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Optimal submergence of horizontal plates for maximum wave energy dissipation
Ocean Engineering 142 (2017) 78–86
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Optimal submergence of horizontal plates for maximum wave energy dissipation
MARK
⁎
Manuel Gerardo Verduzco-Zapataa, , Francisco Javier Ocampo-Torresb, Edgar Mendozac, Rodolfo Silvac, Marco Liñán-Cabelloa, Ernesto Torres-Orozcoa a Facultad de Ciencias Marinas, Universidad de Colima, Carretera Manzanillo-Cihuatlán Km 19.5, Colonia El Naranjo, Manzanillo, Colima CP 28868, Mexico b Departamento de Oceanografía Física, Centro de Investigación Científica y de Educación Superior de Ensenada, Baja California, Carretera EnsenadaTijuana 3918, 22860 Baja California, Mexico c Edificio 5, Ciudad Universitaria, 04510 Coyoacán, Distrito Federal, Mexico
A R T I C L E I N F O
A BS T RAC T
Keywords: Horizontal plate Wave-structure interaction Submerged plate Dissipation coefficient CADMAS-SURF Wave energy dissipation
Rubble-mound breakwaters are widely used for beach protection and generate shelter zones in the water, although their feasibility diminishes as water depth increases, due mainly to construction costs. In this paper an alternative structure, a fixed submerged horizontal plate, is tested in the hope that such a design may be a practical engineering option for the protection of small ports and marinas, and/or providing a sheltered zone for recreational purposes. The interaction of irregular unidirectional-waves with the plates was studied using CADMAS-SURF software and validated experimentally. Several non-dimensional parameters were tested in order to evaluate their contribution in dissipating the incoming waves. Results suggest that energy loss depends to a large degree on the combination of the relative depth, wave steepness, relative thickness, relative length and relative submergence, with the last being the most important, as it significantly affects the wave-plate interaction. The plate produces greatest dissipation when placed at a depth of 3% of the water column. It should therefore be deployed near the surface and in regions with low tidal range conditions. Plate lengths of 20% of the incoming wavelength and a relative thickness of 0.08 are recommended to produce greater energy dissipation with lower transmission and reflection.
1. Introduction Most coastal activities need protection from wave attack in order to be safely carried out. Some activities, such as fishing and navigation, operate better when sheltered by structures or barriers which control wave energy. Erosion protection is also offered to beaches by such structures, which are usually rubble-mound breakwaters constructed of natural or artificial elements, where the stability comes from their weight, friction and interlocking features. Where protection from beach erosion is not the main focus, alternative structures such as floating breakwaters can be considered (Teh, 2013), placed near the mean sea level, where they interact with the surface waves. McCartney (1985) introduced four general categories: Box, Pontoon, Mat and Tethered, stating that some site characteristics, such as poor foundations conditions, deep water, water quality, ice problems and breakwater layout flexibility, are better served by floating breakwaters. These have the main advantage that their cost and construction are independent of water depth, they are more ecologically friendly as they allow water ⁎
Corresponding author. E-mail address: [email protected] (M.G. Verduzco-Zapata).
http://dx.doi.org/10.1016/j.oceaneng.2017.06.068 Received 9 March 2016; Received in revised form 26 June 2017; Accepted 27 June 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.
exchange and they can constitute a temporary and transportable coastal defense solution (Yamamoto, 1981; Ji et al., 2016), being more effective with wave periods of up to 5 s (Isaacson et al., 1995). The plates investigated in this paper are intended for use at small ports or marinas, where relatively short waves have great effect on small boats; and for protecting recreational areas (bathing beaches) (Wang et al., 2016) providing a sheltered zone. The plates are fixed in a pilesupported structure. Where their deployment is in deeper waters, the plate can be moored to the bottom, in which case the results presented here would be an approximation of the actual performance, as the mooring lines are not capable of restraining all degrees of freedom of motion. The idea of using horizontal plates as breakwaters is not new and has been studied from a variety of perspectives, including wave energy reflection and transmission and their dependency on several nondimensional parameters. From studies reported by Stoker (1957), Sendil and Graf (1974), Sundar and Dakshinamoorty (1980), Patarapanich and Cheong (1989), it can be concluded that the most
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and concluded that in general a composite wavy structure gives higher reflection coefficients compared to a flat plate. Nevertheless, more research is needed as their numerical results show some discrepancy with their experimental data, attributed to the intrinsic model limitation, as this does not consider non-linear effects. Medina-Rodríguez et al. (2016) studied another wavy plate configuration, using an asymptotic analysis, and their results agree with those of Yueh et al. (2016), as in general the wavy plate generated higher reflection coefficients. They concluded that when there are a large number of undulations, their performance is similar to the flat plate. Although the wavy plates seem to be a promising design, it is important to note that they work differently from flat plates, inducing more reflection which may not be desirable in some situations. Given the above, the main purpose of this paper is to study the interaction of an irregular wave train (JONSWAP spectrum) with a fixed submerged horizontal flat plate placed in relative water depths (h/Lp) > 0.5, and to understand the combined effects of several non-dimensional parameters on the kd, as wave dissipation is the main goal of this kind of barrier when used to create a sheltered area in a small harbor. The 3D effects are ignored, therefore only waves impacting perpendicular to the plate are considered as this is the most critical condition for plate performance (Cho and Kim, 2013). The scaling of the numerical tests was done by means of the non-dimensional parameters that relate plate dimensions to water depth and the properties of the incoming waves (see Table 1). The numerical wave model CADMAS-SURF was used to perform the experiments and the results were validated with physical modeling in a wave flume. In the following section the setup used in the numerical experiments is given. In Sections 3 and 4 a description of the physical experiments is made and the methods used to process the data are described. Section 5 details and discusses the results, and concluding remarks are presented in Section 6.
important parameters affecting wave transmission are the wave steepness (Hs/Lp), relative water depth (h/Lp), relative submergence (ds/h) and relative length (B/Lp); where Hs is the significant wave height; Lp the wavelength related to the peak period; h the still water depth; ds the vertical distance from top of the structure (crown) to the still water surface; and B the plate length. Other research reports that relative thickness (Z/h) (where Z is the plate thickness) is another important factor that affects the hydrodynamics around the structure. Zheng et al. (2007) and Liu et al. (2009, 2012), who based their work on analytical formulations, concluded that, in general, thinner plates transmit less wave energy, suggesting that the plate should have the minimum thickness, which meets structural strength requirements. On the other hand, using the numerical model CADMAS-SURF (CS), developed by (CDIT, 2001), Verduzco-Zapata et al. (2015) analyzed the interaction between plates and irregular waves, finding that the most important parameters in wave transmission are relative submergence and relative length, while other non-dimensional parameters, such as relative thickness, wave steepness and relative depth alter the effect that ds/h and B/Lp have on the transmission coefficient (kt). This is important as it implies that the influence of each parameter on kt is not always the same, and it depends more on the parameter combination in the experiment rather than the value of just one parameter by itself. These results coincide with those found by Patarapanich and Cheong (1989). Recently, Cho and Kim (2013) used linear potential theory to study the effects of the wave incidence angle, porosity, relative submergence and relative depth on reflection, transmission and dissipation coefficients (kr, kt and kd, respectively), as well as wave forces acting on plates interacting with monochromatic waves. They found that there is an optimum relative submergence of 0.05–0.10 (similar to results reported by Patarapanich and Cheong (1989), where this value was around 0.05–0.15 for regular and 0.10–0.20 for irregular waves, respectively) and an optimum porosity value of around 0.10. Regarding the effects of the wave incidence angle, they found that although there was not a significant effect on high frequency waves, for longer waves the plate performance was greatly improved as the incident angle becomes higher, due to the fact that the effective ratio between B/L increases as the wavelength in the direction perpendicular to the plate becomes smaller. Liu and Li (2014) extended those studies in a wave flume using regular and irregular waves interacting with two layers of porous plates placed at different depths in the water column. Their results suggest that the area of the holes is not a relevant parameter for wave dissipation. Although there have been advances recently concerning the open breakwater plate type, in order to set optimal design values for the plate geometry and position, more extensive investigation is needed to analyze the non-dimensional parameters relevant to wave-structure interaction (Cho and Kim, 2013). As a consequence, several studies related to plate-wave interaction are still on-going. Pinon et al. (2016) studied numerically the formation of vortices and their forces on submerged plates, and found that varying the plate length produces different vortex trajectories at both ends, emphasizing the occurrence of several recirculation cells that modify the vortices. This vortex generation is important in the wave dissipation process (Bung et al., 2008; Poupardin et al., 2012). The longer plates tend to have high vortical activity and to produce an energetic downward jet in the lee-side, impacting the sea bed and feeding the recirculation cell. There is a possibility that the flow and pressures exerted on the bottom may produce sediment movement, even in deep water conditions (Rey et al., 2015), producing scour around the plates. Recently, different types of plates have been studied, such as the multi arc plate (Wang et al., 2016) and wavy plates (Yueh et al., 2016; Medina-Rodríguez et al., 2016). The latter are promising as they try to maximize wave reflection by inducing a resonance condition (Bragg reflection). Yueh et al. (2016), studied the wave scattering produced by this kind of wavy structure, using the dual boundary element method,
2. Numerical model 2.1. Model setup A numerical wave tank with approximately eight wavelengths of effective flume (Fig. 1) was simulated using a 2d hydrodynamic model. There are several 2D numerical models adequate for engineering applications (Liu and Lin, 1997), VOFbreak2 (Troch and De Rouck, 1999), FLOW-3D (Flow-Science, 2014), CADMAS-SURF (CDIT, 2001), etc.). The latter was recognized by coastal engineers to give accurate results (Hieu and Tanimoto, 2006; Hanzawa et al., 2012; VerduzcoZapata et al., 2012, 2015), as well as having the advantage of being an open source code. CADMAS-SURF solves the Navier-Stokes equations (Eqs. (1) and (2)) and continuity (Eq. (3)) equation to find flow velocities and pressure and employs the Volume of Fluid method Table 1 Experimental data with relevant non-dimensional parameters used in wave-structure interaction tests. All combinations of parameters were used for numerical simulation using the CS model, with the purpose of evaluating their effect on energy dissipation. a
Nondimensional parameters
Description
Values
h/Lp Hs/Lp
Relative depth Wave steepness Relative submergence Relative thickness Relative length
0.75 0.045
0.65 0.03
0.55 0.015
0.00
0.03
0.06
0.09
0.02
0.08
0.14
0.20
0.10
0.15
0.20
0.25
ds/h Z/h B/Lp
0.12
0.15
0.18
a Where h, Lp, Hs, ds, Z and B are water depth, wavelength related to peak frequency, incident significant wave height, freeboard, plate thickness and plate length, respectively.
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Fig. 1. (a) Numerical wave flume of approximately eight wavelengths of effective flume plus two damping zones used to treat open boundaries, in addition to a radiation condition at both ends; (b) zoom, with labels used to identify each wave and plate properties.
2.2. Non-dimensional parameters
(VOF) (Hirt and Nichols, 1981), which is a powerful, State of the Art technique for modeling problems involving free surfaces (Eq. (4)). The wave generation source method used was that of Kawasaki (1999) for generating irregular waves inside the numerical domain, using a modified JONSWAP spectrum (Goda, 2000). This method produces waves traveling in both directions, so special care is needed to treat spurious wave reflection from the open boundaries at both flume ends. To reduce numerical reflection, an Orlanski radiation condition (Orlanski, 1976) was used as well as a relaxation zone (Hinatsu, 1992), which numerically dissipates wave energy that may be reflected from the boundaries. The mesh was formed by rectangular cells with Δz = h/100 and Δx = min (Lp/100, 4Δz), where Δz and Δx are the cell height and length, respectively. Several cells were blocked in order to define plate geometry, where a free slip condition was used at the fluidobstacle interface. A small constant time step (Δt) was employed in all the experiments to maintain numerical stability. The Navier-Stokes, continuity and VOF equations are defined as follows:
∂u ∂w + = Sp, ∂x ∂z
Having revised previous studies, it was concluded that the most important parameters affecting the wave-plate interaction are the wave steepness (Hs/Lp), relative water depth (h/Lp), relative submergence (ds/h), relative length (B/Lp); and relative thickness (Z/h). Each of these parameters were therefore assigned values, as shown in Table 1, and were defined in order to represent conditions where open breakwaters could be used for engineering purposes. In all cases small wave steepness were selected to prevent the depth-induced wave breaking before reaching the plate (0.015 ≤ Hs/Lp ≤ 0.045). In conditions with a relative water depth greater than 0.50, these plates have an advantage compared with rubble-mound breakwaters, as the main wave energy is found close to the mean sea level. Thus the plate was placed at relative submergences between 0.0 and 0.18, where optimum submergence is reported to occur. Verduzco-Zapata et al. (2015) found that the relative thickness of the plate contributes to wave transmission, depending on the relative depth and position of the plate; their values were fixed between 2% and 20% of the water column. Finally, the relative length varied from 10% to 25% of the wavelength; smaller plates have a rather poor interaction with incoming waves, and larger plates are considered impractical. All possible combinations (1008 experiments) of these five non-dimensional parameters were used for numerical simulations in order to study their combined influence on wave energy dissipation, providing a complete set of data to develop a robust parametric equation for the design of this kind of structure. Water depth was kept constant for all the numerical experiments, and it was used to calculate all dimensional quantities (Lp, Hs, ds, Z and B) used in the nondimensional parameters of Table 1.
(1)
∂u ∂u ∂u 1 ∂p ∂ ⎧ ⎛ ∂u ⎞⎫ ∂ ⎧ ⎛ ∂u ∂w ⎞⎫ ⎨ν⎜2 ⎟⎬ + ⎨ν⎜ +u +w =− + + ⎟⎬ − βx u ∂t ∂x ∂z ∂x ⎩ ⎝ ∂x ⎠⎭ ∂z ⎩ ⎝ ∂z ∂x ⎠⎭ ρ ∂x + Su,
(2)
∂w ∂w ∂w 1 ∂p ∂ ⎧ ⎛ ∂w ∂u ⎞⎫ ∂ ⎧ ⎛ ∂w ⎞⎫ ⎨ν⎜ +u +w =− + + ⎟⎬ + ⎨ν⎜2 ⎟⎬ ∂t ∂x ∂z ρ ∂z ∂x ⎩ ⎝ ∂x ∂z ⎠⎭ ∂z ⎩ ⎝ ∂x ⎠⎭ − βzw + Sw − g ,
(3)
and
∂F ∂F ∂F +u +w = SF , ∂t ∂x ∂z
3. Experimental tests (4) 3.1. Laboratory model setup
where: u, w S p, S u , S w and SF g ρ, p V F
The experimental tests were carried out in a wave flume (37 × 0.8 × 1.2 m) at the National Autonomous University of Mexico (UNAM), which features a piston-type wave generator equipped with an active wave absorbing system. In this flume rectangular acrylic prisms were used to represent plates, which were fixed to the side walls of the flume with aluminum fasteners (Fig. 2a and b). Wave reflection at the end of the flume was minimized using a gravel passive absorber. Four wave probes were placed along the flume to record water surface elevation. The first three were placed before the plate in a spatial array as proposed by Mansard and Funke (1980) to be able to decompose the
velocity components in x and z direction, respectively; energy input functions with non-zero values only at the wave source; gravity acceleration; fluid density and pressure, respectively; kinematic viscosity; and filled cell volume (with values between 0 and 1, for an empty and full cell, respectively). 80
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Fig. 2. (a) Wave flume configuration (37 × 0.8 × 1.2 m). Water depth: 0.5 m. (b) Left: panoramic view of the wave flume with the fixed plate and four wave sensors; right: plate made of acrylic, fastened to the vertical glass side wall with aluminum strips.
numerical simulation solutions. The ranges of each non-dimensional parameter are shown in Table 2. Incoming waves, corresponding to a JONSWAP spectrum were generated using AWASYS software.
Table 2 Experimental setup used in laboratory and the CS model comparison tests. Transmission and reflection coefficients are plotted in Fig. 3. Exp.
h/Lp
Hs/Lp
ds/h
Z/h
B/Lp
kr_num
kt_num
kr_exp
kt_exp
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0.40 0.28 0.57 0.40 0.28 0.40 0.40 0.28 0.28 0.40 0.40 0.28 0.57 0.40 0.28 0.28 0.40 0.28 0.28 0.57 0.40 0.28 0.40
0.040 0.028 0.057 0.040 0.028 0.060 0.040 0.028 0.042 0.040 0.040 0.042 0.057 0.040 0.028 0.042 0.040 0.028 0.042 0.057 0.040 0.028 0.040
0.00 0.00 0.06 0.06 0.06 0.14 0.00 0.00 0.00 0.10 0.14 0.14 0.00 0.00 0.00 0.00 0.06 0.06 0.06 0.10 0.10 0.10 0.14
0.10 0.10 0.10 0.10 0.10 0.10 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20
0.08 0.06 0.11 0.08 0.06 0.08 0.12 0.08 0.08 0.12 0.12 0.08 0.23 0.16 0.11 0.11 0.16 0.11 0.11 0.23 0.16 0.11 0.16
0.38 0.19 0.21 0.26 0.12 0.14 0.51 0.40 0.34 0.23 0.18 0.08 0.71 0.56 0.44 0.37 0.25 0.23 0.33 0.36 0.39 0.20 0.27
0.71 0.87 0.66 0.67 0.80 0.84 0.52 0.72 0.70 0.72 0.83 0.87 0.26 0.44 0.62 0.65 0.48 0.58 0.55 0.66 0.63 0.71 0.77
0.28 0.28 0.26 0.24 0.15 0.19 0.39 0.37 0.39 0.25 0.25 0.12 0.80 0.46 0.42 0.37 0.33 0.29 0.27 0.37 0.36 0.23 0.24
0.75 0.81 0.68 0.70 0.82 0.88 0.58 0.70 0.71 0.83 0.84 0.92 0.31 0.48 0.60 0.66 0.42 0.63 0.65 0.61 0.68 0.79 0.79
4. Data analysis Surface water elevations were recorded for the experimental and numerical tests using wave gauges placed along the flume. A three sensor arrangement was placed before the plate (Wave staffs 1, 2 and 3, and sensors 2, 3 and 4, for experimental and numerical tests, respectively) in order to be able to separate incoming and reflected spectra, using the method proposed by Mansard and Funke (1980). This method employs a least squares method to decompose the spectra and requires the wave amplitude and phase simultaneously at each sensor. In all the laboratory experiments, the distances between wave gauges 1–2 and 2–3 were 27 and 64 cm, respectively; while for the numerical tests the distances between sensors 2–3 and 3–4 were 0.10 Lp and 0.15 Lp, respectively. These spaces were defined in such a way that distance A, between the first and second sensor, is not a multiple of Lp/2 and similarly distance B, between the second and third sensor, is not a multiple of A. After separating spectra, the total hydrodynamic coefficients (reflection kr; and transmission kt) were calculated. The dissipation coefficient (kd) was estimated using the relation kd = 1 − kr2 − kt2 (Seelig and Ahrens, 1981). 5. Results and discussions
incident and reflected spectra. In order to obtain an experimental matrix in a non-dimensional form, some variables were fixed: for ease of construction three different sizes of acrylic plates were built (10 × 5, 15 × 10 and 20 × 10 cm); and in every test the still water level was set to 50 cm. All the other variables were chosen arbitrarily to provide a set of different configurations, so the results can be compared with the
5.1. Model validation: experimental vs. numerical tests Comparisons were made between the kr and kt values obtained in the experimental tests and those calculated using the CS model. A total of 23 tests were 81
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was expected as the kr calculation is usually more sensitive to deviations from linear theory when using the spectral separation method employed in this study (Isaacson, 1991).
5.2. Influence of non-dimensional parameters on the dissipation coefficient (kd) The results from the numerical tests are shown in Figs. 4–7. In previous works, optimal submergence for the minimum kt was reported to be around 5–10% and up to 20% of the water column (Cho and Kim, 2013; Patarapanich and Cheong, 1989). Nevertheless, VerduzcoZapata et al. (2015) reported that for the deep water case (h/Lp > 0.50), there are non-dimensional parameter combinations where this critical submergence was not found with the ds/h values used in their experiments (0.0, 0.05, 0.10, 0.15, 0.20, 0.25). The present results show that maximum dissipation was found when the plates were submerged to a depth of 3% of the water column (Figs. 4–6). This corresponds to a relative submergence (ds/h) of 0.03. Beyond this point, dissipation rapidly declines. This ds/h value permits three main mechanisms for energy dissipation: a) wave breaking; b) generation of a horizontal vortex on the lee side of the plate (Bung et al., 2008; Poupardin et al., 2012), which induces energy loss in the wake behind the plate; and c) creation of a pulsating flow due to pressure differences between the front and rear edges of the plate (Graw, 1992; Yu, 2002), causing the current below the structure to interfere with incoming waves, leading to wave dissipation (Wang et al., 2012). These energy sinks may act alone or together, depending on the non-dimensional parameters used in each experiment. There were only two cases where critical submergence for
Fig. 3. Comparison between kr and kt obtained in the wave flume and those calculated with the CS model. A strong positive correlation can be seen between experimental and numerical data with a R2 of 0.83 and 0.91 for kr and kt, respectively.
performed and the results are shown in Table 2 and Fig. 3. The scatter plot showed a strong positive correlation between experimental and numerical data with a determination coefficient (R2) of 0.83 and 0.91 for the coefficient of reflection kr and transmission kt, respectively; suggesting that the CS model can accurately reproduce this kind of experiment, with a little less precision for the reflection coefficient. This
Fig. 4. Dissipation coefficient kd as a function of relative submergence at a relative depth of 0.55, with wave steepness of 0.045 and 0.015 (Fig. 4a and b, respectively).
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Fig. 5. Dissipation coefficient kd as a function of relative submergence at a relative depth of 0.65, with wave steepness of 0.045 and 0.015 (Fig. 5a and b, respectively).
kd was not detected (Fig. 7). Both cases had a relative length (B/Lp) of 0.20 but different relative depths (h/Lp = 0.65 and 0.75), relative wave heights (Hs/Lp = 0.015 and 0.045) and relative thicknesses (Z/h = 0.02 and 0.20). For these two cases, there was low wave dissipation, but kt remained low because of wave blocking that maintained a high kr. This is not an optimum performance; it would be better to have minimum wave transmission and reflection with maximum energy dissipation, in order to ensure a sheltered region on the leeside of the plate, and not generate excessive agitation on the seaward side. Figs. 4–6 show how the non-dimensional parameter combinations affect the hydrodynamic coefficient response. Crosses, triangles, squares and circles represent different relative lengths of the plate (0.10, 0.15, 0.20 and 0.25, respectively); the colors red, cyan, blue and black indicate the relative thickness of the plate (0.02, 0.08, 0.14 and 0.20, respectively). Only two wave steepnesses are presented: 0.015 and 0.045 in each figure. Within the range of the experiments, it can be seen that it would be a mistake to assume that longer or thinner plates are always the best option. When placing the plate at the water surface, the best options seem to be a long plate (B/Lp = 0.25, 0.20 or 0.15) with a relative thickness greater than 0.02, as within this configuration kd was greater than 0.60 regardless of the relative water depth (h/Lp) and relative wave steepness (Hs/Lp) of the experiment. On the other hand, when these options are used at a relative submergence of ds/h = 0.03, there is an important overall improvement in wave dissipation for almost all configurations. However, Fig. 7 shows that when the relative plate length is 0.20 and Z/h=0.20, and the wave steepness is 0.045, kd can drop below 0.60. As a result, the suggested configurations for forcing wave dissipation at optimal submergence or less (ds/h ≤ 0.03) are B/Lp
= 0.20 with Z/h between 0.08 and 0.14; or B/Lp = 0.25 or 0.15 with a Z/h greater than 0.02, as these plate configurations will tend to induce more dissipation and their sheltering capabilities will be enhanced. Further extending the analysis of the configurations suggested above, kr and kt are calculated. It was observed that although kt was less than 0.60 for all setups, at ds/h=0, the shortest (B/Lp = 0.15) and thickest (Z/h = 0.20 and 0.14) plates give higher kt values when placed at ds/h=0.03. Moreover, longer or thicker plates tend to induce an excessive kr when placed at ds/h=0. This suggests that kr and kt obtained with these configurations are unbalanced (one is much greater than the other). A further refinement was made and experiments with a difference of 50% between kr and kt (|kt-kr|/kr > 0.50) were filtered out. Results suggest that only the plates with B/Lp = 0.20 and 0.15 with a Z/h = 0.08 fulfill this condition, regardless of the relative water depth and wave steepness, as can be seen in Tables 3 and 4. With these last two configurations, B/Lp = 0.20 has a slightly better performance as overall it produces the smallest kt and a low kr for both ds/h values. As a consequence, the use of plates of B/Lp = 0.20 and Z/h = 0.08 gives kd, kr and kt values of approximately 0.60–0.85, 0.30–0.70 and 0.30–0.55, respectively, when the plate is located near the water surface (ds/h ≤ 0.03) in h/Lp > 0.50. This setup needs the plate to be placed with a submergence (ds) with the same order of magnitude as the tidal range, therefore it is recommended for use in locations with low tidal range conditions ( < 30 cm) (i.e. parts of the Mediterranean, Baltic and Caribbean Seas (Pilkey et al., 2011)). Using simple regression techniques, a first approximation of an empirical equation for plates with the above recommended setup (B/Lp = 0.20 and Z/h = 0.08) was obtained and is presented in Eq. (5) as a 83
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Fig. 6. Dissipation coefficient kd as a function of relative submergence at a relative depth of 0.75, with wave steepness of 0.045 and 0.015 (Fig. 6a and b, respectively).
Fig. 7. Dissipation coefficient kd as a function of ds/h at relative depths of 0.65 and 0.75 with wave steepness of 0.045 and 0.03, respectively.
starting point for future work. The results show a good fit between the calculated kd (using the data obtained with the CS model) and the kd estimated with Eq. (5), with a R2 of 0.91 and a Normalized Root Mean Square Error (NRMSE) of 0.095 (Fig. 8). It is worth mentioning that this equation is valid within the range of the data used for its formulation (h/Lp ≥ 0.55; 0.015 ≤ Hs/Lp ≤ 0.045; and 0 ≤ ds/h ≤ 0.18). More experiments are required in order to validate these results for other sea conditions and a further advanced statistical study is
necessary to develop a complete empirical equation that includes all parameters relevant for the optimal design of plates.
⎧ ds ⎪a h + b kd = ⎨ d e ⎪ c( s − d ) ⎩ h where: 84
ds h ds h
< 0.03 ≥ 0.03
(5)
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Table 3 Results of kr, kt and kd of a plate placed near the water surface (ds/h ≤ 0.03) with 0.15 and 0.08 relative length and relative thickness, respectively. h/Lp
Hs/Lp
ds/h
Z/h
B/Lp
kr
kt
kd
0.75 0.75 0.65 0.65 0.55 0.55 0.75 0.75 0.65 0.65 0.55 0.55 0.75 0.75 0.65 0.65 0.55 0.55
0.03 0.03 0.03 0.03 0.03 0.03 0.015 0.015 0.015 0.015 0.015 0.015 0.045 0.045 0.045 0.045 0.045 0.045
0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03
0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15
0.56 0.69 0.52 0.48 0.5 0.4 0.63 0.47 0.59 0.5 0.55 0.53 0.55 0.64 0.54 0.47 0.53 0.57
0.54 0.35 0.54 0.42 0.54 0.33 0.45 0.57 0.49 0.47 0.52 0.4 0.54 0.35 0.54 0.44 0.54 0.36
0.63 0.63 0.66 0.77 0.68 0.86 0.63 0.67 0.64 0.73 0.65 0.75 0.64 0.68 0.65 0.77 0.65 0.74
0.04 0.49 0.04 0.13 0.08 0.18 0.29 0.21 0.17 0.06 0.05 0.25 0.02 0.45 0.00 0.06 0.02 0.37
mean max min
0.54 0.69 0.40
0.47 0.57 0.33
0.69 0.86 0.63
0.16 0.49 0.00
kt − kr kr
Fig. 8. Coefficient of dissipation kd calculated with numerical data (within the range h/ Lp ≥ 0.55; 0.015 ≤ Hs/Lp ≤ 0.045; and 0 ≤ ds/h ≤ 0.18) from the CS model and the kd estimated with Eq. (5). The dotted grey line corresponds to the values where an exact match would occur.
6. Conclusions The interaction between irregular waves and horizontal submerged plates was studied, using physical tests and CADMAS-SURF software. It was found that almost all tests showed a critical relative submergence (ds/h=0.03), where the energy dissipation is maximum. A greater depth causes the wave energy dissipation to decrease, due to the fact that the wave energy is concentrated near the water surface as the waves are propagating at relative water depths (h/Lp) > 0.5. It was confirmed that relative thickness (Z/h) and relative length (B/Lp) have different contributions to the structure-wave interaction process, and the way they induce energy dissipation will depend on the overall configuration of the plate, incoming wave characteristics and positioning of the plate (relative submergence ds/h and relative depth h/Lp). For instance, when the plate is at the critical submergence or less, based on the experimental range of this study, it is recommended to have 0.15≤B/Lp ≤ 0.20 with a Z/h = 0.08. With these configurations, the latter (B/Lp = 0.20) will produce the highest energy dissipation (0.60 < kd < 0.85) with low reflection (0.30 < kr < 0.70) and transmission (0.30 < kt < 0.55) coefficients, generating a sheltered zone on the landward side while minimizing wave reflection on the seaward side. As the functionality of the plates is very sensitive to relative submergence, the recommended deployment sites are regions with conditions of low tidal ranges ( < 30 cm). More experiments are required in order to validate these results for other sea conditions. It is hoped that the results detailed here can form a database (portal.ucol.mx/geo/plates_data.htm) to help develop empirical formulae for the optimal design of breakwaters of this type.
Table 4 Results of kr, kt and kd of a plate placed near the water surface (ds/h ≤ 0.03) with 0.20 and 0.08 relative length and relative thickness, respectively. h/Lp
Hs/Lp
ds/h
Z/h
B/Lp
kr
kt
kd
0.75 0.75 0.65 0.65 0.55 0.55 0.75 0.75 0.65 0.65 0.55 0.55 0.75 0.75 0.65 0.65 0.55 0.55
0.03 0.03 0.03 0.03 0.03 0.03 0.015 0.015 0.015 0.015 0.015 0.015 0.045 0.045 0.045 0.045 0.045 0.045
0 0.03 0 0.03 0 0.03 0 0.03 0 0.03 0 0.03 0 0.03 0 0.03 0 0.03
0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
0.62 0.46 0.58 0.46 0.57 0.46 0.7 0.43 0.63 0.4 0.62 0.34 0.62 0.53 0.59 0.47 0.61 0.48
0.43 0.39 0.39 0.35 0.42 0.3 0.37 0.54 0.4 0.47 0.44 0.44 0.43 0.34 0.42 0.32 0.4 0.31
0.66 0.8 0.72 0.82 0.71 0.84 0.61 0.72 0.67 0.79 0.65 0.83 0.66 0.78 0.69 0.82 0.68 0.82
0.31 0.15 0.33 0.24 0.26 0.35 0.47 0.26 0.37 0.18 0.29 0.29 0.31 0.36 0.29 0.32 0.34 0.35
mean max min
0.53 0.70 0.34
0.40 0.54 0.30
0.74 0.84 0.61
0.30 0.47 0.15
a = − 7.40 b=
kt − kr kr
Hs + 4.55 Lp
Acknowledgments
H 1.10 s + 0.64 Lp
This work has been developed as part of the research program of the Wave Group at FACIMAR, CICESE and UNAM, and supported by the Fondo Ramón Álvarez-Buylla (FRABA 14-2013). The authors would like to thank CICESE and FACIMAR for the IXACHI and ColimaWS1 server support, respectively, as well as UNAM for access to the wave flume for the experiments. Julian Delgado´s (CICESE) assistance in setting up the numerical model is appreciated; as is that of UNAM lab staff Erick García, Isaac Moreno and David Rosales for their help in conducting the experiments.
H c = − 5.10 s + 0.36 Lp d = − 2.23 e = − 16.47
Hs + 0.06 Lp Hs + 0.09 Lp
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Orlanski, I., 1976. A simple boundary condition for unbounded hyberbolic flows. J. Comput. Phys. 21, 251–269. Patarapanich, M., Cheong, H., 1989. Reflection and transmission characteristics of regular and random waves from a submerged horizontal plate. Coast. Eng. 13, 161–182. Pilkey, O., Neal, W., Cooper, J., Kelley, J., 2011. The World's Beaches: A Global Guide to the Science of the Shoreline. University of California Press, Los Angeles, CA. Pinon, G., Perret, G., Cao, L., Poupardin, A., Brossard, J., Rivoalen, E., 2016. Vortex kinematics around a submerged plate under water waves. Part II: Numer. Comput. Eur. J. Mech. – B/Fluids. Poupardin, A., Perret, G., Pinon, G., Bourneton, N., Rivoalen, E., Brossard, J., 2012. Vortex kinematic around a submerged plate under water waves. Part I: experimental analysis. Eur. J. Mech. B/Fluids 34, 47–55. Rey, V., Touboul, J., Sous, D., 2015. Effect of a submerged plate on the near-bed dynamics underincoming waves in deep water conditions. Appl. Ocean Res. 53, 67–74. Seelig, W., Ahrens, J., 1981. Estimation of Wave Reflection and Energy Dissipation Coefficients for Beaches, Revetments, and Breakwaters. U.S. Army, Corps of Engineers Coastal Engineering Research Center, Fort Belvoir, VA. Sendil, U., Graf, W., 1974. Transmission of regular waves past floating plates. In: Proceedings of the XIV Conference Coastal Engineering. ASCE, pp. 1924–1936. Stoker, J., 1957. Water Waves – The Mathematical Theory with Applications. Publishers, Interscience, USA. Sundar, V., Dakshinamoorty, S., 1980. Wave transmission characteristics of fixed and floating breakwaters. In: Proceedings of the 7th Australasian Hydraulics and Fluid Mechanics Conference. BARTON A.C.T. INSTITUTION OF ENGINEERS, pp. 56–59. Teh, H., 2013. Hydraulic performance of free surface breakwaters: a review. Sains Malays. 42, 1301–1310. Troch, P., De Rouck, J., 1999. Development of two-dimensional numerical wave flume for wave interaction with rubble mound breakwaters. In: Proceedings of the Coastal Engineering 1998. ASCE, pp. 1638–1649. Verduzco-Zapata, M., Ocampo-Torres, F., Osuna, P., Parés-Sierra, a.F., Kawasaki, K., 2012. Practical estimation of wave transmission and reflection from fixed submerged structures. Ocean Eng. 45, 63–74. Verduzco-Zapata, M., Ocampo-Torres, F., Osuna, P., Parés, A., 2015. Descripción y análisis de la transmisión de la energía del oleaje irregular debido a su interacción con obstáculos rectangulares sumergidos. In: Sosa Ávalos, R., Verduzco-Zapata, M. (Eds.), Estudios Acuícolas Y Marinos En El Pacífico Mexicano. Universidad de Colima, Colima, 275–292. Wang, G., Ren, B., Wang, Y., 2016. Experimental study on hydrodynamic performance of arc plate breakwater. Ocean Eng. 111, 593–601. Wang, K., Zhang, Z., Chen, Z., 2012. Flow field analysis of submerged horizontal plate type breakwater. Int. J. Civ. Environ. Struct. Constr. Archit. Eng. 6, 554–562. Yamamoto, T., 1981. Moored floating breakwater response to regular and irregular waves. Appl. Ocean Res. 3, 27–36. Yu, X., 2002. Functional performance of a submerged and essentially horizontal plate for offshore wave control: a review. Coast. Eng. J. 44, 127–147. Yueh, C., Chuang, S., Wen, C., 2016. Wave scattering by submerged composite wavy plate breakwaters using a dual BEM. Ocean Eng. 124, 192–203. Zheng, Y., Liu, P., Shen, Y., Wu, B., Sheng, S., 2007. On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas. Ocean Eng. 34, 436–450.
References Bung, D., Hildebrandt, A., Oertel, M., Schlenkhoff, A., Schlurmann, T., 2008. Bore propagation over a submerged horizontal plate by physical and numerical simulation. In: Proceedings of the 31st International Conference on Coastal Engineering. ASCE, pp. 3542–3553. CDIT, 2001. Research and Development of A Numerical Wave Flume; CADMAS-SURF Report of the Research Group for Development of Numerical Wave Flume for the Design of Maritime Structures. Coastal Development Institute of Technology of Japan, Tokio, Japan. Cho, I., Kim, M., 2013. Transmission of oblique incident waves by a submerged horizontal porous plate. Ocean Eng. 61, 56–65. Flow-Science, 2014. FLOW-3D User Manual. Goda, Y., 2000. Random Seas and Design of Maritime Structures. World Scientific, Singapore. Graw, K., 1992. The submerged plate as a wave filter the stability of the pulsating flow phenomenon. In: Proceedings of the International Conference on Coastal Engineering. ASCE, pp. 1153–1160. Hanzawa, M., Matsumoto, A., Tanaka, H., 2012. Applicability of CADMAS-surf to evaluate detached breakwater effects on solitary tsunami wave reduction. Earth Planets Space 64, 955–964. Hieu, P., Tanimoto, K., 2006. Verification of a VOF-based two-phase flow model for wave breaking and wave-structure interactions. Ocean Eng. 33, 1565–1588. Hinatsu, M., 1992. Numerical simulation of unsteady viscous nonlinear waves using moving grid system fitted on a free surface. J. Kansa. Soc. Nav. Archit., 1–11, (Japan 217, J-STAGE). Hirt, C., Nichols, B., 1981. Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225. Isaacson, M., 1991. Measurement of regular wave reflection. J. Waterw. Port Coast. Ocean Eng. 117, 553–569. Isaacson, M., Whiteside, N., Gardiner, R., Hay, D., 1995. Modelling of a circular-section floating breakwater. Can. J. Civil. Eng. 22, 714–722. Ji, C., Chen, X., Cui, J., Gaidai, O., Incecik, A., 2016. Experimental study on configuration optimization of floating breakwaters. Ocean Eng. 117, 302–310. Kawasaki, K., 1999. Numerical simulation of breaking and post-breaking wave deformation process around a submerged breakwater. Coast. Eng. J. 41, 201–223. Liu, P., Lin, P., 1997. A Numerical Model for Breaking Waves: The Volume of Fluid Method. Center for Applied Coastal Research, Ocean Engineering Laboratory, University of Delaware. Liu, Y., Li, H., 2014. Wave scattering by dual submerged horizontal porous plates: Further results. Ocean Eng. 81, 158–163. Liu, Y., Li, H., Li, Y., 2012. A new analytical solution for wave scattering by a submerged horizontal porous plate with finite thickness. Ocean Eng. 42, 83–92. Liu, Y., Li, Y., Teng, B., 2009. Wave motion over two submerged layers of horizontal thick plates. J. Hydrodyn. 21, 453–462. Mansard, E., Funke, E., 1980. The measurement of incident and reflected wave spectra using a least squares method. In: Proceedings of the 17th International Conference on Coastal Engineering. ASCE, pp. 154–172. McCartney, B., 1985. Floating breakwater design. J. Waterw. Port Coast. Ocean Eng. 111, 304–318. Medina-Rodríguez, A., Bautista, E., Méndez, F., 2016. Asymptotic analysis of the interaction between linear long waves and a submerged floating breakwater of wavy surfaces. Appl. Ocean Res. 59, 345–365.
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Optimal submergence of horizontal plates for maximum wave energy dissipation
MARK
⁎
Manuel Gerardo Verduzco-Zapataa, , Francisco Javier Ocampo-Torresb, Edgar Mendozac, Rodolfo Silvac, Marco Liñán-Cabelloa, Ernesto Torres-Orozcoa a Facultad de Ciencias Marinas, Universidad de Colima, Carretera Manzanillo-Cihuatlán Km 19.5, Colonia El Naranjo, Manzanillo, Colima CP 28868, Mexico b Departamento de Oceanografía Física, Centro de Investigación Científica y de Educación Superior de Ensenada, Baja California, Carretera EnsenadaTijuana 3918, 22860 Baja California, Mexico c Edificio 5, Ciudad Universitaria, 04510 Coyoacán, Distrito Federal, Mexico
A R T I C L E I N F O
A BS T RAC T
Keywords: Horizontal plate Wave-structure interaction Submerged plate Dissipation coefficient CADMAS-SURF Wave energy dissipation
Rubble-mound breakwaters are widely used for beach protection and generate shelter zones in the water, although their feasibility diminishes as water depth increases, due mainly to construction costs. In this paper an alternative structure, a fixed submerged horizontal plate, is tested in the hope that such a design may be a practical engineering option for the protection of small ports and marinas, and/or providing a sheltered zone for recreational purposes. The interaction of irregular unidirectional-waves with the plates was studied using CADMAS-SURF software and validated experimentally. Several non-dimensional parameters were tested in order to evaluate their contribution in dissipating the incoming waves. Results suggest that energy loss depends to a large degree on the combination of the relative depth, wave steepness, relative thickness, relative length and relative submergence, with the last being the most important, as it significantly affects the wave-plate interaction. The plate produces greatest dissipation when placed at a depth of 3% of the water column. It should therefore be deployed near the surface and in regions with low tidal range conditions. Plate lengths of 20% of the incoming wavelength and a relative thickness of 0.08 are recommended to produce greater energy dissipation with lower transmission and reflection.
1. Introduction Most coastal activities need protection from wave attack in order to be safely carried out. Some activities, such as fishing and navigation, operate better when sheltered by structures or barriers which control wave energy. Erosion protection is also offered to beaches by such structures, which are usually rubble-mound breakwaters constructed of natural or artificial elements, where the stability comes from their weight, friction and interlocking features. Where protection from beach erosion is not the main focus, alternative structures such as floating breakwaters can be considered (Teh, 2013), placed near the mean sea level, where they interact with the surface waves. McCartney (1985) introduced four general categories: Box, Pontoon, Mat and Tethered, stating that some site characteristics, such as poor foundations conditions, deep water, water quality, ice problems and breakwater layout flexibility, are better served by floating breakwaters. These have the main advantage that their cost and construction are independent of water depth, they are more ecologically friendly as they allow water ⁎
Corresponding author. E-mail address: [email protected] (M.G. Verduzco-Zapata).
http://dx.doi.org/10.1016/j.oceaneng.2017.06.068 Received 9 March 2016; Received in revised form 26 June 2017; Accepted 27 June 2017 0029-8018/ © 2017 Elsevier Ltd. All rights reserved.
exchange and they can constitute a temporary and transportable coastal defense solution (Yamamoto, 1981; Ji et al., 2016), being more effective with wave periods of up to 5 s (Isaacson et al., 1995). The plates investigated in this paper are intended for use at small ports or marinas, where relatively short waves have great effect on small boats; and for protecting recreational areas (bathing beaches) (Wang et al., 2016) providing a sheltered zone. The plates are fixed in a pilesupported structure. Where their deployment is in deeper waters, the plate can be moored to the bottom, in which case the results presented here would be an approximation of the actual performance, as the mooring lines are not capable of restraining all degrees of freedom of motion. The idea of using horizontal plates as breakwaters is not new and has been studied from a variety of perspectives, including wave energy reflection and transmission and their dependency on several nondimensional parameters. From studies reported by Stoker (1957), Sendil and Graf (1974), Sundar and Dakshinamoorty (1980), Patarapanich and Cheong (1989), it can be concluded that the most
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and concluded that in general a composite wavy structure gives higher reflection coefficients compared to a flat plate. Nevertheless, more research is needed as their numerical results show some discrepancy with their experimental data, attributed to the intrinsic model limitation, as this does not consider non-linear effects. Medina-Rodríguez et al. (2016) studied another wavy plate configuration, using an asymptotic analysis, and their results agree with those of Yueh et al. (2016), as in general the wavy plate generated higher reflection coefficients. They concluded that when there are a large number of undulations, their performance is similar to the flat plate. Although the wavy plates seem to be a promising design, it is important to note that they work differently from flat plates, inducing more reflection which may not be desirable in some situations. Given the above, the main purpose of this paper is to study the interaction of an irregular wave train (JONSWAP spectrum) with a fixed submerged horizontal flat plate placed in relative water depths (h/Lp) > 0.5, and to understand the combined effects of several non-dimensional parameters on the kd, as wave dissipation is the main goal of this kind of barrier when used to create a sheltered area in a small harbor. The 3D effects are ignored, therefore only waves impacting perpendicular to the plate are considered as this is the most critical condition for plate performance (Cho and Kim, 2013). The scaling of the numerical tests was done by means of the non-dimensional parameters that relate plate dimensions to water depth and the properties of the incoming waves (see Table 1). The numerical wave model CADMAS-SURF was used to perform the experiments and the results were validated with physical modeling in a wave flume. In the following section the setup used in the numerical experiments is given. In Sections 3 and 4 a description of the physical experiments is made and the methods used to process the data are described. Section 5 details and discusses the results, and concluding remarks are presented in Section 6.
important parameters affecting wave transmission are the wave steepness (Hs/Lp), relative water depth (h/Lp), relative submergence (ds/h) and relative length (B/Lp); where Hs is the significant wave height; Lp the wavelength related to the peak period; h the still water depth; ds the vertical distance from top of the structure (crown) to the still water surface; and B the plate length. Other research reports that relative thickness (Z/h) (where Z is the plate thickness) is another important factor that affects the hydrodynamics around the structure. Zheng et al. (2007) and Liu et al. (2009, 2012), who based their work on analytical formulations, concluded that, in general, thinner plates transmit less wave energy, suggesting that the plate should have the minimum thickness, which meets structural strength requirements. On the other hand, using the numerical model CADMAS-SURF (CS), developed by (CDIT, 2001), Verduzco-Zapata et al. (2015) analyzed the interaction between plates and irregular waves, finding that the most important parameters in wave transmission are relative submergence and relative length, while other non-dimensional parameters, such as relative thickness, wave steepness and relative depth alter the effect that ds/h and B/Lp have on the transmission coefficient (kt). This is important as it implies that the influence of each parameter on kt is not always the same, and it depends more on the parameter combination in the experiment rather than the value of just one parameter by itself. These results coincide with those found by Patarapanich and Cheong (1989). Recently, Cho and Kim (2013) used linear potential theory to study the effects of the wave incidence angle, porosity, relative submergence and relative depth on reflection, transmission and dissipation coefficients (kr, kt and kd, respectively), as well as wave forces acting on plates interacting with monochromatic waves. They found that there is an optimum relative submergence of 0.05–0.10 (similar to results reported by Patarapanich and Cheong (1989), where this value was around 0.05–0.15 for regular and 0.10–0.20 for irregular waves, respectively) and an optimum porosity value of around 0.10. Regarding the effects of the wave incidence angle, they found that although there was not a significant effect on high frequency waves, for longer waves the plate performance was greatly improved as the incident angle becomes higher, due to the fact that the effective ratio between B/L increases as the wavelength in the direction perpendicular to the plate becomes smaller. Liu and Li (2014) extended those studies in a wave flume using regular and irregular waves interacting with two layers of porous plates placed at different depths in the water column. Their results suggest that the area of the holes is not a relevant parameter for wave dissipation. Although there have been advances recently concerning the open breakwater plate type, in order to set optimal design values for the plate geometry and position, more extensive investigation is needed to analyze the non-dimensional parameters relevant to wave-structure interaction (Cho and Kim, 2013). As a consequence, several studies related to plate-wave interaction are still on-going. Pinon et al. (2016) studied numerically the formation of vortices and their forces on submerged plates, and found that varying the plate length produces different vortex trajectories at both ends, emphasizing the occurrence of several recirculation cells that modify the vortices. This vortex generation is important in the wave dissipation process (Bung et al., 2008; Poupardin et al., 2012). The longer plates tend to have high vortical activity and to produce an energetic downward jet in the lee-side, impacting the sea bed and feeding the recirculation cell. There is a possibility that the flow and pressures exerted on the bottom may produce sediment movement, even in deep water conditions (Rey et al., 2015), producing scour around the plates. Recently, different types of plates have been studied, such as the multi arc plate (Wang et al., 2016) and wavy plates (Yueh et al., 2016; Medina-Rodríguez et al., 2016). The latter are promising as they try to maximize wave reflection by inducing a resonance condition (Bragg reflection). Yueh et al. (2016), studied the wave scattering produced by this kind of wavy structure, using the dual boundary element method,
2. Numerical model 2.1. Model setup A numerical wave tank with approximately eight wavelengths of effective flume (Fig. 1) was simulated using a 2d hydrodynamic model. There are several 2D numerical models adequate for engineering applications (Liu and Lin, 1997), VOFbreak2 (Troch and De Rouck, 1999), FLOW-3D (Flow-Science, 2014), CADMAS-SURF (CDIT, 2001), etc.). The latter was recognized by coastal engineers to give accurate results (Hieu and Tanimoto, 2006; Hanzawa et al., 2012; VerduzcoZapata et al., 2012, 2015), as well as having the advantage of being an open source code. CADMAS-SURF solves the Navier-Stokes equations (Eqs. (1) and (2)) and continuity (Eq. (3)) equation to find flow velocities and pressure and employs the Volume of Fluid method Table 1 Experimental data with relevant non-dimensional parameters used in wave-structure interaction tests. All combinations of parameters were used for numerical simulation using the CS model, with the purpose of evaluating their effect on energy dissipation. a
Nondimensional parameters
Description
Values
h/Lp Hs/Lp
Relative depth Wave steepness Relative submergence Relative thickness Relative length
0.75 0.045
0.65 0.03
0.55 0.015
0.00
0.03
0.06
0.09
0.02
0.08
0.14
0.20
0.10
0.15
0.20
0.25
ds/h Z/h B/Lp
0.12
0.15
0.18
a Where h, Lp, Hs, ds, Z and B are water depth, wavelength related to peak frequency, incident significant wave height, freeboard, plate thickness and plate length, respectively.
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Fig. 1. (a) Numerical wave flume of approximately eight wavelengths of effective flume plus two damping zones used to treat open boundaries, in addition to a radiation condition at both ends; (b) zoom, with labels used to identify each wave and plate properties.
2.2. Non-dimensional parameters
(VOF) (Hirt and Nichols, 1981), which is a powerful, State of the Art technique for modeling problems involving free surfaces (Eq. (4)). The wave generation source method used was that of Kawasaki (1999) for generating irregular waves inside the numerical domain, using a modified JONSWAP spectrum (Goda, 2000). This method produces waves traveling in both directions, so special care is needed to treat spurious wave reflection from the open boundaries at both flume ends. To reduce numerical reflection, an Orlanski radiation condition (Orlanski, 1976) was used as well as a relaxation zone (Hinatsu, 1992), which numerically dissipates wave energy that may be reflected from the boundaries. The mesh was formed by rectangular cells with Δz = h/100 and Δx = min (Lp/100, 4Δz), where Δz and Δx are the cell height and length, respectively. Several cells were blocked in order to define plate geometry, where a free slip condition was used at the fluidobstacle interface. A small constant time step (Δt) was employed in all the experiments to maintain numerical stability. The Navier-Stokes, continuity and VOF equations are defined as follows:
∂u ∂w + = Sp, ∂x ∂z
Having revised previous studies, it was concluded that the most important parameters affecting the wave-plate interaction are the wave steepness (Hs/Lp), relative water depth (h/Lp), relative submergence (ds/h), relative length (B/Lp); and relative thickness (Z/h). Each of these parameters were therefore assigned values, as shown in Table 1, and were defined in order to represent conditions where open breakwaters could be used for engineering purposes. In all cases small wave steepness were selected to prevent the depth-induced wave breaking before reaching the plate (0.015 ≤ Hs/Lp ≤ 0.045). In conditions with a relative water depth greater than 0.50, these plates have an advantage compared with rubble-mound breakwaters, as the main wave energy is found close to the mean sea level. Thus the plate was placed at relative submergences between 0.0 and 0.18, where optimum submergence is reported to occur. Verduzco-Zapata et al. (2015) found that the relative thickness of the plate contributes to wave transmission, depending on the relative depth and position of the plate; their values were fixed between 2% and 20% of the water column. Finally, the relative length varied from 10% to 25% of the wavelength; smaller plates have a rather poor interaction with incoming waves, and larger plates are considered impractical. All possible combinations (1008 experiments) of these five non-dimensional parameters were used for numerical simulations in order to study their combined influence on wave energy dissipation, providing a complete set of data to develop a robust parametric equation for the design of this kind of structure. Water depth was kept constant for all the numerical experiments, and it was used to calculate all dimensional quantities (Lp, Hs, ds, Z and B) used in the nondimensional parameters of Table 1.
(1)
∂u ∂u ∂u 1 ∂p ∂ ⎧ ⎛ ∂u ⎞⎫ ∂ ⎧ ⎛ ∂u ∂w ⎞⎫ ⎨ν⎜2 ⎟⎬ + ⎨ν⎜ +u +w =− + + ⎟⎬ − βx u ∂t ∂x ∂z ∂x ⎩ ⎝ ∂x ⎠⎭ ∂z ⎩ ⎝ ∂z ∂x ⎠⎭ ρ ∂x + Su,
(2)
∂w ∂w ∂w 1 ∂p ∂ ⎧ ⎛ ∂w ∂u ⎞⎫ ∂ ⎧ ⎛ ∂w ⎞⎫ ⎨ν⎜ +u +w =− + + ⎟⎬ + ⎨ν⎜2 ⎟⎬ ∂t ∂x ∂z ρ ∂z ∂x ⎩ ⎝ ∂x ∂z ⎠⎭ ∂z ⎩ ⎝ ∂x ⎠⎭ − βzw + Sw − g ,
(3)
and
∂F ∂F ∂F +u +w = SF , ∂t ∂x ∂z
3. Experimental tests (4) 3.1. Laboratory model setup
where: u, w S p, S u , S w and SF g ρ, p V F
The experimental tests were carried out in a wave flume (37 × 0.8 × 1.2 m) at the National Autonomous University of Mexico (UNAM), which features a piston-type wave generator equipped with an active wave absorbing system. In this flume rectangular acrylic prisms were used to represent plates, which were fixed to the side walls of the flume with aluminum fasteners (Fig. 2a and b). Wave reflection at the end of the flume was minimized using a gravel passive absorber. Four wave probes were placed along the flume to record water surface elevation. The first three were placed before the plate in a spatial array as proposed by Mansard and Funke (1980) to be able to decompose the
velocity components in x and z direction, respectively; energy input functions with non-zero values only at the wave source; gravity acceleration; fluid density and pressure, respectively; kinematic viscosity; and filled cell volume (with values between 0 and 1, for an empty and full cell, respectively). 80
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Fig. 2. (a) Wave flume configuration (37 × 0.8 × 1.2 m). Water depth: 0.5 m. (b) Left: panoramic view of the wave flume with the fixed plate and four wave sensors; right: plate made of acrylic, fastened to the vertical glass side wall with aluminum strips.
numerical simulation solutions. The ranges of each non-dimensional parameter are shown in Table 2. Incoming waves, corresponding to a JONSWAP spectrum were generated using AWASYS software.
Table 2 Experimental setup used in laboratory and the CS model comparison tests. Transmission and reflection coefficients are plotted in Fig. 3. Exp.
h/Lp
Hs/Lp
ds/h
Z/h
B/Lp
kr_num
kt_num
kr_exp
kt_exp
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
0.40 0.28 0.57 0.40 0.28 0.40 0.40 0.28 0.28 0.40 0.40 0.28 0.57 0.40 0.28 0.28 0.40 0.28 0.28 0.57 0.40 0.28 0.40
0.040 0.028 0.057 0.040 0.028 0.060 0.040 0.028 0.042 0.040 0.040 0.042 0.057 0.040 0.028 0.042 0.040 0.028 0.042 0.057 0.040 0.028 0.040
0.00 0.00 0.06 0.06 0.06 0.14 0.00 0.00 0.00 0.10 0.14 0.14 0.00 0.00 0.00 0.00 0.06 0.06 0.06 0.10 0.10 0.10 0.14
0.10 0.10 0.10 0.10 0.10 0.10 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20 0.20
0.08 0.06 0.11 0.08 0.06 0.08 0.12 0.08 0.08 0.12 0.12 0.08 0.23 0.16 0.11 0.11 0.16 0.11 0.11 0.23 0.16 0.11 0.16
0.38 0.19 0.21 0.26 0.12 0.14 0.51 0.40 0.34 0.23 0.18 0.08 0.71 0.56 0.44 0.37 0.25 0.23 0.33 0.36 0.39 0.20 0.27
0.71 0.87 0.66 0.67 0.80 0.84 0.52 0.72 0.70 0.72 0.83 0.87 0.26 0.44 0.62 0.65 0.48 0.58 0.55 0.66 0.63 0.71 0.77
0.28 0.28 0.26 0.24 0.15 0.19 0.39 0.37 0.39 0.25 0.25 0.12 0.80 0.46 0.42 0.37 0.33 0.29 0.27 0.37 0.36 0.23 0.24
0.75 0.81 0.68 0.70 0.82 0.88 0.58 0.70 0.71 0.83 0.84 0.92 0.31 0.48 0.60 0.66 0.42 0.63 0.65 0.61 0.68 0.79 0.79
4. Data analysis Surface water elevations were recorded for the experimental and numerical tests using wave gauges placed along the flume. A three sensor arrangement was placed before the plate (Wave staffs 1, 2 and 3, and sensors 2, 3 and 4, for experimental and numerical tests, respectively) in order to be able to separate incoming and reflected spectra, using the method proposed by Mansard and Funke (1980). This method employs a least squares method to decompose the spectra and requires the wave amplitude and phase simultaneously at each sensor. In all the laboratory experiments, the distances between wave gauges 1–2 and 2–3 were 27 and 64 cm, respectively; while for the numerical tests the distances between sensors 2–3 and 3–4 were 0.10 Lp and 0.15 Lp, respectively. These spaces were defined in such a way that distance A, between the first and second sensor, is not a multiple of Lp/2 and similarly distance B, between the second and third sensor, is not a multiple of A. After separating spectra, the total hydrodynamic coefficients (reflection kr; and transmission kt) were calculated. The dissipation coefficient (kd) was estimated using the relation kd = 1 − kr2 − kt2 (Seelig and Ahrens, 1981). 5. Results and discussions
incident and reflected spectra. In order to obtain an experimental matrix in a non-dimensional form, some variables were fixed: for ease of construction three different sizes of acrylic plates were built (10 × 5, 15 × 10 and 20 × 10 cm); and in every test the still water level was set to 50 cm. All the other variables were chosen arbitrarily to provide a set of different configurations, so the results can be compared with the
5.1. Model validation: experimental vs. numerical tests Comparisons were made between the kr and kt values obtained in the experimental tests and those calculated using the CS model. A total of 23 tests were 81
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was expected as the kr calculation is usually more sensitive to deviations from linear theory when using the spectral separation method employed in this study (Isaacson, 1991).
5.2. Influence of non-dimensional parameters on the dissipation coefficient (kd) The results from the numerical tests are shown in Figs. 4–7. In previous works, optimal submergence for the minimum kt was reported to be around 5–10% and up to 20% of the water column (Cho and Kim, 2013; Patarapanich and Cheong, 1989). Nevertheless, VerduzcoZapata et al. (2015) reported that for the deep water case (h/Lp > 0.50), there are non-dimensional parameter combinations where this critical submergence was not found with the ds/h values used in their experiments (0.0, 0.05, 0.10, 0.15, 0.20, 0.25). The present results show that maximum dissipation was found when the plates were submerged to a depth of 3% of the water column (Figs. 4–6). This corresponds to a relative submergence (ds/h) of 0.03. Beyond this point, dissipation rapidly declines. This ds/h value permits three main mechanisms for energy dissipation: a) wave breaking; b) generation of a horizontal vortex on the lee side of the plate (Bung et al., 2008; Poupardin et al., 2012), which induces energy loss in the wake behind the plate; and c) creation of a pulsating flow due to pressure differences between the front and rear edges of the plate (Graw, 1992; Yu, 2002), causing the current below the structure to interfere with incoming waves, leading to wave dissipation (Wang et al., 2012). These energy sinks may act alone or together, depending on the non-dimensional parameters used in each experiment. There were only two cases where critical submergence for
Fig. 3. Comparison between kr and kt obtained in the wave flume and those calculated with the CS model. A strong positive correlation can be seen between experimental and numerical data with a R2 of 0.83 and 0.91 for kr and kt, respectively.
performed and the results are shown in Table 2 and Fig. 3. The scatter plot showed a strong positive correlation between experimental and numerical data with a determination coefficient (R2) of 0.83 and 0.91 for the coefficient of reflection kr and transmission kt, respectively; suggesting that the CS model can accurately reproduce this kind of experiment, with a little less precision for the reflection coefficient. This
Fig. 4. Dissipation coefficient kd as a function of relative submergence at a relative depth of 0.55, with wave steepness of 0.045 and 0.015 (Fig. 4a and b, respectively).
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Fig. 5. Dissipation coefficient kd as a function of relative submergence at a relative depth of 0.65, with wave steepness of 0.045 and 0.015 (Fig. 5a and b, respectively).
kd was not detected (Fig. 7). Both cases had a relative length (B/Lp) of 0.20 but different relative depths (h/Lp = 0.65 and 0.75), relative wave heights (Hs/Lp = 0.015 and 0.045) and relative thicknesses (Z/h = 0.02 and 0.20). For these two cases, there was low wave dissipation, but kt remained low because of wave blocking that maintained a high kr. This is not an optimum performance; it would be better to have minimum wave transmission and reflection with maximum energy dissipation, in order to ensure a sheltered region on the leeside of the plate, and not generate excessive agitation on the seaward side. Figs. 4–6 show how the non-dimensional parameter combinations affect the hydrodynamic coefficient response. Crosses, triangles, squares and circles represent different relative lengths of the plate (0.10, 0.15, 0.20 and 0.25, respectively); the colors red, cyan, blue and black indicate the relative thickness of the plate (0.02, 0.08, 0.14 and 0.20, respectively). Only two wave steepnesses are presented: 0.015 and 0.045 in each figure. Within the range of the experiments, it can be seen that it would be a mistake to assume that longer or thinner plates are always the best option. When placing the plate at the water surface, the best options seem to be a long plate (B/Lp = 0.25, 0.20 or 0.15) with a relative thickness greater than 0.02, as within this configuration kd was greater than 0.60 regardless of the relative water depth (h/Lp) and relative wave steepness (Hs/Lp) of the experiment. On the other hand, when these options are used at a relative submergence of ds/h = 0.03, there is an important overall improvement in wave dissipation for almost all configurations. However, Fig. 7 shows that when the relative plate length is 0.20 and Z/h=0.20, and the wave steepness is 0.045, kd can drop below 0.60. As a result, the suggested configurations for forcing wave dissipation at optimal submergence or less (ds/h ≤ 0.03) are B/Lp
= 0.20 with Z/h between 0.08 and 0.14; or B/Lp = 0.25 or 0.15 with a Z/h greater than 0.02, as these plate configurations will tend to induce more dissipation and their sheltering capabilities will be enhanced. Further extending the analysis of the configurations suggested above, kr and kt are calculated. It was observed that although kt was less than 0.60 for all setups, at ds/h=0, the shortest (B/Lp = 0.15) and thickest (Z/h = 0.20 and 0.14) plates give higher kt values when placed at ds/h=0.03. Moreover, longer or thicker plates tend to induce an excessive kr when placed at ds/h=0. This suggests that kr and kt obtained with these configurations are unbalanced (one is much greater than the other). A further refinement was made and experiments with a difference of 50% between kr and kt (|kt-kr|/kr > 0.50) were filtered out. Results suggest that only the plates with B/Lp = 0.20 and 0.15 with a Z/h = 0.08 fulfill this condition, regardless of the relative water depth and wave steepness, as can be seen in Tables 3 and 4. With these last two configurations, B/Lp = 0.20 has a slightly better performance as overall it produces the smallest kt and a low kr for both ds/h values. As a consequence, the use of plates of B/Lp = 0.20 and Z/h = 0.08 gives kd, kr and kt values of approximately 0.60–0.85, 0.30–0.70 and 0.30–0.55, respectively, when the plate is located near the water surface (ds/h ≤ 0.03) in h/Lp > 0.50. This setup needs the plate to be placed with a submergence (ds) with the same order of magnitude as the tidal range, therefore it is recommended for use in locations with low tidal range conditions ( < 30 cm) (i.e. parts of the Mediterranean, Baltic and Caribbean Seas (Pilkey et al., 2011)). Using simple regression techniques, a first approximation of an empirical equation for plates with the above recommended setup (B/Lp = 0.20 and Z/h = 0.08) was obtained and is presented in Eq. (5) as a 83
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Fig. 6. Dissipation coefficient kd as a function of relative submergence at a relative depth of 0.75, with wave steepness of 0.045 and 0.015 (Fig. 6a and b, respectively).
Fig. 7. Dissipation coefficient kd as a function of ds/h at relative depths of 0.65 and 0.75 with wave steepness of 0.045 and 0.03, respectively.
starting point for future work. The results show a good fit between the calculated kd (using the data obtained with the CS model) and the kd estimated with Eq. (5), with a R2 of 0.91 and a Normalized Root Mean Square Error (NRMSE) of 0.095 (Fig. 8). It is worth mentioning that this equation is valid within the range of the data used for its formulation (h/Lp ≥ 0.55; 0.015 ≤ Hs/Lp ≤ 0.045; and 0 ≤ ds/h ≤ 0.18). More experiments are required in order to validate these results for other sea conditions and a further advanced statistical study is
necessary to develop a complete empirical equation that includes all parameters relevant for the optimal design of plates.
⎧ ds ⎪a h + b kd = ⎨ d e ⎪ c( s − d ) ⎩ h where: 84
ds h ds h
< 0.03 ≥ 0.03
(5)
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Table 3 Results of kr, kt and kd of a plate placed near the water surface (ds/h ≤ 0.03) with 0.15 and 0.08 relative length and relative thickness, respectively. h/Lp
Hs/Lp
ds/h
Z/h
B/Lp
kr
kt
kd
0.75 0.75 0.65 0.65 0.55 0.55 0.75 0.75 0.65 0.65 0.55 0.55 0.75 0.75 0.65 0.65 0.55 0.55
0.03 0.03 0.03 0.03 0.03 0.03 0.015 0.015 0.015 0.015 0.015 0.015 0.045 0.045 0.045 0.045 0.045 0.045
0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03 0.00 0.03
0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15
0.56 0.69 0.52 0.48 0.5 0.4 0.63 0.47 0.59 0.5 0.55 0.53 0.55 0.64 0.54 0.47 0.53 0.57
0.54 0.35 0.54 0.42 0.54 0.33 0.45 0.57 0.49 0.47 0.52 0.4 0.54 0.35 0.54 0.44 0.54 0.36
0.63 0.63 0.66 0.77 0.68 0.86 0.63 0.67 0.64 0.73 0.65 0.75 0.64 0.68 0.65 0.77 0.65 0.74
0.04 0.49 0.04 0.13 0.08 0.18 0.29 0.21 0.17 0.06 0.05 0.25 0.02 0.45 0.00 0.06 0.02 0.37
mean max min
0.54 0.69 0.40
0.47 0.57 0.33
0.69 0.86 0.63
0.16 0.49 0.00
kt − kr kr
Fig. 8. Coefficient of dissipation kd calculated with numerical data (within the range h/ Lp ≥ 0.55; 0.015 ≤ Hs/Lp ≤ 0.045; and 0 ≤ ds/h ≤ 0.18) from the CS model and the kd estimated with Eq. (5). The dotted grey line corresponds to the values where an exact match would occur.
6. Conclusions The interaction between irregular waves and horizontal submerged plates was studied, using physical tests and CADMAS-SURF software. It was found that almost all tests showed a critical relative submergence (ds/h=0.03), where the energy dissipation is maximum. A greater depth causes the wave energy dissipation to decrease, due to the fact that the wave energy is concentrated near the water surface as the waves are propagating at relative water depths (h/Lp) > 0.5. It was confirmed that relative thickness (Z/h) and relative length (B/Lp) have different contributions to the structure-wave interaction process, and the way they induce energy dissipation will depend on the overall configuration of the plate, incoming wave characteristics and positioning of the plate (relative submergence ds/h and relative depth h/Lp). For instance, when the plate is at the critical submergence or less, based on the experimental range of this study, it is recommended to have 0.15≤B/Lp ≤ 0.20 with a Z/h = 0.08. With these configurations, the latter (B/Lp = 0.20) will produce the highest energy dissipation (0.60 < kd < 0.85) with low reflection (0.30 < kr < 0.70) and transmission (0.30 < kt < 0.55) coefficients, generating a sheltered zone on the landward side while minimizing wave reflection on the seaward side. As the functionality of the plates is very sensitive to relative submergence, the recommended deployment sites are regions with conditions of low tidal ranges ( < 30 cm). More experiments are required in order to validate these results for other sea conditions. It is hoped that the results detailed here can form a database (portal.ucol.mx/geo/plates_data.htm) to help develop empirical formulae for the optimal design of breakwaters of this type.
Table 4 Results of kr, kt and kd of a plate placed near the water surface (ds/h ≤ 0.03) with 0.20 and 0.08 relative length and relative thickness, respectively. h/Lp
Hs/Lp
ds/h
Z/h
B/Lp
kr
kt
kd
0.75 0.75 0.65 0.65 0.55 0.55 0.75 0.75 0.65 0.65 0.55 0.55 0.75 0.75 0.65 0.65 0.55 0.55
0.03 0.03 0.03 0.03 0.03 0.03 0.015 0.015 0.015 0.015 0.015 0.015 0.045 0.045 0.045 0.045 0.045 0.045
0 0.03 0 0.03 0 0.03 0 0.03 0 0.03 0 0.03 0 0.03 0 0.03 0 0.03
0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
0.62 0.46 0.58 0.46 0.57 0.46 0.7 0.43 0.63 0.4 0.62 0.34 0.62 0.53 0.59 0.47 0.61 0.48
0.43 0.39 0.39 0.35 0.42 0.3 0.37 0.54 0.4 0.47 0.44 0.44 0.43 0.34 0.42 0.32 0.4 0.31
0.66 0.8 0.72 0.82 0.71 0.84 0.61 0.72 0.67 0.79 0.65 0.83 0.66 0.78 0.69 0.82 0.68 0.82
0.31 0.15 0.33 0.24 0.26 0.35 0.47 0.26 0.37 0.18 0.29 0.29 0.31 0.36 0.29 0.32 0.34 0.35
mean max min
0.53 0.70 0.34
0.40 0.54 0.30
0.74 0.84 0.61
0.30 0.47 0.15
a = − 7.40 b=
kt − kr kr
Hs + 4.55 Lp
Acknowledgments
H 1.10 s + 0.64 Lp
This work has been developed as part of the research program of the Wave Group at FACIMAR, CICESE and UNAM, and supported by the Fondo Ramón Álvarez-Buylla (FRABA 14-2013). The authors would like to thank CICESE and FACIMAR for the IXACHI and ColimaWS1 server support, respectively, as well as UNAM for access to the wave flume for the experiments. Julian Delgado´s (CICESE) assistance in setting up the numerical model is appreciated; as is that of UNAM lab staff Erick García, Isaac Moreno and David Rosales for their help in conducting the experiments.
H c = − 5.10 s + 0.36 Lp d = − 2.23 e = − 16.47
Hs + 0.06 Lp Hs + 0.09 Lp
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