Using OpenFOAM

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Hydrodynamic Model on OpenFOAM Platform

#### 2.1.1. Continuity and RANS Equations

^{2}between air and water at 20 °C, while ${\kappa}_{\gamma}$ is the surface curvature and $\mathsf{\gamma}$ is a quantity related to the Volume of Fluid method (further information is provided in Section 2.1.2).

#### 2.1.2. Volume of Fluid (VOF) Equations

#### 2.1.3. Turbulence Modelling

#### 2.1.4. Wave Generation and Absorption

#### 2.2. Morphodynamic Model

#### 2.2.1. Bed Load and Sheet Flow

_{50}the median grain size, while ${a}_{w}$, ${a}_{n}$ and $b$ are coefficients given by Camenen and Larson [17] and ${\theta}_{cw,net}$ is calculated by the equation:

_{wc}is the time period (part of the wave period) when the velocities are positive and corresponds to wave crest, while T

_{wt}is the time period (part of the wave period) when the velocities are negative and corresponds to wave trough (T

_{w}= T

_{wc}+ T

_{wt}, in which T

_{w}is the wave period).

_{w,crsf}is the critical velocity for the initiation of the sheet flow, U

_{w}is the wave orbital velocity and w

_{s}the sediment fall velocity.

_{cw}the total wave and current velocity, f

_{cw}the friction coefficient for the wave and current interaction. There is no current in this work, so the friction coefficient f

_{cw}is related to the waves $\left({f}_{w}\right)$ only and is equal to [1]:

_{50}[27], while ${a}_{0}\text{}=\text{}{U}_{w,max}$ × T/2π is the amplitude of the orbital bottom velocity ${U}_{w,max}$.

#### 2.2.2. Suspended Load

#### 2.2.3. Conservation of Sediment Transport

_{b}is the local seabed elevation (for each cell) and q

_{x,t}(= q

_{s,x}+ q

_{b,x}), q

_{y,t}(= q

_{s,y}+ q

_{b,y}) are the total sediment transport (bedload plus suspended load) rates in x and y horizontal directions respectively. The coefficient ${\epsilon}_{w}$ is the slope coefficient (as mentioned above) according to Watanabe [26], which for the cases of flat seabed is equal to zero.

#### 2.3. Coupling of the Two Models—Iterative Method

## 3. Results

#### 3.1. Geometry of the Model—Mesh Generation

#### 3.2. Wave and Sediment Characteristics

_{50}is the median grain size, w

_{f}is the sediment fall velocity (taken from [1]), H is the wave height, T is the wave period and L is the wave length. Fifth-order Stokes regular waves were implemented for all 3 cases. The last column in Table 1 shows the classification of each case according to Xie [1] and the case numbering refers to Xie’s [1] respective experiments.

#### 3.3. Hydrodynamic Results—Standing Wave Formation

#### 3.4. Morphodynamic Results—Scouring Depth and Scouring Patterns

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Surface Elevation at 2 different locations. (

**a**) 13.8 m from the breakwater, just before the slope, where the transition from incident to standing wave is shown. (

**b**) 1 m from the breakwater where fully standing wave formation is depicted.

**Figure 5.**Comparison between wave heights at location 1 m away from the breakwater for different cell sizes (different dx).

**Figure 6.**Comparison between wave heights at location 1 m away from the breakwater for different divergence numerical scheme for the convection terms of RANS equations.

**Figure 7.**Instantaneous Surface Elevation at the position of 1 m before the breakwater. Comparison between model results and experimental data.

**Figure 8.**(

**a**) Surface Elevation and particle velocity distribution at 46.2 s (model results for t = 35 *T). (

**b**) Surface Elevation and particle velocity vectors at 46.53 s (model results for t = 35T + T/2). (

**c**) Standing wave envelope and particle movement (model results).

**Figure 9.**Orbital velocity at the position of the node (35.4 m from the inlet of the flume) at 5 cm above the seabed. Comparison between numerical results and analytical solution (Miche second-order theory).

**Figure 10.**Orbital velocity profiles (

**a**) at the position of the node (35.4 m from the inlet of the flume), and (

**b**) L/8 from the position of the node (35.15 m from the inlet of the flume). Comparison between numerical results and Miche second-order analytical solution.

**Figure 12.**Final numerical seabed profile and standing wave. Comparison with experimental data (test 2a).

**Figure 13.**Seabed profiles and scouring depth—comparison between model results and experimental data/theory (test 2a).

**Figure 15.**Final numerical seabed profile and standing wave—Comparison with experimental data (test 7a).

**Figure 16.**Seabed profiles and scouring depth-comparison between model results and experimental data/theory (test 7a).

**Figure 18.**Final numerical seabed profile and standing wave—Comparison with experimental data (test 23a).

**Figure 19.**Seabed profiles and scouring depth—Comparison between model results and experimental data (test 23a).

Case | D_{50} | w_{f} | H | T | L | H/L | H/gT^{2} | d/L | H/D_{50} | Group |
---|---|---|---|---|---|---|---|---|---|---|

2a | 106 μm | 0.7 cm/s | 7.5 cm | 1.32 s | 2.0 m | 0.0375 | 0.0044 | 0.15 | 707.5 | Fine |

7a | 106 μm | 0.7 cm/s | 5 cm | 2.41 s | 4.0 m | 0.0125 | 0.0009 | 0.075 | 471.7 | Fine |

23a | 780 μm | 11 cm/s | 6.5 cm | 1.53 s | 2.4 m | 0.0271 | 0.0028 | 0.125 | 83.3 | Coarse |

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**MDPI and ACS Style**

Karagiannis, N.; Karambas, T.; Koutitas, C.
Numerical Simulation of Scour Depth and Scour Patterns in Front of Vertical-Wall Breakwaters Using OpenFOAM. *J. Mar. Sci. Eng.* **2020**, *8*, 836.
https://doi.org/10.3390/jmse8110836

**AMA Style**

Karagiannis N, Karambas T, Koutitas C.
Numerical Simulation of Scour Depth and Scour Patterns in Front of Vertical-Wall Breakwaters Using OpenFOAM. *Journal of Marine Science and Engineering*. 2020; 8(11):836.
https://doi.org/10.3390/jmse8110836

**Chicago/Turabian Style**

Karagiannis, Nikolaos, Theophanis Karambas, and Christopher Koutitas.
2020. "Numerical Simulation of Scour Depth and Scour Patterns in Front of Vertical-Wall Breakwaters Using OpenFOAM" *Journal of Marine Science and Engineering* 8, no. 11: 836.
https://doi.org/10.3390/jmse8110836