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Wave interaction with ‘⊥’-type breakwaters
Ocean Engineering 29 (2002) 561–589
Technical note
Wave interaction with ‘Ⲛ’-type breakwaters S. Neelamani a
a,*
, R. Rajendran
b
Department of Ocean Engineering, IIT Madras, Chennai 600 036, India b Chennai Port Trust, Chennai 600 001, India Received 4 October 2000; accepted 3 January 2001
Abstract The wave transmission, reflection and energy dissipation characteristics of ‘Ⲛ’-type breakwaters were studied using physical models. Regular and random waves in a wide range of wave heights and periods and a constant water depth were used. Five different depths of immersion (two emerged, one surface flushing and two submerged conditions) of this breakwater were selected. The coefficient of transmission, Kt, and coefficient of reflection, Kr, were obtained from the measurements, and the coefficient of energy loss, Kl was calculated using the law of balance of energy. It was found that the wave transmission is significantly reduced with increased relative water depth, d/L, whether the vertical barrier of the breakwater is surface piercing or submerged, where ‘d’ is the water depth and ‘L’ is the wave length. The wave reflection decreases and energy loss increases with increased wave steepness, especially when the top tip of the vertical barrier of this breakwater is kept at still water level (SWL). For any incident wave climate (moderate or storm waves), the wave transmission consistently decreases and the reflection increases with increased relative depth of immersion, ⌬/d from ⫺0.142 to 0.142. Kt values less than 0.3 can be easily obtained for the case of ⌬/d=+0.071 and 0.142, where ⌬ is the height of exposure (+ve) or depth of immersion (⫺ve) of the top tip of the vertical barrier. This breakwater is capable of dissipating wave energy to an extent of 50–80%. The overall performance of this breakwater was found to be better in the random wave fields than in the regular waves. A comparison of the hydrodynamic performance of ‘Ⲛ’-type and ‘T’-type shows that ‘T’-type breakwater is better than ‘Ⲛ’-type by about 20– 30% under identical conditions. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Special breakwaters; Wave transmission; Reflection; Energy dissipation; Surface piercing; Submerged barrier
* Corresponding author. Tel.: +91-44-445-8639; fax: +91-44-235-0509. E-mail address: [email protected] (S. Neelamani). 0029-8018/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 1 ) 0 0 0 3 0 - 0
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1. Introduction Rubble mound breakwaters are widely used around the world for the construction of artificial harbors and for shore protection works. When sufficient quantity and good quality rubble are not available in the vicinity of the proposed harbor site, then concrete caissons can be used. Both structures are generally expensive for deeper waters. For such conditions, special types of breakwaters, which require less concrete per unit length and are capable of transmitting less wave energy, must be developed. Since in deeper waters most of the wave energy is concentrated near the water surface, a structure is required, which can effectively dissipate or reflect this energy. Partially immersed horizontal plates were found to be good energy dissipaters due to artificial stimulation of wave breaking on the top of the plates (Patarapanich, 1984). Partially immersed vertical barriers are capable of reflecting the incident wave energy effectively in deeper waters (Ursell, 1947; Wiegel, 1960; Reddy and Neelamani, 1992a). Combining these characteristics of a horizontal and a vertical barrier into a ‘T’-type or a ‘Ⲛ’-type breakwater is expected to improve the performance as a breakwater due to the additive properties of wave breaking and reflection. Hence, this type of breakwater is expected to reduce the energy transmission towards the lee side of the structure. This has motivated the authors to investigate the wave transmission, reflection and energy dissipation characteristics of the ‘T’- and ‘Ⲛ’type breakwaters. Neelamani and Rajendran (2000) reported the hydrodynamic performance of ‘T’-type breakwaters. A good breakwater should allow minimum transmission and maximize wave energy dissipation. In the present study, the depth of submergence of the breakwater is varied in order to find out the appropriate depth of immersion at which the wave transmission is minimum. Ⲛ-type breakwaters can be supported on piles and the environmental loads can be transferred to the seabed through these piles (Fig. 1). This type of breakwater can be mass fabricated and assembled on the land, and quickly installed using floating barges during a fair weather window. This breakwater can be installed as a permanent structure at any desired depth of submergence of the horizontal part of the barrier, depending upon the expected hydrodynamic performance and the aesthetic aspects. Fig. 2 shows a definition sketch of the Ⲛ-type breakwater. The following processes will dissipate a part of the incident wave energy: (a) (b) (c) (d) (e)
Vortex shedding at the top of the vertical barrier of the breakwater during its submerged condition. Wave breaking over the breakwater, due to fast shoaling effect because of a sudden change in the depth of wave propagation. Turbulence developed by the overtopping mass of water on the vertical barrier. Boundary layer formation under the horizontal barrier. Wave trapping by the vertical barrier and a resulting increase in the horizontal velocity gradient of the overtopping water mass.
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Fig. 1.
Schematic view of the Ⲛ-type breakwater.
Fig. 2.
Definition sketch.
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The present experimental investigation with partially submerged Ⲛ-type breakwater has been carried out with the following objectives in mind: (a)
(b)
(c) (d)
To determine the wave transmission, reflection and energy dissipation characteristics of this breakwater for different incident wave characteristics and for different depths of submergence of these breakwaters due to regular and random waves. To analyze the water surface fluctuations in front of these breakwaters for different incident wave characteristics and for different depths of submergence. To compare the hydrodynamic performances due to regular and random waves. To compare the performance of ‘T’- type and ‘Ⲛ’-type barriers under identical test conditions.
Investigations on the characteristics of wave transmission for a wide range of wave conditions and different depths of submergence of the structure are essential for design, once the permissible wave transmission for a particular activity at the lee side of the structure is decided. For example, for given wave conditions (say peak period of 8 s and significant wave height of 2.0 m), if the permissible transmitted wave height is 0.4 m, then it is required to appropriately select the depth of immersion and size of this breakwater to achieve the intended wave conditions at the lee side (see Appendix A, where a worked out example is given to illustrate the utility of the present study). The study of energy dissipation character is important to understand the efficiency of the structure as a breakwater. A breakwater can be considered to be performing well if the energy dissipation capability is high. It is also necessary to understand the wave climate at the seaside of the structure due to wave reflection from the breakwater, since the safe navigation of approaching vessels is a function of this wave climate. Hence, wave histories are also measured at a typical location in the seaside of the breakwater.
2. Literature review Though no previous study has been carried out on ‘Ⲛ’-type breakwaters, literature on some of the special breakwaters more related to the present topic are presented. Ursell (1947) developed a theory for the transmission and reflection of gravity water waves in deep water for a fixed vertical infinitely thin barrier extending from the water surface to some depth below the surface. He proposed an equation for the coefficient of transmission as a function of modified Bessel function. Heins (1948, 1950) presented a solution procedure for the diffraction of water waves by a thin semi-infinite surface horizontal plate in water of finite depth and extended it to the case of a submerged plate in water of finite depth. Using the Wiener–Hopf technique, Greene and Heins (1953) extended these solutions to the case of a thin surface plate
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in water of finite depth. Stoker (1957) obtained simple expressions for the computation of the transmission and reflection coefficients by matching the horizontal velocity and pressures between adjacent regions for the submerged plate. Wiegel (1960) developed a theory based on the wave power transmission past a rigid vertical thin barrier, which extends from above the water surface to some distance below the surface. A satisfactory comparison of the theory with laboratory measurements was noticed for limited sets of data. A consistent trend of decreasing transmission coefficient with increasing wave steepness was noticed from the experimental studies, which was attributed to the excess loss in energy during transmission by separation at the bottom of the barrier. Burke (1964) extended the Wiener–Hopf technique to the case of a thin submerged plate in deep water. The solution procedure is complicated due to the iterative nature of the computation of the coefficients. Ijima et al. (1971) presented a method using eigenfunction expansions of the wave dispersion relation for the case of a surface plate in water of finite depth. Leppington (1972) applied the method of matched asymptotic expansions to the surface plate in deep water. Siew and Hurley (1972) extended this method to solve the problem of a submerged horizontal plate in shallow water. Bai (1975) presented a numerical method using finite elements for solving the linearized water-wave diffraction problem due to submerged objects. Patarapanich (1978) applied this method to the case of a submerged horizontal plate in finite water depth. Nallayarasu et al. (1992) applied the same method for fixed submerged inclined plates in finite water depth. The model tests of Hattori (1975) comprising a horizontal plate, fixed at the still water level (surface plate) and a submerged plate fixed at a quarter and at a half of the water depths exhibited a trend of lower transmission and higher reflection at small submergence depth ratio. These rather limited results led to the conclusion that the surface plate is more effective than the submerged plate in reducing the wave transmission. Experimental results subsequently produced by Dattatri et al. (1977) also showed that the wave transmission decreased with decreasing submergence, but reflection was maximum at ⌬/d=0.07 (where ⌬ is the depth of submergence of the plate and ‘d’ is the water depth) and not when the plate was at the surface. The optimum plate width for minimum transmission was found to be 0.3–0.4 times that of the incident wavelength. Patarapanich (1984) applied the finite element technique and calculated the reflection and transmission coefficients for a submerged horizontal plate from deep to shallow water limits. The conditions for maximum and zero reflection were analyzed. In the theoretical model, zero energy loss is assumed. It was concluded that the wave reflection from a submerged horizontal plate depends on the plate width to wavelength ratio, the relative depth of immersion and the relative water depth. It was found that the reflection coefficient generally increases with decreasing relative depth of immersion, ⌬/d and relative water depth, d/L. Wave transmission at the lee side was found to be minimum when the plate width is about 0.7 times the wavelength. McIver (1985) considered the scattering of surface waves by a moored, submerged, horizontal plate using eigenfunction expansions within the finite domain. Patarapanich and Cheong (1989) investigated experimentally the reflection and transmission
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characteristics of a submerged horizontal plate due to regular and random waves. It was found that the maximum reflection and minimum transmission of regular waves are achieved if the plate width to wavelength ratio is 0.5–0.71, when the relative depth of immersion, ⌬/d is in the range of 0.05–0.15. Parson and Martin (1992) solved the problem of wave scattering by a submerged, horizontal plate using a hypersingular integral equation for the discontinuity in the potential across the plate. Liu and Iskandarani (1989) extended the method of Ijima et al. (1971) to the case of a submerged plate of finite thickness in finite water depth. This solution was found to be unstable in shallow water conditions. Neelamani and Reddy (1992b) experimentally investigated the wave transmission and reflection characteristics of rigid surface fixed as well as submerged horizontal plates. It was found that for a rigid plate, the coefficient of transmission is minimum and the coefficient of reflection is maximum, but the maximum value of energy dissipation occurs for plates submerged closer to SWL. Reddy and Neelamani (1992a) also experimentally investigated the wave transmission and reflection characteristics of a partially immersed rigid vertical barrier. The results of power transmission theory proposed by Wiegel (1960) were compared with the measurements. This theory is found to predict satisfactorily the wave transmission for wave steepness up to 0.05, especially when the depth of immersion of the plate is more than about 20% of the water depth. Abul-Azm (1993) used the eigenfunction expansion method to find the velocity potential around partially immersed horizontal barriers and other configurations. The reflection and transmission coefficients were obtained from the velocity potential. Neelamani and Rajendran (2000) investigated the hydrodynamic performance of ‘T’type breakwaters. It was found that the coefficient of transmission, reflection and dissipation in the range of 0.15–0.55, 0.28–0.58 and 0.75–0.85, respectively, is possible due to random wave interaction, for conditions, (Hi)s/Lp from 0.006 to 0.048, d/Lp from 0.094 to 0.452, ⌬/d from 0 to ⫺0.286 and (Hi)s/d from 0.054 to 0.206. Here (Hi)s is the significant incident wave height, Lp is the local wavelength corresponding to the peak period, ‘d’ is the local water depth and ‘⌬’ is the depth of immersion of the horizontal barrier of the ‘T’ breakwater. A detailed survey of existing literature shows that experimental and theoretical studies pertaining to the performance characteristics of either a partially immersed horizontal plate (or) a partially immersed vertical barrier are available. The performance of a breakwater combining the advantages of a horizontal plate and a vertical barrier has not yet been investigated. Hence, a combination of these two barriers in the form of a ‘Ⲛ’-type barrier is selected for the present investigations.
3. Experimental investigations The wave transmission, reflection and dissipation characteristics of ‘Ⲛ’-type breakwaters are experimentally investigated in a wave flume by using physical models. Different hydrodynamic conditions and five different depths of submergence of these breakwaters are used. Both regular and random waves are used for the present investigation. Details of the experimental investigations are given in this section.
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3.1. Model scale Froude scaling is adopted for physical modelling, which allows for the correct reproduction of gravitational and fluid inertial forces. A scale of 1:25 is chosen for the selection of model dimensions and wave properties in the present study. Table 1 gives details of the proposed prototype conditions and the corresponding model dimensions. The Ⲛ-type breakwater supported on piles can be used for water depths ranging from say 10 to 25 m. However, studies are carried out in the laboratory for a constant water depth of 0.70 m, which corresponds to 17.50 m water depth in the prototype for 1:25 scale. Since the results are presented using normalized inputs and outputs, design curves can also be used for other water depths. A worked out example provided in Appendix A gives the application of the present investigation. 3.2. Model details The breakwater model is fabricated by using a Hylam sheet of thickness 3 mm. The model consists of two portions, namely (1) a horizontal plate and (2) a vertical plate. The size of the horizontal plate is 1.95×1 m2 and the vertical plate is 1.95×0.25 m2. Slotted angles of 50×50×6 mm3 are braced widthwise and lengthwise to ensure strength and rigidity. Eight-millimeter G.I. bolts with washers are used to fix the slotted angles and sheets. The model is fixed rigidly in between the side walls by means of teak wooden planks/wedges. The flume width is 2.0 m, whereas this model width is 1.95 m. The 5 cm clearance is filled with wooden planks and wedges. 3.3. Hydrodynamic testing facility Details of the hydrodynamic test facilities used for the investigation can be found in Neelamani and Rajendran (2000) and Rajendran (1999). 3.4. Instrumentation details Standard conductivity type wave probes are used. Static calibration of the wave probes is carried out every day and at the beginning and at the end of each set of experiments. The calibration constants are found to repeat with a standard deviation Table 1 Prototype and model details Description
Prototype
Model (1:25)
Water depth, d (m) Wave period, T (s) Wave height, H (m) Width of breakwater (m)
17.5 5–15 1.25–7.5 25
0.70 1–3 0.05–0.3 1.0
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of less than 1%. To measure the incident and reflected wave heights from the structure, three wave probes are positioned in front of the structure, 10 m away from the breakwater model (Fig. 3). Three wave probes are used in order to reduce the errors in the measurement of amplitudes and phases due to the non-linearity in the wave in the two-probe arrangements (Mansard and Funke, 1980). The distances between the wave probes are selected as follows. The distance between wave probes No. 1 and No. 3 (i.e. ⌬l13), according to the DHI Manual (1994) should satisfy the criterion, ⌬l13ⱖLmax/20, where Lmax is the maximum wave length to be generated. For the present work, though the maximum length of wave generated is 7.45 m, a wave length of 10.00 m is considered for this calculation to give a leverage for any possible long waves generated in the random waves. Hence, ⌬l13 is selected as 0.50 m. According to the DHI manual, the distance between wave probes 2 and 3, ⌬l23 is given as 0.1 ⌬l13 (or) 0.3 ⌬l13. Hence, ⌬l23=0.15 m is selected. These guidelines are followed in order to avoid singularity in the estimation of incident and reflected wave heights as described by Goda and Suzuki (1976). Fig. 3 shows the positions of the three wave probes WP1, WP2, WP3 used to measure the partial standing waves in front of the structure, from which the incident wave height (Hi) and the reflected wave height (Hr) are estimated. To measure the transmitted wave height, a wave probe (WP5) is fixed 7 m away from the breakwater model at its lee side, which is approximately one wavelength of the longest wave studied. To measure the water surface fluctuation at the leading edge of the model, a wave probe WP4 is fixed just in front of the structure (Fig. 3).
Fig. 3. Details of flume, position of the model and wave gauges (not to scale).
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3.5. Hydrodynamic test conditions 3.5.1. Regular waves Regular waves of heights from 0.05 to 0.3 m and wave periods from 1.0 to 3.0 s are used. Data for each run are acquired for a total duration of 30 s at a sampling frequency of 20 Hz. Waves are generated for a total duration of 60 s. Depending upon the period of wave generated, data collection commences 30–65 s after starting the wave generation. This is to make sure that the data collection is started only after repeatability of the same wave heights at the model location is established. The starting time for data collection is set based on trial runs with different periods of waves. The data collection duration is set at 30 s, based on the following criteria: 1. The regular wave time series should have at least 10 wave cycles. 2. The data collection must be completed before any reflected waves, either from the wave maker or from the beach, affect the measurements around the test section.
3.5.2. Random waves Random waves are generated for a total period of 120 s. Theoretical Pierson– Moskowitz spectra of different peak periods and significant wave heights are selected for random wave generation. The details of starting time for data acquisition, duration of data collection and running time of wave maker are given in Table 2. In total, 2100 data points for each channel are collected and the first 2048 data points are used for FFT analysis. As soon as the run is completed, real time series (which is the time series in volts multiplied by the appropriate calibration constants) are viewed on the monitor as a preliminary check on the quality of the data collected. The data are stored for further analysis. 3.6. Test programme, experimental procedure and data analysis 3.6.1. General The investigations are carried out for five different depths of immersion, ⌬, namely, ⫺0.05, ⫺0.10, 0.0, 0.05 and 0.10 m. ⌬=0.0 corresponds to the case when the top tip of the vertical barrier is kept at SWL. Negative values of ⌬ are when Table 2 Data acquisition details for random waves Peak period Tp (s)
Starting time for data acquisition (s)
Duration of data acquisition (s)
Runtime of the wave maker (s)
1.0 2.0 3.0
50 40 30
105 105 105
120 120 120
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the top of the vertical barrier is projecting above SWL (emerged conditions) and positive values of ⌬ indicate that the top of the vertical barrier is immersed in the water (submerged conditions) (Fig. 2). As stated in the previous section, the breakwater model is subjected to the action of regular waves of heights ranging from 0.05 to 0.30 m with intervals of 0.05 m. Wave periods ranging from 1.0 to 3.0 s with intervals of 0.50 s are used. For a wave period of 1.0 s, only wave heights up to 0.15 m were generated. Wave heights more than this were found breaking at the generation point itself. For a wave period of 3.0 s, it was difficult to generate higher heights, due to large displacements of the paddle. The selected wave height and period combinations were repeated for different depths of immersion of the breakwater, for a constant water depth of 0.7 m. For random waves, significant wave heights (Hs) from 0.05 to 0.15 m and peak periods (Tp) of 1.0, 2.0 and 3.0 s were selected. Details of the various non-dimensional parameters obtained in the present study are as follows. For regular waves: (a) (b) (c) (d)
Range Range Range Range
of of of of
incident wave steepness, Hi/L=0.004–0.133. relative water depth, d/L=0.094–0.452. relative depth of immersion, ⌬/d=⫺0.143 to +0.143. relative wave height, Hi/d=0.037–0.527.
For the calculation of Hi/L and Hi/d, the actually measured Hi was used. For random waves: (a) (b) (c) (d)
Range Range Range Range
of of of of
incident wave steepness, (Hi)s/L=0.006–0.048. relative water depth, d/Lp=0.094–0.452. relative depth of immersion, ⌬/d=⫺0.143 to +0.143. relative wave height, (Hi)s/d=0.054–0.206.
For the calculation of (Hi)s/L and (Hi)s/d, the actually measured (Hi)s values were used. 3.6.2. Analysis of time series of WP1, WP2 and WP3 for incident and reflected waves Frequency domain analysis using the DHI software package is used to carry out the reflection analysis. The response of the three wave probes installed in front of the model and the distance between the probes are the input for this package. The DHI software separates the incident and reflected wave components from the recorded response of these three wave probes. The reflection analysis gives incident significant wave height (Hi) and average reflection coefficient (Kr) based on the spectral analysis. The average reflection coefficient is the ratio of reflected and incident wave height (i.e. Kr=Hr/Hi). Hence, the significant reflected wave height is computed using the relationship Hr=KrHi.
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3.6.3. Analysis of time series of WP4 and WP5 Frequency domain analysis is carried out on the transmitted wave history. The significant transmitted wave height is obtained from the zeroth moment of the transmission spectrum, m0t by using the equation Ht=4.0(m0t)0.5. A similar procedure is adopted for the analysis of the water surface fluctuations in front of the structure. Zero down crossing analysis of the time series WP4 (water surface fluctuations in front of the breakwater) is also carried out and Hmin, Hmax and Hmean are obtained, where Hmin is the minimum, Hmax is the maximum and Hmean is the average wave heights of WP4. A similar analysis is also carried out on the transmitted waves. 3.7. Estimation of coefficient of energy loss, Kl The law of conservation of energy is used for the estimation of the coefficient of energy loss, Kl, since it is not possible to measure it. A breakwater is said to be better if it dissipates most of the incident wave energy. From the law of conservation of energy K 2t⫹K 2r⫹K 2l⫽1
(1)
Kl⫽冑1−K 2t−K 2r
(2)
This formula is used to estimate Kl values where Kt is the coefficient of transmission, Kr is the coefficient of reflection and Kl is the coefficient of dissipation.
4. Results and discussions 4.1. General The interaction of regular and random waves with the ‘Ⲛ’-type breakwater are discussed by using the Kt, Kr, and Kl values. Under regular wave interaction, focus is put on the effect of wave steepness, Hi/L, relative water depth, d/L, and relative depth of immersion ⌬/d on Kt, Kr and Kl. The importance of Hi/L, d/L and ⌬/d for the design of a ‘Ⲛ’-type breakwater is as follows: (a)
(b)
(c)
Investigating the effect of wave steepness, Hi/L on Kt, Kr and Kl is necessary to understand the performance of the breakwater for moderate and storm wave conditions. The effect of relative water depth, d/L, on Kt, Kr and Kl is essential to understand the hydrodynamic characteristics of the present breakwater for coastal and deep water regions. Investigating the effect of relative depth of immersion, ⌬/d, on Kt, Kr and Kl is required to select the appropriate structure configuration (depth of immersion) as required from the hydrodynamic performance and aesthetic considerations.
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For random waves, the effect of ⌬/d is investigated. Then, the wave climate in front of the ‘Ⲛ’-type breakwater is discussed. Here again for the case of regular waves, the effects of Hi/L, d/L and ⌬/d on the wave climate at the leading edge of the breakwater are discussed. For the case of random waves, the influence of ⌬/d on Hmean/(Hi)s and Hmax/(Hi)s is investigated. 4.2. Hydraulic performance under regular wave conditions Experimental investigation with regular waves is essential for basic understanding of the wave–structure interaction problem. 4.2.1. Effect of wave steepness Hi/L on Kt, Kr and Kl The effect of wave steepness on Kt, Kr and Kl for four different d/L values ( d/L=0.115, 0.152, 0.224 and 0.452) for the case of ⌬/d=0.0 is provided in Fig. 4. In general, the Kt value varies with Hi/L, but the trend is not consistent. For d/L=0.224 and 0.452, the transmission coefficient is between 0.15 and 0.25 and it appears to be independent of Hi/L. For d/L=0.115 and 0.152, the Kt values are high (varying from 0.35 to 0.65) and it also varies with increased incident wave steepness, but no specific trend is obtained. This is due to the fact that long waves (smaller d/L values) have high overtopping capacity compared to short waves. The overtopping is not linearly increasing with increases in incident wave energy, due to varying reflec-
Fig. 4.
Effect of wave steepness on Kt, Kr and Kl (regular waves).
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tion and dissipation as shown in Fig. 4. The overtopping wave energy contributes a considerable portion to the generation of transmitted wave height. In general, the Kr value reduces with increased Hi/L. This is due to the increased dissipation of energy for steeper waves. The dissipation coefficient, Kl generally shows an increasing trend with increased wave steepness. The range of Kl is found to vary from 0.50 to 0.90. A similar plot for a submerged condition of the breakwater, ⌬/d=⫺0.142 is provided in Fig. 5. From this plot it can be observed that the increase in wave steepness does not seem to have much influence on Kt, Kr and Kl, the values of which are in the range 0.40–0.80, 0.30–0.57 and 0.25–0.82, respectively. Basically, changes in d/L provide significant differences in the Kt, Kr and Kl values. This is discussed in the following section. 4.2.2. Effect of relative water depth d/L on Kt, Kr and Kl The effect of relative water depth d/L on Kt, Kr and Kl for moderate wave climate (Hi/d=0.054–0.100) and storm wave climate (Hi/d=0.430–0.499), when the vertical barrier of the ‘Ⲛ’-type breakwater emerges above SWL with ⌬/d=+0.142 is provided in Fig. 6. In general, increases in d/L result in a reduction in the Kt value. The reason for this trend is explained as follows. When d/L is increased, most of the energy is concentrated near SWL. The energy available below the horizontal barrier of the ‘Ⲛ’ breakwater reduces with increased d/L, which results in less transmission of energy. The Kl value increases with increased d/L for higher energy levels, which results in a decrease in the Kr value
Fig. 5.
Effect of wave steepness on Kt, Kr and Kl (regular waves).
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Fig. 6. Effect of relative water depth on Kt, Kr and Kl (regular waves).
with increased d/L. In contrast, for moderate energy levels, the Kr value is found to increase with increasing d/L, which is due to less dissipation of energy. The ‘wave trapping’ effect of the breakwater is pronounced for d/L by more than about 0.2, where the transmission coefficient is less than about 0.15. 4.2.3. Effect of relative depth of immersion ⌬/d on Kt, Kr and Kl Fig. 7 shows the effect of relative depth of immersion on Kt, Kr and Kl for four different relative water depths (d/L=0.115, 0.152, 0.224 and 0.452). This plot is provided for moderate wave conditions (Hi/d=0.050–0.100). It is to be noted that a negative value of ⌬/d indicates the submerged condition of the top tip of the breakwater, a positive value is for the emerging condition and ⌬/d=0.0 indicates that the top tip of the vertical barrier is flush with SWL. It is found that the transmission is minimum when the vertical barrier of the ‘Ⲛ’type breakwater emerges above SWL. The reason for the behavior is as follows: (a)
In general, the wave energy is more concentrated near SWL. When the vertical barrier of the ‘Ⲛ’-type breakwater is projecting above SWL, it gives an effective blockage (better wave trapping effect) of wave propagation. If the same ‘Ⲛ’-type breakwater is submerged, it provides free passage of the wave energy (poor wave trapping effect) over its vertical projection. This is the main reason for significant reduction of the Kt value when ⌬/d is increased from ⫺0.142 to +0.142. Hence, it can be concluded that the ‘Ⲛ’-type break-
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Fig. 7.
(b)
(c)
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Effect of relative depth of immersion on Kt, Kr and Kl (moderate wave conditions, regular waves).
water in the emerged condition is efficient in reducing the wave transmission. When the vertical wall of the ‘Ⲛ’-type breakwater is projecting above SWL, it provides better trapping of the concentrated wave energy near SWL, which has to reflect back. This is the reason for an increase in Kr when ⌬/d is increased from ⫺0.142 to +0.142. The variation of Kl is not consistent when ⌬/d is increased from ⫺0.142 to +0.142 for different d/L values. For d/L=0.115, the Kl value increases when ⌬/d is increased from ⫺0.142 to +0.142, whereas for d/L=0.224, it shows a completely different trend. Hence, obtaining a general conclusion for Kl is cumbersome. However, it is found that most of the Kl values are in the range of 0.55–0.85. This shows that the ‘Ⲛ’-type breakwater is also a good wave energy dissipater.
A similar type of plot for storm wave conditions (Hi/d=0.260–0.527) is given in Fig. 8. The trend of Kt with increased ⌬/d is similar to the earlier plot. Kt values of less than 0.10 are obtained when ⌬/d=+0.142 for d/L=0.224 and 0.452. This encourages usage of this new structure for coastal protection works and as a breakwater harbors. It is possible to obtain Kt less than 0.5 for all d/L values for the emerged condition of the barrier. It is found that Kr values in the range of 0.3–0.65 and Kl
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Fig. 8.
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Effect of relative depth of immersion on Kt, Kr and Kl (storm wave conditions, regular waves).
values in the range of 0.45–0.90 are obtained. Hence, it can be concluded that the ‘Ⲛ’-type breakwater can be used not only as a breakwater for purposes like coastal protection, marinas, maintenance, etc., but also as a breakwater for an artificial major harbor. 4.2.4. General inference with regular wave investigations It is very clear that the present breakwater can perform well in the emerged condition of the vertical barrier and closer to deep-water conditions. The effect of wave steepness on transmission, reflection and dissipation is not clear due to the complicated wave–structure interaction. The results of regular wave interactions cannot be directly used for the field conditions, unless the field wave condition is narrow banded. 4.3. Transmission, reflection and dissipation due to random waves 4.3.1. Effect of relative depth of immersion, ⌬/d on Kt, Kr and Kl Fig. 9 shows the effect of ⌬/d on Kt, Kr and Kl for moderate wave conditions ((Hi)s/d=0.056–0.069) and for three different d/Lp conditions (0.094, 0.152 and 0.452). Similar to Figs. 7 and 8 for regular waves, the Kt values reduce and the Kr values increase significantly with increased ⌬/d values. However, the minimum Kt is achieved for emerged condition, ⌬/d=+0.071. For this ⌬/d, the Kr value is
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Fig. 9.
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Effect of relative depth of immersion on Kt, Kr and Kl (moderate wave conditions, random waves).
maximum and the Kl value is minimum, which means that the barrier acts as a better wave trapping system for ⌬/d=+0.071. Hence, if the higher reflection is not a real problem for the application purpose, then the emerged condition with ⌬/d=0.071 is the best configuration among the five different configurations studied. If reflection needs to be less, then the barrier submerged configuration with ⌬/d=⫺0.142 is better, but Kt will be to the order of 0.5. It is interesting to note that the energy dissipation is maximum (Kl=0.85) for the submerged cases (⌬/d=⫺0.142 and ⫺0.071). This is due to high turbulence developed by the wave breaking, turbulence developed by the overtopping water on the lee side of the vertical barrier along with the dissipation by vortex shedding at the top tip of the vertical barrier and losses at the boundary layer around the horizontal barrier. A similar plot for higher incident wave energy levels ((Hi)s/d=0.103–0.184) is given in Fig. 10. The trend of variation of Kt, Kr and Kl is similar to that of Fig. 9. For the complete range of present investigations, the following important conclusions can be drawn. (a)
(b)
For both moderate conditions and storm wave conditions and for d/Lp=0.094–0.452, the value of Kt is less than 0.50. It is possible to obtain Kt even less than 0.30, when the vertical barrier of the ‘Ⲛ’ breakwater emerges above SWL (i.e. for ⌬/d=+0.071 and +0.142). It is not possible to obtain Kt values less than 0.35 for both the input energy
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Fig. 10.
Effect of relative depth of immersion on Kt, Kr and Kl (storm wave conditions, random waves).
(c)
levels when the vertical barrier of the ‘Ⲛ’-type breakwater is submerged with ⌬/d=⫺0.142. A Kl value in the order of 0.70–0.90 can be realized for most cases, which means that this breakwater is capable of dissipating the incident wave energy from 50 to 80%.
4.4. Comparison of Kt, Kr and Kl due to regular and random waves A comparison of the performance of the ‘Ⲛ’-type breakwater for regular and random waves is provided in order to understand whether the performance of this breakwater is better for the random wave case, which is closer to the real field conditions. For this purpose the regular wave results from Figs. 7 and 8 are compared with the random wave results provided in Figs. 9 and 10. The following points summarize the comparison. (a)
The trend of variation of Kt, Kr and Kl due to the variation of all the three normalized input parameters (⌬/d, d/L and Hi/d) remains the same both for regular and random waves.
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(b)
(c)
(d) (e)
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For the case of regular waves, the maximum value of Kt is 0.73, whereas for the random waves, it is only 0.49. This shows that the performance of the ‘Ⲛ’ breakwater is better in the random wave fields. For regular waves, Kt⬍0.30 is possible when the vertical barrier of the breakwater is surface emerged (i.e. ⌬/d=+0.071 and +0.142) and for d/L⬎ 0.152, whereas for the case of random waves, Kt⬍0.30 is achieved for d/Lp⬎0.094 when ⌬/d=+0.071 and +0.142. The ‘Ⲛ’-type breakwater is a better energy dissipater in random waves, particularly for storm wave conditions, compared to regular waves. The overall performance of the ‘Ⲛ’-type breakwater is better for random wave than regular wave interactions. This proves that ‘Ⲛ’-type breakwaters can be used in field applications.
4.5. Wave climate in front of the ‘Ⲛ’-type breakwater due to regular waves 4.5.1. Effect of wave steepness Hi/L on Hmean/Hi The effect of incident wave steepness on Hmean/Hi for five different relative depths of immersion, ⌬/d=+0.142, +0.071, 0.0, ⫺0.071 and ⫺0.142 and for two different d/L values (0.115 and 0.452) is given in Fig. 11. Generally, an increase in Hi/L reduces Hmean/Hi. The range of Hmean/Hi is found to vary from 0.42 to 1.80. The
Fig. 11. Effect of wave steepness on Hmean/Hi for different relative submergence, ⌬/d (regular waves).
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reduction of Hmean/Hi is significant due to the increase in Hi/L for the emerged case (⌬/d=+0.142 and +0.071) compared to the submerged case of the vertical barrier (i.e. ⌬/d=⫺0.142 and ⫺0.071). This is due to the wave trapping effect at the seaside for lower energy levels and significant overtopping of water on the harbor side for higher energy levels. 4.5.2. Effect of relative water depth d/L on Hmean/Hi The effect of d/L on Hmean/Hi for the five different immersions of the ‘Ⲛ’-type breakwater and for two energy levels (Hi/d=0.051–0.099 and 0.224–0.439) are given in Fig. 12. In general, Hmean/Hi reduces with increased d/L. The maximum Hmean/Hi recorded is 1.80 and the minimum is 0.32. 4.5.3. Effect of relative depth of immersion ⌬/d on Hmean/Hi Fig. 13 shows the effect of ⌬/d on Hmean/Hi for two different d/L values (0.094 and 0.452) and for two different Hi/d values (0.053–0.100 and 0.160–0.339). The following points should be noted. (a)
The maximum Hmean/Hi is about 1.84, which occurs when d/L=0.094 for lower energy levels. The minimum Hmean/Hi is 0.40, which occurs when d/L=0.452 for higher energy levels.
Fig. 12. waves).
Effect of relative water depth on Hmean/Hi for different relative submergence, ⌬/d (regular
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Fig. 13.
(b) (c)
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Effect of relative depth of immersion on Hmean/Hi (regular waves).
Hmean/Hi shows an increasing trend when ⌬/d is increased. The value of Hmean/Hi is consistently higher for a low d/L value of 0.094. Increasing input energy levels reduces the value of Hmean/Hi.
4.6. Wave height in front of the ‘Ⲛ’-type breakwater due to random waves 4.6.1. Effect of relative depth of immersion ⌬/d on Hmax/(Hi)s and Hmean/(Hi)s The effect of ⌬/d on Hmax/(Hi)s and Hmean/(Hi)s for two different d/Lp values (0.094 and 0.452) and for two different (Hi)s/d values (0.056–0.061 and 0.103–0.181) is given in Fig. 14. In general, both Hmax/(Hi)s and Hmean/(Hi)s increase with an increase in ⌬/d. The maximum Hmax/(Hi)s recorded is 2.70, whereas the maximum Hmean/(Hi)s is only 1.50.
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Fig. 14. Effect of relative depth of immersion on Hmax/(Hi)s and Hmean/(Hi)s (random waves).
4.7. Comparison of wave height in front of the structure due to regular and random waves Figs. 13 and 14 are compared in order to understand the effect of regular and random waves on the water surface fluctuations at the leading edge of the ‘Ⲛ’-type breakwater. The following points should be noted. (a)
(b)
In the case of regular waves, the maximum Hmean/Hi is achieved for d/L=0.094, whereas for the case of random waves, the maximum value of Hmax/(Hi)s is achieved for a d/Lp value of 0.452. Both for regular and random waves, increases in ⌬/d increase the values of Hmean/Hi, Hmax/(Hi)s and Hmean/(Hi)s, especially for lower energy levels.
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4.8. Comparison of transmission, reflection and dissipation characteristics of ‘T’type and ‘Ⲛ’-type breakwaters 4.8.1. Introduction Neelamani and Rajendran (2000) investigated the performance of ‘T’-type breakwaters for different hydrodynamic and structural conditions. In this section, it is intended to compare the transmission, reflection and dissipation characteristics of ‘T’- and ‘Ⲛ’-type breakwaters in order to compare their hydrodynamic performance under identical test conditions. In order to make the comparison meaningful the following information must be kept in mind. (a)
(b)
The experimental investigations were carried out for five different depths of immersion for both ‘T’-type (⌬/d=0.0, ⫺0.071, ⫺0.142, ⫺0.213 and ⫺0.284) and ‘Ⲛ’-type (⌬/d=+0.142, +0.071, 0, ⫺0.071 and ⫺0.142) breakwaters. Only three different depths of immersion are common for both the breakwaters, namely 0, ⫺0.071 and ⫺0.142. These common ⌬/d values can only be considered for the purpose of comparison. It is required to compare the transmission, reflection and dissipation characteristics of ‘T’- and ‘Ⲛ’-type breakwaters for moderate conditions and storm waves, for smaller and higher values of d/L and for regular and random waves, in order to understand and utilize the results for different ranges of field conditions.
The normalized transmission, reflection and dissipation values to be used and the interpretation for comparing the ‘T’- and ‘Ⲛ’-type breakwaters are given below. (a)
(b)
(Kt)T/(Kt)Ⲛ, where (Kt)T is the transmission coefficient of the ‘T’-type and (Kt)Ⲛ is the transmission coefficient of the ‘Ⲛ’-type breakwater for identical hydrodynamic and structural configurations. If this parameter is equal to 1, then both ‘T’- and ‘Ⲛ’-type breakwaters transmit the same magnitude of wave height. If (Kt)T/(Kt)Ⲛ⬎1, then the ‘Ⲛ’-type breakwater transmits less compared to the ‘T’-type breakwater. If (Kt)T/(Kt)Ⲛ⬍1, then the ‘T’-type breakwater is better at reducing the transmission than the ‘Ⲛ’-type breakwater. (Kr)T/(Kr)Ⲛ, where (Kr)T and (Kr)Ⲛ are the reflection coefficients of the ‘T’and the ‘Ⲛ’-type breakwaters, respectively. If (Kr)T/(Kr)Ⲛ=1, both the ‘T’- and ‘Ⲛ’-type breakwaters are equally good/bad at reflecting the wave energy. If (Kr)T/(Kr)Ⲛ⬎1, then the ‘T’-type breakwater is a better reflector than the ‘Ⲛ’-type breakwater. If (Kr)T/(Kr)Ⲛ⬍1, then the ‘T’-type breakwater is a poorer reflector than the ‘Ⲛ’-type breakwater.
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(c)
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(Kl)T/(Kl)Ⲛ, where (Kl)T and (Kl)Ⲛ are the energy loss coefficients of the ‘T’and ‘Ⲛ’-type breakwaters, respectively. If (Kl)T/(Kl)Ⲛ=1, both the breakwaters are equally good/bad at dissipating the incident wave energy. If (Kl)T/(Kl)Ⲛ⬎1, the ‘T’-type breakwater is better at dissipating the energy than the ‘Ⲛ’-type breakwater. If (Kl)T/(Kl)Ⲛ⬍1, then the ‘Ⲛ’-type breakwater is better at dissipating the energy than the ‘T’-type breakwater.
4.8.2. Comparison of ‘T’- and ‘Ⲛ’-type breakwaters in regular waves Fig. 15 shows a comparison of the ‘T’- and ‘Ⲛ’-type breakwaters for d/L=0.152 and 0.452 for lower and higher input wave climates (Hi/d=0.074–0.099 and 0.142–0.330). A detailed investigation of this plot yields the following information. (a) (b)
In general, the ‘T’-type breakwater is better than the ‘Ⲛ’-type breakwater in reducing wave transmission, on average, by about 20–30%. In general, the ‘T’-type breakwater reflects more energy than the ‘Ⲛ’-type breakwater. The efficiency of the ‘T’-type breakwater in reflecting the wave energy is as high as two times that of the ‘Ⲛ’-type breakwater for
Fig. 15. Effect of relative depth of immersion on (Kt)T/(Kt)Ⲛ, (Kr)T/(Kr)Ⲛ and (Kl)T/(Kl)Ⲛ (regular waves).
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(c)
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d/L=0.152 due to the better wave trapping effect of the submerged ‘T’type breakwater. For moderate wave conditions, the ‘Ⲛ’-type breakwater is a better energy dissipater than the ‘T’-type breakwater, whereas for storm conditions, the ‘T’-type breakwater seems to be the better energy dissipater. This may be due to the following facts. When the wave climate is mild, the ‘T’-type breakwater reflects most of the energy due to wave trapping effects and the dissipation due to breaking on the horizontal plate is less. During storm wave conditions, however, the ‘T’-type breakwater not only reflects more energy due to wave trapping effects but also dissipates a significant part of the input energy due to wave breaking on the horizontal barrier.
Overall for these three submerged ⌬/d values, the ‘T’-type breakwater is better than the ‘Ⲛ’-type breakwater. 4.8.3. Comparison of ‘T’- and ‘Ⲛ’-type breakwaters in regular waves A similar plot for random wave interaction is given in Fig. 16. Three different d/Lp values (0.094, 0.152 and 0.452) and two different (Hi)s/d values (0.056–0.069 and 0.102–0.206) are used.
Fig. 16. Effect of relative depth of immersion on (Kt)T/(Kt)Ⲛ, (Kr)T/(Kr)Ⲛ and (Kl)T/(Kl)Ⲛ (random waves).
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In general, the trends of (Kt)T/(Kt)Ⲛ, (Kr)T/(Kr)Ⲛ and (Kl)T/(Kl)Ⲛ are very similar to the regular wave interaction case. The following important points obtained from this plot should be noted. (a)
(b)
(c)
When ⌬/d=0.0, the ‘Ⲛ’-type breakwater seems to be better than the ‘T’-type breakwater for d/Lp=0.094 and 0.152. For d/Lp=0.452, the ‘T’-type breakwater with ⌬/d=0.0 is better. For the other two ⌬/d values, the ‘T’-type breakwater seems to be better. At ⌬/d=0.0, both types of breakwater are equally good in reflecting the wave energy. With increased depth of immersion, the ‘T’-type breakwater reflects more energy than the ‘Ⲛ’-type breakwater (note that (Kt)T/(Kt)Ⲛ reaches as high as 2.0). The ‘Ⲛ’-type breakwater seems better at dissipating the wave energy to an extent of about 20–30% compared to the ‘T’–type breakwater.
Overall, the ‘T’-type breakwater is found to be better than the ‘Ⲛ’-type breakwater under submerged conditions.
5. Conclusions The hydrodynamic characteristics, such as wave transmission, reflection, energy dissipation and water surface fluctuations at the leading edge of the ‘Ⲛ’-type breakwater are assessed based on physical model studies. Regular and random waves of different combinations of wave heights and periods are used. A constant water depth of 0.7 m is selected. Five different depths of immersion of the breakwaters are selected. The effects of incident wave steepness, relative water depth and relative depth of immersion of this breakwater on the hydrodynamic parameters are investigated. The salient conclusions of this study are as follows. 5.1. Transmission, reflection and energy dissipation characteristics 5.1.1. Regular waves 1. Reflection reduces and energy loss increases with increased wave steepness, especially when the ‘Ⲛ’-type breakwater is used with ⌬/d=0.0 or more. 2. Wave transmission significantly reduces with increased d/L, whether the vertical barrier of the breakwater is surface piercing or submerged. 3. For any incident wave climate (normal or hostile), transmission reduces and reflection increases with increases in ⌬/d from ⫺0.142 to 0.142. The ‘Ⲛ’-type breakwater is more efficient at reducing Kt when the vertical barrier is surface piercing. 4. A Kt value of less than 0.5 is possible to achieve for d/L=0.094–0.452 and for a wide ranging wave climate. 5. A Kt value of less than 0.3 can be obtained when ⌬/d=+0.071 and 0.142.
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6. The ‘Ⲛ’-type breakwater is capable of dissipating wave energy to the order of 50–80%, since Kl=0.70–0.90. 5.1.2. Random waves 1. It is possible to achieve Kt⬍0.5 for d/L in the range of 0.094–0.452. 2. A Kl value of 0.7–0.9 (i.e. energy dissipation of 50–80%) can be achieved using the ‘Ⲛ’-type breakwater. 3. A minimum Kr of 0.20–0.25 can be achieved for the submerged case of the breakwater for conditions closer to shallow water. 5.1.3. Comparison of regular and random waves The ranges of Kt, Kr and Kl in regular and random waves are Coefficient
Regular waves
Random waves
Kt Kr Kl
0.05–0.74 0.3–0.95 0.35–0.90
0.10–0.50 0.25–0.85 0.50–0.95
The results of the regular wave studies are valid for Hi/L=0.004–0.133, d/L=0.094–0.452, ⌬/d=⫺0.143 to +0.143 and Hi/d=0.037–0.527. The random wave results are applicable for (Hi)s/L=0.006–0.048, d/Lp=0.094–0.452, ⌬/d=⫺ 0.143 to +0.143 and (Hi)s/d=0.054–0.206. 5.2. Comparison of ‘T’- and ‘Ⲛ’-type breakwaters for the submerged case Under identical hydrodynamic and structural conditions for the ‘submerged’ case, the ‘T’-type breakwater is superior compared to the ‘Ⲛ’-type breakwater in reducing the wave transmission, increasing the wave reflection and dissipation. 5.3. Wave climate in front of the breakwater 1. For regular waves, the maximum wave height at the leading edge is about 1.84 times the incident wave height. 2. For random waves, the maximum wave height of about 2.7 times the significant incident wave height is achieved. Appendix A. Sample design calculation to illustrate the use of the present study Given: Design water depth, d=9.0 m
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Significant wave height, Hs=1.0 m Design peak period, Tp=8.0 s Find The immersion depth of the ‘Ⲛ’-type breakwaters, if Kt⬍0.40. Solution: Lop=1.56×T 2p=1.56×82=99.84 m d/Lop=9.0/99.84=0.09014 d/Lp=0.1323 (HI)s/d=1/10=0.111 For this d/Lp and (Hi)s/d, use the design curve Fig. 10. This curve is for (Hi)s/d=0.103–0.184. Since d/Lp is available only for three different d/Lp values (0.452, 0.152 and 0.094), proper interpolation is needed. Interpolation of Kt values between d/Lp of 0.094 and 0.152 shows that ⌬/d values of ⫺0.071, 0.0, 0.071 and 0.142 for the breakwater, if adopted, will provide Kt⬍0.40. The corresponding ⌬ values are ⫺0.64, 0.0, 0.64 and 1.28 m, respectively. Out of these four ⌬ values, one is the immersed condition (⌬=⫺0.64 m), one condition corresponds to the flushing of the top tip of the vertical barrier with SWL (⌬=0.0), and two are emerged conditions (⌬=0.64 and 1.28 m). The width of the horizontal barrier is 13.0 m (since the model scale is about 1:13) and the height of the vertical barrier is 3.25 m. For the same peak period of 8.0 s and significant wave height of 1.0 m, if the water depth is 18.0 m (i.e. d/Lp=0.209 and (HI)s/d=1/18=0.056), then to obtain wave transmission of 0.4 m or less, ⌬ can be any value from ⫺1.28 to +2.56 m. The width of the horizontal barrier is worked out as 26 m (since the model scale is 1:26) and the height of the vertical barrier is 6.5 m.
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Technical note
Wave interaction with ‘Ⲛ’-type breakwaters S. Neelamani a
a,*
, R. Rajendran
b
Department of Ocean Engineering, IIT Madras, Chennai 600 036, India b Chennai Port Trust, Chennai 600 001, India Received 4 October 2000; accepted 3 January 2001
Abstract The wave transmission, reflection and energy dissipation characteristics of ‘Ⲛ’-type breakwaters were studied using physical models. Regular and random waves in a wide range of wave heights and periods and a constant water depth were used. Five different depths of immersion (two emerged, one surface flushing and two submerged conditions) of this breakwater were selected. The coefficient of transmission, Kt, and coefficient of reflection, Kr, were obtained from the measurements, and the coefficient of energy loss, Kl was calculated using the law of balance of energy. It was found that the wave transmission is significantly reduced with increased relative water depth, d/L, whether the vertical barrier of the breakwater is surface piercing or submerged, where ‘d’ is the water depth and ‘L’ is the wave length. The wave reflection decreases and energy loss increases with increased wave steepness, especially when the top tip of the vertical barrier of this breakwater is kept at still water level (SWL). For any incident wave climate (moderate or storm waves), the wave transmission consistently decreases and the reflection increases with increased relative depth of immersion, ⌬/d from ⫺0.142 to 0.142. Kt values less than 0.3 can be easily obtained for the case of ⌬/d=+0.071 and 0.142, where ⌬ is the height of exposure (+ve) or depth of immersion (⫺ve) of the top tip of the vertical barrier. This breakwater is capable of dissipating wave energy to an extent of 50–80%. The overall performance of this breakwater was found to be better in the random wave fields than in the regular waves. A comparison of the hydrodynamic performance of ‘Ⲛ’-type and ‘T’-type shows that ‘T’-type breakwater is better than ‘Ⲛ’-type by about 20– 30% under identical conditions. 2002 Elsevier Science Ltd. All rights reserved. Keywords: Special breakwaters; Wave transmission; Reflection; Energy dissipation; Surface piercing; Submerged barrier
* Corresponding author. Tel.: +91-44-445-8639; fax: +91-44-235-0509. E-mail address: [email protected] (S. Neelamani). 0029-8018/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 9 - 8 0 1 8 ( 0 1 ) 0 0 0 3 0 - 0
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1. Introduction Rubble mound breakwaters are widely used around the world for the construction of artificial harbors and for shore protection works. When sufficient quantity and good quality rubble are not available in the vicinity of the proposed harbor site, then concrete caissons can be used. Both structures are generally expensive for deeper waters. For such conditions, special types of breakwaters, which require less concrete per unit length and are capable of transmitting less wave energy, must be developed. Since in deeper waters most of the wave energy is concentrated near the water surface, a structure is required, which can effectively dissipate or reflect this energy. Partially immersed horizontal plates were found to be good energy dissipaters due to artificial stimulation of wave breaking on the top of the plates (Patarapanich, 1984). Partially immersed vertical barriers are capable of reflecting the incident wave energy effectively in deeper waters (Ursell, 1947; Wiegel, 1960; Reddy and Neelamani, 1992a). Combining these characteristics of a horizontal and a vertical barrier into a ‘T’-type or a ‘Ⲛ’-type breakwater is expected to improve the performance as a breakwater due to the additive properties of wave breaking and reflection. Hence, this type of breakwater is expected to reduce the energy transmission towards the lee side of the structure. This has motivated the authors to investigate the wave transmission, reflection and energy dissipation characteristics of the ‘T’- and ‘Ⲛ’type breakwaters. Neelamani and Rajendran (2000) reported the hydrodynamic performance of ‘T’-type breakwaters. A good breakwater should allow minimum transmission and maximize wave energy dissipation. In the present study, the depth of submergence of the breakwater is varied in order to find out the appropriate depth of immersion at which the wave transmission is minimum. Ⲛ-type breakwaters can be supported on piles and the environmental loads can be transferred to the seabed through these piles (Fig. 1). This type of breakwater can be mass fabricated and assembled on the land, and quickly installed using floating barges during a fair weather window. This breakwater can be installed as a permanent structure at any desired depth of submergence of the horizontal part of the barrier, depending upon the expected hydrodynamic performance and the aesthetic aspects. Fig. 2 shows a definition sketch of the Ⲛ-type breakwater. The following processes will dissipate a part of the incident wave energy: (a) (b) (c) (d) (e)
Vortex shedding at the top of the vertical barrier of the breakwater during its submerged condition. Wave breaking over the breakwater, due to fast shoaling effect because of a sudden change in the depth of wave propagation. Turbulence developed by the overtopping mass of water on the vertical barrier. Boundary layer formation under the horizontal barrier. Wave trapping by the vertical barrier and a resulting increase in the horizontal velocity gradient of the overtopping water mass.
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Fig. 1.
Schematic view of the Ⲛ-type breakwater.
Fig. 2.
Definition sketch.
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The present experimental investigation with partially submerged Ⲛ-type breakwater has been carried out with the following objectives in mind: (a)
(b)
(c) (d)
To determine the wave transmission, reflection and energy dissipation characteristics of this breakwater for different incident wave characteristics and for different depths of submergence of these breakwaters due to regular and random waves. To analyze the water surface fluctuations in front of these breakwaters for different incident wave characteristics and for different depths of submergence. To compare the hydrodynamic performances due to regular and random waves. To compare the performance of ‘T’- type and ‘Ⲛ’-type barriers under identical test conditions.
Investigations on the characteristics of wave transmission for a wide range of wave conditions and different depths of submergence of the structure are essential for design, once the permissible wave transmission for a particular activity at the lee side of the structure is decided. For example, for given wave conditions (say peak period of 8 s and significant wave height of 2.0 m), if the permissible transmitted wave height is 0.4 m, then it is required to appropriately select the depth of immersion and size of this breakwater to achieve the intended wave conditions at the lee side (see Appendix A, where a worked out example is given to illustrate the utility of the present study). The study of energy dissipation character is important to understand the efficiency of the structure as a breakwater. A breakwater can be considered to be performing well if the energy dissipation capability is high. It is also necessary to understand the wave climate at the seaside of the structure due to wave reflection from the breakwater, since the safe navigation of approaching vessels is a function of this wave climate. Hence, wave histories are also measured at a typical location in the seaside of the breakwater.
2. Literature review Though no previous study has been carried out on ‘Ⲛ’-type breakwaters, literature on some of the special breakwaters more related to the present topic are presented. Ursell (1947) developed a theory for the transmission and reflection of gravity water waves in deep water for a fixed vertical infinitely thin barrier extending from the water surface to some depth below the surface. He proposed an equation for the coefficient of transmission as a function of modified Bessel function. Heins (1948, 1950) presented a solution procedure for the diffraction of water waves by a thin semi-infinite surface horizontal plate in water of finite depth and extended it to the case of a submerged plate in water of finite depth. Using the Wiener–Hopf technique, Greene and Heins (1953) extended these solutions to the case of a thin surface plate
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in water of finite depth. Stoker (1957) obtained simple expressions for the computation of the transmission and reflection coefficients by matching the horizontal velocity and pressures between adjacent regions for the submerged plate. Wiegel (1960) developed a theory based on the wave power transmission past a rigid vertical thin barrier, which extends from above the water surface to some distance below the surface. A satisfactory comparison of the theory with laboratory measurements was noticed for limited sets of data. A consistent trend of decreasing transmission coefficient with increasing wave steepness was noticed from the experimental studies, which was attributed to the excess loss in energy during transmission by separation at the bottom of the barrier. Burke (1964) extended the Wiener–Hopf technique to the case of a thin submerged plate in deep water. The solution procedure is complicated due to the iterative nature of the computation of the coefficients. Ijima et al. (1971) presented a method using eigenfunction expansions of the wave dispersion relation for the case of a surface plate in water of finite depth. Leppington (1972) applied the method of matched asymptotic expansions to the surface plate in deep water. Siew and Hurley (1972) extended this method to solve the problem of a submerged horizontal plate in shallow water. Bai (1975) presented a numerical method using finite elements for solving the linearized water-wave diffraction problem due to submerged objects. Patarapanich (1978) applied this method to the case of a submerged horizontal plate in finite water depth. Nallayarasu et al. (1992) applied the same method for fixed submerged inclined plates in finite water depth. The model tests of Hattori (1975) comprising a horizontal plate, fixed at the still water level (surface plate) and a submerged plate fixed at a quarter and at a half of the water depths exhibited a trend of lower transmission and higher reflection at small submergence depth ratio. These rather limited results led to the conclusion that the surface plate is more effective than the submerged plate in reducing the wave transmission. Experimental results subsequently produced by Dattatri et al. (1977) also showed that the wave transmission decreased with decreasing submergence, but reflection was maximum at ⌬/d=0.07 (where ⌬ is the depth of submergence of the plate and ‘d’ is the water depth) and not when the plate was at the surface. The optimum plate width for minimum transmission was found to be 0.3–0.4 times that of the incident wavelength. Patarapanich (1984) applied the finite element technique and calculated the reflection and transmission coefficients for a submerged horizontal plate from deep to shallow water limits. The conditions for maximum and zero reflection were analyzed. In the theoretical model, zero energy loss is assumed. It was concluded that the wave reflection from a submerged horizontal plate depends on the plate width to wavelength ratio, the relative depth of immersion and the relative water depth. It was found that the reflection coefficient generally increases with decreasing relative depth of immersion, ⌬/d and relative water depth, d/L. Wave transmission at the lee side was found to be minimum when the plate width is about 0.7 times the wavelength. McIver (1985) considered the scattering of surface waves by a moored, submerged, horizontal plate using eigenfunction expansions within the finite domain. Patarapanich and Cheong (1989) investigated experimentally the reflection and transmission
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characteristics of a submerged horizontal plate due to regular and random waves. It was found that the maximum reflection and minimum transmission of regular waves are achieved if the plate width to wavelength ratio is 0.5–0.71, when the relative depth of immersion, ⌬/d is in the range of 0.05–0.15. Parson and Martin (1992) solved the problem of wave scattering by a submerged, horizontal plate using a hypersingular integral equation for the discontinuity in the potential across the plate. Liu and Iskandarani (1989) extended the method of Ijima et al. (1971) to the case of a submerged plate of finite thickness in finite water depth. This solution was found to be unstable in shallow water conditions. Neelamani and Reddy (1992b) experimentally investigated the wave transmission and reflection characteristics of rigid surface fixed as well as submerged horizontal plates. It was found that for a rigid plate, the coefficient of transmission is minimum and the coefficient of reflection is maximum, but the maximum value of energy dissipation occurs for plates submerged closer to SWL. Reddy and Neelamani (1992a) also experimentally investigated the wave transmission and reflection characteristics of a partially immersed rigid vertical barrier. The results of power transmission theory proposed by Wiegel (1960) were compared with the measurements. This theory is found to predict satisfactorily the wave transmission for wave steepness up to 0.05, especially when the depth of immersion of the plate is more than about 20% of the water depth. Abul-Azm (1993) used the eigenfunction expansion method to find the velocity potential around partially immersed horizontal barriers and other configurations. The reflection and transmission coefficients were obtained from the velocity potential. Neelamani and Rajendran (2000) investigated the hydrodynamic performance of ‘T’type breakwaters. It was found that the coefficient of transmission, reflection and dissipation in the range of 0.15–0.55, 0.28–0.58 and 0.75–0.85, respectively, is possible due to random wave interaction, for conditions, (Hi)s/Lp from 0.006 to 0.048, d/Lp from 0.094 to 0.452, ⌬/d from 0 to ⫺0.286 and (Hi)s/d from 0.054 to 0.206. Here (Hi)s is the significant incident wave height, Lp is the local wavelength corresponding to the peak period, ‘d’ is the local water depth and ‘⌬’ is the depth of immersion of the horizontal barrier of the ‘T’ breakwater. A detailed survey of existing literature shows that experimental and theoretical studies pertaining to the performance characteristics of either a partially immersed horizontal plate (or) a partially immersed vertical barrier are available. The performance of a breakwater combining the advantages of a horizontal plate and a vertical barrier has not yet been investigated. Hence, a combination of these two barriers in the form of a ‘Ⲛ’-type barrier is selected for the present investigations.
3. Experimental investigations The wave transmission, reflection and dissipation characteristics of ‘Ⲛ’-type breakwaters are experimentally investigated in a wave flume by using physical models. Different hydrodynamic conditions and five different depths of submergence of these breakwaters are used. Both regular and random waves are used for the present investigation. Details of the experimental investigations are given in this section.
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3.1. Model scale Froude scaling is adopted for physical modelling, which allows for the correct reproduction of gravitational and fluid inertial forces. A scale of 1:25 is chosen for the selection of model dimensions and wave properties in the present study. Table 1 gives details of the proposed prototype conditions and the corresponding model dimensions. The Ⲛ-type breakwater supported on piles can be used for water depths ranging from say 10 to 25 m. However, studies are carried out in the laboratory for a constant water depth of 0.70 m, which corresponds to 17.50 m water depth in the prototype for 1:25 scale. Since the results are presented using normalized inputs and outputs, design curves can also be used for other water depths. A worked out example provided in Appendix A gives the application of the present investigation. 3.2. Model details The breakwater model is fabricated by using a Hylam sheet of thickness 3 mm. The model consists of two portions, namely (1) a horizontal plate and (2) a vertical plate. The size of the horizontal plate is 1.95×1 m2 and the vertical plate is 1.95×0.25 m2. Slotted angles of 50×50×6 mm3 are braced widthwise and lengthwise to ensure strength and rigidity. Eight-millimeter G.I. bolts with washers are used to fix the slotted angles and sheets. The model is fixed rigidly in between the side walls by means of teak wooden planks/wedges. The flume width is 2.0 m, whereas this model width is 1.95 m. The 5 cm clearance is filled with wooden planks and wedges. 3.3. Hydrodynamic testing facility Details of the hydrodynamic test facilities used for the investigation can be found in Neelamani and Rajendran (2000) and Rajendran (1999). 3.4. Instrumentation details Standard conductivity type wave probes are used. Static calibration of the wave probes is carried out every day and at the beginning and at the end of each set of experiments. The calibration constants are found to repeat with a standard deviation Table 1 Prototype and model details Description
Prototype
Model (1:25)
Water depth, d (m) Wave period, T (s) Wave height, H (m) Width of breakwater (m)
17.5 5–15 1.25–7.5 25
0.70 1–3 0.05–0.3 1.0
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of less than 1%. To measure the incident and reflected wave heights from the structure, three wave probes are positioned in front of the structure, 10 m away from the breakwater model (Fig. 3). Three wave probes are used in order to reduce the errors in the measurement of amplitudes and phases due to the non-linearity in the wave in the two-probe arrangements (Mansard and Funke, 1980). The distances between the wave probes are selected as follows. The distance between wave probes No. 1 and No. 3 (i.e. ⌬l13), according to the DHI Manual (1994) should satisfy the criterion, ⌬l13ⱖLmax/20, where Lmax is the maximum wave length to be generated. For the present work, though the maximum length of wave generated is 7.45 m, a wave length of 10.00 m is considered for this calculation to give a leverage for any possible long waves generated in the random waves. Hence, ⌬l13 is selected as 0.50 m. According to the DHI manual, the distance between wave probes 2 and 3, ⌬l23 is given as 0.1 ⌬l13 (or) 0.3 ⌬l13. Hence, ⌬l23=0.15 m is selected. These guidelines are followed in order to avoid singularity in the estimation of incident and reflected wave heights as described by Goda and Suzuki (1976). Fig. 3 shows the positions of the three wave probes WP1, WP2, WP3 used to measure the partial standing waves in front of the structure, from which the incident wave height (Hi) and the reflected wave height (Hr) are estimated. To measure the transmitted wave height, a wave probe (WP5) is fixed 7 m away from the breakwater model at its lee side, which is approximately one wavelength of the longest wave studied. To measure the water surface fluctuation at the leading edge of the model, a wave probe WP4 is fixed just in front of the structure (Fig. 3).
Fig. 3. Details of flume, position of the model and wave gauges (not to scale).
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3.5. Hydrodynamic test conditions 3.5.1. Regular waves Regular waves of heights from 0.05 to 0.3 m and wave periods from 1.0 to 3.0 s are used. Data for each run are acquired for a total duration of 30 s at a sampling frequency of 20 Hz. Waves are generated for a total duration of 60 s. Depending upon the period of wave generated, data collection commences 30–65 s after starting the wave generation. This is to make sure that the data collection is started only after repeatability of the same wave heights at the model location is established. The starting time for data collection is set based on trial runs with different periods of waves. The data collection duration is set at 30 s, based on the following criteria: 1. The regular wave time series should have at least 10 wave cycles. 2. The data collection must be completed before any reflected waves, either from the wave maker or from the beach, affect the measurements around the test section.
3.5.2. Random waves Random waves are generated for a total period of 120 s. Theoretical Pierson– Moskowitz spectra of different peak periods and significant wave heights are selected for random wave generation. The details of starting time for data acquisition, duration of data collection and running time of wave maker are given in Table 2. In total, 2100 data points for each channel are collected and the first 2048 data points are used for FFT analysis. As soon as the run is completed, real time series (which is the time series in volts multiplied by the appropriate calibration constants) are viewed on the monitor as a preliminary check on the quality of the data collected. The data are stored for further analysis. 3.6. Test programme, experimental procedure and data analysis 3.6.1. General The investigations are carried out for five different depths of immersion, ⌬, namely, ⫺0.05, ⫺0.10, 0.0, 0.05 and 0.10 m. ⌬=0.0 corresponds to the case when the top tip of the vertical barrier is kept at SWL. Negative values of ⌬ are when Table 2 Data acquisition details for random waves Peak period Tp (s)
Starting time for data acquisition (s)
Duration of data acquisition (s)
Runtime of the wave maker (s)
1.0 2.0 3.0
50 40 30
105 105 105
120 120 120
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the top of the vertical barrier is projecting above SWL (emerged conditions) and positive values of ⌬ indicate that the top of the vertical barrier is immersed in the water (submerged conditions) (Fig. 2). As stated in the previous section, the breakwater model is subjected to the action of regular waves of heights ranging from 0.05 to 0.30 m with intervals of 0.05 m. Wave periods ranging from 1.0 to 3.0 s with intervals of 0.50 s are used. For a wave period of 1.0 s, only wave heights up to 0.15 m were generated. Wave heights more than this were found breaking at the generation point itself. For a wave period of 3.0 s, it was difficult to generate higher heights, due to large displacements of the paddle. The selected wave height and period combinations were repeated for different depths of immersion of the breakwater, for a constant water depth of 0.7 m. For random waves, significant wave heights (Hs) from 0.05 to 0.15 m and peak periods (Tp) of 1.0, 2.0 and 3.0 s were selected. Details of the various non-dimensional parameters obtained in the present study are as follows. For regular waves: (a) (b) (c) (d)
Range Range Range Range
of of of of
incident wave steepness, Hi/L=0.004–0.133. relative water depth, d/L=0.094–0.452. relative depth of immersion, ⌬/d=⫺0.143 to +0.143. relative wave height, Hi/d=0.037–0.527.
For the calculation of Hi/L and Hi/d, the actually measured Hi was used. For random waves: (a) (b) (c) (d)
Range Range Range Range
of of of of
incident wave steepness, (Hi)s/L=0.006–0.048. relative water depth, d/Lp=0.094–0.452. relative depth of immersion, ⌬/d=⫺0.143 to +0.143. relative wave height, (Hi)s/d=0.054–0.206.
For the calculation of (Hi)s/L and (Hi)s/d, the actually measured (Hi)s values were used. 3.6.2. Analysis of time series of WP1, WP2 and WP3 for incident and reflected waves Frequency domain analysis using the DHI software package is used to carry out the reflection analysis. The response of the three wave probes installed in front of the model and the distance between the probes are the input for this package. The DHI software separates the incident and reflected wave components from the recorded response of these three wave probes. The reflection analysis gives incident significant wave height (Hi) and average reflection coefficient (Kr) based on the spectral analysis. The average reflection coefficient is the ratio of reflected and incident wave height (i.e. Kr=Hr/Hi). Hence, the significant reflected wave height is computed using the relationship Hr=KrHi.
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3.6.3. Analysis of time series of WP4 and WP5 Frequency domain analysis is carried out on the transmitted wave history. The significant transmitted wave height is obtained from the zeroth moment of the transmission spectrum, m0t by using the equation Ht=4.0(m0t)0.5. A similar procedure is adopted for the analysis of the water surface fluctuations in front of the structure. Zero down crossing analysis of the time series WP4 (water surface fluctuations in front of the breakwater) is also carried out and Hmin, Hmax and Hmean are obtained, where Hmin is the minimum, Hmax is the maximum and Hmean is the average wave heights of WP4. A similar analysis is also carried out on the transmitted waves. 3.7. Estimation of coefficient of energy loss, Kl The law of conservation of energy is used for the estimation of the coefficient of energy loss, Kl, since it is not possible to measure it. A breakwater is said to be better if it dissipates most of the incident wave energy. From the law of conservation of energy K 2t⫹K 2r⫹K 2l⫽1
(1)
Kl⫽冑1−K 2t−K 2r
(2)
This formula is used to estimate Kl values where Kt is the coefficient of transmission, Kr is the coefficient of reflection and Kl is the coefficient of dissipation.
4. Results and discussions 4.1. General The interaction of regular and random waves with the ‘Ⲛ’-type breakwater are discussed by using the Kt, Kr, and Kl values. Under regular wave interaction, focus is put on the effect of wave steepness, Hi/L, relative water depth, d/L, and relative depth of immersion ⌬/d on Kt, Kr and Kl. The importance of Hi/L, d/L and ⌬/d for the design of a ‘Ⲛ’-type breakwater is as follows: (a)
(b)
(c)
Investigating the effect of wave steepness, Hi/L on Kt, Kr and Kl is necessary to understand the performance of the breakwater for moderate and storm wave conditions. The effect of relative water depth, d/L, on Kt, Kr and Kl is essential to understand the hydrodynamic characteristics of the present breakwater for coastal and deep water regions. Investigating the effect of relative depth of immersion, ⌬/d, on Kt, Kr and Kl is required to select the appropriate structure configuration (depth of immersion) as required from the hydrodynamic performance and aesthetic considerations.
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For random waves, the effect of ⌬/d is investigated. Then, the wave climate in front of the ‘Ⲛ’-type breakwater is discussed. Here again for the case of regular waves, the effects of Hi/L, d/L and ⌬/d on the wave climate at the leading edge of the breakwater are discussed. For the case of random waves, the influence of ⌬/d on Hmean/(Hi)s and Hmax/(Hi)s is investigated. 4.2. Hydraulic performance under regular wave conditions Experimental investigation with regular waves is essential for basic understanding of the wave–structure interaction problem. 4.2.1. Effect of wave steepness Hi/L on Kt, Kr and Kl The effect of wave steepness on Kt, Kr and Kl for four different d/L values ( d/L=0.115, 0.152, 0.224 and 0.452) for the case of ⌬/d=0.0 is provided in Fig. 4. In general, the Kt value varies with Hi/L, but the trend is not consistent. For d/L=0.224 and 0.452, the transmission coefficient is between 0.15 and 0.25 and it appears to be independent of Hi/L. For d/L=0.115 and 0.152, the Kt values are high (varying from 0.35 to 0.65) and it also varies with increased incident wave steepness, but no specific trend is obtained. This is due to the fact that long waves (smaller d/L values) have high overtopping capacity compared to short waves. The overtopping is not linearly increasing with increases in incident wave energy, due to varying reflec-
Fig. 4.
Effect of wave steepness on Kt, Kr and Kl (regular waves).
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tion and dissipation as shown in Fig. 4. The overtopping wave energy contributes a considerable portion to the generation of transmitted wave height. In general, the Kr value reduces with increased Hi/L. This is due to the increased dissipation of energy for steeper waves. The dissipation coefficient, Kl generally shows an increasing trend with increased wave steepness. The range of Kl is found to vary from 0.50 to 0.90. A similar plot for a submerged condition of the breakwater, ⌬/d=⫺0.142 is provided in Fig. 5. From this plot it can be observed that the increase in wave steepness does not seem to have much influence on Kt, Kr and Kl, the values of which are in the range 0.40–0.80, 0.30–0.57 and 0.25–0.82, respectively. Basically, changes in d/L provide significant differences in the Kt, Kr and Kl values. This is discussed in the following section. 4.2.2. Effect of relative water depth d/L on Kt, Kr and Kl The effect of relative water depth d/L on Kt, Kr and Kl for moderate wave climate (Hi/d=0.054–0.100) and storm wave climate (Hi/d=0.430–0.499), when the vertical barrier of the ‘Ⲛ’-type breakwater emerges above SWL with ⌬/d=+0.142 is provided in Fig. 6. In general, increases in d/L result in a reduction in the Kt value. The reason for this trend is explained as follows. When d/L is increased, most of the energy is concentrated near SWL. The energy available below the horizontal barrier of the ‘Ⲛ’ breakwater reduces with increased d/L, which results in less transmission of energy. The Kl value increases with increased d/L for higher energy levels, which results in a decrease in the Kr value
Fig. 5.
Effect of wave steepness on Kt, Kr and Kl (regular waves).
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Fig. 6. Effect of relative water depth on Kt, Kr and Kl (regular waves).
with increased d/L. In contrast, for moderate energy levels, the Kr value is found to increase with increasing d/L, which is due to less dissipation of energy. The ‘wave trapping’ effect of the breakwater is pronounced for d/L by more than about 0.2, where the transmission coefficient is less than about 0.15. 4.2.3. Effect of relative depth of immersion ⌬/d on Kt, Kr and Kl Fig. 7 shows the effect of relative depth of immersion on Kt, Kr and Kl for four different relative water depths (d/L=0.115, 0.152, 0.224 and 0.452). This plot is provided for moderate wave conditions (Hi/d=0.050–0.100). It is to be noted that a negative value of ⌬/d indicates the submerged condition of the top tip of the breakwater, a positive value is for the emerging condition and ⌬/d=0.0 indicates that the top tip of the vertical barrier is flush with SWL. It is found that the transmission is minimum when the vertical barrier of the ‘Ⲛ’type breakwater emerges above SWL. The reason for the behavior is as follows: (a)
In general, the wave energy is more concentrated near SWL. When the vertical barrier of the ‘Ⲛ’-type breakwater is projecting above SWL, it gives an effective blockage (better wave trapping effect) of wave propagation. If the same ‘Ⲛ’-type breakwater is submerged, it provides free passage of the wave energy (poor wave trapping effect) over its vertical projection. This is the main reason for significant reduction of the Kt value when ⌬/d is increased from ⫺0.142 to +0.142. Hence, it can be concluded that the ‘Ⲛ’-type break-
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Fig. 7.
(b)
(c)
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Effect of relative depth of immersion on Kt, Kr and Kl (moderate wave conditions, regular waves).
water in the emerged condition is efficient in reducing the wave transmission. When the vertical wall of the ‘Ⲛ’-type breakwater is projecting above SWL, it provides better trapping of the concentrated wave energy near SWL, which has to reflect back. This is the reason for an increase in Kr when ⌬/d is increased from ⫺0.142 to +0.142. The variation of Kl is not consistent when ⌬/d is increased from ⫺0.142 to +0.142 for different d/L values. For d/L=0.115, the Kl value increases when ⌬/d is increased from ⫺0.142 to +0.142, whereas for d/L=0.224, it shows a completely different trend. Hence, obtaining a general conclusion for Kl is cumbersome. However, it is found that most of the Kl values are in the range of 0.55–0.85. This shows that the ‘Ⲛ’-type breakwater is also a good wave energy dissipater.
A similar type of plot for storm wave conditions (Hi/d=0.260–0.527) is given in Fig. 8. The trend of Kt with increased ⌬/d is similar to the earlier plot. Kt values of less than 0.10 are obtained when ⌬/d=+0.142 for d/L=0.224 and 0.452. This encourages usage of this new structure for coastal protection works and as a breakwater harbors. It is possible to obtain Kt less than 0.5 for all d/L values for the emerged condition of the barrier. It is found that Kr values in the range of 0.3–0.65 and Kl
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Fig. 8.
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Effect of relative depth of immersion on Kt, Kr and Kl (storm wave conditions, regular waves).
values in the range of 0.45–0.90 are obtained. Hence, it can be concluded that the ‘Ⲛ’-type breakwater can be used not only as a breakwater for purposes like coastal protection, marinas, maintenance, etc., but also as a breakwater for an artificial major harbor. 4.2.4. General inference with regular wave investigations It is very clear that the present breakwater can perform well in the emerged condition of the vertical barrier and closer to deep-water conditions. The effect of wave steepness on transmission, reflection and dissipation is not clear due to the complicated wave–structure interaction. The results of regular wave interactions cannot be directly used for the field conditions, unless the field wave condition is narrow banded. 4.3. Transmission, reflection and dissipation due to random waves 4.3.1. Effect of relative depth of immersion, ⌬/d on Kt, Kr and Kl Fig. 9 shows the effect of ⌬/d on Kt, Kr and Kl for moderate wave conditions ((Hi)s/d=0.056–0.069) and for three different d/Lp conditions (0.094, 0.152 and 0.452). Similar to Figs. 7 and 8 for regular waves, the Kt values reduce and the Kr values increase significantly with increased ⌬/d values. However, the minimum Kt is achieved for emerged condition, ⌬/d=+0.071. For this ⌬/d, the Kr value is
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Fig. 9.
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Effect of relative depth of immersion on Kt, Kr and Kl (moderate wave conditions, random waves).
maximum and the Kl value is minimum, which means that the barrier acts as a better wave trapping system for ⌬/d=+0.071. Hence, if the higher reflection is not a real problem for the application purpose, then the emerged condition with ⌬/d=0.071 is the best configuration among the five different configurations studied. If reflection needs to be less, then the barrier submerged configuration with ⌬/d=⫺0.142 is better, but Kt will be to the order of 0.5. It is interesting to note that the energy dissipation is maximum (Kl=0.85) for the submerged cases (⌬/d=⫺0.142 and ⫺0.071). This is due to high turbulence developed by the wave breaking, turbulence developed by the overtopping water on the lee side of the vertical barrier along with the dissipation by vortex shedding at the top tip of the vertical barrier and losses at the boundary layer around the horizontal barrier. A similar plot for higher incident wave energy levels ((Hi)s/d=0.103–0.184) is given in Fig. 10. The trend of variation of Kt, Kr and Kl is similar to that of Fig. 9. For the complete range of present investigations, the following important conclusions can be drawn. (a)
(b)
For both moderate conditions and storm wave conditions and for d/Lp=0.094–0.452, the value of Kt is less than 0.50. It is possible to obtain Kt even less than 0.30, when the vertical barrier of the ‘Ⲛ’ breakwater emerges above SWL (i.e. for ⌬/d=+0.071 and +0.142). It is not possible to obtain Kt values less than 0.35 for both the input energy
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Fig. 10.
Effect of relative depth of immersion on Kt, Kr and Kl (storm wave conditions, random waves).
(c)
levels when the vertical barrier of the ‘Ⲛ’-type breakwater is submerged with ⌬/d=⫺0.142. A Kl value in the order of 0.70–0.90 can be realized for most cases, which means that this breakwater is capable of dissipating the incident wave energy from 50 to 80%.
4.4. Comparison of Kt, Kr and Kl due to regular and random waves A comparison of the performance of the ‘Ⲛ’-type breakwater for regular and random waves is provided in order to understand whether the performance of this breakwater is better for the random wave case, which is closer to the real field conditions. For this purpose the regular wave results from Figs. 7 and 8 are compared with the random wave results provided in Figs. 9 and 10. The following points summarize the comparison. (a)
The trend of variation of Kt, Kr and Kl due to the variation of all the three normalized input parameters (⌬/d, d/L and Hi/d) remains the same both for regular and random waves.
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(b)
(c)
(d) (e)
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For the case of regular waves, the maximum value of Kt is 0.73, whereas for the random waves, it is only 0.49. This shows that the performance of the ‘Ⲛ’ breakwater is better in the random wave fields. For regular waves, Kt⬍0.30 is possible when the vertical barrier of the breakwater is surface emerged (i.e. ⌬/d=+0.071 and +0.142) and for d/L⬎ 0.152, whereas for the case of random waves, Kt⬍0.30 is achieved for d/Lp⬎0.094 when ⌬/d=+0.071 and +0.142. The ‘Ⲛ’-type breakwater is a better energy dissipater in random waves, particularly for storm wave conditions, compared to regular waves. The overall performance of the ‘Ⲛ’-type breakwater is better for random wave than regular wave interactions. This proves that ‘Ⲛ’-type breakwaters can be used in field applications.
4.5. Wave climate in front of the ‘Ⲛ’-type breakwater due to regular waves 4.5.1. Effect of wave steepness Hi/L on Hmean/Hi The effect of incident wave steepness on Hmean/Hi for five different relative depths of immersion, ⌬/d=+0.142, +0.071, 0.0, ⫺0.071 and ⫺0.142 and for two different d/L values (0.115 and 0.452) is given in Fig. 11. Generally, an increase in Hi/L reduces Hmean/Hi. The range of Hmean/Hi is found to vary from 0.42 to 1.80. The
Fig. 11. Effect of wave steepness on Hmean/Hi for different relative submergence, ⌬/d (regular waves).
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reduction of Hmean/Hi is significant due to the increase in Hi/L for the emerged case (⌬/d=+0.142 and +0.071) compared to the submerged case of the vertical barrier (i.e. ⌬/d=⫺0.142 and ⫺0.071). This is due to the wave trapping effect at the seaside for lower energy levels and significant overtopping of water on the harbor side for higher energy levels. 4.5.2. Effect of relative water depth d/L on Hmean/Hi The effect of d/L on Hmean/Hi for the five different immersions of the ‘Ⲛ’-type breakwater and for two energy levels (Hi/d=0.051–0.099 and 0.224–0.439) are given in Fig. 12. In general, Hmean/Hi reduces with increased d/L. The maximum Hmean/Hi recorded is 1.80 and the minimum is 0.32. 4.5.3. Effect of relative depth of immersion ⌬/d on Hmean/Hi Fig. 13 shows the effect of ⌬/d on Hmean/Hi for two different d/L values (0.094 and 0.452) and for two different Hi/d values (0.053–0.100 and 0.160–0.339). The following points should be noted. (a)
The maximum Hmean/Hi is about 1.84, which occurs when d/L=0.094 for lower energy levels. The minimum Hmean/Hi is 0.40, which occurs when d/L=0.452 for higher energy levels.
Fig. 12. waves).
Effect of relative water depth on Hmean/Hi for different relative submergence, ⌬/d (regular
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Fig. 13.
(b) (c)
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Effect of relative depth of immersion on Hmean/Hi (regular waves).
Hmean/Hi shows an increasing trend when ⌬/d is increased. The value of Hmean/Hi is consistently higher for a low d/L value of 0.094. Increasing input energy levels reduces the value of Hmean/Hi.
4.6. Wave height in front of the ‘Ⲛ’-type breakwater due to random waves 4.6.1. Effect of relative depth of immersion ⌬/d on Hmax/(Hi)s and Hmean/(Hi)s The effect of ⌬/d on Hmax/(Hi)s and Hmean/(Hi)s for two different d/Lp values (0.094 and 0.452) and for two different (Hi)s/d values (0.056–0.061 and 0.103–0.181) is given in Fig. 14. In general, both Hmax/(Hi)s and Hmean/(Hi)s increase with an increase in ⌬/d. The maximum Hmax/(Hi)s recorded is 2.70, whereas the maximum Hmean/(Hi)s is only 1.50.
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Fig. 14. Effect of relative depth of immersion on Hmax/(Hi)s and Hmean/(Hi)s (random waves).
4.7. Comparison of wave height in front of the structure due to regular and random waves Figs. 13 and 14 are compared in order to understand the effect of regular and random waves on the water surface fluctuations at the leading edge of the ‘Ⲛ’-type breakwater. The following points should be noted. (a)
(b)
In the case of regular waves, the maximum Hmean/Hi is achieved for d/L=0.094, whereas for the case of random waves, the maximum value of Hmax/(Hi)s is achieved for a d/Lp value of 0.452. Both for regular and random waves, increases in ⌬/d increase the values of Hmean/Hi, Hmax/(Hi)s and Hmean/(Hi)s, especially for lower energy levels.
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4.8. Comparison of transmission, reflection and dissipation characteristics of ‘T’type and ‘Ⲛ’-type breakwaters 4.8.1. Introduction Neelamani and Rajendran (2000) investigated the performance of ‘T’-type breakwaters for different hydrodynamic and structural conditions. In this section, it is intended to compare the transmission, reflection and dissipation characteristics of ‘T’- and ‘Ⲛ’-type breakwaters in order to compare their hydrodynamic performance under identical test conditions. In order to make the comparison meaningful the following information must be kept in mind. (a)
(b)
The experimental investigations were carried out for five different depths of immersion for both ‘T’-type (⌬/d=0.0, ⫺0.071, ⫺0.142, ⫺0.213 and ⫺0.284) and ‘Ⲛ’-type (⌬/d=+0.142, +0.071, 0, ⫺0.071 and ⫺0.142) breakwaters. Only three different depths of immersion are common for both the breakwaters, namely 0, ⫺0.071 and ⫺0.142. These common ⌬/d values can only be considered for the purpose of comparison. It is required to compare the transmission, reflection and dissipation characteristics of ‘T’- and ‘Ⲛ’-type breakwaters for moderate conditions and storm waves, for smaller and higher values of d/L and for regular and random waves, in order to understand and utilize the results for different ranges of field conditions.
The normalized transmission, reflection and dissipation values to be used and the interpretation for comparing the ‘T’- and ‘Ⲛ’-type breakwaters are given below. (a)
(b)
(Kt)T/(Kt)Ⲛ, where (Kt)T is the transmission coefficient of the ‘T’-type and (Kt)Ⲛ is the transmission coefficient of the ‘Ⲛ’-type breakwater for identical hydrodynamic and structural configurations. If this parameter is equal to 1, then both ‘T’- and ‘Ⲛ’-type breakwaters transmit the same magnitude of wave height. If (Kt)T/(Kt)Ⲛ⬎1, then the ‘Ⲛ’-type breakwater transmits less compared to the ‘T’-type breakwater. If (Kt)T/(Kt)Ⲛ⬍1, then the ‘T’-type breakwater is better at reducing the transmission than the ‘Ⲛ’-type breakwater. (Kr)T/(Kr)Ⲛ, where (Kr)T and (Kr)Ⲛ are the reflection coefficients of the ‘T’and the ‘Ⲛ’-type breakwaters, respectively. If (Kr)T/(Kr)Ⲛ=1, both the ‘T’- and ‘Ⲛ’-type breakwaters are equally good/bad at reflecting the wave energy. If (Kr)T/(Kr)Ⲛ⬎1, then the ‘T’-type breakwater is a better reflector than the ‘Ⲛ’-type breakwater. If (Kr)T/(Kr)Ⲛ⬍1, then the ‘T’-type breakwater is a poorer reflector than the ‘Ⲛ’-type breakwater.
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(c)
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(Kl)T/(Kl)Ⲛ, where (Kl)T and (Kl)Ⲛ are the energy loss coefficients of the ‘T’and ‘Ⲛ’-type breakwaters, respectively. If (Kl)T/(Kl)Ⲛ=1, both the breakwaters are equally good/bad at dissipating the incident wave energy. If (Kl)T/(Kl)Ⲛ⬎1, the ‘T’-type breakwater is better at dissipating the energy than the ‘Ⲛ’-type breakwater. If (Kl)T/(Kl)Ⲛ⬍1, then the ‘Ⲛ’-type breakwater is better at dissipating the energy than the ‘T’-type breakwater.
4.8.2. Comparison of ‘T’- and ‘Ⲛ’-type breakwaters in regular waves Fig. 15 shows a comparison of the ‘T’- and ‘Ⲛ’-type breakwaters for d/L=0.152 and 0.452 for lower and higher input wave climates (Hi/d=0.074–0.099 and 0.142–0.330). A detailed investigation of this plot yields the following information. (a) (b)
In general, the ‘T’-type breakwater is better than the ‘Ⲛ’-type breakwater in reducing wave transmission, on average, by about 20–30%. In general, the ‘T’-type breakwater reflects more energy than the ‘Ⲛ’-type breakwater. The efficiency of the ‘T’-type breakwater in reflecting the wave energy is as high as two times that of the ‘Ⲛ’-type breakwater for
Fig. 15. Effect of relative depth of immersion on (Kt)T/(Kt)Ⲛ, (Kr)T/(Kr)Ⲛ and (Kl)T/(Kl)Ⲛ (regular waves).
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(c)
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d/L=0.152 due to the better wave trapping effect of the submerged ‘T’type breakwater. For moderate wave conditions, the ‘Ⲛ’-type breakwater is a better energy dissipater than the ‘T’-type breakwater, whereas for storm conditions, the ‘T’-type breakwater seems to be the better energy dissipater. This may be due to the following facts. When the wave climate is mild, the ‘T’-type breakwater reflects most of the energy due to wave trapping effects and the dissipation due to breaking on the horizontal plate is less. During storm wave conditions, however, the ‘T’-type breakwater not only reflects more energy due to wave trapping effects but also dissipates a significant part of the input energy due to wave breaking on the horizontal barrier.
Overall for these three submerged ⌬/d values, the ‘T’-type breakwater is better than the ‘Ⲛ’-type breakwater. 4.8.3. Comparison of ‘T’- and ‘Ⲛ’-type breakwaters in regular waves A similar plot for random wave interaction is given in Fig. 16. Three different d/Lp values (0.094, 0.152 and 0.452) and two different (Hi)s/d values (0.056–0.069 and 0.102–0.206) are used.
Fig. 16. Effect of relative depth of immersion on (Kt)T/(Kt)Ⲛ, (Kr)T/(Kr)Ⲛ and (Kl)T/(Kl)Ⲛ (random waves).
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In general, the trends of (Kt)T/(Kt)Ⲛ, (Kr)T/(Kr)Ⲛ and (Kl)T/(Kl)Ⲛ are very similar to the regular wave interaction case. The following important points obtained from this plot should be noted. (a)
(b)
(c)
When ⌬/d=0.0, the ‘Ⲛ’-type breakwater seems to be better than the ‘T’-type breakwater for d/Lp=0.094 and 0.152. For d/Lp=0.452, the ‘T’-type breakwater with ⌬/d=0.0 is better. For the other two ⌬/d values, the ‘T’-type breakwater seems to be better. At ⌬/d=0.0, both types of breakwater are equally good in reflecting the wave energy. With increased depth of immersion, the ‘T’-type breakwater reflects more energy than the ‘Ⲛ’-type breakwater (note that (Kt)T/(Kt)Ⲛ reaches as high as 2.0). The ‘Ⲛ’-type breakwater seems better at dissipating the wave energy to an extent of about 20–30% compared to the ‘T’–type breakwater.
Overall, the ‘T’-type breakwater is found to be better than the ‘Ⲛ’-type breakwater under submerged conditions.
5. Conclusions The hydrodynamic characteristics, such as wave transmission, reflection, energy dissipation and water surface fluctuations at the leading edge of the ‘Ⲛ’-type breakwater are assessed based on physical model studies. Regular and random waves of different combinations of wave heights and periods are used. A constant water depth of 0.7 m is selected. Five different depths of immersion of the breakwaters are selected. The effects of incident wave steepness, relative water depth and relative depth of immersion of this breakwater on the hydrodynamic parameters are investigated. The salient conclusions of this study are as follows. 5.1. Transmission, reflection and energy dissipation characteristics 5.1.1. Regular waves 1. Reflection reduces and energy loss increases with increased wave steepness, especially when the ‘Ⲛ’-type breakwater is used with ⌬/d=0.0 or more. 2. Wave transmission significantly reduces with increased d/L, whether the vertical barrier of the breakwater is surface piercing or submerged. 3. For any incident wave climate (normal or hostile), transmission reduces and reflection increases with increases in ⌬/d from ⫺0.142 to 0.142. The ‘Ⲛ’-type breakwater is more efficient at reducing Kt when the vertical barrier is surface piercing. 4. A Kt value of less than 0.5 is possible to achieve for d/L=0.094–0.452 and for a wide ranging wave climate. 5. A Kt value of less than 0.3 can be obtained when ⌬/d=+0.071 and 0.142.
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6. The ‘Ⲛ’-type breakwater is capable of dissipating wave energy to the order of 50–80%, since Kl=0.70–0.90. 5.1.2. Random waves 1. It is possible to achieve Kt⬍0.5 for d/L in the range of 0.094–0.452. 2. A Kl value of 0.7–0.9 (i.e. energy dissipation of 50–80%) can be achieved using the ‘Ⲛ’-type breakwater. 3. A minimum Kr of 0.20–0.25 can be achieved for the submerged case of the breakwater for conditions closer to shallow water. 5.1.3. Comparison of regular and random waves The ranges of Kt, Kr and Kl in regular and random waves are Coefficient
Regular waves
Random waves
Kt Kr Kl
0.05–0.74 0.3–0.95 0.35–0.90
0.10–0.50 0.25–0.85 0.50–0.95
The results of the regular wave studies are valid for Hi/L=0.004–0.133, d/L=0.094–0.452, ⌬/d=⫺0.143 to +0.143 and Hi/d=0.037–0.527. The random wave results are applicable for (Hi)s/L=0.006–0.048, d/Lp=0.094–0.452, ⌬/d=⫺ 0.143 to +0.143 and (Hi)s/d=0.054–0.206. 5.2. Comparison of ‘T’- and ‘Ⲛ’-type breakwaters for the submerged case Under identical hydrodynamic and structural conditions for the ‘submerged’ case, the ‘T’-type breakwater is superior compared to the ‘Ⲛ’-type breakwater in reducing the wave transmission, increasing the wave reflection and dissipation. 5.3. Wave climate in front of the breakwater 1. For regular waves, the maximum wave height at the leading edge is about 1.84 times the incident wave height. 2. For random waves, the maximum wave height of about 2.7 times the significant incident wave height is achieved. Appendix A. Sample design calculation to illustrate the use of the present study Given: Design water depth, d=9.0 m
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Significant wave height, Hs=1.0 m Design peak period, Tp=8.0 s Find The immersion depth of the ‘Ⲛ’-type breakwaters, if Kt⬍0.40. Solution: Lop=1.56×T 2p=1.56×82=99.84 m d/Lop=9.0/99.84=0.09014 d/Lp=0.1323 (HI)s/d=1/10=0.111 For this d/Lp and (Hi)s/d, use the design curve Fig. 10. This curve is for (Hi)s/d=0.103–0.184. Since d/Lp is available only for three different d/Lp values (0.452, 0.152 and 0.094), proper interpolation is needed. Interpolation of Kt values between d/Lp of 0.094 and 0.152 shows that ⌬/d values of ⫺0.071, 0.0, 0.071 and 0.142 for the breakwater, if adopted, will provide Kt⬍0.40. The corresponding ⌬ values are ⫺0.64, 0.0, 0.64 and 1.28 m, respectively. Out of these four ⌬ values, one is the immersed condition (⌬=⫺0.64 m), one condition corresponds to the flushing of the top tip of the vertical barrier with SWL (⌬=0.0), and two are emerged conditions (⌬=0.64 and 1.28 m). The width of the horizontal barrier is 13.0 m (since the model scale is about 1:13) and the height of the vertical barrier is 3.25 m. For the same peak period of 8.0 s and significant wave height of 1.0 m, if the water depth is 18.0 m (i.e. d/Lp=0.209 and (HI)s/d=1/18=0.056), then to obtain wave transmission of 0.4 m or less, ⌬ can be any value from ⫺1.28 to +2.56 m. The width of the horizontal barrier is worked out as 26 m (since the model scale is 1:26) and the height of the vertical barrier is 6.5 m.
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