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Cubipod^{®} Armor Design in Depth-Limited Regular Wave-Breaking Conditions

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^{2}

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## Abstract

**:**

^{®}armors in depth-limited regular wave breaking and non-overtopping conditions with horizontal foreshore (m = 0) and armor slope (α) with cotα = 1.5. An experimental methodology was established to ensure that 100 waves attacked the armor layer with the most damaging combination of wave height (H) and wave period (T) for the given water depth (h

_{s}). Finally, for a given water depth, empirical formulas were obtained to estimate the Cubipod

^{®}size which made the armor stable regardless of the deep-water wave storm.

## 1. Introduction

_{s}= H

_{m}

_{0}= 4(m

_{0})

^{0.5}, at the toe of the structure and wave height with a 2% exceedance probability, H

_{2%}, are usually considered to describe the incident wave characteristics in non-breaking wave conditions. H

_{2%}is strongly correlated to H

_{m}

_{0}in deep water when wave heights are Rayleigh-distributed, but this is not the case in depth-limited breaking wave conditions (see Battjes and Groenendijk 2000) [6].

_{s}) is the key variable for designing mound breakwaters because the maximum wave height (H

_{max}) attacking the structure mostly depends on h

_{s}. In these conditions, irregular waves may cause less armor damage than regular waves. In a given wave storm, the highest waves may break on the foreshore while smaller waves cause irrelevant armor damage; only a very low proportion of waves will have a wave height (H) close to the maximum wave height (H ≈ H

_{max}) to be relevant when analyzing the hydraulic stability of the armor layer. By contrast, runs of regular waves with the appropriate wave height (H) and wave period (T) can be generated in 2D physical experiments to attack the structure with many waves, say 100 waves, close to the maximum wave height, which would never be obtained with runs of a few thousand irregular waves propagating on a horizontal foreshore.

_{s}), a test matrix of regular wave runs covering the full range of wave height (H) and wave period (T), guarantees that the structure will be attacked by many of the most damaging waves. In these conditions, armor damage under the full test matrix of regular waves is higher than damage caused by a conventional test matrix of irregular wave runs (1,000 waves). Design rules based on the results using regular waves are on the safe side because the structure in shallow water and horizontal foreshore will never be attacked by such a large number of high waves, regardless of the design storm.

^{®}armored breakwaters on a horizontal foreshore, in depth-limited regular wave breaking and non-overtopping conditions. New empirical formulas depending on the design water depth (h) are proposed to estimate the size of Cubipod

^{®}units needed to design stable armors with cotα = 1.5 on a horizontal bottom (m = 0), for any deep-water wave climate. The paper is structured as follows: Section 2 describes the background of the research. Then, Section 3 and Section 4 describe the experimental methodology and present the experimental results, respectively. Section 5 summarizes and details the proposed design criterion.

## 2. Background

_{n}is the nominal diameter of the units, Δ = (ρ

_{r}− ρ

_{w})/ρ

_{w}is the relative submerged mass density, ρ

_{r}is the mass density of the units, ρ

_{w}is the mass density of the sea water, K

_{D}is the stability coefficient, α is the armor slope, H

_{s}is the significant wave height (H

_{s}= H

_{1/3}= average of one-third highest waves or H

_{s}= H

_{m0}= spectral significant wave height), and N

_{s}is the stability number.

_{D}) to use Equation (1) in breaking wave conditions, reducing K

_{D}values for breaking waves. K

_{D}takes into account the geometry of the armor unit, number of layers, breakwater section (trunk or head), armor slope (α), and also an implicit safety factor for design (see Medina and Gómez-Martín 2012 [9]). Equation (1) does not explicitly consider the duration of the wave storm, the wave period or wave steepness, the permeability of core and filter layers, or other relevant factors affecting the hydraulic stability of the armor layer. Nevertheless, Equation (1) is still widely used by practitioners for preliminary designs and feasibility studies to compare construction costs of different breakwater designs using different armor units.

_{D}for single-layer armors corresponds to an armor damage level much lower than IDa. The values of K

_{D}for single- and double-layer armors are based on small-scale physical tests and may change slightly over time, depending on experience and implicit or explicit safety factors.

_{s}with H

_{2%}/1.4 in Equations (2) and (3), after conducting several physical tests in breaking wave conditions with an m = 1/30 bottom slope and a permeable structure. The relationship H

_{2%}/1.4 = H

_{s}is valid if wave heights are Rayleigh distributed (non-breaking wave conditions) but it is a conservative criterion if the highest waves break before reaching the structure (breaking wave conditions).

_{s}is the stability number, D

_{n}

_{50}is the equivalent cube size or nominal diameter of the units, S is the dimensionless armor damage, ξ

_{mc}= 6.2P

^{0.31}(tanα)

^{0.5})

^{1/(P+0.5)}is the critical breaker parameter, 0.1 ≤ P ≤ 0.6 is a parameter which considers the permeability of the structure, N

_{z}is the number of waves, and ξ

_{m}= tanα/(2πH

_{s}/(gT

_{m}

^{2}))

^{0.5}is the surf similarity parameter based on the mean period, T

_{m}.

_{s}/h

_{s}< 1.11 and 0.15 < H

_{s}/h

_{s}< 0.78, respectively) with different bottom and armor slopes. Furthermore, available formulas require knowing H

_{s}or H

_{2%}at the toe of the structure; however, these formulas are calibrated with waves measured at a certain distance from the structure.

_{s}/h

_{s}< 0.90), considering the observed potential 6-power relationship between the equivalent dimensionless armor damage (S

_{e}) and the spectral significant wave height (H

_{m}

_{0}) at a seaward distance of 3h

_{s}from the breakwater toe, where h

_{s}is the water depth at the toe.

^{TM}(K

_{D}= 15) and other interlocking armor units (K

_{D}= 16), valid for armor slopes with cotα = 1.33 to 1.5 in non-breaking conditions. CLI (2018) [13] as well as USACE (1975, 1984) [7,8] recommended using lower values of K

_{D}for wave breaking conditions. On the contrary, Xbloc

^{®}(2014) [14] recommended K

_{D}= 16 and 14.2 for armor slopes with cotα = 1.33 and 1.5, respectively, in breaking or partially breaking wave conditions (H

_{s}/h

_{s}> 0.4), and K

_{D}= 10.7 and 8.0 for 0.29 < H

_{s}/h

_{s}< 0.4 and H

_{s}/h

_{s}< 0.29, respectively, when cotα = 1.33.

_{D}when characterizing the hydraulic stability of the armor layers, Medina and Gómez-Martín (2012) [9] proposed including explicit safety factors (SF) for IDa and IDe associated to each recommended K

_{D}. Based on the experimental results for each armor unit, the safety factors SF(IDa) and SF(IDe) are the ratio between the stability number to IDa and IDe and the design stability number (N

_{sd}), SF(IDa) = N

_{s}(IDa)/N

_{sd}and SF(IDe) = N

_{s}(IDe)/N

_{sd}, where N

_{sd}is given by Equation (1) when H

_{s}= H

_{sd}and H

_{sd}is the design significant wave height. Medina and Gómez-Martin (2016) [15] recommended K

_{D}values for single- and double-layer Cubipod

^{®}armors in non-overtopping and non-breaking conditions, with the corresponding explicit safety factors.

## 3. Experimental Methodology

_{s}(cm) = 30 and h

_{s}(cm) = 42 at the model area, and between h(cm) = 55 and h(cm) = 67 at the wave generation zone, with a 4% slope bottom transition between the two horizontal wave flume bottoms at different levels. Figure 1 shows a longitudinal cross-section of the LPC-UPV wave flume with the location of the wave gauges used in these tests.

^{®}armors with toe berm and double-layer Cubipod

^{®}armor without toe berm) were tested in breaking wave conditions, using the same core and filter layers (see Vanhoutte and De Rouck 2009) [18]. The breakwater crest elevation was high enough for non-overtopping conditions, and the armor slope was α with cotα = 1.5. The armor porosities were 39% and 40% for single-and double-layer armors. The characteristics of the core, filter and armor layer are specified in Table 1, where D

_{n}

_{50}is the equivalent cube size or nominal diameter of the units, D

_{85}and D

_{15}is the equivalent cube size corresponding to the 85% and 15% percentile, respectively, ρ

_{r}is the mass density, and M is the unit mass.

_{s}), the wave period (T) and the bottom slope (m = 0 in this study). The worst test conditions were those generating runs of regular waves with a variety of wave periods and with series of increasing wave heights (H), for each water depth (h

_{s}). For a given water depth (h

_{s}) and a wave period (T), the energy of the wave attack on the breakwater model increased with increasing H until waves broke in the horizontal foreshore before reaching the structure.

_{s}(cm) = 30, 35, 38, 40 and 42 (h

_{s}(m) = 15.0, 17.5, 19.0, etc. at prototype scale), trains of 100 regular waves with different wave periods T(s) = 0.85, 1.28, 1.70, 2.13 and 2.55 (6 < T(s) < 18 at prototype scale) were generated increasing wave height (H) from no damage up to the armor being severely damaged (IDe) or wave breaking on the foreshore. H was increased in steps of 1 cm within the range 9 ≤ H(cm) ≤ 26 (4.5 < H(m) < 13 at prototype scale). For each water depth, around 3000 waves attacked the structure, which covers the range of periods for Spanish coasts in deep water. For a given water depth (h

_{s}) and wave period (T), if H is too high, the waves break before reaching the structure and there is no damage; if H is too small, the waves will not cause the maximum damage to the structure. The methodology described above ensured that the most damaging waves attacked the structure for a given water level.

^{®}armored breakwater model without toe berm, and Figure 3 shows the cross-sections of single- and double-layer Cubipod

^{®}armors with toe berm. A detail of the toe berm used in these experiments is shown in Figure 4. The toe berm was built with filter material (G1). Equivalent dimensionless armor damage (S

_{e}) was measured with the virtual net method described by Gómez-Martín and Medina (2014) [19], considering armor unit extractions, armor layer sliding as a whole, and heterogeneous packing (HeP) failure modes simultaneously. The Cubipods in the bottom armor layer were painted white to enhance color contrast, while those in the upper armor layer were painted with stripes of different colors to facilitate the visual counting. The armor was photographed before and after each run of 100 regular waves, allowing armor damage to be measured with the virtual net method. The virtual net method was applied to the photographs taken perpendicular to the armor to calculate the corresponding equivalent dimensionless damage parameter, S

_{e}. After each series of tests with constant water depth, the armor damage was obtained, considering the previous armor damage obtained for each water depth. The accumulated armor damage was measured to obtain the maximum armor damage associated to each water depth, increasing water levels from h

_{s}(cm) = 30 to 42, for each model. Different armor damage was observed for each water depth (h

_{s}); the higher the water depth, the greater armor damage.

## 4. Experimental Results

^{®}armor damage observations in non-breaking conditions also follow this relationship. Therefore, the linearized equivalent dimensionless armor damage S

_{e}

^{1/5}is used in this study to show the stability of Cubipod

^{®}armors in depth-limited wave breaking conditions.

_{D}= 3.5. Where H is the wave height corresponding to damage S, and H

_{D}

_{=0}is the design wave height corresponding to dimensionless armor damage parameter, S = 0.7. Based on Equation (5), a general damage function (Equation (6)) for any given armor unit can be calculated taking into account the corresponding K

_{D}value.

_{e}

^{1/5}, measured in single- and double-layer Cubipod

^{®}armor experiments with and without toe berm is represented in Figure 5, as a function of the dimensionless wave height (considering the wave height, H

_{D}

_{=0}, that produces IDa in rough quarrystone (K

_{D}= 3.5) with the Hudson formula). It was observed that the K

_{D}for single- and double-layer Cubipod

^{®}armors is approximately 10 times higher than that for rough quarrystone armors in breaking-wave conditions (USACE 1975 [7]). For damage levels below IDa, the toe berm slightly increases the hydraulic stability of the double-layer Cubipod

^{®}armor; however, for armor damage above IDa, the toe berm has no significant influence on armor damage. The stability coefficient for single- and double-layer Cubipod

^{®}armors with breaking waves is K

_{D}≈ 35 (10 times higher than K

_{D}≈ 3.5 for rough quarrystone). The double-layer Cubipod

^{®}armors with a toe berm were tested up to h

_{s}(cm) = 42 (corresponding to 21 m at prototype scale). The other two armors were tested up to h

_{s}(cm) = 40 (20 m at prototype scale).

_{s}/ΔD

_{n}) and the linearized equivalent dimensionless armor damage (S

_{e}

^{1/5}), Equation (7) is proposed as the dimensionless failure function to describe the hydraulic stability of Cubipod

^{®}armored breakwaters on horizontal bottoms in depth-limited wave breaking conditions.

_{e}is the equivalent dimensionless armor damage, h

_{s}is the water depth at the toe of the structure, Δ is the relative submerged mass density, and D

_{n}is the nominal diameter or equivalent cube side length.

_{e}< 2.0, say S

_{e}= 1.6, then h

_{s}< 7.0(ΔD

_{n}) is the water depth which guarantees that both single- and double-layer armors will not exceed IDa under any deep water wave climate conditions. Furthermore, the tests carried out indicate a relevant safety margin to IDe. Taking a safety factor SF(IDe) = 1.15 for double–layer Cubipod

^{®}armors and SF(IDe) = 1.30 for single–layer Cubipod

^{®}armors, the following design values (Equations (8) and (9)) are obtained for safe Cubipod

^{®}armored breakwaters placed on horizontal seafloors (m = 0):

_{s}< 7.0(ΔD

_{n}) for double-layer Cubipod

^{®}armors

_{s}< 6.2(ΔD

_{n}) for single-layer Cubipod

^{®}armors

_{s}(cm) = 30 and double-layer Cubipod

^{®}armored model without toe berm.

_{max}/h

_{s}≈0.60 with a horizontal seafloor (m = 0). For gentle bottom slopes (m > 0), a criterion similar to that proposed by Goda (2010) [21] and Herrera et al. (2017) [5] is reasonable. The design water depth (h) is calculated at a distance three times the water depth at the toe of the structure (h

_{s}), h = h

_{s}(1 + 3m). This approximation is reasonable for gentle bottom slopes (m ≤ 0.02); for steep bottom slopes (e.g., m = 0.10), the toe berm design is critical, and armor layer and toe berm must be studied simultaneously (see Herrera and Medina 2015 [22] or Herrera et al. 2016 [23]). Small-scale models are highly recommended when breakwaters are placed on steep sea bottoms in depth-limited breaking wave conditions.

_{s}(1 + 3m), the size of Cubipod

^{®}units can be estimated to design safe Cubipod

^{®}armored breakwaters (cotα = 1.5 and perpendicular wave attack) for any deep water wave climate, regardless of the wave height at the toe of the structure, considering the following equations:

_{n}> h/7.0Δ = h

_{s}(1 + 3m)/(7.0Δ) for double-layer Cubipod

^{®}armors

_{n}> h/6.2Δ = h

_{s}(1 + 3m)/(6.2Δ) for single-layer Cubipod

^{®}armors

## 5. Summary and Conclusions

^{®}armored breakwaters on horizontal seafloors in depth-limited breaking wave and non-overtopping conditions. This methodology ensures that the main armor is attacked by runs of 100 regular waves with the most damaging combination of wave height (H) and period (T) for the given water depth (h

_{s}). In wave breaking conditions, the maximum wave height attacking the breakwater (H

_{max}) depends on the water depth at the toe (h

_{s}), the wave period (T) and the bottom slope (m = 0 in this study). This research verifies that, for a given water depth, any irregular wave train caused less armor damage compared to that caused by the series of regular waves corresponding to the experimental methodology used in this study. The worst test conditions were obtained generating regular waves for a range of wave height and periods, which can be observed in nature; for each water depth (h

_{s}) and wave period (T), a series of runs of 100 waves with increasing wave heights (H) was generated until waves broke on the foreshore before reaching the structure. The accumulated equivalent dimensionless armor damage (S

_{e}) was measured to obtain the maximum armor damage associated with each water depth (h

_{s}) and multiple combinations of wave height (H) and wave period (T).

_{e}= 1.6 and a safety factor SF(IDe) = 1.15 and 1.30 for double- and single-layer Cubipod

^{®}armors, respectively, the water depths at the toe of the structure which guarantee that both double- and single-layer armors will not exceed IDa are h

_{s}< 7.0(ΔD

_{n}) and h

_{s}< 6.2(ΔD

_{n}), respectively. Armor unit size can be estimated with Equations (10) and (11) to design safe Cubipod

^{®}armored breakwaters on horizontal seafloors with cotα = 1.5 for any deep-water wave climate. Taking a design water depth (h) at a distance three times the water depth at the toe of the structure, h = h

_{s}(1 + 3m), Equations (10) and (11) can be safely be used to design Cubipod

^{®}armored breakwaters on gentle bottom slopes (0 ≤ m ≤ 0.02) regardless of the wave height at the toe of the structure and deep-water wave climate.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Longitudinal cross-section of the Laboratory of Ports and Coasts at the Universitat Politècnica de València (LPC-UPV) wave flume (dimensions in meters).

**Figure 2.**View of a double-layer Cubipod

^{®}armored breakwater model without toe berm tested at the LPC-UPV.

**Figure 5.**Accumulated linearized equivalent dimensionless armor damage (S

_{e}

^{1/5}) as a function of dimensionless wave height in wave breaking conditions for single- and double-layer Cubipod

^{®}armors with and without toe berm.

**Figure 6.**Linearized equivalent dimensionless armor damage Se

^{1/5}as a function of the dimensionless water depth.

**Figure 7.**H measured at the model area as a function of H measured at the wave generation zone (m = 0).

Materials | D_{n}_{50} (cm) | D_{85}/D_{15} | ρ_{r} (g/cm^{3}) | M (g) |
---|---|---|---|---|

Core (G2) | 0.70 | 2.0 | 2.70 | 0.9 |

Toe berm and filter (G1) | 1.80 | 1.5 | 2.70 | 16 |

Cubipod^{®} | 3.82 | - | 2.30 | 128 |

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## Share and Cite

**MDPI and ACS Style**

Gómez-Martín, M.E.; Herrera, M.P.; Gonzalez-Escriva, J.A.; Medina, J.R.
Cubipod^{®} Armor Design in Depth-Limited Regular Wave-Breaking Conditions. *J. Mar. Sci. Eng.* **2018**, *6*, 150.
https://doi.org/10.3390/jmse6040150

**AMA Style**

Gómez-Martín ME, Herrera MP, Gonzalez-Escriva JA, Medina JR.
Cubipod^{®} Armor Design in Depth-Limited Regular Wave-Breaking Conditions. *Journal of Marine Science and Engineering*. 2018; 6(4):150.
https://doi.org/10.3390/jmse6040150

**Chicago/Turabian Style**

Gómez-Martín, M. Esther, María P. Herrera, Jose A. Gonzalez-Escriva, and Josep R. Medina.
2018. "Cubipod^{®} Armor Design in Depth-Limited Regular Wave-Breaking Conditions" *Journal of Marine Science and Engineering* 6, no. 4: 150.
https://doi.org/10.3390/jmse6040150