Model description

The OpenFOAM (Open Field Operation and Manipulation) software along with the CFD toolbox is a free and open-source software that contains a set of libraries for solving diverse problems, including those related to fluid dynamics ( ^{OpenFOAM, 2015}).

To evaluate the generation and absorption of surface water waves, ^{Jacobsen, Fuhrman, and Fredsøe (2012)} developed a library called waves2Foam using the relaxation zone technique (active sponge layers) for generating waves based on different analytical theories (i.e. regular waves ENT#091;Stokes-I, -II -VENT#093;, waves as a function of flow and cnoidal and irregular waves). Relaxation zones were implemented to avoid the reflection of waves from the domain borders or the interference of waves with the borders of the wave generation zone - the first relaxation zone. Additionally, the wavePorousFoam library developed by ^{Alcérreca-Huerta (2014)} was applied to porous beach revetments to evaluate the propagation of waves through the interior of one or several porous obstacles. This model integrates the capacities of the waves2foam library concerning the generation and absorption of waves and the simulation of flow through explicitly defined porous structures considering the interaction of two immiscible, isothermal and incompressible fluids (e.g. water and air). To achieve this, wavePorousFoam solves the governing equations of volume-averaged Reynolds average Navier-Stokes (VARANS) and employs the VOF technique to define the interface between fluids. These equations can be consulted in ^{Hsu, Sakakiyama and Liu (2002)}; ^{Del Jesus et al. (2012)}, and ^{Jensen et al. (2014)}. The mathematical form of the VARANS equations used in this study was presented in ^{Alcérreca-Huerta and Oumeraci (2016)} as follows:

Continuity equation

Momentum equation

where P is the pressure, g is the gravitational acceleration, n is the porosity, X is the Cartesian coordinates, ρ is the average weighted density of the fluids according to the volume fraction γ occupied by each pass or fluid in a cell and U is the velocity. Turbulent dynamic viscosity μ _eff is calculated as μ _eff = μ _t + μ, where μ _t is the turbulent dynamic viscosity estimated using the k-ε model (^{Nakayama & Kuwahara, 1999}) and μ is the averaged dynamic viscosity. The term σ _t κ∇_𝛶 represents the surface tension between the two phases, where σ is the surface tension and κ is the curvature of the interface: κ = ∇ ∙ (n / ∣n∣), where n = ∇_𝛶.

The term CT; is defined according to ^{van Gent (1995)} as follows:

The coefficients a and b can be estimated by different mathematical formulas. The coefficient c _A was defined by ^{van Gent (1993)} for stationary and oscillatory flow through porous media as follows, where coefficient 𝛶_vG is equal to 0.34:

Geometry of the transversal section

The calculation of the dimensions of the transversal section of the RMBR-S was performed according to the method proposed by ^{Mendoza et al. (2010)}, which is summarizedat following (Figure 1):

Figure 1 Cross section of the RMBR-S design proposed by ^{Mendoza et al. (2010)} with three slopes and their corresponding values. 

In section a), the restriction of 2 < R2/H < 5 in conjunction with the slope value selected for zone R2 ensures that the highest point of R2 is not above the still water level.

Based on the geometrical design described above, four-wave heights (H = 4, 5, 6 and 7 m) were used to generate the dimensions of four RMBR-Ss. The transversal sections of these designs are shown in Figure 2. The selected slope values for the three regions of R1, R2 and R3 can also be observed.

Figure 2 Numerically modeled dimensions for four RMBR-Ss based on wave heights (H) of 4, 5, 6 and 7 m. 

Porosity

Different values of porosity were considered. In the first stage of the numerical work, only one porosity value (n = 0.45) was considered to obtain values for the reflection coefficient K _r as a function of the Iribarren number ξ to subsequently compare the results with available experimentally measured data. In the second stage of the numerical work, different porosity values were considered, including the porosity mentioned above value of 0.45, to observe the effect of this parameter on the K _r value. At following, the work carried out during these two stages is described in greater detail.

Wave characteristics

The simulated waves were monochromatic and represented by wave heights (H) of 4, 5, 6 and 7 m. Four-wave periods (T) of 9, 10, 11 and 12 s were modeled as well as a complimentary case with a T = 8 s and a H = 7 m. For all sea states, the most appropriate analytical representation of the waves was Stokes fifth-order waves.

Accordingly, a total of 17 sea states were generated from the combination of four-wave heights with four periods in addition to the sea state of H = 7 m and T = 8 s. As previously mentioned, the four-wave heights determine the dimensions of the transversal sections of the RMBR-Ss, which are shown in Figure 2.

Length of the calculation domain

To optimize the simulation time, a sensitivity analysis concerning the length of the calculation domain was performed considering that the time required for modeling increases as the length of the computational domain increases. Different lengths, including 310, 400 and 500 m, were tested under the condition that at least one wavelength be visible, which means that at least one crest and one trough would be present in the domain. The case selected for this analysis had a T = 9 s (shortest wavelength) and a H = 7 m (highest wave height). Also, 40 m were added to the windward side of the breakwater to have enough space to estimate the transmitted waves and to incorporate the relaxation zones. The simulation time for each case was 120 s.

The results were compared according to the dynamic pressure, velocity, and water level values on the face of the RMBR-Ss. The position of the level sensors and the sampling line are shown in Figure 3. The results for dynamic pressure and velocity are shown in Figure 4, and the corresponding results for water level are shown in Figure 5.

Figure 3 Lengths of the calculation domains and positions of the level sensors.  

Figure 4 A comparison of the dynamic pressure at t = 61.5 s along the S profile for three proposed domain lengths is shown in the upper panel. A comparison of the horizontal component of velocity along the S profile is shown in the lower panel.  

Figure 5 Comparison of the results of the level sensors for the proposed calculation domain lengths.  

The results shown in Figures 4 and 5 do not indicate significant variations in the spatial and temporal evolution of pressure, velocity and water level. However, the relative errors for the water surface values at the domain lengths of 440 and 540 m compared to the water surface elevation at a domain length of 350 m varied between 8% and 20%. Based on this observation, it was decided that the final calculation length would be 350 m. The cell size was maintained constant all three analyzed lengths. In particular, the cell size was 1 m at the beginning of the domain and 0.5 m in the vicinity of the breakwater.

Characteristics of the porous medium

To numerically represent a RMBR-S as a continuous porous medium, the values of the coefficients a and b of equation (3) must be known. Several formulas are available in the literature to calculate these coefficients (e.g.^{Ergun, 1952}; ^{Engelund, 1953}; ^{Koenders, 1985}; ^{Shih, 1990}). Version 2.1.0 of OpenFOAM uses the formula of ^{Engelund (1953)} , which was also considered acceptable in the present study.

Equivalent slope

Lastly, the dimensionless parameter that best describes the behavior of the reflection coefficient is the Iribarren number, or the surf parameter ξ (-). To estimate this number in the present case, it is necessary to define the value of an equivalent slope that is representative of the hydrodynamics of the S-shaped profile of the RMBR-S. For this estimation, the procedure proposed by ^{van der Meer (1992)} was used, which considers the incident weight height (H _i ) as a reference measurement unit. The slope of a distance equal to this unit that extends both above and below the average water level is determined. First, lines are drawn parallel lines to the average water level. These lines cross the s-shaped profile, and their points of crossing serve as a reference for drawing the line whose slope is the required equivalent. In Figure 7, an example of this procedure is provided, and an equivalent slope value of 1:2.67 was obtained for the four transversal sections of the RMBR-Ss.

Figure 7 Definition of the equivalent slope for the four proposed transversal sections.  

Reflection coefficient as a function of the Iribarren number

The estimated K _r values are shown in Figure 8 as a function of the Iribarren number.

Figure 8 Reflection coefficient as a function of the Iribarren number.  

From Figure 8, the following can be concluded:

Reflection coefficient as a function of porosity

In Figure 9, the results of K _r are shown as a function of porosity. The porosity was varied from nearly impermeable (n = 0.05) to very permeable (n = 0.85). In the obtained results, it is clear the K _r value is higher at lower porosity values.

Figure 9 Reflection coefficient as a function of porosity.  

Discussion of the numerical results

The obtained numerical results are compared to the reflection coefficients experimentally estimated in ^{Quiñones (2006)} and those presented in ^{Zanuttigh and van der Meer (2008)} in Figure 10. The numerical results are within the range of variation of those obtained in the physical model. For the majority of the numerical results, the associated wave breakage type is plunging, while for the experimental data, mostly surging breakers are indicated. All numerical data in Figure 10 were obtained given a porosity value of 0.45.

Figure 10 The numerical results are compared to those of ^{Quiñones (2006}), and ^{Zanuttigh and van der Meer (2008)} in the left panel. The lowest Iribarren numbers are presented in the right panel.  

The combination of the two numerical cases allowed the Iribarren number and porosity to be included in the calculation of reflection coefficient, which is ultimately dependent on these prior two variables. The consideration of porosity is important given that this value can change throughout the useful life of a breakwater as a result of the movement of pieces in the primary layer or the clogging of interstitial spaces with material traveling in suspension as well as saltation. Hence, the results for the best-fit curves of the models (linear equations were selected) were obtained and drawn to establish a basis for a family of equations that would enable the reflection coefficient of a RMBR-S to be determined as a function of the Iribarren number and porosity (Figure 11). The slope and y-intercept values of this family of lines are shown in Table 1.

Figure 11 Adjustment curve of Kr = f(ξ). The coefficients of the linear model are shown in Table 1. 

To evaluate the performance of the linear equations, the obtained results are compared in Figure 12 with those of other expressions available in the literature, namely those of ^{Losada and Gimenez-Curto (1981)} (eq. (5)); ^{Seelig and Ahrens (1981)} (eq. (6)); ^{Postma (1989)} (eq. (7)), and ^{Zanuttigh and van der Meer (2008)} (eq. (8)), which are presented as following:

The equation of ^{Losada and Giménez-Curto (1981)} generated reflection values that were slightly lower than those of the line estimates in Table 1 (Figure 12, panel A). These latter authors considered porosity but applied their formula to a straight and homogeneous slope. In this case, an average error and standard deviation of 26.1% and 13.5%, respectively, were found. Meanwhile, the equations of ^{Seelig and Ahrens (1981)} generated slightly higher reflection values (see Figure 12, panel B), but the adjustment was acceptable. In this case, an average error and standard deviation of 24.6% and 16.3%, respectively, were found. The equation of ^{Postma (1989)} was similarly compared to the family of lines (see Figure 12 panel C). In this case, an average error and standard deviation of 34.7 and 37%, respectively, were found. The linear equations estimated higher values for the reflection coefficient, yet the data dispersion was not too large. Finally, the equation of ^{Zanuttigh and van der Meer (2008)} was compared to the family of lines. In this case, an average error and standard deviation of 24.3% and 28.5% were observed. This comparison showed the least difference between the generated values.

Figure 12 Comparison of the reflection coefficient values given by the linear equations (Table 1) with those given by equation 5 (panel A), equation 6 (panel B), equation 7 (panel C) and equation 8 (panel D). 

From the observation of Figure 12, it can be concluded that the family of lines summarised in Table 1 is a useful tool for estimating the reflection coefficient as a result of reflection from RMBR-S and other types of breakwaters.