Ocean Engineering 38 (2011) 1269–1276
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Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
Short Communication
Wave reflection by submerged vertical and semicircular breakwaters
D. Morgan Young, Firat Y. Testik n
Civil Engineering Department, 110 Lowry Hall, Clemson University, Clemson, SC 29634 0911, USA
a r t i c l e i n f o
abstract
Article history:
Received 30 April 2010
Accepted 4 May 2011
Editor-in-Chief: A.I. Incecik
Available online 28 May 2011
This short manuscript presents a laboratory investigation on the effects of submerged vertical and
semicircular breakwaters on local wave characteristics, particularly with the aim of determining the
parameterizations for the wave reflection coefficients for submerged vertical and semicircular breakwaters. Experiments were conducted with normally incident monochromatic waves breaking at the
breakwater on both sloping and horizontal sandy bottoms. The reflection coefficient (Cr) is observed to
rely mainly on the dimensionless submergence parameter, a/Hi (a—the breakwater’s depth of
submergence and Hi—the height of the incident wave at the breakwater). Two semi-empirical
parameterizations are proposed to predict reflection coefficients for submerged vertical and semicircular breakwaters. While both parameterizations share the same functional dependency on a/Hi, the
functions have different constant coefficients. For the limiting case when a approaches zero (breakwater crest is at the mean water level), the Cr value tends toward 0.53 for both breakwaters. However,
as a increases, the submerged vertical breakwaters reflect more energy than submerged semicircular
breakwaters for the same a/Hi value. Results of this study are expected to be of use to coastal engineers
for preliminary feasibility and desk design of submerged vertical and semicircular breakwaters.
& 2011 Elsevier Ltd. All rights reserved.
Keywords:
Wave reflection
Submerged breakwater
Vertical breakwater
Semicircular breakwater
1. Introduction
Offshore breakwaters are regularly employed to provide
defense to important coastal areas such as marinas, ports, and
beaches from energetic ocean waves. Upon breakwater impact,
the incident wave undergoes three separate decompositions:
reflection from the breakwater, dissipation on the breakwater,
and transmission through (or over) the breakwater (Chakrabarti,
1999). As some part of the wave energy is dissipated and reflected
out to sea, less energy is transmitted through the breakwater and
imparted on the beach. A significant problem associated with the
transmission of incoming waves onto natural coasts and existing
coastal structures is beach erosion and sediment scour, which can
lead to a dramatic loss in beach material around the foundation of
the structures, which may result in their subsequent destabilization (Davidson et al., 1996; Young and Testik, 2009; Sumer et al.,
2001, 2005; Sumer and Fredsoe, 2005). Traditionally, emerged
breakwaters (i.e., breakwaters with crests piercing the mean
water level) have been used to minimize these problems. The
construction of submerged breakwaters, or breakwaters that lie
entirely beneath the mean water level, has become more common
in recent years (Ming and Chiew, 2000). Submerged breakwaters
serve as a defense by inducing partial reflection-transmission
n
Corresponding author. Tel.: þ1 864 656 0484; fax: þ1 864 656 2670.
E-mail address: [email protected] (F.Y. Testik).
0029-8018/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2011.05.003
and/or breaking of large waves (Grilli et al., 1994). According to
Hur and Mizutani (2003), submerged breakwaters have become
ever more popular because they are often more esthetically
pleasing than emerged breakwaters, which is critical to the
tourism industry in most coastal areas (Johnson, 2006). Another
advantage of submerged breakwaters is their capacity to maintain
the landward flow of water, which may be important for
water quality considerations (Kobayashi et al., 2007). In addition,
there is growing interest in the concept that the layout and crosssection of submerged coastal protection structures can be optimized
to also enhance local surfing conditions (Ranasinghe and Turner,
2006). Conversely, submerged breakwaters usually dissipate less
wave energy than emerged breakwaters.
Various types of submerged breakwaters include vertical
breakwaters, semicircular breakwaters, rubble mound (porous)
breakwaters, and geosynthetic breakwaters. This study concentrates on the examination and evaluation of submerged vertical
and semicircular breakwaters. Submerged vertical breakwaters
typically exist as a robust vertical wall while submerged semicircular breakwaters are composed of a precast reinforced concrete
structure built with a semicircular vault and bottom slab (Yuan
and Tao, 2003). The concrete structure is placed over a formed
rubble mound foundation. While submerged vertical breakwaters
typically reflect more incident wave energy than submerged
semicircular breakwaters, submerged semicircular breakwaters
are oftentimes more stable under wave forcing, thus decreasing
the potential for failure.
1270
D.M. Young, F.Y. Testik / Ocean Engineering 38 (2011) 1269–1276
number, h—still water depth) on Cr and found no functional trend
between the two. Grilli et al. (1994) investigated submerged
trapezoidal breakwater performance under solitary waves and
found that results depended heavily on breakwater height and
incident wave height. They also determined that the wave
transmission consistently fell between 55% and 90%.
It is important to note that few studies exist, which introduce
reflection parameterizations for submerged structures. Parameterizations are introduced in Van der Meer et al. (2005) and
Davidson et al. (1996) but can only be applied to rubble mound
breakwaters. Stamos et al. (2002) investigated reflection, transmission, and energy loss effects of submerged rectangular and
semicircular breakwaters. They concluded that, for rigid breakwaters, the rectangular models are more effective than the
semicircular models in terms of reflecting incident wave energy;
a conclusion shared by the authors of the present study.
A scaling analysis that accounts for the sediment scour and wave
field characteristics for two-dimensional submerged vertical and
semicircular breakwaters is conducted by Young and Testik (2009).
Omitting the details, this analysis indicates the following five
dimensionless parameters relevant to the wave field characteristics
around the submerged breakwaters considered in this present study:
breakwater Reynolds number, Re ¼ ððHi p=TÞWbw Þ=n; KeuleganCarpenter number, KC ¼ ðHi p=Wbw Þ; dimensionless submergence
depth, a/Hi; dimensionless wave height, Hi/Wbw; and dimensionless
water depth, hbw/Hi. Here, T is the wave period, n is the kinematic
viscosity of the fluid, hbw is the water depth at the breakwater, and a
is the submergence depth of the breakwater crest. Note that the
selected flow parameters (i.e., Hi and T) are surface observables that
have been monitored and recorded for a long time for various
coastlines worldwide and, if needed, can be measured relatively
easily through various means including remote sensing methods.
Our preliminary experiments indicate that, among these five relevant
dimensionless parameters, the dimensionless submergence depth,
a/Hi, is the sole governing dimensionless parameter for wave
reflection characteristics.
This manuscript is organized as follows: experimental setup,
methodology, and data processing are described in Section 2;
results for wave field alterations around submerged breakwaters
are presented in Section 3; and discussions and conclusions are
given in Sections 4 and 5, respectively.
Upon the introduction of a structure into a flow field, wave
transformations around the structure occur. For breakwaters, an
important wave transformation characteristic is the wave reflection and, thereby, the reflection coefficient (Cr) as defined in
Eq. (1), which is a measure of the incident wave energy that is
reflected out to sea.
Hr
Cr ¼
ð1Þ
Hi
where Hr represents the reflected wave height and Hi represents
the incident wave height. As Cr increases, less incident wave
energy is available to be imparted on the shoreline. Therefore,
estimation of the Cr value is crucial in determining the efficacy of
a breakwater.
Several experimental studies have investigated wave reflection
characteristics for submerged breakwaters. Christou et al. (2008),
investigating the interaction of surface water waves with a
rectangular submerged breakwater, concluded that the reflection
of waves from a submerged breakwater is fundamentally linear,
even if the incident waves are nonlinear. Losada et al. (1996),
studying the effects of both regular and irregular waves on Cr for
submerged breakwaters, found that submerged breakwaters
under the influence of irregular waves induce smaller Cr values
than submerged breakwaters tested under regular waves.
However, the difference in Cr values between the two scenarios
did not exceed 5%, indicating that the findings on wave reflection
characteristics under regular wave conditions in the present
study is also a representative for wave reflection characteristics
under irregular wave conditions. Several studies (e.g., Huang and
Chao, 1992; Twu and Chieu, 2000) have shown that the breakwater
width (or thickness) (Wbw, see definition schematic in Fig. 1) is
an important variable in the design of permeable submerged
breakwaters (e.g., rubble mound breakwaters). Huang and Chao
(1992) found that the reflection coefficient decreases with the
increase of porosity of the breakwater. In the case of permeable
breakwaters, an increase in thickness or layers of porous material
of the breakwater causes more wave energy dissipation, thereby
lowering the reflection coefficient. Losada et al. (1996) determined
that the breakwater width factors into Cr calculation up to a
critical Wbw value, after which Cr is not affected by Wbw. In the
present study, the reflection coefficient did not rely on the
breakwater width as the breakwaters used in the present study
are impermeable; any energy dissipation advantages due to
breakwater thickness or layering are negated. Stamos and Hajj
(2001) and Stamos et al. (2002), studying wave reflection for rigid
and flexible breakwater cases, reported that the reflection coefficient increases with increase in the stiffness of the breakwater.
These two studies also investigated the effects of kh (k—wave
2. Experimental setup, methodology, and data processing
The experiments are carried out in a wave tank (12 m 0.6 m 0.6 m) that mimics the oceanic coastal zone (see Fig. 1
for a schematic of the wave tank). The tank consists of a beach
(1)
(6)
ε, T
Hi
z
(2)
(4)
(3)
a
Wbw
h
A
(5)
x
d1 d2
Fig. 1. Wave tank schematic: (1) linear actuator and motor; (2) breakwater; (3) wave paddle; (4) sloping beach; (5) wave absorber; and (6) moveable cart assembly with
wave gages and acoustic Doppler velocimeter (ADV). Symbols: a—depth of submergence, Wbw—breakwater crest width, A—breakwater height, Hi—incident wave height,
e—amplitude of wave paddle excursion, T—wave period, and h—still water depth at the paddle. Notes: d1 ¼ 0.5 m, d2 ¼0.4 m.
D.M. Young, F.Y. Testik / Ocean Engineering 38 (2011) 1269–1276
Table 1
Breakwater dimensions.
1271
Table 2
Experimental conditions.
Breakwater name
Type
Wbw (m)a
A (m)b
Radius (m)
Exp #a
BW nameb
h (m)
T (s)
Hi (m)c
Li (m)d
a/Hi
Cr
SC-1
SC-2
V-1
V-2
V-3
V-4
NB
Semicircular
Semicircular
Vertical
Vertical
Vertical
Vertical
No breakwater
0.28
0.50
0.15
0.08
0.17
0.30
–
0.23
0.30
0.23
0.30
0.30
0.30
–
0.15
0.30
–
–
–
–
–
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
SC-1
SC-2
V-1
V-3
V-4
NBf
SC-1
SC-2
V-1
V-3
V-4
NBf
SC-1
SC-2
V-1
V-3
V-4
NBf
SC-1
SC-2
V-1
V-3
V-4
NBf
SC-2
V-3
V-4
NBf
V-1
V-1
SC-1
SC-1
V-1
V-1
V-1
V-1
V-3
V-3
V-3
V-3
V-3
V-3
V-3
V-3
SC-2
SC-2
SC-2
SC-2
SC-2
SC-2
SC-2
0.40
0.40
0.40
0.40
0.40
1.33
1.33
1.33
1.33
1.33
1.33
2.00
2.00
2.00
2.00
2.00
2.00
1.33
1.33
1.33
1.33
1.33
1.33
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.20
2.50
3.00
3.50
2.00
2.00
2.00
2.00
2.00
2.00
2.00
0.139
0.139
0.139
0.139
0.139
0.139
0.099
0.099
0.099
0.099
0.099
0.099
0.130
0.130
0.130
0.130
0.130
0.130
0.139
0.139
0.139
0.139
0.139
0.139
0.080
0.080
0.080
0.080
0.077
0.077
0.077
0.077
0.093
0.091
0.099
0.086
0.061
0.065
0.065
0.063
0.084
0.073
0.056
0.057
0.061
0.065
0.065
0.063
0.091
0.090
0.088
2.02
2.02
2.02
2.02
2.02
2.02
3.48
3.48
3.48
3.48
3.48
3.48
1.65
1.65
1.65
1.65
1.65
1.65
2.66
2.66
2.66
2.66
2.66
2.66
3.01
3.01
3.01
3.01
2.77
2.75
2.77
2.75
2.76
2.76
2.76
2.76
2.59
2.58
2.58
2.57
3.36
3.71
4.30
4.63
2.59
2.58
2.58
2.57
2.85
3.09
3.08
1.2
0.7
1.2
0.7
0.7
0.04
0.10
0.11
0.25
0.25
1.7
1.0
1.7
1.0
1.0
0.02
0.07
0.04
0.16
0.16
1.3
0.8
1.3
0.8
0.8
0.04
0.08
0.09
0.26
0.20
1.2
0.7
1.2
0.7
0.7
0.02
0.10
0.06
0.22
0.25
1.3
1.3
1.3
0.05
0.12
0.13
1.3
1.8
1.3
1.8
0.4
0.5
0.6
0.9
0.0
0.2
0.3
0.5
1.2
1.4
1.8
1.8
0.0
0.2
0.3
0.5
0.4
0.7
0.9
0.07
0.05
0.02
0.01
0.25
0.20
0.18
0.16
0.50
0.40
0.32
0.28
0.14
0.09
0.07
0.06
0.49
0.37
0.28
0.22
0.18
0.18
0.14
a
b
Wbw—breakwater crest width.
A—breakwater height.
with adjustable sandy slope (0–1:20), a wave generator assembly,
and walls composed of 0.01 m thick Plexiglas for visualization.
The wave generator consists of a computer-controlled linear
actuator coupled with a wave paddle. The wave generator can
achieve accelerations up to 10 ms 2 and velocities up to
1.5 ms 1. The precision of the wave paddle motion is 0.0001 m.
A computer code in LabView is written to control the wave
generator. Two submerged semicircular breakwaters and four
submerged vertical breakwaters are used in this study. The
vertical breakwaters are constructed of oriented strand board
and the semicircular breakwaters of PVC pipe (see Table 1 for
breakwater dimensions). The width of each breakwater is set
equal to the width of the tank due to the two-dimensional nature
of the study. The breakwaters are also built to allow for a height
adjustment in order to provide a larger range of experimental
parameters.
Several experimental apparatuses are used to collect information regarding flow field characteristics. The principle measurements of interest are water surface profiles and flow velocities.
Water surface elevation data are collected by three capacitancetype wave gages that are capable of sampling data at a rate of
50 Hz with an accuracy of 0.001 m and a measurement range of
0.005–1 m. Each wave gage is located at a different location along
the slope and voltage readings from each gage are acquired
simultaneously. Following a standard procedure, these readings
are then converted to water surface elevations using a calibration
curve. Flow velocity measurements are taken using a 10 MHz
acoustic Doppler velocimeter (ADV) from Sontek/YSI. The ADV
provides three-dimensional velocity components at the sampling
volume 0.05 m below the probe tip using a physical principle
called the Doppler effect. The ADV is capable of a sampling rate of
25 Hz with an accuracy of 1%.
Before each experiment, several preparatory tasks are
completed to ensure consistency and accuracy. First, the appropriate
breakwater is installed at a specific distance from the wave
paddle and the beach is formed as either a 1:20 sloping beach
or a flat beach. The tank is then filled with water to a specific
depth (h¼0.27–0.40 m, see Table 2) and the wave gages’ initial
voltages are recorded from the computer to be used as the
reference to the still water level in the tank. Waves are then
generated and once the wave field is developed, water particle
velocities and wave elevations are measured after approximately
100 waves. Wave elevation measurements are collected for two
purposes: (i) to calculate wave reflection coefficients, and (ii) to
spatially monitor wave profile. For wave reflection coefficient
calculations, continuous wave elevation measurements using two
simultaneously recording wave gages are conducted for over 200
wave periods. The two wave gages are separated by 0.4 m (see
Fig. 1). Goda and Suzuki (1976) determined that wave resolution
may be performed in the range of 0.05o Dl/Lo0.45; where Dl
represents the distance between the wave gages and L represents
the incident wavelength. The value of Dl¼0.4 m is chosen
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.40
0.33
0.37
0.33
0.37
0.27
0.28
0.29
0.31
0.30
0.31
0.32
0.33
0.40
0.40
0.40
0.40
0.30
0.31
0.32
0.33
0.34
0.36
0.38
e
mg
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
0
0
0
0
0
0
0
0
0
0
0
0
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
1:20
a
Exp. #—experiment number.
BW name—breakwater name.
c
Hi—incident wave height at 50 cm offshore of the location of the breakwater’s
offshore face.
d
Li—incident wavelength at 50 cm offshore of the location of the breakwater’s
offshore face.
e
Cr—reflection coefficient.
f
NB—no breakwater installed.
g
m—beach slope. The wave tank background reflection coefficient ranged from
0.05 to 0.07.
b
because for this value of Dl, Dl/L values in all our experiments
conform to the wave gage separation distance criterion. Additionally, the onshore gage is positioned at 0.5 m offshore of the
breakwater face based on additional conclusions introduced by
Goda and Suzuki (1976) when it was determined that the wave
gage can be placed as near as 0.1L to the reflective surface. In the
present study, a distance of 0.5 m from the onshore wave gage to
the offshore face of the breakwater corresponds to a distance
1272
D.M. Young, F.Y. Testik / Ocean Engineering 38 (2011) 1269–1276
larger than 0.1L for all our experiments, conforming to the second
criterion while allowing us more reliable data collection due to
the stability of the waves at this point (i.e., before wave breaking
point). The goal was to acquire the data as close as possible to the
breakwater without compromising the data quality while adhering to rules outlined in previous work by Goda and Suzuki (1976).
For spatial monitoring of wave profiles, wave elevation measurements are conducted at x¼ 0.5, 1, 1.5, and 2 m from the breakwater face (x—horizontal coordinate along the tank, positive
being offshore, see Fig. 1) for over 40 wave periods at each
location, and a computer code is used to period-average the data
for 40 wave periods. The ADV is then used to collect over 40 wave
periods of velocity data 0.1 m above the sand-water interface at
the same locations, which is then period-averaged for 40 wave
periods.
The incident wave height, Hi, is determined from the water
surface elevation recordings in the absence of a breakwater using
the wave gage that is located 0.50 m offshore of the location
where the offshore breakwater face would be located (see Fig. 1).
This measured value of Hi may be considered to represent the
field installation conditions. For each set of experimental conditions, the background wave reflection coefficient and wavelength
are calculated using the procedure outlined in the following
paragraph. It should be noted that in the oceanic coastal zone,
typical reflection coefficients for similar beach profiles are
approximately 0.02. Cotter and Chakrabarti (1994) stated that
for an efficient experimental beach, the reflection coefficient
should be consistently less than 0.1 and preferably less than
0.05. For both horizontal and sloping (1:20) sandy beaches, the
wave tank used in the present study consistently produces
reflection coefficients of approximately 0.05–0.07 without breakwaters, indicating the suitability of our experimental tank in this
study to simulate the oceanic coastal zone.
The wave reflection coefficients are calculated using a MATLAB
code using a method introduced by Goda and Suzuki (1976). As
waves are reflected from the breakwater, they propagate towards
the wave paddle and reflect once again; a process that is ongoing
throughout the experiments. Thus, the wave system can be
regarded as a superposition of a number of waves propagating
in the positive and negative x-direction down the length of the
wave tank. In the method by Goda and Suzuki (1976), the incident
and reflected wave spectra are constructed from water surface
elevation recordings at two adjacent stations and then the ratio of
incident and reflected wave energies is employed in estimating
the reflection coefficient. For an in-depth analysis of this wave
harmonic separation technique, refer to Goda and Suzuki (1976).
Though the method introduced by Goda and Suzuki (1976)
centers on wave reflection estimations over a flat bed, the authors
decided to proceed with reflection coefficient estimations using
this method for the present study’s experiments over a sloping
bed. There are other studies that use this method for experiments
over a mildly sloping bed (e.g., Rathbun et al., 1998). To verify the
accuracy of this method for our sloping bed experiments, experiments were conducted to determine wave reflection from a rigid,
impermeable, vertical wall for both flat and sloping (1/20) beds.
Estimated values of the reflection coefficient at different distances
from the vertical wall were similar for both horizontal and sloping
beds, starting from close to the theoretical value of 1 and
decreasing approximately with the same trend as the distance
from the wall increases. Therefore, the method introduced by
Goda and Suzuki (1976) is adequate for our experimental configuration. There are several new methods proposed for reflection
coefficient determination for sloped-bottom experiments (e.g.,
Chang, 2002; Chang and Hsu, 2003; Wang et al. 2008), which
utilize different assumptions/simplifications to consider wave
shoaling and phase shift effects. However, to the author’s
knowledge, these methods still await thorough experimental
verification and are therefore not utilized in this study.
The incident wavelength, Li, is calculated by determining the
time, t, required for a wave crest to travel the known distance,
d ( ¼0.4 m in our experiments), between the two adjacent wave
gages that are located 0.5 m offshore of the breakwater face (see
Fig. 1). Once t is determined, wavelength (Li ¼ cT) is calculated as
the product of the wave celerity (c ¼d/t) and the wave period (T)
(see also Cotter and Chakrabarti, 1994).
As mentioned earlier, wave reflection characteristics are
observed to be mainly determined by the dimensionless breakwater submergence. Consequently, experiments were conducted
for a wide range of dimensionless submergence depth to elucidate the functional dependences of wave reflection on the
dimensionless submergence depth. Experimental conditions are
summarized in Table 2. Note that in the calculations of the
dimensionless submergence depth, experimental incident wave
height measurements at x¼ 0.5 m in the absence of breakwaters
are employed.
3. Flow field around submerged breakwaters
This section, presenting the results on the effect of a breakwater on the flow field, is arranged into two subsections: wave
elevations and water particle velocities, and wave reflection
coefficients (Cr).
3.1. Wave elevations and water particle velocities
As waves approach shore and begin to shoal, wave heights and
underlying water particle velocities increase and wavelengths
decrease due to decrease in the water depth. Fig. 2 presents data
from two experimental runs under the same conditions one
without a breakwater (Experiment # 6 in Table 2, Fig. 2a) and
the other with breakwater V-4 installed (Experiment # 5 in
Table 2, Fig. 2b). As can be seen, the wave elevations do not
follow the same shoaling trend. In the absence of a breakwater
the wave heights increase steadily as the water depth decreases.
However, in the presence of a breakwater a partial standing wave
field develops due to breakwater-induced wave reflection. A
direct result of this phenomenon can be seen in Fig. 2b as the
occurrence of larger wave heights at antinodes (x E1.0 and 2.0 m)
than those at nodes (x E0.5 and 1.5 m).
A typical envelope of the partial standing wave induced by a
breakwater (Experiment # 5 in Table 2) is shown in Fig. 3. In this
figure, maximum and minimum wave elevations measured at
different spatial positions along the tank at 0.2 m intervals from
x¼0.1 to 2.5 m are given. In a partial standing wave field, nodes
and antinodes alternate spatially at x-locations at increments of
ðLi =4Þ. This is seen in Fig. 3: nodes—ðx=Li Þ ¼0.25, 0.75, 1.25;
antinodes—ðx=Li Þ ¼0.5, 1.0. Similarly, in Fig. 3, where Li ¼2.02 m,
antinodes occur at approximately x¼1.0 and 2.0 m, and nodes
occur approximately at x¼0.5 and 1.5 m. Since only some
percentage of the wave energy is reflected from a submerged
breakwater, the spatial wave profile will always be that of a
partial standing wave system. In a partial standing wave system,
the envelope height at the antinodes is the result of the incident
wave height plus the partial reflected wave height ðHi ð1 þ Cr ÞÞ
whereas the envelope height at the nodal points is Hi ð1Cr Þ.
Figs. 4 and 5 present the horizontal and vertical water particle
velocity profiles at different locations along the slope from two
experimental runs under the same conditions as in Figs. 2 and 3
(Experiments #5 and 6 in Table 2). Figs. 4a and 5a display
horizontal and vertical particle velocity data from an experiment
without a breakwater installed while Figs. 4b and 5b exhibit
D.M. Young, F.Y. Testik / Ocean Engineering 38 (2011) 1269–1276
10
30
8
20
6
10
u (cm/s)
η (cm)
4
2
0
-2
0
0.5
1
1.5
1
1.5
1
1.5
20
6
10
2
-2
0.5
0
1
1.5
u (cm/s)
4
0
t (s)
30
8
η (cm)
0.5
-30
t (s)
10
0
0
0.5
-10
-20
-4
-6
-30
t (s)
Fig. 2. Spatial wave elevations, Z: (a) no breakwater installed (Cr ¼ 0.06), (b) V-4
breakwater installed (Cr ¼ 0.25). (&) x¼ 0.5 m, (’) x ¼1.0 m, (J) x¼ 1.5 m, and
(K) x ¼2.0 m.
10
Fig. 4. Horizontal water particle velocities, u: (a) no breakwater installed,
(b) breakwater V-4 installed. (&) x¼ 0.5 m, (’) x ¼1.0 m, (J) x¼ 1.5 m, and (K)
x¼ 2.0 m.
figure, near-bed velocity data collected spatially at 0.2 m intervals
from x¼ 0.1 to 2.50 m are given. As can be seen in this figure, the
maximums and minimums spatially alternate at distances of
x ¼ ðLi =4Þ, similar to the partial standing wave height envelope.
The maximum horizontal (minimum vertical) velocities occur at
nodes whereas minimum horizontal (maximum vertical) velocities occur at antinodes.
5
η (cm)
0
-20
-6
0
0.25
0.5
0.75
1
1.25
1.5
-5
-10
0
-10
-4
0
1273
x/Li
Fig. 3. Partial standing wave envelope for experiment #5 in Table 2. Incident
wavelength, Li ¼2.02 m.
information from an experiment with breakwater V-4 installed on
the sandy bed. The effect of breakwater V-4 as shown in Fig. 4
causes significant disturbances in the spatial horizontal and
vertical velocity profiles. The maximum horizontal velocities
occur at distances of x ¼0.5 and 1.5 m (nodes) while the minimum
velocities occur at distances of x ¼1.0 and 2.0 m (antinodes).
Unlike the horizontal velocity field, the maximum vertical
velocities occur at distances of x¼1.0 and 2.0 m (antinodes) while
the minimum vertical velocities occur at distances of x ¼0.5 and
1.5 m (nodes). Typical horizontal and vertical velocity envelopes
of the partial standing wave field are shown in Fig. 6. In this
3.2. Wave reflection
To quantify the efficacy of submerged breakwaters in reflecting
wave energy out to the sea, the reflection coefficient, Cr(¼Hr/Hi), is
investigated for both vertical and semicircular breakwaters. Experimentally measured reflection coefficients are tabulated in Table 2 and
plotted in Fig. 7.
Laboratory experiments indicate that the only dimensionless
parameter that governs the wave reflection is ða=Hi Þ, the dimensionless submergence depth. Using a linearized solution for the
flow field around an infinitely long, rigid, wide, submerged,
impermeable, and vertical breakwater, Abul-Azm (1994) parametrically studied the effect of breakwater width and the relative
water depth (ratio of the water depth to the wavelength) and
reported Cr dependency on these parameters. In the present
study, we did not observe a noticeable dependence of the
reflection coefficient on the breakwater width and the relative
water depth for the experimental parameter range studied. The
discrepancy between the results of Abul-Azm and our experimental observations may be due to the highly nonlinear wave
behavior in our experiments whereas the theoretical treatment of
Abul-Azm considers linear waves. Moreover, it is important to
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D.M. Young, F.Y. Testik / Ocean Engineering 38 (2011) 1269–1276
4
v (cm/s)
2
0
0.5
0
1
1.5
1
1.5
-2
-4
t (s)
4
v (cm/s)
2
0
0.5
0
-2
-4
t (s)
Fig. 5. Vertical water particle velocities, v: (a) no breakwater installed,
(b) breakwater V-4 installed. (&) x¼ 0.5 m, (’) x¼ 1.0 m, (J) x¼ 1.5 m, and (K)
x¼ 2.0 m.
30
for estimation of the reflection coefficient for normally incident
waves to vertical (Eq. (2)) and semicircular (Eq. (3)) submerged
breakwaters.
u, v (cm/s)
20
10
0
-10
-20
-30
Fig. 7. Relationship between ða=Hi Þ and Cr for (a) vertical breakwaters and
(b) semicircular breakwaters. Solid lines—estimate by Eq. (2) for (a) and estimate
by Eq. (3) for (b); (’)—measured for (a), (J)—measured for (b).
0
0.25
0.5
0.75
x/Li
1
1.25
1.5
Fig. 6. Horizontal (u, solid line) and vertical (v, dashed line) velocity envelope for
the partial standing wave field for experiment # 5.
note that since the breakwaters used in the present study are
impermeable, we did not observe any energy dissipation advantages due to breakwater thickness or layering as in permeable
breakwaters. These observations are consistent with Van der
Meer et al. (2005). Measured wave reflection coefficients for
vertical (see Fig. 7a) and semicircular (see Fig. 7b) breakwaters
share the same exponential functional dependency on ða=Hi Þ.
Based on these experimental observations and scaling arguments,
we propose the following two semi-empirical parameterizations
Crvertical ¼ 0:53eð0:85ða=Hi ÞÞ
ð2Þ
Crsemicircular ¼ 0:53eð1:4ða=Hi ÞÞ
ð3Þ
Here, subscripts, vertical and semicircular, indicate the submerged breakwater type. The correlation coefficients (R2 value)
for the vertical and semicircular breakwaters are 0.92 and 0.97,
respectively, indicating a good statistical fit.
Dependency of measured and estimated Cr values for vertical
(Fig. 7a) and semicircular (Fig. 7b) breakwaters on ða=Hi Þ values
are shown in Fig. 7. This figure illustrates that as ða=Hi Þ decreases
(for fixed Hi and decreasing a), Cr increases for both types of
breakwaters. As ða=Hi Þ becomes zero (breakwater’s crest at the
still water surface), Cr reaches its maximum value, 0.53. For the
asymptotic case when ða=Hi Þ approaches infinity, Cr approaches
zero. In order for ða=Hi Þ to approach infinity, either a approaches
infinity (finite breakwater height at infinite water depth) while Hi
remains finite or Hi approaches zero (absence of waves) while a
remains finite.
4. Discussion
For comparison purposes, Fig. 8 displays measured and estimated
reflection coefficients for vertical and semicircular breakwaters in the
D.M. Young, F.Y. Testik / Ocean Engineering 38 (2011) 1269–1276
1275
5. Conclusion
Fig. 8. Measured and estimated reflection coefficients for vertical and semicircular
breakwaters. Solid squares represent measured Cr for vertical breakwaters and
open circles represent measured Cr for semicircular breakwaters. Upper solid line
corresponds to vertical breakwater Cr estimations using Eq. (2) and the lower solid
line corresponds to semicircular breakwater Cr estimations using Eq. (3).
same graph. In this figure, it is observed that the ratio of the reflection
coefficients of the two breakwater types approaches towards unity as
the breakwater’s crest approaches the still water surface (i.e., ða=Hi Þ
approaches zero). This finding indicates that for relatively small
submergence depths breakwater shape does not play a significant
role in the wave reflection. However, as submergence depth
increases, the difference between wave reflection by vertical and
semicircular breakwaters becomes pronounced.
Comparing the collected data from the sloping beach experiments and a limited number of (only 10) flat beach experiments
(see Table 2), it is curious that the presence of a sloping bottom
did not seem to have an effect on the reflection coefficient. A
possible cause of this result is due to the nature of the experimental wave-tanks in which background reflections exist due to
various components of the wave-tanks (e.g., tank walls, wave
paddle, bed slope) even in the absence of the reflecting structure.
In most setups, including this small-scale setup, background
reflection, while being much smaller than the reflection due to
the reflecting structure, may be much larger than the reflection
due to the mildly sloping beach alone. This may render the
reflection effects of the sloping beach negligible. However, it is
important to note that as the beach slope increases beyond the
tested range the potential for an effect on the reflection coefficient increases likewise.
Since breakwaters are employed to protect the coastal line or
coastal structures, both the estimation of the reflected wave
energy and the estimation of the transmitted wave energy
are crucial in determining the efficacy of a breakwater. Reflected
and transmitted wave energies are related through the dissipation losses. However, due to the difficulty in obtaining the
dissipation losses, transmitted and reflected wave energy information cannot simply be related in practice. Therefore, studies
centered on reflection and transmission of waves around
breakwaters are complementary. In such a complementary
study to our present study, Van der Meer et al. (2005) compiled
a variety of datasets to investigate the transmission characteristics of submerged breakwaters. Based on these datasets, Van der
Meer et al. (2005) reported a decrease in the transmission
coefficient (indicating an increase in the reflection coefficient)
as the relative submergence depth (a/Hi) decreases. This conclusion agrees with the findings of the present study as can be seen
in Figs. 7 and 8.
This note presents the results of a laboratory investigation on
the effects of submerged vertical and semicircular breakwaters on
the local wave field. The primary goal of the conducted research is
to provide accurate parameterizations for estimating the wave
reflection coefficient.
The conducted experiments, centering on the wave reflection
characteristics of submerged breakwaters, led to the conclusion
that the dimensionless submergence depth (a/Hi) is the sole
dimensionless flow parameter for determining the Cr value for
the range of parameters studied. Semi-empirical parameterizations for the reflection coefficients for submerged breakwaters of
vertical and semicircular shape are developed (Eqs. (2) and (3),
respectively). The reflection coefficient parameterizations for both
breakwater types were found to share the same functional
dependency on a/Hi with different constant empirical coefficients.
For both types of breakwaters, maximum Cr value is 0.53 and
occurs for the dimensionless submergence depth value of zero.
The Cr values decrease as the dimensionless submergence depth
increases, with a larger decrease for semicircular breakwaters
indicating their reduced efficiency compared to their vertical
counterparts.
The importance of breakwaters cannot be overstated. The
monetary benefits of establishing these breakwaters as part of
an all-inclusive beach nourishment plan, together with their
obvious use as ecological and marine support systems, can
positively impact susceptible coastlines and harbors in times of
need. Submerged breakwaters serve both these purposes while
remaining concealed beneath the water surface and maintaining
attractive esthetics. Along with conclusions developed in similar
studies centered on breakwater transmission coefficients, results
of this study are expected to be useful to coastal engineers
in preliminary feasibility studies and conceptual designs of
submerged semicircular and vertical breakwaters.
Acknowledgments
This research was supported by the funds provided by College
of Engineering and Science at Clemson University to the second
author. This work is part of the M.Sc. thesis of the first author
conducted under the guidance of the second author. We would
like to thank Mr. Mathew Hornack for his help in the laboratory.
The authors are grateful to the anonymous referees for their
valuable comments.
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