Advance in wave re ection estimation for rubble mound breakwaters: the importance of the relative water depth

Advance in wave reflection estimation for rubble mound breakwaters: the importance of the relative water depth Pilar D´ıaz-Carrasco1∗ , Mads Røge Eldrup2, Thomas Lykke Andersen2 1Department of Mechanical Engineering, Universit´e de Sherbrooke, 2500 boulevard de l’Universit´e, J1K 2R1, Sherbrooke (QC), Canada 2Department of the Built Environment, Aalborg University, Thomas Manns Vej 23, Aalborg Ø, DK-9220, Denmark Abstract The main objective of this research is to present a new formula to estimate the reflection coefficient (KR) for rubble mound breakwaters. New physical model tests were performed for this purpose and existing data was also considered. The evaluation of existing prediction formulas for KRbased on the Iribarren number (ξ) shows that the experimental scatter increases with increasing ξ. This is caused by the wavelength having much greater influence on the reflection than the wave height and thus the use of the wave steepness is inappropriate. A dimensional analysis is performed and the influence of the different dimensionless parameters on wave reflection is studied. Based herein the major dimensionless parameters were found to be the relative water depth (h/L) and the structure front slope angle (α). A new formula to estimate wave reflection for rubble mound breakwaters based on these two parameters is developed. Keywords: Breakwaters, wave reflection, laboratory tests, wavelength, slope angle ∗Corresponding author. E-mail address: [email protected] (P. D´ıaz-Carrasco) Preprint submitted to Elsevier December 30, 2020 1. Introduction Wave reflection from coastal structures causes many problems, such as, dangerous sea states close to harbour entrances, influence on ship navigation in entrance channels and ports, and sediment erosion, which can induce failure modes in maritime structures. It is common in engineering practice to characterize the magnitude of wave reflection by means of the reflection coefficient KR, which is defined as the ratio of the reflected to the incident spectral significant wave height. A considerable number of studies were carried out to investigate wave reflection of normally incident waves on smooth structures, rock slopes (with permeable and impermeable core), and slopes with all kind of artificial armour units. A detailed discussion is given in Zanuttigh and Van der Meer (2006, 2008). Most of the formulas are based on the Iribarren number ξ(Iribarren and Nogales, 1949) for all types of slopes in design conditions. For smooth slope, Battjes (1974); Ursell et al. (1960); Seeling and Ahrens (1981) improved the Miche (1951)’s empirical equation to estimate KRby redefining it in terms of the peak Iribarren number ξp=tan(α) √Hm0/L0,p . Zanuttigh and Van der Meer (2008) refitted the coefficient (a1,b1) in the Seeling and Ahrens (1981) formula using an Iribarren number as a function of the spectral energy wave period at the structure toe, Tm,−1,0, instead of the peak period. For rough permeable slopes, the formula depend not only on Iribarren number (Losada and Gim´enez-Curto, 1981; Postma, 1989), but also on some geometric parameters of the breakwater. For example, Seeling and Ahrens (1981) assumed that wave reflection is also a function of the toe depth, the offshore seabed slope, the armour characteristics and the number of armour layers; Van der Meer (1988, 1992, 2005) introduced the notational permeability factor, P, and the slope angle in the analysis of the wave reflection; Davidson et al. (1996) developed a prediction scheme that weights the contributions of wavelength, wave height, water depth, slope angle and diameter of the main armour layer. To improve the wave reflection predictions, Zanuttigh and Van der Meer (2006, 2008) refitted the coefficient in the Seeling and Ahrens (1981) formula according 2 to the slope type and the amour units, using the spectral energy period in the expression of Iribarren number. They also developed a new formula with the roughness factor included and thus reflection and overtopping processes were linked. Furthermore, Zanuttigh and Lykke Andersen (2010) examined the dependence of the reflection coefficient on wave directional spreading and obliquity. The formulas to estimate the KRbased on Iribarren number fit the general tendency of the reflection coefficient, but the scatter tends to increase as the value of ξincreases. Despite this scattering, wave reflection estimations for the design of rouble mound breakwaters are usually based on these prediction formula and thus a high degree of uncertainty is introduced. Over the last thirty years, research studies have questioned the dependence of KR=f(ξ). Hughes (1993) and Sutherland and O’Donoghue (1998) applied the parameter xm/L (where xm=h/ tan(α)) to also quantify the phase of the reflected wave train. Benedicto (2004) analysed wave reflection, depending on h/L and grain size. V´ılchez et al. (2016a) modifies the Iribarren number to incorporate, grain diameter, and the width and depth of the breakwater core in a single parameter. Finally, D´ıaz-Carrasco et al. (2020) collated the dependence of wave energy transformation processes (reflection, transmission and dissipation) with Iribarren number. For that they applied a dimensional analysis, which showed that the scattering of the experimental results with ξmight partly be explained by missing the effects of the porous media, the wave breaking on the slope and the wave dissipation in the main armour layer (Clavero et al., 2020). Consequently, in order to extend the knowledge of wave reflection on rubble mound breakwaters, the present paper addresses the following steps: (1) evaluation of the existing formulas that predict the wave reflection coefficient based on the Iribarren number, (2) to study the dependence of the wave reflection with dimensionless parameters that includes the wave train parameters and the geometric configuration of the breakwater, and (3) to develop a new method to estimate the wave reflection coefficient that improves the existing ones and reduces the scattering. In order to fulfil these objectives laboratory experiments were carried out in the wave flume of Aalborg University. Different physical 3 models were tested to cover the parameters that describe the wave motion and the geometric configuration of the breakwater. 2. The experimental dataset 2.1. The tests Four configurations of a conventional rubble mound breakwater were performed in the Wave Flume of Aalborg University. The models were built with a main armour layer of rocks or cubes and a core composed by finer rock and most models also included a filter layer (see Fig. 1). Table 1 summarizes the variations between the four breakwaters models, which mainly collect changes in: (a) the total area of the rubble mound, (b) the material of the main amour layer and the core, (c) the existence or not of a filter layer, and (d) two front slope angles. All the configurations had a core height of FMT,c = 0.55 m and the landward slope was armoured by rocks with Dn50 = 43.4 mm and a slope angle cot(β) = 1.5. The density of cubes and rocks were ρs= 2.3 and 2.62 tn/m3, respectively. The thickness of the armour and filter layers were 2Dn50, being Dn50 the characteristic diameter of each layer, respectively. FMT,c 1 1.5 Bb h=0.4 m β α Dn50a Dn50f Dn50c FMT PRODUCED BY AN AUTODESK STUDENT VERSION PRODUCED BY AN AUTODESK STUDENT VERSION PRODUCED BY AN AUTODESK STUDENT VERSION PRODUCED BY AN AUTODESK STUDENT VERSION Figure 1: Model cross-section of the conventional rubble mound breakwater tested. The model 1A was the base-model and the modifications made on it were in order to quantify the influence on the wave reflection from the breakwater configuration: the amour layer (model 1B), the filter and core size (model 1C), the porous media area (model 2A), and the angle of the front slope (model 3A). Irregular wave conditions were tested and shown in Table 2. Water depth was kept constant and equal to h= 0.4 m. Tests were done in series with 4 Configurations Bb(m) cot(α) Armour Core Dn50,c (mm) Filter Dn50,f (mm) Aeq (m2) Model 1A 3Dn50,a 1.5 Rock Dn50,a = 43.9 mm 5.8 15 0.6140 Model 1B 3Dn50,a 1.5 Cubes Dn= 40 mm 5.8 15 0.6034 Model 1C 3Dn50,a 1.5 Rock Dn50,a = 43.9 mm 15 No filter 0.5780 Model 2A 9Dn50,a 1.5 Rock Dn50,a = 43.9 mm 5.8 15 0.7194 Model 3A 1.25Dn50,a 3 Rock Dn50,a = 43.9 mm 5.8 15 0.8640 Table 1: Geometric configurations of the rubble mound breakwaters tested. Bbis the armour crown width of the breakwater; αis the front armour slope angle; Dn50,a,Dn50,c and Dn50,f are the nominal diameter of the armour, core and filter layers, respectively; Dnis the cube side length; and Aeq is the area of the porous media under the still water level. Test Conditions Sop Hm0(m) Tp(s) Wave generation S1 0.04 1.6 2nd order S2 0.06 1.96 2nd order S3 0.01 0.08 2.26 Ad-hoc unified S4 0.10 2.53 Ad-hoc unified S5 0.12 2.77 Ad-hoc unified S6 0.04 1.13 2nd order S7 0.06 1.39 2nd order S8 0.02 0.08 1.6 2nd order S9 0.10 1.79 2nd order S10 0.12 1.96 Ad-hoc unified S11 0.04 0.86 2nd order S12 0.06 1.05 2nd order S13 0.035 0.08 1.46 2nd order S14 0.10 1.35 2nd order S15 0.04 0.75 2nd order S16 0.045 0.06 0.92 2nd order S17 0.08 1.17 2nd order Table 2: Overview of the wave conditions tested in the Wave Flume. Sop is the wave steepness, Hm0is the significant spectral wave height and Tpis the peak wave period. 2nd order generation refer to Sch¨affer (1996) with modification by Eldrup and Andersern (2019), while ad-hoc unified generation refer to Zhang et al. (2007) . 1,000 waves per test and using the JONSWAP spectrum spectrum with a peak enhancement factor of γp= 3.3. In each series the wave height was increased in steps of 0.02 m, but maintaining a constant wave steepness Sop =Hs/Lop. Waves were non-breaking on the foreshore and thus breaking only occurred due to wave-breakwater interaction. The structures were non-overtopped and 5 armour damage was limited (SD<3). 2.2. Wave measurements and data processing The model tests were performed in a wave flume of dimensions 22.1 x 1.5 x 1.5 m (length x width x depth) at Aalborg University. An array of five resistance type wave gauges were placed with a distance of 0.4Lp(Lpbeing the peak wavelength) (Klopman and Van der Meer, 1999) to the structures to separate incident and reflected waves and an additional wave gauge was located at the toe of the breakwater (see Fig. 2). Irregulars waves were generated by a piston wavemaker with steering signals generated by the AwaSys7 software using the random phase method (Aalborg University, 2007). Surface elevation signals were bandpass filtered to remove energy outside 0.05 Hz to 5 Hz in model scale. The method of Eldrup and Lykke Andersen (2019a), which is the extension of Lykke Andersen et al. (2017)’s method to irregular waves, was applied to calculate the incident and reflected wave spectra and the time domain incident and reflected wave trains using the WaveLab3 software package (Aalborg University, 2015). Figure 2: Longitudinal section of the wave flume. The dimensions are in cm. 3. Analysis of the results The analysis of the results has been organized in the following steps: first, a comparison of the experimental data with selected existing formulas is presented; then, the dimensionless parameters that describe the wave-breakwater interaction are redefined following V´ılchez et al. (2016a) and D´ıaz-Carrasco et al. 6 (2020). Finally, according to the previous analysis, a new formula is proposed in Section 4. 3.1. Application of exiting wave reflection formulas In this section, the experimental results of KRare compared with the formulas collected in Table 3. As mentioned in the Introduction, most of the formulas are based on the Iribarren number with the peak wave steepness, ξp. However, preliminary investigations on the use of ξ0=f(T−1,0) in the reflection analysis were carried out by Van der Meer (2005); Dekker et al. (2009). They noticed that the parameter ξwith the energy period is useful for representing bi-modal spectra or shallow water with flat spectra, where for instance a peak period is not well defined. Reference KRCoefficients Seeling and Ahrens (1981) a1·ξ2 p ξ2 p+b1a1= 0.6 for rough slope b1= 6.6 Van der Meer (1992) P= 0.4 with filter 0.07(P−0.08 +ξp,vdm) Van der Meer (1992) P= 0.5 without filter Seeling and Ahrens (1981) a1·ξ2 0 ξ2 0+b1a1= 0.75 refitted by Zanuttigh and Van der Meer (2006) b1= 15 Zanuttigh and Van der Meer (2006, 2008) tanh(a·ξb 0)a= 0.12 for rock and cubes armor layer b= 0.87 Muttray et al. (2006) ∗∗ 1 1.3+3h2π L0p for rubble mound breakwaters Calabrese et al. (2010) ∗∗ KR0+r r =m·exp(−30.82S0p−1.3P) for rock armor layer KR0=f(m, h/L0p) Table 3: Formulas selected for the prediction of wave reflection coefficient KR.ξp= tan(α)/q2πHm0/gT 2 pand ξ0= tan(α)/q2πHm0/gT 2 −1,0are the Iribarren numbers obtained from the peak wave period and the spectral energy period, respectively; ξp,vdm = tan(α)0.62/(Hm0/L0p)0.46 is the peak Iribarren number adapted to the formula proposed by Van der Meer (1992); Pis the notational permeability factor defined by Van der Meer (1988); m= tan(α) is the slope; and L0pis the deep water peak wavelength. (**) The formulas of Muttray et al. (2006) and Calabrese et al. (2010) are compared in section 5. Fig. 3 show the comparison of KRexperimental data results for all the breakwaters models tested with the formulas written in Table 3. By looking at 7 1 1.5 2 2.5 3 3.5 4 p 0 0.2 0.4 0.6 0.8 1 KR Model 3A S&A 1981 345678 p 0 0.2 0.4 0.6 0.8 1 KR Model 1A Model 1B Model 1C Model 2A S&A 1981 (a.1) (a.2) 1 1.5 2 2.5 3 3.5 4 0 0 0.2 0.4 0.6 0.8 1 KR Model 3A Z&VdM 2006 S&A 1981 refitted by Z&VdM 345678 0 0 0.2 0.4 0.6 0.8 1 KR Model 1A Model 1B Model 1C Model 2A Z&VdM 2006 S&A 1981 refitted by Z&VdM (b.2) (b.1) 1 1.5 2 2.5 3 3.5 4 p,vdm 0 0.2 0.4 0.6 0.8 1 KR Model 3A VdM filter 345678 p,vdm 0 0.2 0.4 0.6 0.8 1 KR Model 1A Model 1B Model 2A VdM filter (c.1) (c.2) 345678 p,vdm 0 0.2 0.4 0.6 0.8 1 KR Model 1C VdM no filter (c.3) Figure 3: Comparison of KRexperimental data results for all the breakwaters tested with the formulas from existing studies for rough permeable slopes based on: (a) Iribarren number with the peak wave period, Tp, (b) Iribarren number with with the energy period, T−1,0, and (c) Iribarren number adapted to the formula proposed by Van der Meer (1992). 8 Fig. 3 one can observe that, •Seeling and Ahrens (1981) overestimate the measured values of KRfor Iribarren numbers ξ > 2, leading to R2= 0.42, ERMS = 10.3% for rocks armour units (models 1A, 1C and 2A), R2= 0.43, ERMS = 8.71% for cubes armour units (model 1B) and R2= 0.43, ERMS = 7.52% for the slope angle 1:3 (model 3A). •Van der Meer (1992) also tends to overestimate KRbut for higher Iribarren numbers than observed for the Seeling and Ahrens (1981)’s formula. The agreement both for the structures with and without filter is very poor (Fig. 3a), being for the slope 1:1.5: R2= 0.48, ERMS = 7.51% with filter and R2= 0.33, ERMS = 8.45% without filter. For the slope 1:3 with filter the agreement is R2= 0.46, ERMS = 3.95%. •The agreements are better with ξ0but Fig. 3b shows that, in this case, the formulas slightly underestimate the measured values of KR. The fit of Zanuttigh and Van der Meer (2006, 2008)’s formula has R2= 0.65, ERMS = 7.71% for rocks armour units, R2= 0.42, ERMS = 9.15% for cubes armour units and R2= 0.21, ERMS = 4.93% for the slope angle 1:3. The Seeling and Ahrens (1981)’s formula refitted by Zanuttigh and Van der Meer (2006, 2008) improved the agreement for rock and cubes armour units (R2= 0.53, ERMS = 6.62% and R2= 0.42, ERMS = 6.99%, respectively), but not for the slope angle 1:3 (R2≈0, ERMS = 8.82%). It is clearly observed that the formulas for predicting the wave reflection coefficient do not take into account the composition of the breakwater and the wave parameters. All formulations show a poor agreement both for ξpand ξ0. As pointed out in Clavero et al. (2018); D´ıaz-Carrasco et al. (2020), the Iribarren number or surf similarity parameter (Iribarren and Nogales, 1949; Battjes and Groenendijk, 1999) does not serve as the sole determining factor of the wave transformation on rubble mound breakwater. In fact, Fig. 3 shows that the scattering increases with increasing ξ. Consequently, next section presents 9 •Its shape can reproduce the reflection from all the tested rubble mound breakwater geometries. •KRrapidly decreases with the wavelength (or period) until a constant values is achieved (saturation regime), where other processes acquire importance inside the breakwater; that is, wave propagation and transmission inside the porous media. •The fit parameters a,γ,KR0and KR1depend on the geometry configuration of the breakwater. •Table 4 show the good agreement between the wave reflection coefficients and the relative water depth. KR0KR1a γ R2ERMS Model 1A 0.29 0.8 0.104 3.47 0.97 2.1 % Model 1B 0.31 0.8 0.108 3.68 0.98 1.6 % Model 1C 0.29 0.8 0.091 3.75 0.96 2.8 % Model 2A 0.26 0.8 0.095 3.70 0.99 2.0 % Model 3A 0.20 0.8 0.06 3.47 0.88 1.8 % Table 4: Values of the fit sigmoid function parameters for each breakwater model tested. The determination coefficient, R2, and the root-mean-square error, ERM S , are calculated to evaluate to goodness-of-fit. Fig. 6 and Table 4 highlight that it is possible to fit a sigmoid function for each armour front slope slope angle (Fig. 7). The optimal γvalues are almost identical for all tested models, but KR0and adepend on the slope angle and to less degree the configuration of the armour layer and porous media (see Table 5). KR0KR1a γ R2ERMS slope 1:1.5 0.3 0.8 0.098 3.73 0.93 3.1 % slope 1:3 0.20 0.8 0.06 3.47 0.88 1.8 % Table 5: Values of the fit sigmoid function parameters for each slope angle tested. The determination coefficient, R2, and the root-mean-square error, ERMS , are calculated to evaluate to goodness-of-fit. The sigmoid function gets a global improvement to predict KR=f(h/L) for 16 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 KR 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Model 1A Model 1B Model 1C Model 2A Model 3A Sigmoid function 1:1.5 Sigmoid function 1:3 h/L Figure 7: The new method to predict KR: sigmoid function that fits the wave reflection coefficients against h/L for each slope angle tested tested. rubble mound breakwaters with respect to the existing formulas. It seems that the number of Iribarren is not a good indicator to estimate the wave reflection coefficient since the relative water depth influences the reflection much more than the wave steepness. The main dimensionless parameter seems instead to be h/L and tan(α). However, as indicated above, only a constant water depth has been tested. The experimental tests of Eldrup et al. (2019); Eldrup and Lykke Andersen (2019b) are analysed below to further verify the sigmoid function with h/L as the independent variable, and to study the dependency of the parameters aand γwith the breakwater geometry. 4.1. Effect of water depth and front slope Eldrup et al. (2019) provided notional permeability factors, P, (Van der Meer, 1988) for seven layers compositions of rubble mound breakwaters. In these tests, the water depth was constant, h= 0.5 m, and three front slope angles were tested, cot(α) = 1.5, 2 and 3. Eldrup and Lykke Andersen (2019b) studied the effect of nonlinear waves on rock armour stability. For that, they tested five different cross-sections of rubble mound breakwater with an armour layer and a filter layer, two front slopes angles, cot(α) = 2 and 3, and several water depths, h=[0.29, 0.5, 0.55, 0.6, 0.64, 0.7] m. 17 Fig. 8 gathers the wave reflection coefficients for the experimental tests of Eldrup et al. (2019) and Eldrup and Lykke Andersen (2019b) against the relative water depth, h/L. It can be seen that for all the water depths tested, h, and also between different laboratory tests, a sigmoid function with the independent variable h/L is fitting well to each slope angle. Thus, this verifies h/L as the proper dimensionless wavelength and not the other parameters studied. The fitted sigmoid function parameters for each slope angle are also marked in Fig. 8. 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 0.8 1 KR Slope 1:1.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.2 0.4 0.6 0.8 1 KR Slope 1:2 0 0.05 0.1 0.15 0.2 0.25 0.3 h/L 0 0.2 0.4 0.6 0.8 1 KR Slope 1:3 a = 0.11 𝛾 = 3.16 𝑅!= 0.92 RMS = 4% a = 0.09 𝛾 = 3.55 𝑅!= 0.96 RMS = 2.7% a = 0.064 𝛾 = 3.29 𝑅!= 0.93 RMS = 3% 𝐾"# = 0.2 𝐾"$ = 0.8 𝐾"# = 0.2 𝐾"$ = 0.8 𝐾"# = 0.2 𝐾"$ = 0.8 Figure 8: Wave reflection coefficients obtained from Eldrup and Lykke Andersen (2019b) (circles) and Eldrup et al. (2019) (triangles) for each slope angle tested against the relative water depth, h/L. The solid line represents the sigmoid function fitted for each slope angle tested by grouping the tests of Eldrup and Lykke Andersen (2019b); Eldrup et al. (2019). The analysis of the sigmoid functions fitted to the experimental tests in this work (Tables 4 and 5) and the tests of Eldrup et al. (2019) and Eldrup and 18 Lykke Andersen (2019b) (Fig. 8), shows that a single γparameter can be set for all the slope angles tested; but, is it possible to fit a single sigmoid function for all the experimental results of KR. 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1 KR 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.2 0.4 0.6 0.8 1 KR a = 0.15 𝛾 = 3.50 𝑅!= 0.90 RMS = 4% a = 0.18 𝛾 = 3.55 𝑅!= 0.96 RMS = 2.7% 𝐾"# = 0.27 𝐾"$ = 0.8 𝐾"# = 0.2 𝐾"$ = 0.8 Slope 1:1.5 Slope 1:2 a = 0.19 𝛾 = 3.46 𝑅!= 0.94 RMS = 4% 𝐾"# = 0.2 𝐾"$ = 0.8 Slope 1:2 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 KR Slope 1:3 h/(L tan( )) Figure 9: Wave reflection coefficients obtained from the experimental tests of this study (red circles), and the tests of Eldrup et al. (2019) and Eldrup and Lykke Andersen (2019b) (grey triangles) against h/(Ltan(α)) for each slope angle tested. The solid lines represent the sigmoid functions. Fig. 9 gathers the wave reflection coefficients, KR, obtained in this work and also the KRdata of Eldrup et al. (2019) and Eldrup and Lykke Andersen (2019b) 19 against a parameter that includes the relative water depth and the slope angle: h/(Ltan(α)). This parameter is a measure of the phase difference of reflections from the lower and upper part of the slope. Fig. 9a shows that by including the slope in the independent variable, the parameter avaries slightly between slopes tested, but KR0depends on the front slope angle, α. This asymptote will also define the shape of the sigmoid function, γ. Hence, it is possible to simplified in one aand γconstants for all the experimental tests with the independent variable h/(Ltan(α)) but not for the asymptote KR0=f(α). Fig. 9 also shows that for small wavelength values (short periods) the reflection achieved a near constant value (saturation regime) and the dissipation dominates by wave breaking on the slope. Wave breaker type depends on the slope angle, α, which defines the asymptote of the sigmoid curve KR0. 4.2. Effect of the porous media The presence of a porous media is relevant to the hydrodynamic performance of the breakwater because it determines the phase lag between the incident and reflected wave trains and its impact on breaker type (D´ıaz-Carrasco et al., 2020). The propagation of a wave train (regular or irregular) through a porous media is well analysed both theoretically (Sollitt and Cross, 1972; Dalrymple et al., 1991) and through physical and numerical experimentation. The dual behaviour of the reflection-dissipation system during train propagation in the porous medium has four aspects relevant: (1) reflection increases and dissipation decreases as a function of the relative core diameter and the core Reynolds number (Dn50,c/L, Rec); (2) the magnitude of the reflected flow tends to saturation as a function of its relative characteristic width, B∗/L (Requejo et al., 2002; Benedicto, 2004); (3) the Forcheimer equation reasonably well represents the momentum equation in the porous media, with coefficients depending on the Reynolds and Keulegan- Carpenter numbers (Rec,KC) (Van Gent, 1995); and (4) the application of Lorentz’s hypothesis allows the analysis of scale effects (P´erez-Romero et al., 2009), with a single friction coefficient that depends on Dn50c/L. Fig. 10 represents the wave reflection coefficients against the relative water 20 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0 0.2 0.4 0.6 0.8 1 KR Slope 1:1.5 Core Dn50,c=5.8 mm Core Dn50,c=15 mm (a) 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.2 0.4 0.6 0.8 1 KR Slope 1:2 Core Dn50,c=5.8 mm Impermeable Core (b) 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 h/L 0 0.2 0.4 0.6 0.8 1 KR Slope 1:3 Core Dn50,c=5.8 mm Impermeable Core (c) Figure 10: Wave reflection coefficient for all the experimental tests (tests of this study, Eldrup et al. (2019) and Eldrup and Lykke Andersen (2019b)) against h/L as a function of the diameter of porous media, Dn50,c, and the slope angle. depth for each slope angle and separated by the permeable core diameter, Dn50,c and the impermeable core. Fig. 10 shows that for a permeable core, data are not segregated by core size although some dispersion is observed; especially when reflection and dissipation dominate, called the transition region, in which more breaker types are observed (D´ıaz-Carrasco et al., 2020). The KRdata have different behaviour if the core is permeable or impermeable. Although two different sets of data are observed, the sigmoid function that best fits the permeable and impermeable core is the same. 21 5. Proposed formula: the logistic sigmoid function A summary of the proposed formula, fsigmoid(h/L, α), and the fit parameters for the experimental tests are given below. KR= (KR1−KR0)h1 + xs aγi−1 +KR0;xs>0 (2) where xsis the independent variable of the sigmoid function, xs=h/(Ltan(α)), and the fit sigmoid parameters are gathered in Table 6. KR0KR1a γ Slope 1:1.5 0.27 0.8 0.15 3.50 Slope 1:2 0.2∗∗ 0.8 0.18 3.55 Slope 1:3 0.2 0.8 0.19 3.46 Table 6: Values of the fit sigmoid function parameters for each slope angle with xs= h/(Ltan(α)). (∗∗ ) Note that the tests of Eldrup et al. (2019) and Eldrup and Lykke Andersen (2019b) cover longer periods, that is, lower values of h/L (x-axis). Hence, the value to which KR0tends for each slope tested may not be collected. KR0depends on the front slope and maybe also the wave steepness; the best fit values from Table 6 is suggested for this parameter. The other parameters may be regarded as constant as the variation with front slope is small and may be natural scatter. Thus for application of the present method KR1= 0.8, a= 0.18 and γ= 3.5 is suggested. Muttray et al. (2006) and Calabrese et al. (2010) proposed two different formulas that also related the wave reflection coefficient with the relative water depth (see Table 3). Fig. 11 presents KRmeasured in laboratory against KRcalculated with Muttray et al. (2006)’s formula, Calabrese et al. (2010)’s formula and the proposed sigmoid function with h/(Ltan(α)) (Figs. 11a, 11b and 11c, respectively). The formula of Muttray et al. (2006) underestimate KR for the models with slope 1:1.5 and overestimate those with the slope 1:3. By contrast, Calabrese et al. (2010)’s formula underestimate the values of KRfor 22 the breakwater models with main armour layer composed by rocks. Fig. 11c includes the last accepted formula for estimating KR; that is, the formula of Zanuttigh and Van der Meer (2006, 2008) based on ξ. It can be observed that although Zanuttigh and Van der Meer (2006, 2008)’s formula better estimates KRvalues compared to Muttray et al. (2006) and Calabrese et al. (2010), the dispersion of the data is large with an underestimation. 0 0.2 0.4 0.6 0.8 1 KR from Muttray and Oumeraci (2006) 0 0.2 0.4 0.6 0.8 1 KR,measured 0 0.2 0.4 0.6 0.8 1 KR from Calabrese et al. (2010) 0 0.2 0.4 0.6 0.8 1 KR,measured slope 1:1.5 slope1:2 slope 1:3 (a) (b) 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 KR, measured 0 0.2 0.4 0.6 0.8 1 KR from Zanuttigh and Van der Meer (2006, 2008) 0 0.2 0.4 0.6 0.8 1 KR, measured (c) (d) KR1 = 0.8 a = 0.18 K from sigmoid function h/(Ltan( )) R = 3.50 Figure 11: Comparison of KRexperimental data with KRestimated by: (a) Muttray et al. (2006)’s formula, for all the breakwater models; (b) Calabrese et al. (2010)’s formula, for rocks armour models 1A, 2A and 3A, and tests of Eldrup et al. (2019) and Eldrup and Lykke Andersen (2019b); (c) Zanuttigh and Van der Meer (2006, 2008)’s formula based on ξ; and (d) the proposed sigmoid function for all the breakwater models with the proposed design parameters. Fig. 11d shows that the proposed sigmoid function, f(h/L, α) fits the data very well and much better than the other predicting formulas. However, there is 23 still a small dispersion of the data may be due to: (i) the flow inside the porous media, f(Dn50,c, porosity); (ii) the breaker type on the slope, with greater dispersion in the transition zone, when both reflection and dissipation processes dominate (D´ıaz-Carrasco et al., 2020), and (iii) the roughness and turbulent flow inside the armour layer. 6. Conclusions The main objective of this research is to present a new formula to estimate the reflected wave energy in rubble mound breakwaters based on the analysis of dimensionless parameters that includes the wave train and the wave-structure interaction. For that, four physical configurations of a non-overtoppable rubble mound breakwater were performed in the 2D wave flume of Aalborg University. In addition, the experimental data of (Eldrup et al., 2019) and (Eldrup and Lykke Andersen, 2019b) were collected for this work to quantify the effect of the front slope angle and the water depth. The existing formulas that predict the wave reflection coefficient based on Iribarren number were also evaluated with the experimental results. The following conclusions can be derived from this study: 1. When considering the Iribarren number as the main parameter for the reflection coefficient the experimental scatter is significant and increases with increasing Iribarren number ξ(lower values of wave steepness, H/L). 2. The analysis of the wave train and wave-breakwater interaction parameters showed that there is a relationship between the reflection coefficient and the relative water depth, h/L and the front slope angle. The wave steepness, H/L, included in the Iribarren number, seems only to be relevant when the energy dissipation dominates. 3. A sigmoid function with the independent variable h/L fits reasonably well the experimental results of KRfor each slope angle tested. 4. The experimental tests of Eldrup et al. (2019) and Eldrup and Lykke Andersen (2019b) show that the sigmoid function with h/L fits the results of 24 KRfor all the water depths tested and also between different laboratory tests. 5. The inclusion of slope angle in the sigmoid function as the independent variable h/(Ltan(α)) allows setting constants fit parameters, aand γ, for all experimental results, but KR0depends on the slope angle. A wider set of slope angles needs to be tested to give a prediction formula for this parameter. 6. The slope is a dimensionless parameter that characterizes the structure and orders the results, i.e. it is a plotting parameter. Hence, using both the sigmoid function with h/L for each slope or with h/(Ltan(α)) are valid. Acknowledgements The work of the first author was funded by the Spanish Ministry of Education, Culture and Sports (Research Contract FPU14/03570). References Aalborg University, 2007. AwaSys 5 homepage. http://www.hydrosoft.civil.auc.dk/ AwaSys. Aalborg University, 2015. Wave Data Acquisition and Analysis Software - WaveLab 3, Aalborg University. Department of Civil Engineering. http://www.hydrosoft.civil.aau.dk/wavelab/). Battjes, J., 1974. Surf similarity. Proc. XIV International Conference of Coastal Engineering, ASCE , 466–480. Battjes, J., Groenendijk, H., 1999. Shallow foreshore wave height statistics, in: Coastal Structures. Benedicto, M., 2004. Comportamiento y evoluci´on de la aver´ıa de los diques de abrigo frente a la acci´on del oleaje. Ph.D. thesis. University of Granada. 25