alexa Laboratory Investigation on the Behavior of Reef Breakwaters | OMICS International
ISSN: 2473-3350
Journal of Coastal Zone Management
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Laboratory Investigation on the Behavior of Reef Breakwaters

Christos Antoniadis*

Coastal and Port Engineer/Rogan & Associates S.A., Athens, Greece

*Corresponding Author:
Christos Antoniadis
Coastal and Port Engineer/Rogan & Associates S.A.
Athens, Greece
Tel: +306979708081
E-mail: [email protected]

Received Date: June 24, 2014; Accepted Date: July 28, 2014; Published Date: August 06, 2014

Citation: Antoniadis C (2014) Laboratory Investigation on the Behavior of Reef Breakwaters. J Coast Dev 17:389. doi: 10.4172/1410-5217.1000389

Copyright: © 2014 Antoniadis C. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

An experimental investigation into the hydrodynamic behavior of reef breakwaters was carried out. The experiment aimed to provide full scale measurements of the main wave features at the front and rear of the breakwater, which have been analysed with variation of frequency and freeboard, for constant crest width and porosity. A series of tests was carried out at a 2-D wave flume. The analysis has shown that the reflection coefficient was a much more linear process than the transmission coefficient. The transmission coefficient was influenced mostly by the variation of the freeboard, in particularly, as the model became submerged.

Keywords

Reef breakwater; Submerged; Transmission; Dissipation; Reflection; Freeboard

Introduction

In this paper the performance of reef breakwaters was investigated. A reef breakwater is a low-crested homogeneous pile of stones without a filter layer or core and is allowed to be reshaped by wave attack. The initial crest height is just above the water level. Under severe wave conditions the crest height adjusts to a new equilibrium crest height. This equilibrium crest height and the corresponding transmission are the main design parameters [1].

Reef breakwaters are generally detached and parallel to the shore, with much overtopping. The mound of (graded) stones from which they are composed allows for the development of a dynamic stable profile, as opposed to low-crest or submerged breakwaters, which are nothing more than conventional, statically and stable rubble mounds [2-4]. Unfortunately, the performance of low-crested rubble mound structures, and particularly a reef breakwater, is not well documented or understood [5-9].

Since the cost of rubble mound increases exponentially with the height of the crest, the economic advantage of a low-crested structure, over a traditional breakwater that is infrequently overtopped, is apparent. Because the reef type breakwater is the state of the art in design simplicity it emerges as the optimum structure for many situations [10,11].

The geometric design of a reef breakwater is largely determined by the fact that marine equipment is normally required for construction. Sometimes construction is carried out with land-based equipment via a (temporary) causeway, but this approach is not favored, as it requires substantially more material handling. Reef breakwaters are generally built as part of a coastal defence scheme, and are mostly constructed in shallow water, with the consequences of shallow water breakwater design and construction [12].

In this paper, the behavior of reef type breakwaters was investigated by means of laboratory model. The importance of laboratory experiment is well known for scientific research, since experiments give rise to the opportunity to check on the accuracy of theoretical models, and also improve on the understanding of the physical processes involved in the theoretical model [13].

The main goal of the report was to investigate the influence of the variation of mean water level (submerged and not submerged model) and the variation of frequency, for random wave conditions, at wave reflection and transmission for reef type breakwaters. An observation of the behavior of the dissipation of the reef type breakwaters was also included in this study.

Experimental Methods and Procedures

Wave flume

Three experiments with a total of 14 tests were carried out in a wave flume in the Brunel Laboratory. The flume had a length of 17.65 m, width of 0.90 m and was used with a working depth of 0.8 m. It had a 1:10 sloping section at the front and then followed by a 3.50 m flatting section reaching a final 1:10 sloping section. The flume bed was made by plastic. The sloping bed at the rear of the flume was reinforced by a wooden section in order to hold the weight of the model. Waves were generated by a hydraulic driven wave paddle system controlled via computer software. The system was capable of generating series of random waves. It has been shown that generating the wave in deep water reduces the magnitude of free log waves (induced by wave paddle) to the minimal. The wave motion generated by the wave paddle was repeatable therefore data collection at different location can be obtained through separate runs of each of the wave cases. Wave absorption was fully software driven using digital filters.

The reef breakwater model

The tests were conducted using a reef breakwater model. The model was placed, as mentioned above, at the end of the flume. It had a 14.27 m distance from the wave paddle system. The model had a front slope of 1:2, a crown height of 0.25 m, a crest width of 0.25 m and a rear slope of 1:1. The rocks had a diameter (Dn50) approximately 5 to 6 cm and an individual weight of 0.45 Kgr. Moreover, to improve the stability of the structure, a cottage wire was placed above the model. The dimensions details of the model and the wave flume can be seen in Figure 1.

coastal-development-wave-flume-model

Figure 1: Dimensions details of the wave flume and the model

During the tests, two white cylinders were used. They had 1.2 m length each with an inner and an outer diameter of 19 mm and 21 mm respectively. They were placed sidelong the model for checking if the water would pass through the model and stabilize behind the structure without flowing back again, after it reflected at the sloping bed, causing increase in the volume of water behind the model and errors in readings. To check whether that would happen, a red dye was used. The red dye was inserted into the water and found that the water flew through the cylinders from the “landward” to the “seaward” direction. As a result, the model proved safe for use.

The experimental tests

The reef breakwater model was tested using three mean water levels (m.w.l.), 0.756 m (Experiment 1), 0.795 m (Experiment 2) and 0.856 m (Experiment 3) for random wave conditions and for various frequencies. The crest width and the crest height were kept constant.

The peak frequencies (fp) that were used, for each different mean water level, were the followings: 0.4 Hz, 0.6 Hz, 0.8 Hz, 0.9 Hz and 1.0 Hz. At 0.756 m m.w.l, the peak frequency equal to 1.0 Hz could not be used for the tests due to the fact that the wave paddle could not generate waves of this peak frequency for the given mean water level. Furthermore, the duration of the readings taken for each peak frequency was 10 minutes. The generated wave height was adjusted to be 0.1 m. Table1 summarize all the experimental test conditions.

Experiment 1 Frequency(Hz) Hi (m) M.W.L. (m) Freeboard (m)
Test 1 0.4 0.1 0.756 -0.044
Test 2 0.6
Test 3 0.9
Experiment 2 Frequency(Hz) Hi (m) M.W.L. (m) Freeboard (m)
Test 1 0.4 0.1 0.795 -0.005
Test 2 0.6
Test 3 0.8
Test 4 0.9
Test 5 1
Experiment 3 Frequency(Hz) Hi (m) M.W.L. (m) Freeboard (m)
Test 1 0.4 0.1 0.856 0.056
Test 2 0.6
Test 3 0.8
Test 4 0.9
Test 5 1

Table 1: Experimental test conditions

Instrumentation/Calibration

The vertical elevation of the water surface was measured using standard surface piercing wave gauges. To analyze wave reflection and transmission, five wave gauges were positioned near the structures registered with a frequency of 100 Hz (0.01 sample interval (sec)). Three gauges (namely R1, R2, and R3) were installed at the front of the structure at a distance of 1.11 m, 0.78 m and 0.65 m respectively. The three gauges were used to measure the directional spectrum of waves composed of incident and reflected waves. Two wave gauges (R4, R5) were also installed at the rear of the structure to measure the directional spectrum of waves composed of transmitted waves. R4 and R5 had a 0.22 m and 0.52 m distance respectively, from the model. The location of the gauges can be seen in Figure 1. All the gauges were calibrated for all the tests. The method that was used for the calibration of the gauges was the following: Data was collected for duration of 100 seconds for the still position of the gauges (zero level), then for the upward position (moved 50 mm upwards) and at last for the downward position (moved 50 mm downwards from the zero level). The collected data was measured in voltage units. The calibration of the gauges was carried out and checked for its accuracy, by plotting graphs of Elevation vs. Voltage, every time that the mean water level was changed.

Results

The data obtained from the experiment was analysed and the results are presented here. The reflection coefficient (Kr) as a function of frequency was estimated from the spectral densities of the incident and the reflected wave fields. The reflection coefficient was investigated at the gauges 1-3. The transmission coefficient (Kt) was estimated similarly (incident and transmitted wave fields). The transmission coefficient was investigated at the gauges 4-5.

Effects of various parameters on reflection coefficient

The numerical presentation of Kr and the various parameters, for each mean water level, can be seen in Tables 2-4. At Tables 2-4, the peak frequency (fp), the wavelength for deep water L0, the water depth at the gauges (d), the wavelength (L), the water depth measured at the toe of the structure (dt), the crest freeboard (F), the incident wave height (Hsi), the iribarren number (ξ), relative crest freeboard (F/Hsi), relative depth of crest freeboard (F/dt), wave steepness (Hsi/L) and celerity (C) can be also seen.

Mean Water Level at 0.756m (gauges1,2,3)
fp (Hz) Lo (m) d (m)     L(m) dt(m) F (m) Hsi(m)      
0.488 6.549 0.563 0.086 0.129 4.373 0.356 -0.044 0.061 1.033 0.369 -0.01
0.586 4.548 0.563 0.124 0.161 3.491 0.356 -0.044 0.091 0.688 0.306 -0.013
0.781 2.558 0.563 0.22 0.242 2.326 0.356 -0.044 0.084 0.553 0.274 -0.019
0.879 2.021 0.563 0.278 0.293 1.921 0.356 -0.044 0.085 0.487 0.214 -0.023
F/Hsi C (m/s) F/dt Hsi/L  
-0.717 2.135 -0.124 0.014
-0.486 2.046 -0.124 0.026
-0.527 1.817 -0.124 0.036
-0.517 1.689 -0.124 0.044

Table 2: Numerical presentation of Kr and various parameters for mean water level at 0.756 m.

Mean Water Level at 0.795m (gauges1,2 and 3)
fp (Hz) Lo (m) d (m) d/L0 d/L L(m) dt(m) F (m) Hsi (m) ξ Kr F/L
0.488 6.556 0.602 0.092 0.134 4.498 0.395 -0.005 0.055 1.091 0.359 -0.001
0.586 4.547 0.602 0.132 0.168 3.573 0.395 -0.005 0.087 0.721 0.203 -0.001
0.781 2.56 0.602 0.235 0.255 2.357 0.395 -0.005 0.078 0.573 0.164 -0.002
0.879 2.021 0.602 0.298 0.31 1.942 0.395 -0.005 0.069 0.54 0.152 -0.003
0.977 1.636 0.602 0.368 0.374 1.608 0.395 -0.005 0.069 0.488 0.131 -0.003
F/Hsi C (m/s) F/dt Hsi/L  
-0.091 2.195 -0.013 0.012
-0.057 2.093 -0.013 0.024
-0.064 1.841 -0.013 0.033
-0.072 1.707 -0.013 0.036
-0.073 1.571 -0.013 0.043

Table 3: Numerical presentation of Kr and various parameters for mean water level at 0.795 m.

Mean Water Level at 0.795m (gauges1,2,3)
fp (Hz) Lo (m) d (m) d/L0 d/L L(m) dt(m) F (m) Hsi (m) ξ Kr F/L
0.488 6.556 0.663 0.101 0.142 4.663 0.456 0.056 0.053 1.114 0.408 0.012
0.586 4.547 0.663 0.146 0.18 3.685 0.456 0.056 0.091 0.708 0.235 0.015
0.781 2.56 0.663 0.259 0.276 2.402 0.456 0.056 0.075 0.584 0.203 0.023
0.879 2.021 0.663 0.328 0.337 1.964 0.456 0.056 0.071 0.533 0.158 0.029
0.977 1.636 0.663 0.405 0.409 1.618 0.456 0.056 0.069 0.486 0.142 0.035
F/Hsi C (m/s) F/dt Hsi/L  
1.062 2.276 0.123 0.011
0.616 2.159 0.123 0.025
0.747 1.876 0.123 0.031
0.788 1.726 0.123 0.036
0.808 1.581 0.123 0.043

Table 4: Numerical presentation of Kr and various parameters for mean water level at 0.856 m.

The reflection coefficient is affected from various parameters, most notably from the Iribarren number (ξ). The reflection coefficient was plotted against the Iribarren number using results from different depth at the toe of the structure and consequently different mean water level (shown in Figure 2). It was found that Kr was proportional to ξ. Moreover, a large scatter of the data was been noticeable at the initial values of Iribarren number.

coastal-development-reflection-coefficient-iribarren

Figure 2: Reflection coefficient vs. Iribarren number (ξ).

The behavior of the reflection coefficient was observed in Figures 3-4 where it was plotted initially against the deep water wavelength (Lo) and then against the wavelength (L) at the gauges 1, 2 and 3. Kr was systematically increasing in direct relationship with the increase of the deep water wavelength-Lo (and wavelength-L). The maximum values of Kr for different m.w.l. were close to each other with a maximum value of Kr at 0.856 m m.w.l.

coastal-development-reflection-coefficient-wavelength

Figure 3: Reflection coefficient vs. Deep-water Wavelength (Lo)

coastal-development-reflection-coefficient-wavelength

Figure 4: Reflection coefficient vs. Wavelength (L).

Inspection of Figures 5 and 6 showed that the relationship between the reflection coefficient and the depth of the toe of the structure and also with the relative crest freeboard had similar results. In Figure 5, Kr seemed to decrease as dt increased until it reached a value of dt (dt=0.395 m) where the model became submerged. After this value Kr started to increase. The same results can also be seen in Figure 6 for different frequencies where the value of dt=0.395 m corresponded to the value of F=-0.05.

coastal-development-reflection-coefficient-toe

Figure 5: Reflection coefficient vs. Depth at the toe of the structure (dt).

coastal-development-reflection-coefficient-freeboard

Figure 6: Reflection coefficient vs. Crest Freeboard (F).

The reflection was also plotted against the significant incident wave height (Hsi) (in Figure 7) to identified the relationship between them. It was found that Kr was almost proportional to Hsi. The reflection coefficient increased as the significant incident wave height increased, with the exception of values of Hsi which corresponded to the frequency equal to 0.4 Hz.

coastal-development-coefficient-incident-wave

Figure 7: Reflection coefficient vs. significant incident wave height (Hsi).

The relationship of the reflection coefficient with the wave steepness (Hsi/L) can be described as inverse proportional. The reflection coefficient decreased as the wave steepness increased. Similarity between 0.795 m and 0.856 m m.w.l. was obtained when their values of Kr against Hsi/L were very close to each other. In Figure 8, the plot of the reflection coefficient against the wave steepness for the three mean water levels can be seen.

coastal-development-coefficient-wave-steepness

Figure 8: Reflection coefficient vs. Wave Steepness (Hsi/L).

The relationship of Kr and F/Hsi can be seen in Figure 9. The reflection coefficient seemed to be inverse proportional to F/Hsi with an exception at f=0.4 Hz (at 0.856 m m.w.l.) where Kr took a very large value. It can be noticed that when F/Hsi became approximately zero, all influence of the significant wave height was lost which lead to a large scatter in the graph at F/Hsi ≈ 0. Similar results were obtained for F/L (Figure 10) where the increase of Kr after F/L ≈ 0 became smoother as the frequencies increased.

coastal-development-coefficient-crest-freeboard

Figure 9: Reflection coefficient vs. Relative crest Freeboard (F/Hi).

coastal-development-reflection-coefficient

Figure 10: Reflection coefficient vs. F/L.

Effects of various parameters on transmission coefficient

The numerical presentation of Kt and the various parameters, for each mean water level and for each transmission gauge (R4, R5), can be seen in Tables 4-8.

Mean Water Level at 0.756m (gauge 4)
fp (Hz) Lo (m) d (m) d/L0 d/L L(m) dt(m) F (m) Hsi (m) ξ Kt F/L
0.488 6.556 0.235 0.036 0.078 2.997 0.395 -0.005 0.055 1.091 0.357 -0.002
0.586 4.547 0.235 0.052 0.096 2.446 0.395 -0.005 0.087 0.721 0.289 -0.002
0.781 2.56 0.235 0.092 0.134 1.752 0.395 -0.005 0.078 0.573 0.306 -0.003
0.879 2.021 0.235 0.116 0.155 1.514 0.395 -0.005 0.069 0.54 0.302 -0.003
0.977 1.636 0.235 0.144 0.178 1.319 0.395 -0.005 0.069 0.488 0.28 -0.004
F/Hsi C (m/s) F/dt Hsi/L  
-0.091 1.462 -0.013 0.018
-0.057 1.434 -0.013 0.036
-0.064 1.369 -0.013 0.045
-0.072 1.331 -0.013 0.046
-0.073 1.288 -0.013 0.052

Table 5: Numerical presentation of Kt and various parameters for mean water level at 0.795 m.

Mean Water Level at 0.856m (gauge 4)
fp (Hz) Lo (m) d (m) d/L0 d/L L(m) dt(m) F (m) Hsi (m) ξ Kt F/L
0.488 6.556 0.296 0.045 0.089 3.328 0.456 0.056 0.053 1.114 0.455 0.017
0.586 4.547 0.296 0.065 0.109 2.705 0.456 0.056 0.091 0.708 0.48 0.021
0.781 2.56 0.296 0.116 0.155 1.913 0.456 0.056 0.075 0.584 0.459 0.029
0.879 2.021 0.296 0.146 0.181 1.639 0.456 0.056 0.071 0.533 0.475 0.034
0.977 1.636 0.296 0.181 0.209 1.418 0.456 0.056 0.069 0.486 0.491 0.039
F/Hsi C (m/s) F/dt Hsi/L  
1.062 1.624 0.123 0.016
0.616 1.585 0.123 0.034
0.747 1.494 0.123 0.039
0.788 1.441 0.123 0.043
0.808 1.385 0.123 0.049

Table 6: Numerical presentation of Kt and various parameters for mean water level at 0.856 m.

Mean Water Level at 0.795m (gauge 4)
fp (Hz) Lo (m) d (m) d/L0 d/L L(m) dt(m) F (m) Hsi (m) ξ Kt F/L
0.488 6.556 0.185 0.028 0.069 2.681 0.395 -0.005 0.055 1.091 0.789 -0.002
0.586 4.547 0.185 0.041 0.084 2.205 0.395 -0.005 0.087 0.721 0.596 -0.002
0.781 2.56 0.185 0.072 0.116 1.589 0.395 -0.005 0.078 0.573 0.518 -0.003
0.879 2.021 0.185 0.092 0.133 1.389 0.395 -0.005 0.069 0.54 0.539 -0.004
0.977 1.636 0.185 0.113 0.152 1.219 0.395 -0.005 0.069 0.488 0.549 -0.004
F/Hsi C (m/s) F/dt Hsi/L  
-0.091 1.309 -0.013 0.021
-0.057 1.292 -0.013 0.04
-0.064 1.241 -0.013 0.049
-0.072 1.221 -0.013 0.05
-0.073 1.191 -0.013 0.056

Table 7: Numerical presentation of Kt and various parameters for mean water level at 0.795 m.

Mean Water Level at 0.856m (gauge 5)
fp (Hz) Lo (m) d (m) d/L0 d/L L(m) dt(m) F (m) Hsi (m) ξ Kt F/L
0.488 6.556 0.246 0.038 0.081 3.052 0.456 0.056 0.053 1.114 0.792 0.018
0.586 4.547 0.246 0.054 0.098 2.505 0.456 0.056 0.091 0.708 0.235 0.022
0.781 2.56 0.246 0.096 0.138 1.785 0.456 0.056 0.075 0.584 0.781 0.031
0.879 2.021 0.246 0.122 0.159 1.545 0.456 0.056 0.071 0.533 0.763 0.036
0.977 1.636 0.246 0.15 0.184 1.339 0.456 0.056 0.069 0.486 0.762 0.042
F/Hsi C (m/s) F/dt Hsi/L  
1.062 1.489 0.123 0.017
0.616 1.468 0.123 0.036
0.747 1.394 0.123 0.042
0.788 1.358 0.123 0.046
0.808 1.308 0.123 0.052

Table 8: Numerical presentation of Kt and various parameters for mean water level at 0.856 m.

The government parameters related to transmission are: the structure geometry, principally, the relative crest freeboard, crest width and water depth, permeability, and on the wave conditions, principally, the wave height and period. Due to the fact that the structure geometry, crest width and permeability remained constant, the transmission and especially its coefficient would fluctuate in relationship with the rest parameters. Consequently, the relationship between Kt (for both gauges- R4 & R5) and incident wave height (Hi), wave period (T) (already explained previously), relative water depth (d/L), wavelength (L) at R4 and R5, deep water wavelength (Lo), wave steepness (Hi/L), relative crest freeboard (F/Hi), F/L, crest freeboard (F) and relative depth of crest freeboard (F/dt) was investigated.

The relationship between the transmission coefficient and the Significant Incident Wave Height (Hsi) is shown in Figures 11-13. It can be observed that the transmission coefficient was proportional to the increase of the significant incident wave height. However, when the model became submerged (0.856 m m.w.l.) Kt became inversely proportional to the increase of Hsi.

coastal-development-transmission-coefficient-wave

Figure 11: Transmission coefficient vs. Incident Wave Height (Hsi) at 0.756 m m.w.l.

coastal-development-transmission-coefficient-wave

Figure 12: Transmission coefficient vs. Incident Wave Height (Hi) at 0.795 m m.w.l.

coastal-development-transmission-coefficient-wave

Figure 13: Transmission coefficient vs. Incident Wave Height (Hi) at 0.856 m m.w.l.

In Figures 14-16 the transmission coefficient was plotted against the relative water depth (d/L). At 0.756 m m.w.l. the Kt was inversely proportional to the increase of the d/L. This behavior continued at 0.795 m m.w.l. as well, with an exception from the results taken from gauge 5 at which the Kt started to increase after d/L=0.138. At 0.856 m m.w.l. the results taken from gauge 4 showed that Kt start to increase as d/L increased. However, this was inverse proportional with the results taken from gauge 5 (Kt decreased as d/L increased).

coastal-development-transmission-coefficient-wave

Figure 14: Transmission coefficient vs. Relative Water Depth (d/L) at 0.756 m m.w.l.

coastal-development-transmission-coefficient-water

Figure 15: Transmission coefficient vs. Relative Water Depth (d/L) at 0.795 m m.w.l.

coastal-development-transmission-coefficient-water

Figure 16: Transmission coefficient vs. Relative Water Depth (d/L) at 0.856 m m.w.l.

Furthermore, the effect of deep water wavelength (Lo) and wavelength (L) at gauges 4 and 5, on the transmission coefficient showed to be the same. Tables 4-8 showed that Kt was, in general, proportional with the increase of L and Lo until the model became submerged. Therefore at 0.856 m m.w.l. the transmission coefficient became inversely proportional to both of these parameters. The only exception to this behavior was the value of Kt (at gauge 5) corresponding to f=0.6 Hz for 0.856 m m.w.l.

On the other hand, the effect of the transmission coefficient on the wave steepness (Hsi/L) showed inverse results. As it is shown in the numerical presentation, the Kt was inversely proportional to Hsi/L for both gauges for 0.756 m and 0.795 m m.w.l. At 0.856 m m.w.l. the results taken from gauge 5 show that Kt continued to have an inversely proportional effect to the wave steepness, however the results taken from gauge 4 showed Kt to be smoothly proportional to Hsi/L.

Observation of the behavior of Kt with respect to crest freeboard (F), relative depth of crest freeboard (F/dt), relative crest freeboard (F/Hsi) and F/L lead to similar conclusions. Kt was proportional to the increase of those parameters. As these parameters increased, the transmission coefficient increased having a large scatter at values where F was approximately equal to zero.

A comparison between the numerical presentation of both coefficients (Kr, Kt) reveals that the celerity was inversely proportional to the increase of frequency and also greater at gauges 1, 2, 3 than at gauges 4 and 5. Finally, the values of celerity increased as the mean water level increased.

Effects of various parameters on dissipation coefficient

It is known that the equation of energy balance can be derived by the coefficients of reflection, transmission and dissipation [14]:

Kr2 + Kt2 + KD =1        (1)

where,

Kr = reflection coefficient (Hr/Hi);

Kt =transmission coefficient (Ht/Hi); and

KD =dissipation coefficient (Ed/Ei)

where,

Hi =incident wave height at the seaside; and

Ht = transmitted wave height at the lee side.

Hr = reflected wave height

The dissipation coefficient was calculated for all the different frequencies and all the different mean water levels and it was observed that to have high values. The numerical results of that calculation (for both R4 and R5 separately) can be seen in Table 9. Figures 17-18 showed, for both gauges, that KD had a smooth increase as frequency increased. Moreover, it can be concluded that after a point (f=0.6 Hz) was reached, that the increase stabilized to an almost constant value of Kt with respect to the increase of frequency. However, the dissipation coefficient, which was calculated from the results taken from gauge 5, shows one changing point of its behavior at f=0.6 Hz at 0.856 m m.w.l.

Mean Water Level at 0.756m (gauge 4)
Frequencies Reflection coeff. Transmission coeff. Dissipation coeff. Freeboard (m) ξ
0.488 0.369 0.403 0.838 -0.044 1.033
0.586 0.306 0.296 0.905 -0.044 0.688
0.781 0.274 0.297 0.915 -0.044 0.553
0.879 0.214 0.28 0.936 -0.044 0.487
Mean Water Level at 0.795m (gauge 4)
Frequencies Reflection coeff. Transmission coeff. Dissipation coeff. Freeboard (m) ξ
0.488 0.359 0.357 0.862 -0.005 1.091
0.586 0.203 0.289 0.936 -0.005 0.721
0.781 0.164 0.306 0.938 -0.005 0.573
0.879 0.152 0.302 0.941 -0.005 0.54
0.977 0.131 0.28 0.951 -0.005 0.488
Mean Water Level at 0.795m (gauge 5)
Frequencies Reflection coeff. Transmission coeff. Dissipation coeff. Freeboard (m) ξ
0.488 0.359 0.789 0.499 -0.005 1.091
0.586 0.203 0.596 0.777 -0.005 0.721
0.781 0.164 0.518 0.84 -0.005 0.573
0.879 0.152 0.539 0.828 -0.005 0.54
0.977 0.131 0.549 0.825 -0.005 0.488
Mean Water Level at 0.856m (gauge 4)
Frequencies Reflection coeff. Transmission coeff. Dissipation coeff. Freeboard (m) ξ
0.488 0.408 0.455 0.792 0.056 1.114
0.586 0.235 0.48 0.845 0.056 0.708
0.781 0.203 0.459 0.865 0.056 0.584
0.879 0.158 0.475 0.866 0.056 0.533
0.977 0.142 0.491 0.859 0.056 0.486
Mean Water Level 0.856m (gauge 5)
Frequencies Reflection coeff. Transmission coeff. Dissipation coeff. Freeboard (m) ξ
0.488 0.408 0.792 0.453 0.056 1.114
0.586 0.235 0.235 0.943 0.056 0.708
0.781 0.203 0.781 0.591 0.056 0.584
0.879 0.158 0.763 0.627 0.056 0.533
0.977 0.142 0.762 0.632 0.056 0.486

Table 9: Numerical results of the analysis of the dissipation coefficient (KD) for diff. m.w.l.

coastal-development-dissipation-coefficient-frequency

Figure 17: Dissipation coefficient (KD) vs. Frequency (f) for different m.w.l. (gauge 4).

coastal-development-dissipation-coefficient-frequency

Figure 18: Dissipation coefficient (KD) vs. Frequency (f) for different m.w.l. (gauge 5).

From the observation of Figures 19-20 where the dissipation coefficient was plotted against the relative crest freeboard, it can be seen that KD increased as F increased until a value of F, approximately equal to 0, was reached after which it started to decrease. Once more, the dissipation coefficient from gauge 5 showed one changing point of its general behavior at f=0.6 Hz.

coastal-development-dissipation-coefficient-frequency

Figure 19: Dissipation coefficient (KD) vs. Crest Freeboard (F) for different frequencies (gauge 4).

coastal-development-dissipation-coefficient-frequencies

Figure 20: Dissipation coefficient (KD) vs. Crest Freeboard (F) for different frequencies (gauge 5).

Finally, in Figures 21-22 where KD was plotted against Iribarren number (ξ), it can be seen that the dissipation coefficient was inversely proportional to the increase of ξ. Furthermore, at gauge 5 there is again at f=0.6 Hz a changing point at the behavior of KD.

coastal-development-dissipation-coefficient-iribarren

Figure 21: Dissipation coefficient (KD) vs. Iribarren number (ξ) for different m.w.l. (gauge 4).

coastal-development-dissipation-coefficient-iribarren

Figure 22: Dissipation coefficient (KD) vs. Iribarren number (ξ) for different m.w.l. (gauge 5).

Discussion

As the results were presented and were described in the previous section of this paper, their graphical presentation was not a helpful tool for analysis. The most accurate presentation, which can help to analyze and inform the discussion of the results, is the numerical presentation.

The low or high values of the reflection, transmission and dissipation coefficient was dependent on the fact that some waves (observed in the lab) were broken before reaching the model. The waves were broken at the two slopes before the model, and that was more noticeable at waves with high frequencies. As a result, they had hit the model with less energy than they had had in the beginning, due to the fact that it was already lost at the slopes. In summary, that energy was lost because of the bed friction and because of the production of turbulence. The bed friction was varied into the wave channel due to the fact that one part of the bed was made by wood (the sloping bed where the model was placed) and the rest by plastic.

In addition, the waves that broke in front and behind the model at its sloping bed created set-up and consequently produced an increase in the depth of the water above the still water level. This was marked more at the back side of the model when it was submerged. As well, the classic picture of a large wave curling over and breaking on the structure was rarely observed during this study. Possibly because of the sloping bed and the porous nature of the breakwater, waves appeared to be partly absorbed into the structure before they could break on it. This observation is in agreement with the Ahrens’s (1984) observation at reef type breakwaters.

Furthermore, in the lab it was noticed that the waves which hit the slope behind the model were reflected back to the model causing some errors in the readings. The greatest errors were observed at waves with frequency equal to 0.4 Hz, which were the largest waves of the test that were generated, and these errors were shown also in the numerical presentation of the results for both wave conditions. It was observed that an amount of volume of water flew out of the wave channel and this was not reflected at the slope at all.

From the investigation of the results taken from every gauge, it can be distinguished that the heights of waves at gauge 1 were slightly different from the gauges 2 and 3. This happened because of the fact that gauge 1 was placed at the flat bed of the wave channel instead of the gauges 2 and 3 which were placed at the sloping bed.

Additionally, the results from the general characteristics of waves at the gauges showed that celerity was inversely proportional with the increase of frequency and had greater values when the mean water level was raised. Also, when the waves reached the slope the celerity started to increase as they approached the model, in agreement with that reported by Yukiko T and Nobuhisa K [15].

Conclusion

Laboratory tests (experiment) were conducted to observe the performance of a reef breakwater model. The wave reflection, transmission and dissipation characteristics of the model were measured, with more interesting giving in the first two characteristics (Kr and Kt). The tests were carried out using constant crest width and height for a series of random wave conditions, with different mean water levels and different frequencies.

Furthermore, it can be seen from the results that the reflected and incident energy, as well as transmitted and incident energy, approximately cover the same frequency range. In general, the peak frequency of the reflected and transmitted waves appeared to be lower than the incident waves.

From the analysis, reflection coefficient has been shown to be a much more linear process than transmission. Reflection coefficient showed a systematic increase with the Iribarren number (ξ). However, the larger scatter in plots of Kr versus ξ (especially at regular waves) indicated that the Iribarren number does not represent the optimal mean for the description of the reflection process. In general, from the results of the experiment, the reflection coefficient was also proportional to wave period (T), the deep-water wavelength (Lo), wavelength (L) and the significant incident wave height (Hsi). Furthermore, Kr was inversely proportional to crest freeboard (F) and depth at the toe of the structure (dt) as the model became submerged.

Transmission coefficient has shown that was not as much linear as reflection coefficient was, depending more on the crest freeboard (F). From the results, it can be observed that Kt was proportional with F, relative depth of crest freeboard (F/dt), relative crest freeboard (F/H) and to F/L. However, in some cases the behavior of the transmission coefficient was changed as the model became submerged.

The Kt was proportional with the relative water depth (d/L) as the model became submerged. This is expected because larger values of d/L correspond to relatively deeper water waves, where in more energy is concentrated near the surface. When the relative depth of crest freeboard (F/dt) is also large, this energy concentrated near the surface is easily transmitted across the structure.

The transmission coefficient was also proportional with frequency (f) and wave steepness (H/L) as the model became submerged. Nevertheless, the Kt became inversely proportional, as the model became submerged, with significant incident wave height (Hsi), deepwater wavelength (Lo) and wavelength (L).

As far as dissipation is concerned, its value depends on the Kr and Kt since KD=1-Kr2-Kt2. It was found that KD was proportional to the frequency (f) and inversely proportional to the Iribarren number (ξ). However, the influence of the dissipation coefficient by the crest freeboard parameter was having exactly inverse results.

Acknowledgement

I would especially like to thank Professor G. N. Bullock for his encouragement and support.

References

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