# An Integrated Numerical Model for the Design of Coastal Protection Structures

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## Abstract

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## 1. Introduction

## 2. Model Description

#### 2.1. Nearshore Wave Transformation Module–WAVE_L

**U**is the mean velocity vector

_{w}**U**= (U

_{w}_{w}, V

_{w}); d is the depth,

**Q**=

_{w}**U**h

_{w}_{w}= (Q

_{w}, P

_{w}); h

_{w}is the total depth (h

_{w}= d + η); c is the celerity; and c

_{g}is the group velocity (c

_{g}= (gd)

^{0.5}). The term ν

_{h}is a horizontal eddy viscosity coefficient introduced in order to include breaking effects based on the formulation of [25]:

_{m}is the maximum wave height; ρ is the water density; f is the wave frequency; and Q

_{b}is the probability of a wave breaking at a certain depth, expressed as (1 − Q

_{b})/(lnQ

_{b}) = (H

_{rms}/H

_{m})

^{2}according to [26]. The mean square wave height H

_{rms}is calculated from H

_{rms}= 2(<2η

^{2}>)

^{1/2}, with the brackets denoting a time-mean quantity. It should be noted that—since linear wave models are not capable of describing waves in the swash zone—in WAVE_L, the water depth from the rundown point (i.e., depth equal to R/4; R is the runup height) and up to the runup point (i.e., depth equal to −R) is considered to be constant and equal to R/4.

- A sponge layer boundary condition is used to absorb the outgoing waves at the four sides of the domain [27].
- The presence of vertical structures is incorporated by introducing a total reflection boundary condition (
**U**= (U_{w}_{w}. V_{w}) = 0 normal to the boundary, where U_{w}is the mean velocity vector; for a rectilinear grid, the above is equivalent to U_{w}= 0 or V_{w}= 0). - The presence of submerged structures is incorporated as in [31].
- The presence of floating structures is incorporated as in [32].

#### 2.2. Wave-Induced Circulation Module—WICIR

_{xx}, S

_{yy}, and S

_{xy}are the radiation stresses; h = d + ζ (ζ being the mean water elevation); U and V are the depth-averaged current velocities; and τ

_{bx}and τ

_{by}are the bottom shear stresses. Based on linear wave theory, Copeland [33] derived the equations for radiation stresses (S

_{ij}) without the typical assumption of progressive waves, expressed as:

**τ**= (τ

_{b}_{bx}, τ

_{by}) in WICIR are calculated based on the formulae proposed by Kobayashi et al. [34]:

_{b}is the bottom friction factor, σ

_{T}is the standard deviation of the oscillatory horizontal velocity, and |U| = (U

^{2}+ V

^{2})

^{0.5}.

_{u}is the undertow velocity in the direction normal to the shore, ξ = z/(h − ζ

_{t}), h = d + ζ (ζ being the mean water elevation), ζ

_{t}is the wave trough level, dR/dy = 0.14ρgdh/dy, τ

_{s}is the shear stress at the wave trough level, M is the wave mass flux above trough level (including surface roller effects), Θ is the direction of wave propagation (Θ = arctan[(<Q

_{w}

^{2}>/<P

_{w}

^{2}>)

^{1/2}]), and ν

_{τ}is the eddy viscosity coefficient according to De Vriend and Stive [36]:

_{R}at the shoreline is determined according to Baba and Camenen [37] as:

_{0}ξ

_{0}, where H

_{0}is the deep water wave height and ξ

_{0}is the Iribarren number) and Θ is the wave direction near the rundown point at depth d = R/4. The longshore velocity V

_{R}is presumed constant within the swash zone, the width of which is considered as extending from d = R/4 (i.e., the rundown point) to d = −R. The above velocity is indirectly introduced in the model by increasing the radiation stresses in the swash zone, based on the rationale described in the following. Longshore velocity can be expressed analytically by:

_{b}and α

_{b}are the water depth and incident wave angle at the breaking point, respectively. Assuming a linear variation of $\overline{U}$, the velocity at the shoreline can be approximated as:

_{s}is the water depth at the shoreline. A comparison of Equation (20) to Equation (22) shows that the square of the ratio does not deviate significantly from an empirical factor, a

_{s}, expressed as:

_{0}is the Iribarren number, and H

_{0}and L

_{0}are the wave height and wavelength, respectively, for deep water conditions. Accordingly, the aforementioned increase in radiation stresses in the swash zone is achieved by multiplying them by the factor a

_{s}.

- if (d + ζ)
_{i,j}> h_{cr}and (d + ζ)_{i}_{− 1,j}≤ h_{cr}and U_{i,j}> 0 → U_{i,j}= 0 - if (d + ζ)
_{i},_{j}> h_{cr}and (d + ζ)_{i,j}_{ − 1}≤ h_{cr}and V_{i,j}> 0 → V_{i,j}= 0 - if (d + ζ)
_{i,j}≤ h_{cr}and (d + ζ)_{i}_{− 1,j}≤ h_{cr}→ U_{i,j}= 0 - if (d + ζ)
_{i,j}≤ h_{cr}and (d + ζ)_{i,j}_{− 1}≤ h_{cr}→ V_{i,j}= 0 - if (d + ζ)
_{i,j}≤ h_{cr}and (d + ζ)_{i}_{− 1,j}> h_{cr}and U_{i,j}< 0 → U_{i,j}= 0 - if (d + ζ)
_{i,j}≤ h_{cr}and (d + ζ)_{i,j}_{− 1}> h_{cr}and V_{i,j}< 0 → V_{i,j}= 0

_{cr}is a terminal depth below which drying is assumed to occur (e.g., in WICIR this depth is set to h

_{cr}= 0.001 m).

#### 2.3. Sediment Transport Module—SEDTR

_{b}) is estimated with a quasi-steady, semi-empirical formulation, developed by Camenen, and Larson [19,20] for an oscillatory flow combined with a superimposed current under an arbitrary angle:

_{s}/ρ) is the relative density between the sediment (ρ

_{s}) and water (ρ); g is the acceleration due to gravity; d

_{50}is the median grain size; a

_{w}, a

_{n}, and b are empirical coefficients; θ

_{cw}

_{,m}and θ

_{cw}are the mean and maximum Shields parameters due to the wave-current interaction, respectively; θ

_{cn}is the current-related Shields parameter in the direction normal to the wave direction, and θ

_{cr}is the critical Shields parameter for the inception of transport. The net Shields parameter θ

_{cw}

_{,net}in Equation (24) is given by:

_{cw,on}and θ

_{cw,off}are the mean values of the instantaneous Shields parameter over the two half “periods” T

_{wc}(crest-onshore) and T

_{wt}(trough-offshore), α

_{pl,b}is a coefficient for the phase-lag effects [19], and α

_{α}is a coefficient for the acceleration effects [39]. The Shields parameter θ

_{cw}is defined by:

_{cw}being the wave and current velocity and f

_{cw}the friction coefficient taking into account the wave and current interaction, while the subscript j should be replaced either by onshore or offshore. In the above formulation (since linear wave theory cannot be used), the estimation of nonlinear time-varying near-bottom wave velocities is also needed. For the incorporation of nonlinear velocity characteristics (i.e., skewness and asymmetry) in SEDTR, the parameterisation proposed by Isobe and Horikawa [40] is adopted.

_{R}is the reference concentration at the bottom [19], w

_{s}is the sediment fall velocity, and β

_{d}is a coefficient calculated based on [20] by:

## 3. Model Applications

#### 3.1. Comparison with Experimental Data

_{50}= 0.25 mm. The duration of the tests was approximately 15 h (which was the duration needed for the beach to reach an equilibrium state). Three different cases were reproduced numerically and are presented in the following, for normally incident waves of H

_{0}= 0.05 m deep water wave height and T = 0.85 s wave period. The test cases, presented in Table 1, differed in breakwater length (B) and breakwater distance from the initial shoreline (X), in order to cover a wide range of B/X ratios resulting in both tombolo and salient formation behind the breakwaters.

_{50}= 0.12 mm sand grains. The beach—without the presence of the groins—was exposed to wave action for a duration of 4 h until the formation of a nearly stable bathymetry (clear offshore bar trough/step formation). The installation of the groins followed, and the tests continued thereafter in 2 h cycles. In this work, the case of a single groin was modelled, exposed to waves of H

_{s}

_{0}= 0.08 m deep water significant wave height, T

_{p}= 1.15 s peak period, and θ

_{0}= 11.6° deep water incident wave angle, for a total duration of 12 h after groin installation (Test NT2, NRCC test series).

#### 3.2. Application to Paralia Katerinis Beach (Greece)—Coastal Protection with Submerged Breakwaters

_{t}≈ 0.4). In addition, a beach nourishment project was also designed and applied to restore the beach to its previous condition. Bathymetry measurement data for the area are available for the period right after the completion of coastal works (beach nourishment and construction of the submerged breakwaters), as well as for three years later [46].

## 4. Results and Discussion

#### 4.1. Comparison with Experimental Data

#### 4.2. Application to Paralia Katerinis Beach (Greece)—Coastal Protection with Submerged Breakwaters

#### 4.3. General Discussion

_{s}(see Equation (23)) and the region of its application (i.e., the swash zone as defined in Section 2.2); (b) the use of the bottom friction formulae proposed by Kobayashi et al. [34]; and (c) the use of the Camenen and Larson [19,20] formulae for the calculation of bed and suspended load (instead of other approaches).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References and Note

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**Figure 1.**Location and satellite image of the study area at Paralia Katerinis, Greece ([47]; privately processed).

**Figure 2.**(

**a**) Initial wave-induced current velocity field (contours represent the initial bathymetry) and (

**b**) comparison between the computed and measured shoreline evolution (contours represent the final computed bathymetry) for Test 11 of Ming and Chiew [21].

**Figure 3.**Comparison between the computed and measured shorelines evolution (contours represent the final computed bathymetry) for: (

**a**) Test 3 and (

**b**) Test 10 of Ming and Chiew [21].

**Figure 4.**(

**a**) Initial breaking wave-induced current velocity field (contours represent the initial bathymetry) and (

**b**) comparison between the computed and measured shoreline evolution (contours represent the final computed bathymetry) for Test NT2 of Baidei et al. [22].

**Figure 5.**Breaking wave-induced current field for the prevailing SE waves at Paralia Katerinis beach (Greece) after the construction of the detached submerged breakwaters.

**Figure 6.**(

**a**) Initial bed morphology of Paralia Katerinis beach, (

**b**) comparison between the computed and measured bed morphology evolution, and (

**c**) satellite image of the same area ([47]; privately processed).

**Table 1.**Test conditions for the numerically reproduced experiments of Ming and Chiew [21].

Test | B = Breakwater Length (m) | X = Distance from the Initial Shoreline (m) | B/X | Formation of Salient/Tombolo |
---|---|---|---|---|

3 | 1.5 | 0.6 | 2.50 | tombolo |

10 | 1.2 | 1.2 | 1.00 | salient |

11 | 1.5 | 1.2 | 1.25 | tombolo |

**Table 2.**Characteristics of the three representative waves used for the Paralia Katerinis beach runs.

Wave Direction | Significant Wave Height H_{s} (m) | Peak Wave Period T_{p} (s) | Annual Frequency of Occurence f (%) |
---|---|---|---|

SE-S | 1.34 | 5.4 | 12.70 |

E | 1.09 | 4.4 | 1.12 |

NE | 0.94 | 4.6 | 1.17 |

© 2017 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 (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Karambas, T.V.; Samaras, A.G.
An Integrated Numerical Model for the Design of Coastal Protection Structures. *J. Mar. Sci. Eng.* **2017**, *5*, 50.
https://doi.org/10.3390/jmse5040050

**AMA Style**

Karambas TV, Samaras AG.
An Integrated Numerical Model for the Design of Coastal Protection Structures. *Journal of Marine Science and Engineering*. 2017; 5(4):50.
https://doi.org/10.3390/jmse5040050

**Chicago/Turabian Style**

Karambas, Theophanis V., and Achilleas G. Samaras.
2017. "An Integrated Numerical Model for the Design of Coastal Protection Structures" *Journal of Marine Science and Engineering* 5, no. 4: 50.
https://doi.org/10.3390/jmse5040050