# Bulk Wave Dissipation in the Armor Layer of Slope Rock and Cube Armored Breakwaters

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

## 3. Experimental Methodology

#### 3.1. Experimental Setup

- Water depth was kept constant and equal to $h=0.4$ m.
- Wave-breaking was only caused by wave–breakwater interaction and the experiments were under non-overtopping and non-damage conditions.
- The AwaSys software package [42] was used to generate waves with the simultaneously active absorption of reflected waves.
- Irregular waves were generated with a Jonswap spectrum defined by a spectral wave height, ${H}_{m0,target}$, a peak wave period, ${T}_{p,target}$, and the peak enhancement factor of 3.3.
- Each test was performed with runs of 1000 waves by two methods: (a) keeping constant the wave period (IISTA-UGR); and (b) keeping constant the steepness, ${S}_{p,\phantom{\rule{0.166667em}{0ex}}target}={H}_{m0}/{L}_{p}$, being ${L}_{p}$, the peak wave length (AAU). In both laboratories, tests were carried out increasing the wave height by steps of 0.02 m.
- Six resistance wave gauges (G1 to G6) were located along the wave flume of IISTA-UGR and were used to measure the free surface elevations with a sampling frequency of 20 Hz. At AAU, six resistance type wave gauges were placed near the structure to separate incident and reflected waves and one wave gauge was set at the toe of the breakwater.
- At IISTA-UGR, the incident and reflected wave train were separated by Baquerizo [43]’s method, and the reflection and transmission coefficients (${K}_{R}^{2}$, ${\varphi}_{R}$, ${K}_{T}^{2}$) were calculated with the data measured by gauges G1, G2 and G3. Transmission coefficient (${K}_{T}^{2}$) was computed directly from gauge G6. At AAU, the method of Eldrup and Andersen [44] was applied to calculate the incident and reflected wave spectrum. The SIRW method of Frigaard and Brirsen [45] was used to calculate the time domain incident and reflected wave trains. All analyses of wave signal were performed with the Wave-Lab3 software package [46].

#### 3.2. The Log-Experimental Space Based on Dimensional Analysis

## 4. Results

#### 4.1. Observed Wave Breaker Type and the Non-Dimensional Parameter ${H}_{M0}/{D}_{A}$

#### 4.2. Dependence of the Bulk Dissipation on the Armor Unit

#### 4.3. Comparative Bulk Dissipation between Different Sizes and Isolines ${H}_{M0}/{D}_{A}$ and ${N}_{S}$

## 5. Discussion

#### 5.1. Assimilation of Data from Different Laboratories

#### 5.2. The Dependence on the Core: Characteristic Width and Grain Size

#### 5.3. Bulk Dissipation and Armor Stability

#### 5.4. Dissipation in the Main Layer and the Notional Permeability P

## 6. Conclusions

- The bulk dissipation depends on the turbulent regime generated in the main armor layer, which in turn depends on $(h/L)({H}_{m0}/L)$, the relative size of the armor unit, ${D}_{a}/H$, the relative thickness, $e/L$, the shape and specific placement criterion, the characteristics of the porous core, ${B}^{*}/L$, ${D}_{50,p}/L$, and the slope angle of the breakwater.
- The dissipation of the incident energy in the main armor layer with different sizes of cubes is relevant at specific intervals of $(h/L)({H}_{m0}/L)$, related to the transition region and the breaker type, from weak bore to strong plunging. The difference is negligible in the reflective and dissipative regions.
- The dissipation of the incident wave train in the armor layer composed of rocks and in the armor layer with cubes have, in practice, the same bulk dissipation over the entire range of $(h/L)({H}_{m0}/L)$.
- For a given breakwater, that is, the slope angle, ${D}_{50,p}$ and ${B}^{*}$ are constant, and if the armor layer is built with the same thickness and placement criterion, the dimensional analysis provides a functional relationship between the number of stability of the armor unit and the bulk dissipation in the armor layer.
- For the experimental tests performed in the wave flume of IISTA, University of Granada, the experimental spaces are organized around lines parallel to the x-axis based on a constant $h/L$ value. The constant ${H}_{m0}/{D}_{a}$ lines determine an evolution of the breaker type in the sense of surging to weak plunging. Lower values of ${H}_{m0}/{D}_{a}$ offer a higher probability to observe all the breaker types.
- The experimental technique of Aalborg University of keeping the Iribarren number constant is represented in the experimental space by trajectories parallel to the y-axis, that is, $ln(h/L)$. In addition, the physical limitations of the generation system and its control, make it difficult to comply with the constant Iribarren number requirement, so it cannot be assured that the experimentation collects all breaker types.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## List of Symbols

${A}_{eq}$ | Porous area per unit section under the mean water level |

B | Width of the caisson |

${B}_{b}$ | Width of the top of the mound breakwater |

${B}^{*}$ | Characteristic width of the breakwater $={B}_{b}+(0.5{F}_{MT}cot\left(\alpha \right))$ |

${C}_{g,i}$ | Linear theory wave group speed |

${d}_{j}^{*}$ | Source process of wave energy dissipation ($j$ = 1, 2, 3) |

${D}_{a}$ | Diameter of the main armor layer |

${D}_{eq}$ | Equivalent diameter of the main armor layer |

${D}_{50,f}$ | Filter diameter |

${D}_{50,p}$ | Granular core diameter |

${D}^{{}^{\prime}*}$ | Mean bulk dissipation |

${D}^{*}$ | Mean dissipation rate |

e | Thickness of the armor layer $\approx {n}_{l}{D}_{a}$ |

${E}_{i}$ | Wave energy per unit surface |

${F}_{c}$ | Freeboard |

${F}_{i}$ | Mean energy flow |

${F}_{MT}$ | Porous medium height |

g | Gravity acceleration |

h | Water depth |

${h}_{b}$ | Caisson foundation depth |

${H}_{I}$ | Wave height |

${H}_{m0}$ | Spectral incident wave height |

${I}_{r}$ | Number of Iribarren |

${I}_{r}^{*}$ | Number of Modified Iribarren |

${k}_{F}$ | Subset of independent quantities constant in a test |

${k}_{\pi}$ | Subset of independent quantities |

${K}_{R}^{2}$ | Reflected energy coefficient |

${K}_{T}^{2}$ | Transmitted energy coefficient |

l | Size of the cubes of the armor layer (also known as Nominal Diameter ${D}_{n}$) |

L | Wavelength related to ${T}_{z}$ |

${L}_{p}$ | Peak wavelength |

${m}_{0}$ | Zero-order momentum |

n | Number of independent quantities |

${n}_{F}$ | Number of independent quantities constant in a test |

${n}_{l}$ | Real number |

${n}_{p}$ | Core porosity |

${N}_{S}$ | Stability number |

P | Notional permeability |

${Q}_{c}$ | Overflow rate |

${R}_{d}$ | Run-down |

${R}_{u}$ | Run-up |

${R}_{e,{D}_{a}}$ | Armor Reynolds number |

${S}_{p}$ | Wave steepness related to ${T}_{p}$ |

${T}_{p}$ | Peak wave period |

${T}_{z}$ | Mean wave period |

x | Horizontal axes - origin of coordinates at the breakwater toe |

z | Vertical axis - origin of coordinates at S.W.L. |

$\alpha $ | Seaward slope angle |

$\beta $ | Landward slope angle |

$\mathsf{\Delta}$ | Relative density |

$\mu $ | Water viscosity |

$\nu $ | Kinematic water viscosity |

$\mathsf{\Psi}$ | Similarity function |

$\rho $ | Water density |

${\rho}_{s}$ | Soil density |

${\rho}_{s,a}$ | Armor piece density |

${\rho}_{s,p}$ | Core density |

$\theta $ | Incident angle |

Subscripts | |

$i=$I, R, T | Incident, reflected and transmitted, respectively |

$target$ | Theoretical wave parameters generated |

$wb$ | Wave-breaking |

## Appendix A. Dimensional Analysis

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**Figure 1.**Scheme of the wave flumes of: (

**a**) IISTA—University of Granada—23 × 0.65 × 1 m (dimensions in meters), and (

**b**) Aalborg University—21.5 × 1.2 × 1.5 m (dimensions in centimeters). The location of wave gauges positioned in each laboratory is included.

**Figure 2.**Physical model of the breakwater tested: (

**a**) IISTA-UGR, rubble-mound breakwater with crown wall; (

**b**) Aalborg University, a conventional rubble-mound breakwater.

**Figure 3.**The log-experimental space [$ln(h/L)$, $ln({H}_{m0}/L)$] of (

**a**) IISTA—University of Granada, and (

**b**) Aalborg University. Dashed lines represent the experimental limits for wave generation, and non-overtopping and non-damage conditions in the laboratories.

**Figure 4.**The log-experimental space [$ln(h/L)$, $ln({H}_{m0}/L)$] of the experimental results obtained from (

**a**) IISTA, University of Granada, and (

**b**) Aalborg University. The dash line represents the best fit of ${H}_{m0}/{D}_{a}$ and ${H}_{m0}/\left(\mathsf{\Delta}{D}_{a}\right)$ for the data of IISTA-UGR and Aalborg University, respectively. The trajectory and the most likely breaker type of some tests are also identified by numbers with the subindex $wb$: 1 = surging, 2 = weak bore; 3 = strong bore; 4 = strong plunging; 5 = weak plunging; 6 = spilling. See Appendix A for details.

**Figure 5.**Bulk dissipation results against the log-transformation [($h/L$)(${H}_{m0}/L$)] for two sizes of cubes, IISTA-UGR data: (

**a**) Size of $l=25$ mm and equivalent diameter ${D}_{eq}={D}_{a}=31.0$ mm, (

**b**) Size of $l=65$ mm and equivalent diameter ${D}_{eq}={D}_{a}=80.6$ mm. The solid lines represent the fit spline curve and the dashed lines marks the estimated asymptotic trend of bulk dissipation for very small values of $ln[(h/L)({H}_{m0}/L)$].

**Figure 6.**Bulk dissipation results against the log-transformation [($h/L$)(${H}_{m0}/L$)] for two types of unit pieces, Aalborg University data: (

**a**) Rocks ${D}_{a}=44$ mm, (

**b**) Cubes ${D}_{eq}={D}_{a}=40$ mm. The solid lines represent the fit spline curve and the dashed lines marks the estimated asymptotic trend of bulk dissipation for very small values of $ln\left[(h/L)({H}_{m0}/L)\right]$.

**Figure 7.**The bulk dissipation results against the log-transformation of the product ($h/L$)(${H}_{m0}/L$) for all the cubes tested in the wave flume of IISTA-UGR according to (

**a**) isolines of ${H}_{m0}/{D}_{a}=1.00$, (

**b**) isolines of ${H}_{m0}/{D}_{a}=1.20$. The solid lines represent the fit spline curve and the dashed lines marks the estimated values of bulk dissipation for a small values of $ln[(h/L)({H}_{m0}/L)$]. The solid lines with values of ${D}^{*}<0.3$ represent the dissipation difference between the armor constructed with the larger size, $l=65$ mm, and the other four sizes of cubes.

**Figure 8.**The bulk dissipation results against the log-transformation of the product ($h/L$)(${H}_{m0}/L$) for cubes and rocks tested in the wave flume of Aalborg University according to (

**a**) isolines of ${N}_{s}={H}_{m0}/\left(\mathsf{\Delta}{D}_{a}\right)=0.6$, (

**b**) isolines of ${N}_{s}={H}_{m0}/\left(\mathsf{\Delta}{D}_{a}\right)=0.8$. The solid lines represent the fit spline curve and the dashed lines marks the estimated values of bulk dissipation for a small values of $ln[(h/L)({H}_{m0}/L)$]. The green solid line represents the dissipation difference between the rocks and the cubes.

**Table 1.**Geometric parameters of (A) the rubble mound breakwater with crown wall tested in the laboratory of IISTA-UGR, and (B) the conventional rubble mound breakwater tested in the laboratory of Aalborg University. ${B}_{b}$ is the width of the top of the breakwater; ${F}_{MT}$ is the height of the porous core; ${D}_{eq}$ is the equivalent diameter of the main armor layer, where the cube volume is equated to the volume of a sphere; $\alpha $ and $\beta $ are the seaward and landward slopes of the breakwater, respectively; ${\rho}_{s,a}$ and ${\rho}_{s,p}$ are the densities of the armor units and core, respectively; ${F}_{c}$ is the free-board; B is the width of the caisson; ${h}_{b}$ is the caisson foundation depth; ${n}_{p}$ is the porosity of the core according to CIRIA et al. [41]; and ${D}_{50,f}$ is the diameter of the filter. The values of core porosity, ${n}_{p}$, and densities, ${\rho}_{s,a}$, ${\rho}_{s,p}$, for the conventional rubble mound breakwater were provided by Aalborg University.

(A) Breakwater model of IISTA-UGR | |||||||||||

Armor unit cubes size (mm) | ${D}_{a}={D}_{eq}$ (mm) | ${B}_{b}$ (m) | ${\rho}_{s,a}$ (t/m${}^{3}$) | ${F}_{c}$ (m) | ${h}_{b}$ (m) | $cot\left(\alpha \right)$ | B (m) | ${F}_{MT}$ (m) | ${D}_{50,p}$ (mm) | ${\rho}_{s,p}$ (t/m${}^{3}$) | ${n}_{p}$ |

l = 25 mm | 31. 0 | 0.25 | 2.07 | 0.25 | 0.10 | 1.5 | 0.5 | 0.55 | 12 | 2.83 | 0.391 |

l = 33 mm | 40.9 | 2.18 | |||||||||

l = 38 mm | 47.1 | 2.2 | |||||||||

l = 44 mm | 54.6 | 2.21 | |||||||||

l = 65 mm | 80.6 | 2.27 | |||||||||

(B) Breakwater model of Aalborg University | |||||||||||

Armor unit | ${D}_{a}={D}_{eq}$ (mm) | ${B}_{b}$ (m) | ${\rho}_{s,a}$ (t/m${}^{3}$) | Filter ${D}_{50,f}$ (mm) | $cot\left(\alpha \right)$ | $cot\left(\beta \right)$ | ${F}_{MT}$ (m) | ${D}_{50,p}$ (mm) | ${\rho}_{s,p}$ (t/m${}^{3}$) | ${n}_{p}$ | |

Cubes l = 40 mm | 49.6 | $3{D}_{a}$ | 2.30 | 15 | 1.5 | 1.5 | 0.55 | 5.8 | 2.80 | 0.37 | |

Rocks | 44 0 | $3{D}_{a}$ | 2.62 | 15 | 1.5 | 1.5 | 0.55 | 5.8 | 2.80 | 0.37 |

**Table 2.**Wave conditions tested in (A) the laboratory of IISTA-UGR, and (B) the laboratory of Aalborg University. Target parameters for irregular waves generated with a Jonswap spectrum.

(A) Wave conditions of IISTA-UGR | ||

Armor unit cubes size (mm) | ${T}_{p,\phantom{\rule{0.166667em}{0ex}}target}$ (s) | ${H}_{m0,\phantom{\rule{0.166667em}{0ex}}target}$ (m) |

l = 25 mm | [1.05–3] | [0.04–0.08] |

l = 33 mm | [1.05–3] | [0.04–0.10] |

l = 38 mm | [1.05–3] | [0.04–0.10] |

l = 44 mm | [1.05–3] | [0.04–0.10] |

l = 65 mm | [1.05–3] | [0.04–0.12] |

(B) Wave conditions of Aalborg University | ||

Armor unit | ${S}_{p,\phantom{\rule{0.166667em}{0ex}}target}$ | ${H}_{m0,\phantom{\rule{0.166667em}{0ex}}target}$ (m) |

0.01 | [0.04–0.12] | |

Cubes | 0.02 | [0.04–0.12] |

and | 0.035 | [0.04–0.10] |

rocks | 0.045 | [0.04–0.08] |

© 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**

Clavero, M.; Díaz-Carrasco, P.; Losada, M.Á.
Bulk Wave Dissipation in the Armor Layer of Slope Rock and Cube Armored Breakwaters. *J. Mar. Sci. Eng.* **2020**, *8*, 152.
https://doi.org/10.3390/jmse8030152

**AMA Style**

Clavero M, Díaz-Carrasco P, Losada MÁ.
Bulk Wave Dissipation in the Armor Layer of Slope Rock and Cube Armored Breakwaters. *Journal of Marine Science and Engineering*. 2020; 8(3):152.
https://doi.org/10.3390/jmse8030152

**Chicago/Turabian Style**

Clavero, María, Pilar Díaz-Carrasco, and Miguel Á. Losada.
2020. "Bulk Wave Dissipation in the Armor Layer of Slope Rock and Cube Armored Breakwaters" *Journal of Marine Science and Engineering* 8, no. 3: 152.
https://doi.org/10.3390/jmse8030152