# Numerical Analysis of Wind Effect on Wave Overtopping on a Vertical Seawall

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- -
- to verify the ability of these numerical models to correctly reproduce the influence of wind on the overtopping rate, even though simplified wind modelling has been used;
- -
- to clarify how the wind acts and how its effect on the enhancement of the overtopping rate changes depending on the overtopping regime.

## 2. Literature

_{NO-WIND}) by a factor of 3.2.

_{NO-WIND}< 10

^{−2}m

^{3}/s/m, approximately. Furthermore, by gathering field and laboratory data the following formula was proposed, which relates the wind transport factor f

_{WIND}to the overtopping regime:

_{WIND}ranges between 1 (negligible effect of wind) and 4.

_{WIND}between 1.3 and 1.5.

## 3. Numerical Models

#### 3.1. FLOW-3D

_{t}is the eddy viscosity and g is the gravity acceleration. Turbulence in the flow is accounted for with the RNG k-ε model [28].

_{t}in RANS equations is determined through the RNG k-ε model, with the equations for the turbulent kinetic energy k and the specific turbulent dissipation ε, as follows:

^{n}is automatically adjusted to:

_{s}is a user defined sampling rate which depends on the frequency spectrum of the phenomenon under study, and ${\Delta \mathrm{t}}_{\mathrm{CON}}^{\mathrm{n}}$ is a convergence time step size that is needed to avoid numerical instabilities. Since the advective fluxes have been computed using a simple first order donor cell, ${\Delta \mathrm{t}}_{\mathrm{CON}}^{\mathrm{n}}$ is required to meet the following criterion:

_{CFL}is the time step to satisfy the Courant–Friedrichs–Lewy (CFL) stability criterion, and the second quantity at the right-hand side of Equation (7) ensures surface waves cannot propagate more than one cell in one time step (a

_{z}indicates vertical acceleration).

#### The CFD Wind Stress Model

_{10}at an elevation of 10 m above sea level, and the sea surface drag coefficient C

_{D10}, and reads:

- ρ
_{a}is the density of air (1.225 kg m^{−3}), - U
_{10}is the wind velocity at 10 m above the water surface, - C
_{D10}is the wind shear coefficient (or drag coefficient).

_{10}) and wind shear parameters (ρ

_{_a}; C

_{D10}).

#### 3.2. SWASH

_{s}and w

_{b}are the velocity in z-direction at the free surface and at the bottom respectively, ζ is the free-surface elevation from the still water level, d is the still water depth and h the total depth, p

_{b}is the non-hydrostatic pressure at the bottom, g is the gravitational acceleration and c

_{f}is the dimensionless bottom friction coefficient.

#### The SWASH Wind Stress Model

_{D}and wind velocity). Major details can be found in the SWASH user manual [32]. In this study, a constant drag coefficient value was adopted.

## 4. Experiments

#### 4.1. Geometry of the Structure and Foreshore

#### 4.2. Wind Model Implementation

_{D10}, reported in Table 2 and Table 3, was determined with the predictive equation suggested by [36]:

#### 4.3. CFD Numerical Setup

_{E}(also indicated as relative error between two consecutive grids), defined as follows:

_{f,finer}represents the final overtopping volume relative to the finer grid and V

_{f,coarser}is the analogous value for the coarser grid, which was used as reference value.

_{E}, on the ordinates. The plot indicates three main aspects: an area of increasing error as the grid decreases ($\sqrt{\Delta \mathrm{x}\Delta \mathrm{z}}$ between 1.8 and 1.2), a progressively decreasing phase of the error with reducing grid size ($\sqrt{\Delta \mathrm{x}\Delta \mathrm{z}}$ between 1.2 and 0.9), and, finally, a grid independence area ($\sqrt{\Delta \mathrm{x}\Delta \mathrm{z}}$ smaller than 0.9). The initial increasing trend could be interpreted as a consequence of the grid being too coarse to properly represent wave overtopping. Mesh refinement in the x direction is important for water flow even if refinement in the z direction is also important for capturing free surface position.

#### 4.4. SWASH Numerical Setup

_{c}= +3.96 m and +6.0 m. Numerical instabilities occurred with the other wave conditions listed in Table 1. Therefore, further tests with a monochromatic wave generation were added to investigate different magnitudes of overtopping discharge. The wave characteristics of these additional tests are listed in Table 6, where TEST_1S was performed with Rc = +6.0 m, while the other tests were performed with Rc = +3.96 m. The wind conditions examined are those listed in Table 3.

^{−1/3}s was used. The default values of breaking parameters [31] were adopted in the numerical simulations. The non-hydrostatic pressure term was applied with a Keller-box scheme. An adaptive time step was implemented to satisfy the Courant–Friedrichs–Lewy (CFL) condition, with a time step restriction set to a maximum Courant number of 0.5.

## 5. Results

#### 5.1. Physical Processes and Basic Definitions

_{D}is the domain’s length and h

_{0}denotes the offshore water depth.

_{xx}/dx, caused by variation in the wave profile characteristics. According to [48], the gradient is seawards for non-breaking waves, while it is directed towards the shore for breaking waves.

#### 5.2. On the Role of the Mean Water Level

_{D}= 410 m) is too small for a significant wind setup to occur. Using the maximum wind speed (U

_{10}= 31.2 m/s) and drag coefficient (C

_{D}= 0.0026) in Table 2, the parameter A equals 4.7 × 10

^{−4}, indicating that the role of shear forces can be assumed to be negligible.

^{2}

_{WIND}/σ

^{2}

_{NO-WIND}. The latter represents the change in wave energy caused by the wind and is hence proportional to the variation of the radiation stress.

^{2}

_{WIND}/σ

^{2}

_{NO-WIND}< 1 implies a reduction in dS

_{xx}/dx, which, being seawards, leads the mean water level to increase (less set-down).

^{2}

_{WIND}/σ

^{2}

_{NO-WIND}also changes in sign (Figure 5b); this is except for two outliers circled in the figure, for which $\overline{\eta}$ reduces due to a reforming/rebreaking process that occurs in the innermost part of the foreshore (not shown here for brevity’s sake).

_{c}= 2.23 m. More discussion on this point is given in Section 5.6.

#### 5.3. Wind Stress, Wave Profiles and Overtopping Rates

#### 5.4. Numerical Experiments vs. Pullen et al.’s Physical Model Data

_{NO-WIND}greater than 100 l/s/m).

_{WIND}indicated that the results were affected by the wind stress term, but not appropriately. SWASH appears unsuitable to reproduce the phenomenon dealt with in this study. This is probably due to the structure of SWASH, which is a depth-integrated model that cannot simulate the water separation phase in front of the wall. Therefore, it does not reproduce the increase in the overtopping rate due to the wind typical of the lower overtopping regime, where the spray blown over the wall is the predominant mechanism. This is clearly shown in the next section.

#### 5.5. The Wind Factor f_{WIND}

_{WIND}, i.e., the mean overtopping discharge with wind and that without wind ratio.

_{WIND}is equal to 1 when the discharge is greater than 10

^{2}l/s/m; then, a transition zone is discernible, where the wind has an unsystematic effect and can even reduce the overtopping discharge (between 10

^{1}and 10

^{2}l/s/m); and finally, in a zone with a lower overtopping regime, the data clearly demonstrated an increase in the overtopping rate due to the wind (f

_{WIND}> 1).

^{2}l/s/m, 10

^{0}l/s/m and 10

^{−1}l/s/m.

_{NO-WIND}> 100 l/s/m), there were no significant differences in the horizontal velocity and thickness of the overtopping layer when comparing wind and no wind conditions (Figure 14). The wind effect seems to be negligible as compared to the momentum of overtopping water, so that the mean overtopping discharge is not influenced by the wind. Thus, f

_{WIND}is almost 1.

^{0}l/s/m, the comparison in Figure 15 shows that the wind effect cannot be neglected. The wind stress lead to a greater height of the up-rushing jet, the difference of height being about 0.3 m (the water spray was not considered). Moreover, unlike the no wind condition, the fluid stream and the water spray were characterized by a landward velocity (Figure 15b), which ensured that the water crossed the wall instead of falling back into the numerical flume (Figure 15a). Therefore, the presence of wind increased the overtopping rate by inducing a greater height of the up-rushing jet and by transporting water spray over the seawall. The overtopping rates of the order of 10

^{0}l/s/m are thus characterized by a wind factor f

_{WIND}greater than 1.

^{−1}l/s/m, the main mechanism was the advection of the spray by the wind, i.e., “white water” overtopping. In fact, as shown in Figure 16, the overtopping occurred when the spray crossed the seawall. The water spray may be carried over the wall under its own momentum without wind, otherwise it falls back into the sea (Figure 16a). Onshore wind, on the other hand, significantly increased the amount of spray that crossed the wall (Figure 16b) and thus played a key role in the overtopping process. The difference between wind and no wind conditions was remarkable. Therefore, the wind factor reached its maximum value, as reported in Figure 17.

#### 5.6. Quantitative Analysis

_{NO-WIND}. Equation (1) was drawn as well.

^{−1}< q

_{NO-WIND}< 10

^{1}l/s/m, while they remain close to unity at higher rates.

_{WIND}compared to Equation (1); the wind factor increases rather fast and reaches a value in the order of 10 for q

_{NO-WIND}= 10

^{−1}l/s/m, while Equation (1) predicts 2. On the other hand, the overtopping process is unaffected by the presence of wind from q

_{NO-WIND}= O (10

^{2}), rather than O (10

^{1}).

_{c}), f

_{WIND}= 1.

_{WIND}that largely exceeded Equation (1), being significantly larger than unity also for values of q

_{NO-WIND}of the order of 10 l/s/m. Like in Figure 17, the maximum measured enhancement factor is of the order of 10.

_{deep}equals offshore H

_{m}

_{0}in the case of random waves.

## 6. Discussion and Conclusions

- -
- Deformation of the run-up wedge;
- -
- Advection of the droplets formed in the uprush phase;
- -
- Variation in the breaking point with wind.

- -
- the wind enhancement factor reaches the value of 10 rather than a maximum of 4;
- -
- The effect of wind keeps on resenting for overtopping regimes as intense as 100 l/s/m, rather than vanishing at 1 l/s/m.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Characteristics investigated of the Havana foreshore and Malecón seawall. Panel (

**a**) sketch of bathymetry; panel (

**b**) different crests freeboard of vertical wall.

**Figure 4.**Relative volume error as a function of the geometric mean of two dimensions of the cell. TEST_1 (H = 8 m T = 10 s).

**Figure 5.**Variation of $\overline{\eta}$ as a function of σ

^{2}

_{WIND}/σ

^{2}

_{NO-WIND}ratio. Panel (

**a**): results for non-breaking waves; panel (

**b**): results for breaking waves.

**Figure 6.**Up-rushing jet on wave overtopping for different simulation at the time 177.5 s, TEST_3 and Rc = +8.50 m. Panel (

**a**) no wind; panel (

**b**) wind velocity U

_{10}= 19 m/s.

**Figure 7.**Effects of wind stress on wave overtopping, TEST_3 and Rc +8.50 m. Panel (

**a**) wind effects on overtopping discharge; panel (

**b**) wind effects on cumulative wave overtopping volume.

**Figure 8.**CFD numerical results marked with different wind speeds; red data represent breaking waves, while grey data represent non-breaking waves.

**Figure 10.**Cumulative overtopping volumes and wave overtopping discharges for different wind velocity, TEST_1 and Rc = +10.00 m.

**Figure 11.**Breaking waves under wind speed 31 m/s. Panel (

**a**) TEST_1 and Rc = +11.00 m; panel (

**b**) TEST_4 and Rc = +11.00 m.

**Figure 12.**Numerical results vs HRW laboratory experiments of mean overtopping discharges [5] with and without wind.

**Figure 18.**CFD wind factors in function of the ratio between the variation in $\overline{\eta}$ due to the wind and the crest freeboard.

**Figure 19.**Comparison between numerical and literature [27] results.

ID | H (m) | T (s) | Wave Celerity c (m/s) | |
---|---|---|---|---|

Deep Water c_{0} | On 20.45 m Water Depth c | |||

TEST_1 | 8 | 10 | 15.61 | 12.21 |

TEST_2 | 8 | 12 | 18.74 | 12.81 |

TEST_3 | 1.5 | 10 | 15.61 | 12.21 |

TEST_4 | 5.4 | 10 | 15.61 | 12.21 |

U_{10} | C_{D10} | β |
---|---|---|

30 | 0.0025 | 0.62 |

18.7 | 0.0021 | 1.00 |

9.4 | 0.0011 | 1.99 |

6.1 | 0.0009 | 3.07 |

1.9 | 0.0010 | 9.86 |

U_{10} | C_{D10} | β |
---|---|---|

31.2 | 0.0026 | 0.50 |

19 | 0.0021 | 0.82 |

12.5 | 0.0015 | 1.25 |

5.2 | 0.0009 | 3.00 |

3.1 | 0.0009 | 5.04 |

Parameter | Setting |
---|---|

Fluid | Water (20 °C), incompressible |

Turbulence | RNG |

Pressure Solver | GMRES |

VOF advection | Split Lagrangian method (TruVof) |

Time step control | Automatic (stability and convergence) |

ID GRID | Δx (m) | Δz (m) |
---|---|---|

A20 | 2.9 | 1.2 |

A30 | 1.9 | 0.8 |

A40 | 1.5 | 0.6 |

A60 | 1.0 | 0.4 |

A80 | 0.7 | 0.3 |

A160 | 0.4 | 0.15 |

C278 | 0.25 | 0.30 |

M278 | 0.25 | 0.15 |

ID | H (m) | T (s) |
---|---|---|

TEST_1S | 0.95 | 10 |

TEST_2S | 0.8 | 10 |

TEST_3S | 0.7 | 10 |

TEST_4S | 0.6 | 10 |

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**MDPI and ACS Style**

Di Leo, A.; Dentale, F.; Buccino, M.; Tuozzo, S.; Pugliese Carratelli, E. Numerical Analysis of Wind Effect on Wave Overtopping on a Vertical Seawall. *Water* **2022**, *14*, 3891.
https://doi.org/10.3390/w14233891

**AMA Style**

Di Leo A, Dentale F, Buccino M, Tuozzo S, Pugliese Carratelli E. Numerical Analysis of Wind Effect on Wave Overtopping on a Vertical Seawall. *Water*. 2022; 14(23):3891.
https://doi.org/10.3390/w14233891

**Chicago/Turabian Style**

Di Leo, Angela, Fabio Dentale, Mariano Buccino, Sara Tuozzo, and Eugenio Pugliese Carratelli. 2022. "Numerical Analysis of Wind Effect on Wave Overtopping on a Vertical Seawall" *Water* 14, no. 23: 3891.
https://doi.org/10.3390/w14233891