Hydraulic performance of different non-overtopped breakwater types under 2D wave attack

Coastal Engineering 107 (2016) 34–52

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Hydraulic performance of different non-overtopped breakwater types under 2D wave attack Montse Vílchez ⁎, María Clavero, Miguel A. Losada Grupo de Dinámica de Flujos Ambientales, Instituto Interuniversitario del Sistema Tierra en Andalucía (IISTA), University of Granada, CEAMA, Avda. del Mediterráneo s/n, 18006 Granada, Spain

a r t i c l e

i n f o

Article history: Received 20 April 2015 Received in revised form 6 October 2015 Accepted 7 October 2015 Available online xxxx Keywords: Breakwater Coastal structures Hydraulic performance Wave reflection Wave transmission Wave energy dissipation

a b s t r a c t The objective of this research was to develop a method to calculate the hydraulic performance resulting from the interaction of perpendicularly impinging water waves on various types of breakwater. Our study was based on data obtained from physical tests in a wave flume with irregular waves. Based on this information, it was possible to derive the complex wave reflection and transmission coefficients in terms of non-dimensional parameters representing the breakwater geometry, granular materials and incoming wave train. The overall dissipation rate caused by the structure was estimated by applying the energy conservation equation to a control volume, which included the breakwater section. The logistic sigmoid function was used to describe the variation in the modulus and phase of the reflection and transmission coefficients (as well as the energy dissipation rate). Remarkably, the sigmoid function was able to define the domain of the hydraulic performance of the most common breakwaters. It is shown that the sigmoid function depends primarily on a 2D scattering parameter Aeq/L2, where Aeq is the area of a porous medium under the mean water level and L is the wavelength, and on the relative grain size of the porous medium, Dk, where k is the wave number. The logistic sigmoid curves help to include the phase of the reflection coefficient when defining the wave regime in front of, inside, and leeward of the breakwater. Practical examples of how these results can be applied are also included. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Maritime structures and particularly breakwaters must frequently be designed to control wind–wave action. According to Takahashi (1996), there are three structural types of breakwater: (1) sloping or mound breakwaters; (2) vertical breakwaters (including vertical upright as well as composite and horizontal composite breakwaters); and (3) special non-gravity breakwaters. A gravity breakwater is composed of three main sections (ROM, 1.009, 2009): (a) a foundation, which determines the way in which the structure transmits forces to the seabed; (b) a central (or main) body that controls the transformation of incident wave energy and transmits the result of these actions to the foundation; and (c) a superstructure that controls the wave overtopping rate, and, if necessary, provides an access path. Depending on their typology, breakwaters reflect, dissipate, transmit, and radiate incident wave energy. This alters the distribution of the components of the frequency and directional wave spectra (Klopman and van del Meer, 1999; Losada et al., 1993b, 1997b). Partial standing wave patterns are likely to occur at all types of breakwater. Thus, wave transformation (i.e. wave reflection, wave transmission, ⁎ Corresponding author. Tel.: +34 958241781. E-mail addresses: [email protected] (M. Vílchez), [email protected] (M. Clavero), [email protected] (M.A. Losada).

http://dx.doi.org/10.1016/j.coastaleng.2015.10.002 0378-3839/© 2015 Elsevier B.V. All rights reserved.

modulus and phase, and wave energy dissipation) plays an important role in defining the wave regime in front of, near (seaward and leeward), and inside the breakwater (Hughes and Fowler, 1995; Losada et al., 1997a; Sutherland and O'Donoghue, 1998). In the case of regular waves, a pattern of nodes and antinodes in the wave height occurs in front of the structure. However, in the case of random waves, significant wave height only appears as a partial standing wave pattern close to the structure. According to Lamberti (1994), this pattern is a consequence of the effect of coherence between incident and reflected components, which become evanescent as their distance from the reflecting surface increases. Thus, the superposition of incident waves and of those generated and transformed by the presence of breakwaters constitutes the set of oscillation patterns that affects the hydraulic performance of the structure. This evidently signifies that the prediction of such patterns is an important issue in breakwater design (Hughes and Fowler, 1995; ROM, 1.0-09, 2009). The dependence of wave regimes on the complex reflection and transmission coefficients and on the wave energy dissipation rate has been the focus of theoretical considerations, dimensional analysis, and experiments (Dalrymple et al., 1991; Losada et al., 1993a; Pérez-Romero et al., 2009). The bulk reflection and transmission coefficients (Hughes and Fowler, 1995) can be defined, similarly to regular waves, as a complex number with information regarding the amplitude, KR and KT, and phase, ϕR and ϕT, of reflection and transmission, respectively. In addition, the wave energy balance in a control volume

M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

including the breakwater is a convenient engineering method for describing the interaction between phase-averaged wave motion and the structure. It provides the dissipation rate, D*, and works as a closure condition of the problem. The calculated wave energy dissipation rate is a phase-averaged quantity. Since the pioneering work of Iribarren and Nogales (1949), which introduced the Iribarren number in the analysis of the stability of pffiffiffiffiffiffiffiffi ffi mound breakwaters (Ir ¼ tanα T = H=L, where αT is the seaward slope angle of the breakwater and H is the wave length), and its application by Battjes (1974) to the analysis of flow characteristics on sloping structures, many journal and conference papers have focused on the calculation of wave reflection by breakwaters (e.g. Ahrens and Mc Carney, 1975; Allsop and Channell, 1989; Altomare and Gironella, 2014; Losada and Gimenez-Curto, 1981; Zanuttigh and Van der Meer, 2008). Moreover, Medina (1999), Iglesias et al. (2008), and Zanuttigh et al. (2013) demonstrate the efficiency of an artificial neural network (ANN) for the prediction of the wave reflection coefficient for a wide range of coastal and harbor structures. Unfortunately, most of these formulas and ANN methods fit the reflection coefficient modulus obtained without simultaneously evaluating the other hydraulic processes participating in the wave–structure interaction, namely, the transmission coefficient and the energy dissipation rate. Furthermore, most databases, such as EU-project DELOS (www.delos.unibo.it) and CLASH (www.clash-eu.org), only provide the modulus of the reflection coefficient but not its phase. Actually, most databases do not supply the phase at all, and this type of information is extremely difficult to recover. However, as previously mentioned, the phase is essential in the evaluation of the flow on breakwaters. Flow characteristics include run-up, run-down, and overtopping, among other engineering magnitudes, which are crucial for breakwater design. The main objective of this research was to calculate the hydraulic performance resulting from the interaction of wave–breakwater by calculating the complex wave reflection and transmission coefficients (modulus and phase) as well as the overall dissipation rate caused by different non-overtopped structures. The paper is organized as follows. Section 2 summarizes the principal non-dimensional parameters controlling the interaction between water waves and different breakwater typologies and defines the parameter list. The experimental setup is also described. Section 3 explains the data analysis. Next, Section 4 focuses on the governing function (i.e. the sigmoid function) used to describe the wave regime in front, inside, and leeward of the breakwater. Section 5 presents the hydraulic performance for the different breakwater typologies and its dependence on the relevant parameters. Section 6 explains the methodology of

35

application of the hydraulic performance curves obtained and includes some examples. This is followed by Section 7, which discusses the limitations of this research study. The paper ends with a summary of the most important conclusions that can be derived from this research. Finally, Appendix A provides the data used for comparison with other formulas. 2. Breakwater types, parameter list, and experimental setup Fig. 1 shows the breakwater types studied and identifies their geometric parameters. The following typologies are considered (some of which were taken from Kortenhaus and Oumeraci, 1998): (A) porous vertical breakwater (PVB); (B) composite breakwater (CB); (C) mixed breakwater with a berm below or at SWL [FM/h ≤ 1.1] (low and high mound breakwaters, LMB and HMB respectively, and a high mound composite breakwater, HMCB); (D) mixed breakwater with a berm above SWL [FM/h N 1.1] (rubble mound breakwater with crown walls, RMB-CW); and (E) a rubble mound breakwater, plane slope (RMB). Following Kortenhaus and Oumeraci (1998), this research designed a parameter list. In addition to the parameters describing the wave motion, it includes the geometric parameters necessary to unambiguously characterize gravity breakwater types (Fig. 1), • h/L, relative water depth where L is the wavelength • HI/L, wave steepness of the incident wave train (non-breaking) • Ht/L, wave steepness of the total wave height (incident and reflected wave trains) • Dk (or D/L), relative grain diameter • B/L, relative width of the caisson (1D scattering parameter) • Aeq/L2, relative area of the porous medium under the still water level, SWL (2D scattering parameter) • hb/h, relative caisson foundation depth • (Fc + h)/h, relative height of the breakwater • Bb/h, relative berm width • FM/h, relative berm height • αT, seaward slope angle • Sp, type of unit, placement density, and the number of layers beneath the armor layer • St, slope profile • Rep, pore Reynolds number • KCp, pore Keulegan–Carpenter number. Based on the theoretical background as well as numerical and physical experimental works, the 2D hydraulic performance of different non-

Fig. 1. Breakwater types: A) porous vertical breakwater (PVB); B) composite breakwater (CB); C) low and high mound breakwaters (LMB and HMB) and high mound composite breakwater (HMCB) [FM/h ≤ 1.1]; D) rubble mound breakwater with crown walls [FM/h N 1.1] (RMB-CW); E) rubble mound breakwater (RMB).

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M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

overtopped breakwater types (Y) can be expressed by the following function:   Aeq HI Y ¼ f Dk; 2 ; breakwater typology; L L

ð1Þ

where Y characterizes the wave regime resulting from the interaction of the breakwater and the incoming wave train. Y is described by the modulus of the reflection coefficient (KR), the modulus of the transmission coefficient (KT), the phase of the reflection coefficient (ϕR), the phase of the transmission coefficient (ϕT), and the wave energy dissipation (D*). D* can be calculated by means of the energy conservation equation (D* = 1 − KR2 − KT2) (see Section 3). The relative grain diameter, Dk, controls the flow regime inside the porous medium, whereas k is the wave number, k = 2π/L. A general description of the flow regimes in a porous medium is given in Burcharth and Andersen (1995). The ratio B/L is a scattering parameter that controls 1D wave propagation in the porous medium (Dalrymple et al., 1991; Losada et al., 1993a; Pérez-Romero et al., 2009; Clavero et al., 2012, among others). However, for breakwaters consisting of different parts and units (some of which are composed of granular material), it is more convenient to define a 2D scattering parameter, Aeq/L2, where Aeq is the area per unit section under the mean water level. This choice is coherent with the application domain of linear theory, which extends from the sea bottom to the still water level. The scattering parameter is related to the averaged transformation of the wave inside the porous section of the structure. For a vertical porous breakwater (Type A), Aeq is simply B · h, and for a constant depth, the scattering parameter is reduced to B/L, which is the relative breakwater width. Breakwater typology includes the non-dimensional parameters that describe the breakwater geometry and includes, hb/h, FM/h, Bb/h, Sp, and St. Most of the analyzed breakwater types are non-overtopped structures, and the slope profile was kept constant. In the case of sloping breakwaters, the Iribarren number (Battjes, 1974; Iribarren, 1938; Losada and Gimenez-Curto, 1981) is generally acknowledged to be an effective parameter that can be used to control the type of wave breaking against the breakwater slope. Wave breaking can vary by modifying the wave steepness and maintaining the breakwater slope angle, by modifying the slope angle and maintaining the wave steepness, or by modifying both slope angle and wave steepness. This study applied the first option and the slope angle remained constant (cotαT = 1.5). The hydraulic performance of the different breakwater types can be described in terms of the non-dimensional parameters given in Table 1. 2.1. Experimental setup Based on the parameter list, an experimental setup was designed to test different breakwater typologies. Most of the experiments were performed in the wave flume at the CEAMA — University of Granada (23 × 0.65 × 1 m), though other labs such as CITEEC (33.8 × 0.58 × 0.80 m) (Benedicto, 2004; Benedicto and Losada, 2002) contributed data for some of the tests. The experiments were performed during a

Table 1 Parameter list for different breakwater typologies. Typology Type A Type B Type C and Type D

Type E

Parameters PVB CB LMB HMB HMCB RMB-CW RMB

Aeq/L2, Dk, HI/L Aeq/L2, Dk, hb/h, HI/L Aeq/L2, Dk, hb/h, Bb/h, FM/h, HI/L

Aeq/L2, Dk, HI/L, Sp, St

time period of over ten years with a total of 1575 experiments. The data for the B and E typologies were taken from Clavero et al. (2012) and Benedicto (2004) respectively. To widen the intervals of the parameter list, additional specific tests were carried out for A, C, and D typologies. Table 2 includes the variation intervals of the experimental parameters: (a) geometry of the breakwater as well as its parts and elements; (b) porous medium; and (c) incident wave action and sea level. In this work, L and k are the wave length and wave number, respectively, which are associated with the mean period (Tz). Furthermore, the theoretical spectral wave parameters are included, where Tp is the peak period; HIs is the significant incident wave height; and sp is the wave steepness calculated as sp = 2πHIs/gTp2 and s0p = 2πHIs/gT2m − 1.0. For single-peak spectra, it is approximately Tm − 1.0 = Tp/1.1 (Goda, 1985). The caisson used in the experiments had a rectangular parallelepiped shape, and its width was equal to that of its porous base. The granular materials for Types A, B, C and D were classified according to CIRIA/CUR/CETMEF (2007). Their characteristics are shown in Table 3. A Fourier asperity roughness parameter (Kf) was assigned to each material depending on the axial dimensions of the number of single units. Finally, based on Kf, the average stone mass (m) and dry density (ρ0), the characteristic diameter, D, was calculated with the following equation: D = Kf · (m/gρ0)1/3. For a more detailed description, see Scarcella et al. (2006) and Pérez-Romero et al. (2009). For Type E, two breakwater sections were tested. The first breakwater was composed solely of a core (core). The second was built with the same core material but was protected by a main layer of armor stones (core + armor layer). The core material was fine gravel with a median size, D50 = 6.95 mm, and porosity n = 0.42. The relation D85/D15 was 1.60 and the density was ρ0 = 2.7 tn/m3. The armor units were angular stones with nominal diameter, Dn = 2.95 cm, and the armor layer consisted of two layers of units. The porosity of the armor layer was ns = 0.48. The stability of the breakwater elements was not the focus of this research. Therefore, when necessary, the stability of the armor stones in all of the breakwater types was assured by means of a fine wire mesh, which did not modify their hydrodynamic behavior. The flow regime in the cases analyzed was evaluated by using the diagram proposed in Gu and Wang (1991) and subsequently in Van Gent (1995). These authors established the relative importance of resistance forces in the porous medium by means of the Reynolds number, defined as Rep = U · D/(n · ν), and the Keulegan–Carpenter number, KCp = U · T/(n · D), where U is the volume-averaged ensemble-averaged velocity over the control volume (the breakwater section); n is the porosity; and ν is the kinematic viscosity. The seepage velocity, U, was calculated as U ~ nHgT/2L following the methodology proposed by Pérez-Romero et al. (2009), where H is the average of the mean square wave height at the entry and exit of the porous medium; T is the mean period; and L is a characteristic wavelength. Fig. 2 shows the results. In all cases, Rep/KCp was larger than 10. Thus, it was not strictly necessary to include Rep, KCp in the parameter list. Moreover, an in-depth analysis of the influence of Dk on the hydraulic behavior of a breakwater can be found in Pérez-Romero et al. (2009). Tests were performed in which irregular waves were generated with the wave absorption system (AWACS®) activated. They then impinged perpendicularly onto the breakwater. These irregular waves were generated with a Jonswap type spectrum, and a peak enhancement factor of 3.3. Water depth was kept constant and equal to 0.4 m except in the case of Type E in which it was 0.5. The experiments were for nonovertopping conditions. Waves only broke when the wave train impacted against the breakwater wall or when they broke on the berm or slope because of the change in depth. In no case did depth-limited breaking occur, and so the waves did not break before reaching the breakwater. Ten resistance wave gauges (S1 to S10) were located along the flume and used to measure free surface elevations with a sampling frequency of 20 Hz. The distances between the gauges are shown in Fig. 3. The

M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

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Table 2 Range of variables in experimental tests. Typology

Breakwater geometry Aeq/L2

A B

PVB CB Clavero et al. (2012) LMB, HMB and HMCB

C

D

RMB-CW

E

RMB Benedicto (2004)

Porous medium characteristics

hb/h FM/h Bb/h

0.056–0.200 – – 5.05 × 10−4–0.36 0.25 – 0.50 1.00 0.008–0.26 0.50 0.50 0.75 1.00 0.018–0.330 0.50 1.25 1.50 0.040–0.300 – 1.36 1.60

Incident wave and sea level

cotαT Dk

ns

– –

0.01–0.43 0.01–0.53

– –

0.011–0.048 0.10–0.37 1.05–3.00 0.04–0.08 0.0028–0.0465 0.0034–0.0563 0.003–0.05 0.07–0.39 1.05–3.00 0.04–0.08 0.0028–0.0465 0.0034–0.0563

0.250 1.5 0.625

0.02–0.28

–

0.005–0.04

0.09–0.36 1.05–2.50 0.04–0.06 0.0041–0.0349 0.0050–0.0422

0.250 1.5 0.625 0.48 1.5 0.202

0.02–0.27

–

0.005–0.03

0.09–0.36 1.05–2.50 0.04–0.06 0.0041–0.0349 0.0050–0.0422

– –

HIrms/L

h/L

Tp (s)

0.004–0.009 – 0.0052–0.026 0.12–0.30 1.40–2.7 0.48

HIs (m)

sp

s0p

0.12–0.20 0.0106–0.0654 0.0128–0.0792

positions of gauges S1, S2, S3, and S8 were the same for all the typologies. Gauges S9 and S10 were used only for typologies A and B in order to remove the reflection from the dissipation ramp. Gauges S4, S5, and S6 were only applied in the case of typologies C, D, and E because typologies A and B did not have a protection berm. In regard to typology E, gauges S6 and S7 were located at the beginning and the end of the berm, respectively.

Fi, Ei, and Cg,i are phased-averaged quantities and EiCg [KR] and EiCg [KT] represent the reflected (off the front), and transmitted (leeward) wave energy flux per unit section perpendicular to the breakwater. The complex reflection and transmission coefficients (Hughes and Fowler, 1995) can be defined, similarly to regular waves, as a complex number with information regarding the amplitude and phase of reflection and transmission:

3. Analysis of experimental data

K R ¼ K R eiϕR

ð5Þ

The wave energy balance in a control volume including the breakwater is a convenient engineering method for describing the interaction between phase-averaged wave motion and the structure. When linear theory is applied, wave energy is expressed per unit of horizontal surface and unit time as the most energetic wave frequency of the spectrum, and, when the transfer of wave energy to higher harmonics is considered negligible (Losada et al., 1997b), the wave energy balance reads as follows:

K T ¼ K T eiϕT

ð6Þ

F I −F R −F T −D0

¼0

ð2Þ

where KR and KT are the moduli of the reflection and transmission coefficient, defined as follows: KR ¼

H Rrms HIrms

ð7Þ

KT ¼

H Trms H Irms

ð8Þ

and ϕR and ϕT are the reflection and transmission phase, respectively.

where,   F i ¼ EC g i ¼

  1 ρgH2rms C g 8 i

ð3Þ

where g is the gravitational acceleration; and Hrms is the root-meanpffiffiffiffiffiffiffiffiffi square wave height, defined as H rms ¼ 8mo (m0 is the zero-order moment) and i = I (Incident), R (reflected), and T (transmitted). The linear theory group speed is the following: 

Cg

 i

¼

   1 2kh C 1þ 2 sinhð2khÞ i

ð4Þ

where Ci, ki, and h are the wave celerity, wave number, and water depth, respectively. (Cg)i approximates 0.5(C)i in deep water and (C)i in shallow water (Dean and Dalrymple, 2001).

Table 3 Material characteristics: porosity, Fourier asperity roughness parameter, characteristic diameter, sorting parameter, and dry density. Material, D (mm)

n

Kf

D50 (mm)

D85/D15

ρ0 (tn/m3)

110 52 40 26 12

0.471 0.474 0.473 0.462 0.391

1.03 1.03 1.03 1.02 1.02

115 75 45 30 10

1.21 1.21 1.44 1.54 1.71

2.57 2.57 2.69 2.84 2.83

Fig. 2. Importance of the resistance forces (Gu and Wang, 1991) in the tests performed.

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M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

Fig. 3. Scheme of the wave flume and location of wave gauges.

D* is the rate of wave energy dissipation, produced by the breakwater, D ¼

D0 1 ρgC g H 2I;rms 8

¼ Da þ Dp :

ð9Þ

Here all dissipation is assumed to be the result of wave breaking on the front of the breakwater, including the turbulent flow through the voids of the armor units, Da⁎, and the shear stresses inside the porous medium, Dp⁎. Eq. (2) can be expressed in terms of KR, KT, and D* as shown in the energy conservation equation (constant depth): K 2R þ K 2T þ D ¼ 1:

ð10Þ

KR, KT and ϕR were obtained from the experimental data. The complex reflection coefficient modulus and phase were calculated with the data measured by gauges S1, S2, and S3. The incident and reflected wave trains were separated by applying Baquerizo (1995). This method is based on the three-gauge method in Mansard and Funke (1987), but it resolves the mathematical inconsistence of minimizing the complex variable with which it was formulated. This method is based in linear theory. When wave energy transmission through the breakwater was significant (Types A and B, and KT N 0.5), the transmitted root-meansquare wave height (HTrms) was calculated by applying the Baquerizo method to the data obtained from gauges S8, S9, and S10, and by separating the reflected energy flow on the ramp. In the rest of the cases, since HTrms was small and the transmission phase was not relevant, it was obtained only with gauge S8. The transmission coefficient was computed as the ratio of the incident root-mean square wave height, HIrms, and the transmitted root-mean-square wave height, HTrms (Eq. (8)). By

applying Eq. (10), the overall energy dissipation, D*, was calculated for each type of breakwater tested. Gauge S4, located at the toe of the structure (x = 0), and gauge S7, located at the seaward side of the superstructure, provided the rootmean-square total wave height (due to the interaction of the incident and reflected wave trains), Htrms (x = 0) and Hwrms, respectively. In between, gauges S5 and S6 were used to record the water surface displacement on the breakwater face. This information was used to compare these measurements with the values calculated at those points, based on the data from gauges S1, S2, and S3. 4. The logistic sigmoid function Following Churchill and Usagi (1972), if Y(X) is a physical entity describing a transport phenomenon and Y0(X), Y1(X) are known asymptotes to Y(X) for small and large values of the independent variable X,   γ −1 X þ Y0 Y ðX Þ ¼ ðY 1 −Y 0 Þ 1 þ aX

XN0

ð11Þ

which describes a uniform transition between the asymptotes with γ, a blending coefficient, and aX, a parameter of the process inherent to the sigmoid shape. The Churchill–Usagi method has been successfully used to describe various transport phenomena in fluid mechanics, heat transfer, and chemical engineering (Sivanesapillai et al., 2014). Curve definition requires four parameters: Y0, Y1, aX, and γ. Fig. 4 shows how the variation of these parameters affects the form of the sigmoid curve. In this research, the phase-averaged quantities [KR, KT, ϕR, D*] are physical entities describing a wave energy transport phenomenon. Following Churchill and Usagi (1972), and for a given breakwater type, these quantities should adapt to a sigmoid shape for a specific

Fig. 4. Sigmoid curve variation based on parameters γ (left panel) and aX (right panel).

M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

independent variable and other non-dimensional parameters, as given by Eq. (1) and summarized in Table 1. A logistic sigmoid curve should fix uniform transitions of Yi, between Yi0 and Yi1, "

Aeq =L2 Y i ¼ ðY i1 −Y i0 Þ 1 þ aX;i

!γi #−1 þ Y i0

Aeq =L2 N 0 Y i0 b Y i b Y i1 for i ¼ K R and K T Y i1 b Y i b Y i0 for i ¼ D and ϕR

ð12Þ where i is the index denoting the modulus of the reflection coefficient (KR), transmission coefficient (KT), phase of the reflection coefficients (ϕR), or the wave energy dissipation rate (D*). The selected independent variable is the relative volume of granular material per unit of breakwater width, or scattering parameter, X = Aeq/L2. The other two fit parameters of the curve (γi and aX,i) depend on Dk, the type of breakwater, and incoming wave train characteristics. For the reflection process, the largest and smallest values of KR (KR1 and KR0, respectively) are associated with the maximum and minimum values of the reflection coefficient. The value of KR1 defines the breakwater response when the reflection process is dominant, and when the wave energy dissipated, mostly by porous friction, is negligible. It is usually associated with large wave periods (largest wavelength) and the smallest wave steepness. On the other hand, the smallest value, KR0, determines the reflection coefficient when the energy dissipation rate tends to be maximal. Generally speaking, it is associated with short wave periods (shortest wavelength) and the largest wave steepness. Moreover, under such conditions the dissipation process can dominate the wave–structure interaction. The wave transmission (KT) for non-overtoppable permeable breakwaters decreases with X = Aeq/L2. For large Aeq/L2, KT0 ➔ 0, whereas for Aeq/L2 ➔ 0, KT1 must fulfill the energy conservation equation and can be approximated by means of K2T1 ~ 1 − K2R1. Depending on the breakwater type and the incoming wave train, the wave energy dissipation could occur due to one or all of the following mechanisms: porous friction, wave breaking onto the structure, and turbulence in the main layer. Strictly speaking, the wave energy dissipation rate, D*, is the complementary value of the sum of the squares of the two sigmoid curves. Its performance can also be represented by a sigmoid curve, as explained in the following sections. Based on the experimental data, the least squares method can be applied to obtain the four fit parameters (Y0i, Y1i, γi, aX,i) of the sigmoid curve for the phase-averaged quantities [KR, KT, ϕR, D*] for each breakwater typology. The error in the experimental data, yo, and theoretical data, ye, (o = observed, e = estimated) is calculated by means of the coefficient of determination, R2:

2 X yoðiÞ −yeðiÞ R ¼ 1− X

2 : yoðiÞ −yo 2

ð13Þ

In the case of breakwater types with reflection coefficients that experience little variation and whose value is practically constant in regard to the main variable, the quantification of the error is based on the following mean quadratic error, ε: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u

2 u u yoðiÞ −yeðiÞ t : ε¼ y2oðiÞ

ð14Þ

5. Hydraulic performance of different breakwater types This section presents the fitted curve for the modulus and phase of the reflection and transmission coefficients as well as the energy dissipation rate obtained from the experimental data pertaining to irregular wave action for non-overtoppable breakwaters (Fig. 1). Because of

39

length restrictions, only the results of two types of breakwater are shown and analyzed in this work: (i) Type C, a mixed breakwater with a berm below or at SWL [FM/h ≤ 1.1] (low and high mound breakwaters, LMB and HMB respectively, and high mound composite breakwater, HMCB) and (ii) Type E, rubble mound breakwater (RMB). However, a complete report of this research with the experimental data for the rest of typologies can be freely downloaded (Vílchez et al., 2015). 5.1. Type C: mixed breakwater with a berm below or at SWL [FM/h ≤ 1.1] (LMB, HMB and HMCB) The construction of a porous submerged berm with width Bb and a relative height from the seabed of FM/h ≤ 1.1 (Type C), increases the relative width of the breakwater and reinforces its dissipation power with respect to the mixed breakwater without a berm. This can eventually cause the wave to break on the front face of the breakwater and on the berm. The reflection (modulus and phase) and transmission coefficients as well as the energy dissipation rate vary, depending on the geometric characteristics of the berm (the experimental berm was constructed with granular material of the same diameter as that in the foundation), in accordance with Eq. (15):

Y ¼ ½K R ; K T ; ϕR ; D ¼ f Aeq =L2 ; Dk; hb =h; F M =h; Bb =h; H I =L :

ð15Þ

Fig. 5 portrays the variation of Y with the scattering parameter; one caisson foundation depth, hb/h; two berm widths, Bb/h; three berm heights, FM/h; one range of relative diameter, Dk; and two incident wave steepness, HIrms/L. To analyze the influence of the relative diameter, Dk, Fig. 6 shows the variation of Y with the scattering parameter for one berm height, FM/h = 0.75 (HMB), with two berm widths, Bb/h, and two ranges of the relative diameter, Dk. The variation of ϕR (represented by the non-dimensional phase, x0/L) with the scattering parameter is shown for both two ranges of Dk in Fig. 5. They reflect the following: 1. For a relative berm height, FM/h = 0.50 (LMB), and with no wave breaking, the relative berm width (Bb/h), the relative diameter of its granular material (Dk), and the relative height of the superstructure foundation (hb/h) determine the dissipation rate, mostly by friction, and the variation of the reflection and transmission coefficients as X = Aeq/L2 increases. It should be highlighted that for Aeq/L2 N 0.05 the variation of the reflection and transmission coefficient modulus, and energy dissipation rate is negligible. Thus, from an engineering point of view, KR can be regarded as practically constant with a value resulting from the largest expected steepness of incoming waves. 2. For a relative berm height, FM/h = 0.75 (HMB), the dissipation rate increases because few waves break (largest wave steepness). This causes a decrease in the modulus of the reflection coefficient. However, the dissipation rate is mostly due to friction, and the widest berm is more dissipative than the shortest one. In this case, the modulus of KR decreases more rapidly (the dissipation rate increases). Again, for Aeq/L2 N 0.05, KR can be considered constant although its value is lower than the corresponding non-wave-breaking value (FM/h = 0.50). The relative berm geometry, height and width, and the relative diameter control the value of Yi0 and the other parameters of the sigmoid curve. It can be seen (Fig. 6) that the modulus of KT increases with Dk whereas the modulus of the reflection coefficient decreases slightly. 3. For relative berm height FM/h = 1.0 (HMCB), almost all the waves break onto the slope or the berm, which enhances the wave energy dissipation rate by breaking and turbulence. As Aeq/L2 grows, there is a rapid decrease in the modulus of the reflection coefficient (and a corresponding increase in the energy dissipation rate). For Aeq/L2 N 0.15, KR can be considered constant, but its value is much lower than the corresponding non-wave-breaking value (FM/h = 0.50) and strongly depends on the relative berm width. Thus, the

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Fig. 5. Type C (LMB, HMB and HMCB). KR, KT, D* and x0/L as compared to the scattering parameter, Aeq/L2, depending on Bb/h and FM/h, for Dk b 0.06 and hb/h = 0.50. Experimental data and best-fitted curves.

relative berm width controls the value of Yi0 and the shape of the sigmoid curve. Once again, this behavior is weakly dependent on wave steepness. It can be assumed that an HMCB breakwater, con FM/h = 1.0 and Bb/h N 0.625, is totally dissipative for Aeq/L2 N 0.123, independently of wave steepness and with negligible wave reflection and transmission. 4. The general behavior of the non-dimensional phase, x0/L = ϕR/4π, in regard to Aeq/L2 is approximately the same as that of breakwaters with no berm, and follow the same pattern as the modulus. However,

the height of the berm and, to a lesser extent, its width, regardless of wave breaking, tempers the increase of the phase with the scattering parameter. Accordingly, x0/L is the non-dimensional distance between the toe of the breakwater and a reflection point within, which produces the same characteristics as the reflected wave train. Following Sutherland and O'Donoghue (1998), the points in the figure are the experimental values that have been obtained by adding multiples of 2π to the phase though without modifying the results obtained because of the cyclical nature of the phase.

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41

Fig. 6. Type C (HMB). KR, KT and D* as compared to the scattering parameter, Aeq/L2, depending on Bb/h, Dk and hb/h = 0.50. Experimental data and best-fitted curves.

Tables 4 and 5 give the best fit value of the sigmoid curve parameter in terms of the relative berm width (Bb/h), relative berm height (FM/h) and the relative diameter (Dk). Table 6 shows the goodness of fit.

crown wall, the modulus and phase of the reflection and transmission coefficients, as well as the energy dissipation rate, are determined with Eq. (16) (see Fig. 7),

Y ¼ ½K R ; K T ; ϕR ; D ¼ f Aeq =L2 ; Dk; H I =L; Sp ; St :

5.2. Type E: undefined slope breakwater without crown wall (RMB, plane slope) The tilt of the front face of the breakwater may eventually cause the wave to break onto the structure, thus increasing the wave energy dissipation rate. For an undefined plane–slope breakwater without a

ð16Þ

Fig. 7 shows the variation of Y with the scattering parameter; a seaward slope, St, a slope angle, cotαT, and two intervals of incident wave steepness, HIrms/L. Results of two breakwaters are shown: one built

Table 4 Sigmoid curve parameters for obtaining KR and KT. Type C (LMB, HMB and HMCB). Coefficients Parameter

KR

KT

Expression

Bb/h = 0.625

γR

γR ¼ a1R ð F M =hÞ

aX,R

aX;R ¼ b1R ð F M =hÞ

KR0

K R0 ¼ c1R ð F M =hÞ

KR1 γT aX,T

– –

KT,0

K T0 ¼ c1T ðF M =hÞ

KT,1

–

a2R

þ a3R

b2R

c2R

b2T

aX;T ¼ b1T ðF M =hÞ

c2T

þ b3R

þ c3R

þ b3T

þ c3T

Bb/h = 0.250

Dk b 0.06

Dk ≥ 0.06

Dk b 0.06

Dk ≥ 0.06

a1R = 1.512 a2R = 1.820 a3R = 0 b1R = 0.023 b2R = 3.621 b3R = 0 c1R = −0.738 c2R = 5.360 c3R = 0.768 0.95 1 b1T = 0.0053 b2T = 1 b3T = 0.0038 c1T = 0.1332 c2T = −0.1637 c3T = −0.1281 0.30

a1R = 1.767 a2R = 5.777 a3R = 0 b1R = 0.021 b2R = 15.484 b3R = 0 c1R = −0.620 c2R = 4.247 c3R = 0.783 0.95 1.5 b1T = 0.0233 b2T = 1 b3T = 0.0125 c1T = 0.1334 c2T = −0.1635 c3T = −0.1182 0.30

a1R = 1.495 a2R = 4.150 a3R = 0 b1R = 0.026 b2R = 1.866 b3R = 0 c1R = −0.530 c2R = 6.808 c3R = 0.775 0.95 1 b1T = 0.0061 b2T = 1 b3T = 0.0038 c1T = 0.1116 c2T = −0.2290 c3T = −0.1036 0.30

a1R = 1.567 a2R = 6.941 a3R = 0 b1R = 0.028 b2R = 14.524 b3R = 0 c1R = −0.516 c2R = 6.704 c3R = 0.775 0.95 1.5 b1T = 0.0237 b2T = 1 b3T = 0.0118 c1T = 0.1116 c2T = −0.2290 c3T = −0.0936 0.30

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Table 5 Sigmoid curve parameters for obtaining x0/L. Type C (LMB, HMB and HMCB). Parameter x0 L

Expression γϕ aX,ϕ

1.75

(x0/L)0 (x0/L)1

2.15 0

bϕ1 ¼ 0:0412 bϕ2 ¼ 0:0090

aX;ϕ ¼ bϕ1 ð FhM Þ þ bϕ2

Table 6 Goodness of fit of the sigmoid curves for obtaining KR, KT and ϕR. Type C (LMB, HMB and HMCB). Bb/h

FM/h

Dk

εR

R2K T

R2ϕR

0.625

0.50

Dk b 0.06 Dk ≥ 0.06 Dk b 0.06 Dk ≥ 0.06 Dk b 0.06 Dk ≥ 0.06 Dk b 0.06 Dk ≥ 0.06 Dk b 0.06 Dk ≥ 0.06 Dk b 0.06 Dk ≥ 0.06

0.010 0.033 0.024 0.011 0.120 0.012 0.020 0.030 0.007 0.009 0.060 0.070

0.63 0.84 0.71 0.93 0.78 0.92 0.74 0.87 0.90 0.71 0.92 0.87

0.91

0.75 1.00 0.250

0.50 0.75 1.00

0.90 0.91 0.91 0.90 0.91

only with the core, and the other one with the same core material but protected with a main layer of armor stones. They reflect the following: 1. For the undefined sloping breakwater only composed of the core, the modulus of the reflection coefficients decrease with the scattering parameter, Aeq/L2, and for values larger than 0.20, KR and D* can be

regarded as constant. This region is associated with the largest wave steepness and thus waves break on the slope. 2. If one or more upper layers are added to the breakwater section, wave dissipation increases because of the turbulence flow through the armor layer. The addition of a main layer causes the final fit value of the reflection (KR0) to decrease. Again, for Aeq/L2 N 0.20, KR can be regarded as constant, but its value is lower for the protected slope with an armor layer than for the slope built only with the core. Correspondingly, the rate of energy dissipation is also constant but greater for the protected slope. It was found that the experimental values for the interval of the greatest wave steepnesses were slightly lower than those obtained for the smaller wave steepness intervals. Nevertheless, the difference was not sufficiently significant to be taken into account in the practical application. 3. Since in this study wave transmission in these breakwater types was practically negligible, it was not analyzed and the wave energy dissipation rate was computed by using KT = 0. 4. The behavior of the non-dimensional phase follows the same pattern as the previous types of breakwater, but depends on the characteristics of the armor layer, as can be observed in Fig. 7. The presence of the main layer modifies the phase for Aeq/L2 N 0.05, by changing its rate of variation as the scattering parameter grows, at least until Aeq/L2 b 0.4. Larger values of this parameter are usually associated with larger relative breakwater widths. Moreover, dissipation by friction also increases. 5.2.1. Effect of main layer units and porosities in the hydraulic performance For the sake of completeness, this study analyzed the performance of non-overtoppable breakwaters with an armor layer composed of different units and placement density. Two data sources were analyzed: (i) dolosse breakwater (Ruíz et al., 2013) with ns = 0.49 and (ii) cube and cubipod breakwaters (Medina and Gómez-Martín, 2007) with ns = 0.37 and 0.40, respectively. Fig. 8 shows these experimental data

Fig. 7. Type E (RMB with core and core + armor layer). KR, D* and x0/L as compared to the scattering parameter, Aeq/L2, and section type: core (quarry run) and core + armor layer (core protected with material of a uniform grain-size distribution). Experimental data and best-fitted curves.

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43

Fig. 8. Type E (RMB). KR and D* as compared to the scattering parameter, Aeq/L2, and the type of armor unit with the following parameters: core (Benedicto, 2004); S = cubes and S = cubipods (Medina and Gómez-Martín, 2007); and S = armor stones (Benedicto, 2004); S = Dolosse (Ruíz et al., 2013). Experimental data and best-fitted curves, depending on the porosity of the armor layer.

and the best-fitted curves (modulus of the reflection coefficient and wave energy dissipation rate), along with those of the section composed only of a core (Core) or of a core and a rubble layer (Core + armor layer) with armor stones (see Fig. 7). As can be observed, increasing the porosity of the main layer caused the final fit value of KR (KR0) to decrease. Although some of these data depart from the fitted curve, it is also true that the data regarding cubes and cubipods include all the results obtained, namely, those for plane slopes as well as those for fault (or non-planar) slopes. Moreover, Tables 7 and 8 give the best fit value of the sigmoid curve parameters for Type E (RMB) in terms of the presence of the main layer and its porosity (or type of armor unit). Table 9 shows the determination coefficient of the modulus of the reflection coefficients and of the phase of the reflection (when this value is available). These results confirm that for armored non-overtoppable slope breakwaters, the reflection coefficient (modulus and phase) and the wave energy dissipation rate (by wave breaking, turbulence, and porous friction) are accurately represented by a sigmoidal function with a principal variable, the 2D scattering parameter. Moreover the sigmoid curve parameters mainly depend on the porosity and placement density of the armor unit, and weakly depend on the incident wave steepness. 5.3. Comparison with other formulas It is not a simple task to compare the results of this study with those obtained by other researchers. There are two reasons for this. Firstly, the phase of the reflection coefficients is rarely provided in the results of such work. Secondly, some research studies do not show the complete breakwater section, which hinders the calculation of the 2D scattering parameter, Aeq/L2, and blocks the application of the wave energy conservation equation. For undefined sloped breakwaters (non-overtopped), Losada and Gimenez-Curto (1979) and Zanuttigh and van der Meer

(2006) showed that the modulus of the reflection coefficient roughly depends on the Iribarren number, Ir, and type of armor unit. All the formulas predict a larger reflection coefficient for the largest Iribarren number. Moreover, the deviation between the predicted and measured values also increases with Ir. Fig. 9 shows a comparison of the measured and predicted values of KR for an RMB with different armor units. Data with an Iribarren number greater than 4 were considered. Eqs. (17) and (18) show the formulae given by Zanuttigh and van der Meer (2008) and Seelig and Ahrens (1981) respectively. Data with a wave steepness (s0p) value of s0p ≥ 0.01 were selected because Zanuttigh and van der Meer's formula is restricted to this condition.

b K R ¼ tanh a  ξ0

ð17Þ

2

KR ¼

a1  ξ0 2 ξ0

ð18Þ

þ b1

ξ0 is the breaker parameter (Iribarren number) based on the spectral wave period at the breakwater toe. The values of the coefficients are the following: (1) for permeable rock, a = 0.12, b = 0.87, a1 = 0.49 and b1 = 5.456 and (2) for permeable rock with armor units, a = 0.105, b = 0.87, a1 = 0.49 and b1 = 5.456. The data used for the

Table 8 Sigmoid curve parameters for obtaining x0/L. Type E (RMB). Parameter γϕ aX,ϕ (x0/L)0 (x0/L)1

x0 L

Core

Core + armor layer

3.25 0.096 2.15 0

3.25 0.129 2.15 0

Table 7 Sigmoid curve parameters for obtaining KR depending on the presence of main layer and their porosity. Type E (RMB). Table 9 Goodness of fit of the sigmoid curves for obtaining KR and x0/L. Type E (RMB).

Coefficients Parameter

KR

γR aX,R KR0 KR1

Core

Core + armor layer

Dk b 0.06 ns = 0.37 (Cubes) 1.80 0.04 0.35 0.90

ns = 0.40 (Cubipods)

1.60 1.60 aX,R = 0.056 ⋅ ns + 0.023 KR0 = −0.40 ⋅ ns + 0.39 0.85 0.85

Layer ns = 0.48 (Armor stones) 1.60

0.80

ns = 0.49 (Dolosse)

Core Core + armor layer

1.60

0.75

a

The phase value is not available.

R2K R n = 0.37 n = 0.40 n = 0.48 n = 0.49

0.96 0.90 0.90 0.96 0.93

R2ϕ 0.99 a a

0.95 a

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parametric list in a very short time. It also makes it possible to do the following: (i) analyze the sensitivity of the breakwater type to different agents, materials, and geometries and (ii) determine its viability. This section describes the methodology for obtaining the reflection coefficients (modulus and phase), transmission, and dissipation curves, based on the hydraulic performance curves obtained in this research. For this purpose, a practical example is presented, which explains how performance curves can be used to verify certain project requirements or conditioning factors related to the oscillatory regime because of wave–breakwater interaction for non-overtoppable or occasionally overtoppable maritime structures.

6.1. Sequence of application The general application sequence involves doing the following (Fig. 10): Fig. 9. Comparison of KR obtained with the sigmoid function (SF) and the formulas of Zanuttigh and van der Meer (Z&M) and Seelig and Ahrens (S&A). Typology E (RMB) composed of: 1) core and 2) core + armor units.

comparison are included in Appendix A (Table 19). As can be observed, with the exception of the sigmoid curve, none of the formulas matches the experimental values. In fact, they predict an almost constant value. 6. Application of the hydraulic performance curves The application of the logistic sigmoid curve considerably simplifies the characterization of breakwater hydraulic performance in response to wave action. It permits the cost-effective exploration of the entire

1. Selecting the pair of values representative of the design requirements from the joint probability distribution of (HIrms,Tz) at the breakwater location, and choosing the most probable values of the smallest and largest incident wave steepness. 2. Designing a preliminary section of the chosen breakwater type, calculating the equivalent area, Aeq, the corresponding values of scattering parameter, Aeq/L2, and subsequently selecting the nondimensional parameter specific to the breakwater type and to the wave action of the parameter list. 3. Selecting the sigmoid curve parameters for the reflection coefficient (KR1, KR0, γR, aX,R), non-dimensional phase ((x0/L)1, (x0/L)0, γϕ, aX,ϕ), and transmission coefficient (KT1, KT0, γT, aX,T), and then calculating

Fig. 10. Methodological sequence for applying the curves to obtain coefficients, KR, ϕR, KT, and D*.

M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

45

Table 10 Values of the non-dimensional parameter.

Fig. 11. Geometric dimensions.

4.

5.

6. 7. 8.

the evolution of their values, depending on Aeq/L2, and other nondimensional parameters. Calculating the most probable values of the smallest and largest total (incident plus reflected) wave steepness values, [(Aeq/L2)min, (Htrms/L)min], [(Aeq/L2)max, (Htrms/L)max], at the toe of the breakwater. Verifying wave breaking at the front of the breakwater, on the berm, or on the slope, by applying an appropriate breaking criterion, and adjusting, if necessary, the largest total wave steepness values and the sigmoid curve parameters. Verifying that the breakwater is impervious to overtopping (limit freeboard). Calculating the transmitted waves based on the expressions of KT, using Eq. (12). Calculating the dissipation rate of the waves and the variation of other state variables, for example, the root mean square velocity, or pressure, in any location at the front of the breakwater.

Sea state

h/L

HIrms/L

Aeq/L2

hb/h

FM/h

Bb/h

Dk

HIrms, Tzmin HIrms, Tzmax

0.233 0.115

0.044 0.022

0.059 0.015

0.50 0.50

0.50 0.50

0.625 0.625

0.027 0.014

Steps 4 and 5 Largest wave steepness and breaking criterion at the front of the breakwater and on the berm. The values of Htrms are obtained by using the reflection coefficient and phase for each sea state as calculated with the following equation (Losada et al., 1997a): Htrms ðxÞ ¼ HIrms 

ð20Þ

In this work, the maximum wave steepness at the front of the breakwater is estimated by applying the following breaking criterion for partial standing wave trains (ROM, 1.0-09, 2009), ðHtrms =LÞcr ≤ ½ar þ br  fð1−K R Þ=ð1 þ K R Þg  tanhðkhÞ ð21Þ where ar = 0.08 and br = 0.03. Since the critical wave steepness value is never exceeded, there are no breaking waves at the front of the breakwater (Table 13). By following the same procedure, wave breaking on the berm is verified. However, it is necessary to previously calculate the wavelength on the berm. Here the method in Losada et al. (1997b) based on the effective depth of the wave train is used. Values for n = 0.40, s = 1 and f = 3.17 and 6.29 are chosen for sea states 1 and 2, respectively (s is the inertial coefficient and f is the linear friction coefficient [Sollitt and Cross, 1972]). The results are given in Table 14. It is plausible that the sea state with the minimum period causes waves to break on the berm. In regard to the LMB, breaking waves do not significantly modify the reflection coefficient. Nonetheless, this calculation can help to predict if the breaker can reach the caisson and produce impulsive loads. Step 6 Verification of wave overtopping. In this work, to verify that the design breakwater is not overtopped by the waves, the following criterion is applied: the freeboard of the breakwater should be larger than the root-mean-square vertical total displacement of the free surface ηtrms on the wall affected by a coefficient λ (Fc − ληtrms N 0) (Clavero, 2007). The value of ηtrms is calculated with Eq. (20) as Htrms/2. The fact that the wall is located at x = −(Bb + FM · cotαT) from the toe of the breakwater is also taken into account. A value of λ = 2 is used to ascertain whether overtopping occurs in the two sea states. The results are given in Table 15. Step 7 Calculation of transmission coefficient with Eq. (12) and

6.2. Example of application to a non-overtoppable low-mound breakwater (Type C, LMB) In this section, the sigmoid curve is used to determine the wave transformation of the incident wave train resulting from the interaction with a given (preliminary) breakwater design, Step 1 Selecting the pair of values representative of the design requirements from the joint probability distribution of (HIrms,Tz) at the breakwater location. The following sea states impinging on the breakwater were selected: • State 1: HIrms = 3 m; Tzmin = 7 s; normal incidence • State 2: HIrms = 3 m; Tzmax = 12 s; normal incidence. Step 2 Preliminary breakwater design, scattering parameter, and other non-dimensional parameters. An LMB is planned at a depth of h = 16 m, with the following geometric dimensions (preliminary design) (see Fig. 11): hb = 8 m; B = 19 m; FM = 8 m; Bb = 10 m; Fc = 3 m; cotαT = 1.5; D = 0.30 m. The equivalent area, Aeq (red-colored section in Fig. 11), which corresponds to the area of porous material below the still water level, is calculated as follows:   F MT  cotα T  F MT ¼ 280 m2 : ð19Þ Aeq ¼ B  hb þ Bb þ 2 Table 10 shows the scattering parameter and other nondimensional parameters (Aeq/L2, hb/h, FM/h, Bb/h, Dk) for the two sea states, namely, the probable maximum period, Tz (Tzmax), and probable minimum period, Tz (Tzmin), their wavelengths, and the respective steepness of the incident wave train. Step 3 Sigmoid curve parameters for the reflection coefficient, modulus and phase, and transmission coefficient (see Tables 4 and 5). The sigmoid curve parameters are shown in Table 11. The values of KR and ϕR are determined for the two sea states with Eq. (12) (Table 12).

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ K 2R þ 2K R cosð2kx þ ϕR Þ:

Table 11 Sigmoid curve parameters KR, ϕR and KT. Variable

Sigmoid parameter

KR

γR 0.4282 γϕ

aX,R 0.0019 aX,ϕ

1.75 γT 1

0.0296 aX,T 0.0065

x0/L KT

KR0 0.75 ðxL0 Þ0 2.15 KT0 0.0211

KR1 0.95 ðxL0 Þ1 0 KT1 0.30

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M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

Table 12 Values obtained for KR and ϕR of the base curve.

Table 15 Analysis of the presence of overtopping.

Sea state

KR

ϕR

Sea state

ηtrms

Fc − ληtrms

Fc − ληtrms N 0

HIrms, Tzmin HIrms, Tzmax

0.787 0.809

20.84 6.00

HIrms, Tzmax HIrms, Tzmin

1.354 1.115

0.291 0.771

Yes ➔ non-overtopping Yes ➔ non-overtopping

parameters selected in step 3 (Table 16). Step 8 Calculation of wave energy dissipation. D* is calculated with the energy conservation equation (Eq. (10)). The results are shown in Table 17.

Table 16 Sigmoid curve parameters of KT. Sea states

KT

HIrms, Tzmax HIrms, Tzmin

0.0485 0.1071

6.3. Reflection coefficient modulus of various breakwaters in Spain Fig. 12 shows the type and location of various breakwaters along the Spanish coastline defined in Table 18. Their geometric and wave characteristics were obtained from the Atlas de Diques Españoles (Puertos del Estado, 2012). The four parameters needed to define the curves, Y0, Y1, aX, and γ, were taken from the sigmoid curves proposed in this work. Fig. 13 shows the breakwaters and the sigmoid curves for each of the breakwater types analyzed. After the curve values obtained were assigned to the reflection coefficients, it was found that the range of the scattering parameter covered most of the values characteristic of real breakwaters. For the set of breakwaters analyzed, the scattering parameter showed a wide range of variation, {0.005 b Aeq/L2 b 0.4}. The two upper graphs correspond to the design wave period, Td (safety limit states). The two lower graphs were calculated with half that period and are representative of operational limit states. As can be observed, for design conditions, sloping breakwaters (Type E, RMB) could presumably reflect more than 40% of the incident root-mean-square wave height, whereas in normal operational conditions, reflection would lessen to 20–30% of the root-mean-square wave height. Low and high berm breakwaters with a superstructure (LMB and HMB, Type C) would reflect more than 70% of the root-mean-square wave height for design (safety) conditions and operationality. 6.4. Variation of Htrms at the front of the different breakwater types: importance of the phase value The application of the results of this study can be used to determine the variation of the root mean square total wave height (incident and reflected waves) in front of the breakwater, depending on its typology. For that purpose, knowledge of the phase value is essential. Two incident sea states are considered. One state is associated with breakwater safety, ultimate limit state: Tz,s = 14 s, and HIrms,s = 4 m. In contrast, the

Table 13 Analysis of the presence of wave breaking at the front of the breakwater. Sea state

Htrms/L

(Htrms/L)cr

Htrms/L N (Htrms/L)cr

HIrms, Tzmin HIrms, Tzmax

0.043 0.039

0.075 0.052

No ➔ no breaking waves No ➔ no breaking waves

Table 14 Analysis of the presence of wave breaking on the berm.

less energetic state is associated with operational conditions and operational limit state: Tz,o = 7 s, and HIrms,o = 1 m. In both cases, the water depth is h = 15 m and the seabed slope is almost horizontal. The root-mean-square total wave height is calculated at two locations in front of the breakwater: (x = 0), toe of the breakwater; and (x = −20 m), a distance of 20 m of the breakwater toe for each of the breakwater types. All breakwater types have the same stone diameter D = 1 m. The freeboard is such that the possibility of overtopping the breakwater is negligible. As shown in Fig. 14, the breakwater types are represented on the x-axis, whereas the incident and reflected root mean square wave heights at the front of the breakwater are represented on the y-axis. For the sea state with the shortest period, Tz,o = 7 s, and the two locations, the root mean square wave height is slightly higher than that of the impinging waves, except in the case of the rubble mound breakwater (Type E). For the most energetic sea state with the longest period, Tz,s = 14 s, the root mean square wave height at the front of the breakwater at x = 0 decreases to approximately half the HIrms in the case of the Types C and D. However, for Types B and E, there is an increase in the root mean square wave height of roughly 50%. The value of Htrms changes significantly at the other locations (x = −20 m), except in the case of Types A and B, where the breakwater do not have berm and the position have not been changed. This result highlights the importance of the phase value and of its incidence in determining the total oscillation at the front of the breakwater, especially since it is responsible for actions on the structure, such as its overtopping behavior.

7. Discussion This research was based on many other studies that were carried out with the same methodology. For this purpose, the breakwater section was placed in the same location at a distance from the boundaries. The different breakwater types were all made of the same material; the vertical displacement of the free water surface was measured at the same points of the wave flume; and the same separation method was applied to analyze the data. This unity of criteria allowed us to calculate the complex reflection and transmission coefficients, based on simultaneous measurements, which were used to obtain the energy dissipation rate due to wave–structure interaction. Although other researchers have performed similar experiments, to the authors' knowledge, this is the first time that this methodology has been applied to Table 17 Value of D*.

Sea state

kr

hef

Htrms/L

(Htrms/L)cr

Htrms/L N (Htrms/L)cr

Sea state

D*

HIrms, Tzmax HIrms, Tzmin

0.112 0.610

8.323 8.124

0.0530 0.0522

0.061 0.040

No ➔ no breaking waves Yes ➔ breaking waves

HIrms, Tzmax HIrms, Tzmin

0.3781 0.3344

M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

47

Fig. 12. Spanish breakwater location and classifications.

such a wide range of common breakwater types. Because of space constraints, the methodology was not applied to other typologies such as the S-shaped breakwater (Clavero, 2007). For the same reason, only the results for some of breakwater types are shown. The complete list of values of the sigmoid curve parameter of each breakwater type can be downloaded from Vílchez et al. (2015). Despite the large number of experimental tests and the wide range of breakwater types covered, this work is not the end of the journey, but rather the beginning since it opens new avenues of research. Evidently, the results of our study also have limitations stemming from the theoretical background applied to analyze the results, the perpendicular wave incidence, and the experimental setup. Some of these limitations are succinctly addressed in what follows. All of the breakwater experiments were performed with the same slope gradient, cotαT = 1.5. It is well known that the reflection coefficient is only dependent on Ir when the main wave transformation is due to wave breaking by plunging and spilling, generally speaking, when Ir b 1.5. Indeed, it can be assumed that in this range, the modulus

of the reflection coefficient linearly increases with Ir (Losada, 1990). For larger values of Ir, wave reflection becomes the dominant process of wave transformation. As shown in this research, under such conditions, other parameters play a major role in controlling wave–structure interaction. When Ir b 1.5 and breakwater slope 1.5 b cotαT b 2.0, the incident wave steepness HI/L0 ≤ (tan(αT)/Ir)2 is physically limited by generation or by wave instability. As reflected in the results of previous work, for large values of Aeq/L2, when the slope is modified within the previously mentioned interval, a small variation in the value of KR0 should be expected. This can be observed in Fig. 15 (taken from Benedicto, 2004) which shows how the reflection coefficient modulus varies with the relative depth (for a given breakwater typology and constant depth, it is equivalent to parameter Aeq/L2) for different types of wave breaking calculated with the Iribarren number. The experiments were performed on an RMB (core) breakwater impinged by regular waves. Consequently for the analysis of the hydraulic performance of a slope breakwater, the information obtained with the slope cotαT = 1.5 is valid for milder slopes, possibly up to cotαT b 2.5. It should be highlighted that

Table 18 Spanish breakwater characteristics. Number id.

Breakwater name

Typology

Harbor

Sea

Geographic coordinates

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Punta Langosteira breakwater Prioriño breakwater cape Osa breakwater Punta Lucero breakwater Zierbera breakwater North breakwater West breakwater South breakwater Botafoc breakwater Botafoc breakwater South breakwater East breakwater Exterior breakwater Reina Sofía breakwater

Low mound breakwater (LMB) Low mound breakwater (LMB) Rubble mound breakwater (RMB) Rubble mound breakwater (RMB) high mound breakwater (HMB) Rubble mound breakwater (RMB) Low mound breakwater (LMB) Rubble mound breakwater (RMB) Low mound breakwater (LMB) Low mound breakwater (LMB) Rubble mound breakwater (RMB) Low mound breakwater (LMB) Rubble mound breakwater (RMB) Low mound breakwater (LMB)

A Coruña Ferrol Port Gijón Bilbao Bilbao Barcelona Tarragona Valencia Eivissa Eivissa Alicante Motril Algeciras Las Palmas

Atlantic Ocean Atlantic Ocean Cantabric Sea Cantabric Sea Cantabric Sea Mediterranean Sea Mediterranean Sea Mediterranean Sea Mediterranean Sea Mediterranean Sea Mediterranean Sea Mediterranean Sea Mediterranean Sea Atlantic Ocean

43°20′14.3″N 8°29′58.2″W 43°27′15.9″N 8°19′51.5″W 43°33′21.3″N 5°40′53.7″W 43°22′14.8″N 3°05′41.4″W 43°21′31.4″N 3°02′43.9″W 41°21′58.4″N 2°11′22.5″E 41°05′13.7″N 1°12′31.7″E 39°25′25.6″N 0°18′45.0″W 38°54′12.8″N 1°26′55.8″E 38°54′12.8″N 1°26′55.8″E 38°19′16.2″N 0°29′45.2″W 36°42′58.8″N 3°30′54.0″W 36°07′02.8″N 5°26′07.7″W 28°07′26.6″N 15°24′17.0″W

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M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

Fig. 13. Location of the Spanish breakwaters on the fitted curves for the breakwater types analyzed. Aeq/L2 obtained: a) for the design period, Td (upper graphs) and b) for a time period equal to 0.5 · Td (lower graphs).

because of costs and construction-related considerations, the majority of breakwaters designed for harbor protection have steep slopes 1.3 b cotαT b 2.0. Most of the experiments were performed in a wave flume with a horizontal bed. The flume had a wave generation system that controlled wave reflection, but not secondary waves. The sea states were generated with no depth-limited breaking waves and individual waves of low steepness, H/L b b0.1. In these conditions, the wave trains that impinge on and are reflected by the breakwater are reasonably sinusoidal and their interaction is linear. Nevertheless, the description of the sea in this

research is within the framework of linear theory, which uses the concept of the mean energy per unit horizontal surface area of each component. The slope of the continental shelf at the front of the breakwater determines the conditions in which incident and reflected wave trains are propagated. When the slope of the continental shelf is mild, oscillatory movement at the front of the breakwater is the result of the interaction of the incident and reflected wave trains. Klopman and van del Meer (1999) performed experiments in a wave flume with a length of 45 m and width of 1 m. The reflective structure was located at the end of a smooth concrete 1:50 slope in a water depth of 0.50 m. Their

Fig. 14. Comparison of the root mean square wave height at the front of the breakwaters for two sea states and two locations: a) x = 0 and b) x = −20, depending on breakwater type.

M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

49

Fig. 15. Variation of KR with the wave breaking type (Benedicto, 2004).

experiment showed that the changes observed in the wave spectrum and in the significant wave height near the reflective structure agreed with the results of linear theory. Therefore, from an engineering perspective, the experimental data of this research study can be safely applied if the seabed slope in front of the breakwater is milder than 1:60 within a distance of one wavelength from the breakwater. For slopes milder than 1:80, the representativeness of the results should extend to two or more wavelengths in front of the breakwater. This research is based on long-crested incident waves impinging perpendicular to the breakwater. Generally speaking, the influence is slight in the case of a small wave incidence angle of ± 15°, although this depends on the type of breakwater and armor layer (Losada and Gimenez-Curto, 1981). Van Gent (2014) confirmed this result in his analysis of mound breakwater stability under oblique wave attack. This limitation cannot be too restrictive because the main alignment of the breakwater should be oriented parallel to the design wave crests. In addition, perpendicular incidence provides the highest expected value of the reflection coefficients. For an incidence angle in the interval [15° b θ b Brewster angle] which is not for overly oblique waves, it is well known that the reflection coefficient modulus decreases (Dalrymple et al., 1991), depending on the porosity of the medium. When one of the media is dissipative, there is a minimum reflection coefficient, KR,θ, which is known as the principal angle of incidence (Mathieu, 1975). For wave incidence more oblique than the Brewster angle, the linear reflection process does not hold, and the wave crest propagates along the breakwater, and diffraction inside the porous structure takes place (Dalrymple, 1992). This study was carried out on non-overtoppable breakwaters. Nevertheless, generally speaking, breakwaters in port installations are designed so as not to be overtoppable. For this reason, the study of their interaction with incident waves is a necessity in practical engineering. Furthermore, it is well known that overtopping should be significant for it to affect the reflection coefficient (Van der Meer et al., 2005). The curves proposed can be specified in the initial phase of breakwater design based on its hydraulic performance. In this way, it is possible to analyze the sensitivity of the breakwater type to different agents, materials, and geometries. However to determine its viability, and estimate investment costs, it is necessary to calculate the structural response of the different sections and elements of the breakwater in relation to the action resulting from wave–structure interaction (López et al., 2001). This information makes it possible to design efficient breakwaters. 8. Conclusions This work focused on the evaluation of the hydraulic performance resulting from the interaction of perpendicularly impinging water waves on various types of non-overtoppable breakwater. An experimental

setup in a wave flume was designed to test these breakwater types, based on a list of non-dimensional parameters that represent the breakwater geometry, construction materials, and the incoming irregular wave train. This provided smooth performance transitions between different breakwater types and permitted the discrimination of the full spectrum of wave oscillatory regimes resulting from wave– structure interaction for the most common breakwater typologies: (A) porous vertical breakwater (PVB); (B) composite breakwater (CB); (C) mixed breakwater with a berm below or at SWL [FM / h ≤ 1.1] (low and high mound breakwaters, LMB and HMB respectively, and high mound composite breakwater, HMCB); (D) mixed breakwater with a berm above SWL [F M/h N 1.1] (rubble mound breakwater with crown walls, RMB-CW); and (E) rubble mound breakwater, plane slope (RMB). Based on this information, the following conclusions can be derived: (1) The complex wave reflection and transmission coefficients as well as the overall dissipation rate caused by the structure are engineering quantities that can be used to evaluate and compare the hydraulic performance of common types of nonovertoppable breakwaters typologies. The data confirm that the variation of those coefficients (as well as the energy dissipation rate) depends on the dimensions and properties of the parts and elements of the breakwater as well as on the wave characteristics. (2) In this paper, the logistic sigmoid function (Churchill and Usagi, 1972) has been found to define the domain of the hydraulic performance of the most common breakwaters. It relates the modulus and phase of the reflection coefficients, the modulus of the transmission coefficients and the overall energy dissipation rate to the non-dimensional parameters that represent the breakwater geometry, construction materials, and the incoming irregular wave train. It provides smooth performance transitions between different breakwater types and discriminates the full spectrum of wave oscillatory regimes resulting from wave–structure interaction. (3) It is shown that the principal variable of the sigmoid function is a 2D scattering parameter Aeq/L2, where Aeq is the area of a porous medium under the mean water level and L is the wavelength. (4) The curve is defined by the following four parameters: Y0, Y1, aX, and γ. For the reflection process, the largest and smallest values of KR (KR1 and KR0, respectively) are associated with the maximum and minimum values of the reflection coefficient. The value of KR1 defines the breakwater response when the reflection process is dominant. It is usually associated with large wave periods (largest wavelength) and the smallest wave steepness. On the other hand, the smallest value, KR0, determines the reflection coefficient when the energy dissipation rate tends to be large. Generally speaking it

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is associated with short wave periods (shortest wavelength) and the largest wave steepness. The value of Aeq/L2 after which KR can be considered approximately constant and equal to KR0 depends on the typology. Parameters aX, and γ depend on the relative grain size of the porous medium, Dk, and on the nondimensional parameter that represent the breakwater geometry. (5) For a mixed breakwater with a berm below or at SWL, FM/h b =1.1 (Type C), the relative berm geometry, height and width, and the relative diameter Dk control the values of the sigmoid curve parameters. This behavior is weakly dependent on wave steepness. From a practical perspective, it can be admitted that within Type C, a breakwater with a berm height close to SWL (HMCB, FM/h ~ 1.0 and Bb/h N 0.625) will be totally dissipative for Aeq/L2 N 0.123, independently of wave steepness and with negligible wave reflection and transmission. (6) For non-overtoppable slope breakwaters (Type E), the reflection coefficient (modulus and phase) and the wave energy dissipation rate (by wave breaking, turbulence, and porous friction) are accurately represented by a sigmoid function with the 2D scattering parameter as the main variable. Moreover, the sigmoid curve parameters mainly depend on the porosity and placement density of the armor unit, and, weakly, on the incident wave steepness. (7) The application of this method to breakwaters along the Spanish coast showed that the logistic sigmoid curve, along with the values of the relevant parameters can be used for preliminary breakwater design. This considerably simplifies the characterization of the hydraulic performance of the breakwater in response to wave action, and permits the cost-effective exploration of an ample number of alternatives in a very short time. List of notations a Zanuttigh and van der Meer (2008) formula coefficient Seelig and Ahrens (1981) formula coefficient a1 a1i, a2i, a3i fit parameters for γi aX,i inflection point of the sigmoid curve (i = KR, KT, ϕR) coefficients (wave breaking) ar, br area per unit section of a porous medium under the mean Aeq water level relative area of the porous medium under the still water level Aeq/L2 (2D scattering parameter) b Zanuttigh and van der Meer (2008) formula coefficient Seelig and Ahrens (1981) formula coefficient b1 b1i, b2i, b3i fit parameters for aX,i B structure width berm width (Types A, B, C and D)/mound breakwater coronaBb tion width (Type E) B/L relative width of the caisson (1D scattering parameter) relative berm width Bb/h c1i, c2i, c3i fit parameters for Yi0 Cg linear theory wave group speed C wave phase speed D grain size D* wave energy dissipation rate dissipation due to wave breaking including the turbulent flow Da* through the voids of the armor units dissipation due to the shear stresses inside the porous medium Dp* wave energy dissipation rate per unit of time D⁎′ Dk (D/L) relative grain diameter mean special energy per unit of the horizontal area Ei f friction coefficient of the porous medium freeboard Fc wave energy flux (i = I, R, T: incident, reflected, transmitted) Fi berm height (Type C)/distance between bottom and breakFM water coronation (Types D, E) (Fc + h/h) relative height of the breakwater relative berm height FM/h

g H hb hef hb/h h/L H HI HI/L HR Ht HT Hw Ir (Ht/L)cr k KCp Kf KR KR KT KT L m m0 n ns R2 Rep s s0 s0p Sp St T Td Tm − 1.0 U x x0/L yo ye Yi0, Yi1 Yi αT βT γi ε2 ξ0 ϕ, ϕR ϕT η λ ν ρ0 ρ

gravitational acceleration water depth caisson foundation depth effective depth relative caisson foundation depth relative water depth wave height incident wave height wave steepness of the incident wave height reflected wave height total wave height at the front of the structure transmitted wave height total wave height on the wall of the breakwater Iribarren number critical wave steepness to determine wave breaking wave number local Keulegan–Carpenter number of the porous medium Fourier asperity roughness parameter modulus of the reflection coefficient complex reflection coefficient modulus of the transmission coefficient complex transmission coefficient wave length stone mass zero-order moment porosity of the central body of the breakwater porosity of the main layer of the breakwater determination coefficient pore Reynolds number for oscillatory flow inertial coefficient of the porous medium wave steepness related to Tp wave steepness related to Tm − 1,0 type of unit and placement density parameter and number of layers parameter slope profile parameter wave period design wave period spectral wave period at the structure toe volume-averaged ensemble-averaged velocity horizontal axes with origin of coordinates at the toe of the structure non-dimensional phase experimental data (o = observed) theoretical data (e = estimated) final and initial values of the sigmoid curve (i = KR, KT, ϕR) phase-averaged quantity of wave–breakwater interaction (i = KR, KT, ϕR) seaward slope angle leeward slope angle blending coefficient of the sigmoid curve (i = KR, KT, ϕR) mean relative error breaker parameter (Iribarren number) reflection phase transmission phase vertical displacement of the free surface empiric coefficient (overtopping) kinematic viscosity of the fluid stone density water density

Subscripts — Wave statistics max Maximum min Minimum rms Root-mean square s Significant z Mean

M. Vílchez et al. / Coastal Engineering 107 (2016) 34–52

p

Peak

Subscripts — sigmoid function 0 final value of the curve 1 initial value of the curve Breakwater types CB composite breakwater HMB high mound breakwater HMCB high mound composite breakwater LMB low mound breakwater PVB porous vertical breakwater RMB rubble mound breakwater RMB-CW rubble mound breakwater with crown wall Acknowledgments This research was partially funded by the Spanish Ministry of Economy and Competitiveness and the European Union (research project, Método unificado para el diseño y verificacion de los diques de abrigo, REF: BIA2012-37554), the Spanish Ministry of Public Works (research project, Fiabilidad de las estructuras portuarias. REF: P50/08) and the Spanish Ministry of Education and Science (research project, Desarrollo tecnológico de diques en talud con tipología de maxima estabilidad REF: CIT-380000-2007-47). Significant improvements to the original manuscript were suggested by an anonymous reviewer. Appendix A Table 19 Test used for comparison with the formulas of Zanuttigh and van der Meer (Z&M) and Seelig and Ahrens (S&A). Hts (m) (x = 0)

HIs (m)

KR (Z&M)

KR (S&A)

KR (FS)

KR (exp)

Permeable rock 1.5 2.600 1.5 2.559 1.5 1.868 1.5 1.857 1.5 1.429 1.5 1.416 1.5 1.420

0.090 0.104 0.078 0.087 0.037 0.052 0.087

0.058 0.065 0.059 0.068 0.027 0.037 0.064

0.548 0.517 0.455 0.435 0.491 0.432 0.353

0.453 0.448 0.422 0.413 0.435 0.417 0.377

0.632 0.623 0.454 0.452 0.379 0.377 0.378

0.609 0.609 0.472 0.465 0.396 0.408 0.370

Armor units 1.5 1.811 1.5 1.802 1.5 1.617 1.5 1.593 1.5 1.413 1.5 1.395 1.5 1.388 1.5 1.369

0.071 0.078 0.070 0.056 0.080 0.037 0.059 0.049

0.057 0.064 0.068 0.058 0.066 0.034 0.051 0.055

0.411 0.394 0.377 0.407 0.322 0.431 0.359 0.381

0.421 0.413 0.393 0.402 0.372 0.420 0.391 0.383

0.321 0.319 0.279 0.275 0.244 0.242 0.241 0.238

0.355 0.352 0.282 0.282 0.223 0.226 0.252 0.262

cot(αT)

Tp (s)

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