# Wave and Hydrodynamic Processes in the Vicinity of a Rubble-Mound, Permeable, Zero-Freeboard Breakwater

^{*}

## Abstract

**:**

## 1. Introduction

_{r}. Specifically for ZFBs, the formula in [1] becomes:

_{f})) and b = 1.49(γ

_{f}− 0.38)

^{2}+ 0.86 are calibration parameters related to the roughness factor γ

_{f}, C < 1 is a reduction parameter necessary for LCBs, ξ

_{0}= tana(gT

^{2}/2πH

_{i})

^{1/2}is the breaker parameter, tana is the seaward slope of the ZFB, T is the wave period of the incident waves, and H

_{i}is the wave height of the incident waves at the seaward toe of the ZFB. In terms of wave transmission in the leeward region of LCBs, empirical formulas for the prediction of the transmission coefficient:

_{t}is the wave height at the leeward toe of the LCB, derived in [2,3,4,5]. Specifically for ZFBs with rock armor, the corresponding empirical formulas are:

_{S}is the water depth at the seaward toe of the ZFB, and λ

_{0}= gT

^{2}/2π is the deep-water wavelength. The empirical formulas in [2,3] were derived using experimental datasets of LCBs with impermeable core, while the empirical formulas in [4,5] used practically the same datasets as reported in [4]. In all cases of Equation (3), the important effect of ξ

_{0}and/or B on the wave transmission is noted.

_{q}is an adjustment factor that accounts for scale effect corrections. In terms of the wave setup, δ, at the leeward toe of rubble-mound, permeable LCBs, experimental data were presented in [7,8]. In both studies, several cases of ZFBs were included, and the following empirical formulas were derived:

_{50}is the mean diameter of the rubble rocks and λ

_{i}is the wavelength at the seaward toe of the ZFB. Semi-analytical models for the prediction of δ were also presented [9,10], but they refer to submerged barriers and are not considered here where the focus is on ZFBs.

_{t}, while increasing the permeability of the breakwater, i.e., increasing the porosity or the size of the rubble rocks, results in the increase of K

_{t}and the reduction of K

_{r}. The Navier-Stokes equations were solved numerically in [12] to simulate the interaction between a solitary wave and a permeable submerged bar. It was found that the increase of porosity from 0.4 to 0.52 results in the reduction of K

_{t}, while if the porosity is further increased up to 0.7, the transmission coefficient increases, indicating that an optimum porosity value seems to exist.

_{s}= 25 cm, B = 100 cm, while the median diameter, D

_{50}, of the rocks was 4.80 cm in the armor layer and 1.47 cm in the core. Several regular wave cases were examined, which for the ZFB case corresponded to the ranges 0.20 ≤ H

_{i}/d

_{s}≤ 0.78 and 9.38 ≤ λ

_{i}/d

_{s}≤ 19.63. It was observed that breaking waves collapsed on the seaward edge of the ZFB crest inducing a strong vortex cell in this zone, a strong mean shoreward current developed over the ZFB crest, a primary vortex cell was formed in the leeward region of the ZFB, a weak mean seaward current developed in the ZFB trunk, and a secondary vortex cell was formed near the seaward toe of the ZFB.

_{c}= 25 cm, d

_{s}= 30 cm resulting into crest elevation R

_{c}= −5 cm, and R

_{c}/H

_{i}= −0.52. They performed numerical simulations using the COBRAS model and found that the resulting K

_{r}does not depend on B, while the shear stress field attains its maximum values along the armor layer of the crest. Losada et al. [15] also studied numerically a submerged breakwater case at a prototype scale with tana = 2/3, d

_{S}= 5 m, B = 5 m, R

_{C}= −0.5 m. The trunk of the breakwater was homogeneous with D

_{50}= 1.44 m, while the incident wave parameters at the seaward toe of the breakwater were H

_{i}= 0.97 m (R

_{c}/H

_{i}= −0.52) and T = 6 s. It was found that the velocity of the flow over the crest was about one order of magnitude larger than the one in the permeable trunk of the breakwater.

_{c}= 1.59 m, B = 1.825 m, while the median diameter, D

_{50}, of the rocks was 10.82 cm in the armor layer and 3 ± 1 cm in the core. No ZFB case was considered; the only submerged case had R

_{c}= −0.13 m, H

_{i}= 0.286 cm (R

_{c}/H

_{i}= −0.45) and T = 2.18 s. The transmitted wave height in the leeward region of the LCBs was, in general, under-predicted by the numerical simulations in comparison to the experimental data; this discrepancy was attributed to several factors, including the inaccurate replication of the experimental LCB geometry in the numerical model.

## 2. Materials and Methods

#### 2.1. Formulation and Numerical Implementation

_{1}= x and x

_{2}= z are the streamwise and vertical coordinates, respectively, t is the time, u

_{i}is the velocity field, P is the total pressure, ρ is the normalized fluid density, μ is the normalized fluid dynamic viscosity, Fr is the Froude number, δ

_{ij}is the Kronecker’s delta, Re is the Reynolds number, τ

_{ij}are the modeled subgrid-scale (SGS) stresses, and c

_{A}is the added mass coefficient in the form [20]:

_{50}is the median diameter of the rocks forming the ZFB trunk, a

_{p}and β

_{p}are calibration constants, KC is the Keulegan-Carpenter number representing the ratio of the characteristic length scale of fluid particle motion to that of the porous media [21]:

_{i}is a term associated with the implementation of boundary conditions on solid surfaces using the Immersed Boundary (IB) method. The velocity components in Equations (6) and (7) are the resolved ones for the external flow, based on the LES approach in [18], and the spatially-averaged ones for the porous flow in the ZFB trunk, based on the model in [19].

_{s}= 0.1 is the model parameter, Δ = is the filter length scale of the grid, and |S| = (2S

_{ij}S

_{ij})

^{1/2}is the magnitude of the resolved-scale, and the strain-rate tensor is:

^{−3}and μ = 1.8 × 10

^{−2}in the air.

_{i}in Equation (7). This term is zero on all grid points except the so-called “forcing points,” which are the ones in the fluid phase that have at least one neighboring grid point in the solid phase. The value of f

_{i}on the forcing points is computed so that it enforces the non-slip boundary condition on the sea bed.

^{®}11.0 library.

#### 2.2. Wave Generation

_{R}∈ [0:1] is defined so that a

_{R}= 0 at the left boundary of the computational domain (l

_{R}= 1) and a

_{R}= 1 at the end of the relaxation zone (l

_{R}= 0).

#### 2.3. Validation

_{S}= 0.4 m before reaching the bar. The crest depth was equal to d

_{C}= 0.1 m. Using d

_{S}as the characteristic length and (g/d

_{S})

^{1/2}as the characteristic velocity scale, the Reynolds number was Re

_{d}= 800,000. For the simulations (Figure 1), to achieve both adequate resolution in the wall boundary layer and a reasonable computational cost, the experiment was reproduced with d

_{s}= 0.15 m using Froude scaling, leading to a Reynolds number Re

_{d}= 160,000, i.e., five times smaller than the one in the experiments. A grid independence study was performed to select the appropriate grid size that resolves all important flow scales, i.e., vortices generated during wave breaking and flow structure in the wave boundary layer. After trial-and-error, the selected computational grid had a uniform size of Δx/d

_{S}= 0.02 along x (460 grid points per wavelength of the incident wave), and a non-uniform size in z; Δz/d

_{S}= 0.005 in the water and increasing to Δz/d

_{S}= 0.01 in the air. The corresponding values in the water with respect to the Stokes length, δ, were Δx/δ = 4.5 and Δz/δ = 1.13, which were sufficient to get a good resolution with at least five grid points in the wave boundary layer over the bed.

^{−2}and 7 × 10

^{−4}, respectively:

_{S}above the SWL (after trial and error). The trapezoidal bar was impermeable; therefore, the no-slip condition on its surface was imposed using the IB method. Simulations were performed in the Greek supercomputer ARIS, deployed and operated by GRNET (Greek Research and Technology Network). ARIS consists of 532 computational nodes separated by four ‘islands’. We deployed only one of the ‘islands’: the thin nodes, which consist of 426 nodes and 8520 CPU cores. In this case, the simulation time was 1 wave period per 6 h on ARIS. Comparisons of the free-surface elevation between the numerical results and the experimental data in [27] at five locations (Figure 1) over the bar during the last five wave periods are presented in Figure 2. A good agreement is observed; the r.m.s. (root mean square) relative error between the numerical results and the experimental data at each location is reported in the caption of Figure 2.

_{F}= 0.4 m, which is followed by a beach of slope tanβ = 1/20 seawards of the breakwater. The depth at the breakwater toe was d

_{S}= 0.3 m, while the crest depth was d

_{C}= 0.05 m. As mentioned in the Introduction for these experiments, the trunk of the rubble-mound breakwater was permeable with a two-layer armor (D

_{50}= 4.8 cm and n = 0.53) and a core with smaller rocks (D

_{50}= 1.47 cm and n = 0.49). For the simulations (Figure 3), to achieve both adequate resolution in the wall boundary layer and a reasonable computational cost, the experiment was reproduced with d

_{S}= 0.12 m using Froude scaling, leading to a Reynolds number Re

_{d}= 130,000, i.e., six times smaller than the one in the experiments. A mesh independence study was also performed. In order for the mesh not to affect the solution, the computational grid was selected to have a uniform size of Δx/d

_{F}= 0.02 along x (355 grid points per wavelength of the incident wave), and a non-uniform size in z; Δz/d

_{F}= 0.005 in the water and increasing to Δz/d

_{F}= 0.01 in the air. The corresponding values in the water with respect to the Stokes length, δ, were Δx/δ = 4.2 and Δz/δ = 1.06, which were sufficient to get a good resolution with at least five grid points in the wave boundary layer over the bed. The CFL and VSL criteria were equal to 2 × 10

^{−2}and 7 × 10

^{−4}, respectively. In order for the flow field not to be affected by the height of the air layer in the computational domain, this was selected to be 1.5d

_{S}above the SWL (after trial and error). The calibration values α

_{p}= 1000 and β

_{p}= 1.1, according to [19,20], were used in Equation (7); the values β

_{p}= 0.8 in the armor layer and β

_{p}= 1.2 in the core were also used, according to [13], but with negligible differences in the results.

_{F}. The numerical model captures adequately the main characteristics of wave propagation over the submerged breakwater. More specifically, in the seaward region of the structure, the generation of a partially standing wave due to wave reflection is captured precisely by the model. In the vicinity of the crest and near the leeward region of the structure, the model captures adequately the wave dissipation due to wave breaking over the crest and due to filtration through the permeable trunk, while the wave height is under-predicted in the far leeward region. This last behavior is similar to the one observed in [16]. The maximum r.m.s. relative error of the wave height between the numerical results and the experimental data is reported in the caption of Figure 4.

## 3. Results

_{S}, 2d

_{S}, and 3d

_{S}, respectively, where d

_{S}is the constant water depth between the wavemaker and the seaward ZFB toe. Then, the effect of the incident wave period (cases 1 and 4 in Table 1) and the ZFB trunk permeability (cases 1 and 5 in Table 1) were also investigated for the ZFB case with B = d

_{S}.

_{S}was selected to be equal to 0.4 m, which corresponds to Reynolds number Re

_{d}= 800,000. The geometrical details and the incident wave parameters for all cases are summarized in Table 1. For all cases, a grid independence study was performed, and the selected computational grid had a uniform size of Δx/d

_{S}= 0.02 along x (300–450 grid points per wavelength of the incident waves), and a non-uniform size in z; Δz/d

_{S}= 0.0025 in the water and increasing to Δz/d

_{S}= 0.01 in the air. The corresponding values in the water with respect to the Stokes length, δ, were Δx/δ = 10.1 ÷ 12.1 and Δz/δ = 1.27 ÷ 1.51 according to the wave length, which were sufficient to get a good resolution with at least four grid points in the wave boundary layer over the bed. The small flow scales occurring during breaking at the ZFB seaward slope are modeled by the SGS eddy-viscosity model. In all simulations, the CFL and VSL criteria were equal to 2.0 × 10

^{−2}and 5 × 10

^{−4}, respectively. In order for the flow field not to be affected by the height of the air layer in the computational domain, this was selected to be 1.5d

_{S}above the SWL (after trial and error). The calibration values α

_{p}= 1000 and β

_{p}= 1.1, according to [19,20], were used in Equation (7).

_{0}. Finally, reducing trunk permeability (cases 1 and 5) also increased the wave reflection and decreased the wave transmission when crest width and incident waves did not change. Wave energy transmission was affected by both the overtopping over the ZFB crest and the porous flow in the ZFB trunk. In case 5, the less permeable ZFB trunk inhibited the transmission of kinetic energy through the trunk from the seaward to the leeward region of the ZFB in comparison to case 1, thus, the resulting reflection coefficient, K

_{r}, was larger and the transmission coefficient, K

_{t}, was smaller.

_{f}= 0.40, for the fully-permeable ZFB cases 1–4, and γ

_{f}= 0.5, for the partially-permeable ZFB case 5, according to [1]. To achieve the best possible agreement to the computed results, the reduction parameter was set to C = 0.43 instead of the value C = 0.67 suggested in [1] for ZFBs. For wave transmission, the empirical formulas of Equation (3) were used. Overall, better prediction seems to be achieved by the empirical formula in [5].

_{lc}(t) is the instantaneous water level at the leeward end of the ZFB crest. The mean overtopping discharge comprises the discharge through the ZFB trunk and the crest discharge over the ZFB crest. The corresponding results are also shown in Table 3. The wave setup, δ, was computed as the period-mean (of 10 wave periods after 20 wave periods of simulation) free-surface level over the SWL at the leeward ZFB toe.

_{0}in ZFB case 4, in comparison to case 1, also resulted in the weak decrease of the mean overtopping discharge. In ZFB case 5, both the trunk discharge and the one over the crest strengthened in comparison to the ones in case 1, but the mean overtopping discharge decreased. To apply Equation (4) in the present cases, which were laboratory-scale ones since d

_{S}= 0.4 m, an adjustment factor of f

_{q}= 8.2 was used, which is lower than the maximum value of 11 suggested in EurOtop [6] for tana = 1/2. The mean overtopping discharge predicted by Equation (4) was constant for all ZFB cases, as shown in Table 3, reflecting the strong dependence of q on H

_{i}but not the one on B, ξ

_{0}, and permeability as the present results suggest.

_{0}in ZFB case 4, in comparison to case 1, resulted in the decrease of the wave setup, while the decrease of permeability in ZFB case 5 increased the wave setup. The corresponding wave setup predicted by the empirical formulas of Equation (5) overestimated δ (Table 3) since they were both based on experiments without the recirculation system described in the Wave Generation subsection.

## 4. Discussion

_{0}, and the ZFB trunk permeability on wave and flow processes were significant both in the seaward and leeward regions of ZFBs. As ξ

_{0}decreased, the wave reflection, wave transmission, magnitude of currents, mean overtopping discharge, and wave setup all decreased; this observation is in accordance with the relevant empirical formulas (Table 2 and Table 3) considered in this study. Finally, decreasing the ZFB trunk permeability led to the increase of wave reflection, the magnitude of currents, and wave setup, but to the decrease of wave transmission and mean overtopping discharge.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Zanuttigh, B.; van der Meer, J.W. Wave reflection from coastal structures in design conditions. Coast. Eng.
**2008**, 55, 771–779. [Google Scholar] [CrossRef] - Seelig, W.N. Two-Dimensional Tests of Wave Transmission and Reflection Characteristics of Laboratory Breakwaters. Coast. Eng. Res. Cent.
**1980**. [Google Scholar] [CrossRef] [Green Version] - Seabrook, S.R.; Hall, K.R. Wave transmission at submerged rubble mound breakwaters. In Proceedings of the 26th International Conference on Coastal Engineering, Copenhagen, Denmark, 22–26 June 1998. [Google Scholar] [CrossRef]
- Van der Meer, J.W.; Briganti, R.; Zannuttigh, B.; Wang, B. Wave transmission and reflection at low-crested structures: Design formulae, oblique wave attack and spectral change. Coast. Eng.
**2005**, 52, 915–929. [Google Scholar] [CrossRef] - Buccino, M.; Calabrese, M. Conceptual Approach for Prediction of Wave Transmission at Low-Crested Breakwaters. J. Waterw. Port Coast. Ocean Eng.
**2007**, 133, 213–224. [Google Scholar] [CrossRef] - Van der Meer, J.W.; Allsop, N.W.H.; Bruce, T.; De Rouck, J.; Kortenhaus, A.; Pullen, T.; Schüttrumpf, H.; Troch, P.; Zanuttigh, B. EurOtop, Manual on Wave Overtopping of Sea Defences and Related Structures. An. Overtopping Manual Largely Based on European Research, but for Worldwide Application; Envrionment Agency, ENW, KFK: Bristol, UK, 2018. [Google Scholar]
- Diskin, M.H.; Vajda, M.L.; Amir, I. Piling-up behind low and submerged permeable breakwaters. J. Waterw. Harb. Coast. Eng. Div.
**1970**, 96, 359–372. [Google Scholar] [CrossRef] - Loveless, J.H.; Debski, D.; McLeod, A.B. Sea level set-up behind detached breakwaters. In Proceedings of the 26th International Conference on Coastal Engineering, Copenhagen, Denmark, 22–26 June 1998; pp. 1665–1678. [Google Scholar] [CrossRef]
- Bellotti, G. A simplified model of rip currents systems around discontinuous submerged barriers. Coast. Eng.
**2004**, 51, 323–335. [Google Scholar] [CrossRef] - Calabrese, M.; Vicinanza, D.; Buccino, M. 2D Wave setup behind submerged breakwaters. Ocean Eng.
**2008**, 35, 1015–1028. [Google Scholar] [CrossRef] - Mitzutani, N.; Mostafa, A.M.; Iwata, K. Nonlinear regular wave, submerged breakwater and seabed dynamic interaction. Coast. Eng.
**1998**, 33, 177–202. [Google Scholar] [CrossRef] - Huang, C.J.; Chang, H.H.; Hwung, H.H. Structural permeability effects on the interaction of a solitary wave and a submerged breakwater. Coast. Eng.
**2003**, 49, 1–24. [Google Scholar] [CrossRef] - Garcia, N.; Lara, J.L.; Losada, I.J. 2-D numerical analysis of near-field flow at low-crested permeable breakwaters. Coast. Eng.
**2004**, 51, 991–1020. [Google Scholar] [CrossRef] - Hsu, T.J.; Sakakiyama, T.; Liu, P.L.F. A numerical model for wave motions and turbulence flows in front of a composite breakwater. Coast. Eng.
**2002**, 46, 25–50. [Google Scholar] [CrossRef] - Losada, I.J.; Lara, J.L.; Christensen, E.D.; Garcia, N. Modelling of velocity and turbulence fields around and within low-crested rubble-mound breakwaters. Coast. Eng.
**2005**, 52, 887–913. [Google Scholar] [CrossRef] - Lara, J.L.; Garcia, N.; Losada, I.J. RANS modelling applied to random wave interaction with submerged permeable structures. Coast. Eng.
**2006**, 53, 395–417. [Google Scholar] [CrossRef] - Kramer, M.; Zanuttigh, B.; van der Meer, J.W.; Vidal, C.; Gironella, X. Laboratory experiments on low-crested breakwaters. Coast. Eng.
**2005**, 52, 867–885. [Google Scholar] [CrossRef] - Dimas, A.A.; Koutrouveli, I.T. Wave Height Dissipation and Undertow of Spilling Breakers over Beach of Varying Slope. J. Waterw. Port Coast. Ocean Eng.
**2019**, 145. [Google Scholar] [CrossRef] - Liu, P.L.F.; Pengzhi, L.; Chang, K.; Sakakiyama, T. Numerical modeling of wave interaction with porous structures. J. Waterw. Port Coast. Ocean Eng.
**1999**, 125, 322–330. [Google Scholar] [CrossRef] - Van Gent, M.R.A. Wave Interaction with Permeable Coastal Structures. Ph.D. Thesis, Delft University, Delft, The Netherlands, 1995. [Google Scholar]
- Keulegan, G.H.; Carpenter, L.H. Forces on cylinders and plates in an oscillating fluid. J. Res. Nat. Bur. Stand.
**1958**, 60, 423–440. [Google Scholar] [CrossRef] - Smagorinsky, J. General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weather Rev.
**1963**, 91, 99–164. [Google Scholar] [CrossRef] - Yang, J.; Stern, F. Sharp interface immersed-boundary/level-set method for wave–body interactions. J. Comput. Phys.
**2009**, 228, 6590–6616. [Google Scholar] [CrossRef] - Balaras, E. Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations. Comput. Fluids
**2004**, 33, 375–404. [Google Scholar] [CrossRef] - Hughes, S.A. Physical Models and Laboratory Techniques in Coastal Engineering; World Scientific: Singapore, 1993; pp. 367–379. [Google Scholar] [CrossRef]
- Jacobsen, N.G.; Fuhrman, D.R.; Fredsoe, J. A wave generation toolbox for the open-source CFD library: Open Foam
^{®}. Int. J. Numer. Methods Fluids**2011**, 70, 1073–1088. [Google Scholar] [CrossRef] - Beji, S.; Battjes, J.A. Numerical simulation of nonlinear wave propagation over a bar. Coast. Eng.
**1994**, 23, 1–16. [Google Scholar] [CrossRef] - Mansard, E.P.D.; Funke, E.R. The measurement of incident and reflected spectra using a least squares method. In Proceedings of the 17th International Conference on Coastal Engineering, Sydney, Australia, 23–28 March 1980; pp. 154–172. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Sketch of the computational domain and part of the Cartesian grid, as well as definitions of parameters for the numerical simulation of wave propagation over a submerged impermeable trapezoidal bar [27]. Note that the axes are not to scale. The distance from the wavemaker to the seaward toe of the bar is 27 d

_{S}, while the distance from the leeward toe of the bar to the toe of the absorbing beach is 5 d

_{S}.

**Figure 2.**Variation of the free-surface elevation during five wave periods, after 20 wave periods of simulation, at five locations over a submerged impermeable trapezoidal bar. These five locations correspond, respectively, to the stations 2, 3, 4, 5, and 6 of Figure 1. The lines correspond to the present numerical results while the symbols to the experiments in [27]. The r.m.s. (root mean square) relative error of the free-surface elevation between numerical results and experimental data is: (

**a**) 0.09, (

**b**) 0.13, (

**c**) 0.16, (

**d**) 0.19, and (

**e**) 0.18.

**Figure 3.**Sketch of the computational domain and part of the Cartesian grid, as well as definitions of the parameters for the numerical simulation of wave propagation over submerged permeable breakwater [13]. Note that the axes are not to scale. The distance from the wavemaker to the toe of the 1/20 slope is 19.3d

_{F}, while the distance from the leeward toe of the breakwater to the toe of the absorbing beach is 4.5d

_{F}.

**Figure 4.**Envelope of the free-surface elevation of waves passing over submerged permeable breakwater. The lines correspond to the present numerical results while the symbols to the experiments in [13]. The maximum r.m.s. relative error of the wave height between numerical results and experimental data is less than 0.05 in the seaward region and 0.32 in the far leeward region.

**Figure 5.**Sketch of the computational domain and part of the Cartesian grid for the permeable ZFB cases. Note that the axes are not to scale. The distance from the wavemaker to the seaward ZFB toe is 23d

_{S}, while the distance from the leeward ZFB toe to the toe of the absorbing beach is 9d

_{S}.

**Figure 7.**Vorticity field in the seaward region of ZFB case 1 at four instants during the 30th wave period: (

**a**) T/3, (

**b**) 3T/8, (

**c**) 5T/8, and (

**d**) T.

**Figure 8.**Vorticity field in the seaward region of the ZFB case 4 at four instants during the 30th wave period: (

**a**) T/3, (

**b**) 3T/8, (

**c**) 5T/8, and (

**d**) T.

**Figure 9.**Vorticity field near the crest and in the ZFB trunk (case 5) at one instant (T/8) during the 30th wave period.

**Figure 10.**Period-mean velocity field in the seaward region of the ZFBs in cases: (

**a**) 1, (

**b**) 2, and (

**c**) 3. Velocity vectors are shown non-dimensionalized by (gd

_{S})

^{1/2}in the water phase under the wave crest envelope.

**Figure 11.**Period-mean velocity field in the leeward region of the ZFBs in cases: (

**a**) 1, (

**b**) 2, and (

**c**) 3. Velocity vectors are shown non-dimensionalized by (gd

_{S})

^{1/2}in the water phase under the wave crest envelope.

**Figure 12.**Period-mean velocity field for ZFB case 4 in: (

**a**) the seaward region, and (

**b**) the leeward region. Velocity vectors are shown non-dimensionalized by (gd

_{S})

^{1/2}in the water phase under the wave crest envelope.

**Figure 13.**Period-mean velocity field for ZFB case 5. Velocity vectors are shown non-dimensionalized by (gd

_{S})

^{1/2}in the water phase under the wave crest envelope.

**Table 1.**Geometrical characteristics of the examined ZFB cases, the parameters of the incoming waves and the breaking ones on the seaward slope of the ZFB.

Case | Geometry | Incoming Waves | Breaking Waves on Seaward ZFB Slope | ||||||
---|---|---|---|---|---|---|---|---|---|

tana | B/d_{S} | Armor D _{50}/d_{S} | Core D _{50}/d_{S} | H_{i}/d_{S} | T(g/d_{S})^{1/2} | λ/d_{S} | d_{b}/d_{S} | H_{b}/d_{S} | |

1 | 1/2 | 1 | 0.31 | 0.31 | 0.2 | 9.8 | 9 | 0.178 | 0.205 |

2 | 2 | 0.31 | 0.31 | 0.175 | 0.201 | ||||

3 | 3 | 0.31 | 0.31 | 0.180 | 0.202 | ||||

4 | 1 | 0.31 | 0.31 | 6.9 | 6 | 0.179 | 0.197 | ||

5 | 1 | 0.31 | 0.031 | 9.8 | 9 | 0.171 | 0.217 |

**Table 2.**Reflection and transmission coefficients for all the ZFB cases of Table 1.

Case | ξ_{0} | K_{r} | K_{t} | |||||
---|---|---|---|---|---|---|---|---|

Present | Zanuttigh and Van der Meer (2008) | Present | Seeling (1980) | Seabrook and Hall (1998) | Van der Meer et al. (2005) | Buccino and Calabrese (2007) | ||

1 | 4.37 | 0.176 | 0.174 | 0.416 | 0.40 | 0.196 | 0.345 | 0.356 |

2 | 4.37 | 0.171 | 0.174 | 0.238 | 0.29 | 0.103 | 0.187 | 0.206 |

3 | 4.37 | 0.171 | 0.174 | 0.127 | 0.18 | 0.070 | 0.073 | 0.097 |

4 | 3.08 | 0.106 | 0.132 | 0.171 | 0.40 | 0.196 | 0.305 | 0.288 |

5 | 4.37 | 0.207 | 0.195 | 0.290 | 0.40 | 0.196 | 0.345 | 0.356 |

**Table 3.**Mean overtopping discharge and wave setup for all the ZFB cases (Table 1).

Case | q/(g/d_{S}^{3})^{1/2} | δ/d_{S} | |||||
---|---|---|---|---|---|---|---|

Trunk Discharge | Crest Discharge | Overtopping Discharge | EurOtop (2018) | Present | Diskin (1970) | Loveless (1998) | |

1 | −0.0015 | 0.002580 | 0.001080 | 0.000976 | 0.01466 | 0.0735 | 0.0393 |

2 | −0.000035 | 0.000989 | 0.000954 | 0.000976 | 0.01629 | 0.0735 | 0.0785 |

3 | 0.00052 | 0.000377 | 0.000897 | 0.000976 | 0.01427 | 0.0735 | 0.1178 |

4 | −0.00032 | 0.001308 | 0.000988 | 0.000976 | 0.01002 | 0.0735 | 0.0194 |

5 | −0.00219 | 0.003055 | 0.000865 | 0.000976 | 0.02346 | 0.0735 | 0.0393 |

© 2020 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/).

## Share and Cite

**MDPI and ACS Style**

Koutrouveli, T.I.; Dimas, A.A.
Wave and Hydrodynamic Processes in the Vicinity of a Rubble-Mound, Permeable, Zero-Freeboard Breakwater. *J. Mar. Sci. Eng.* **2020**, *8*, 206.
https://doi.org/10.3390/jmse8030206

**AMA Style**

Koutrouveli TI, Dimas AA.
Wave and Hydrodynamic Processes in the Vicinity of a Rubble-Mound, Permeable, Zero-Freeboard Breakwater. *Journal of Marine Science and Engineering*. 2020; 8(3):206.
https://doi.org/10.3390/jmse8030206

**Chicago/Turabian Style**

Koutrouveli, Theofano I., and Athanassios A. Dimas.
2020. "Wave and Hydrodynamic Processes in the Vicinity of a Rubble-Mound, Permeable, Zero-Freeboard Breakwater" *Journal of Marine Science and Engineering* 8, no. 3: 206.
https://doi.org/10.3390/jmse8030206