Available online at www.sciencedirect.com
Journal of Hydro-environment Research 3 (2010) 179e185
www.elsevier.com/locate/jher
Transmission of solitary waves through slotted barriers: A laboratory
study with analysis by a long wave approximation
Zhenhua Huang a,b,*, Zhida Yuan a
a
School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639651
b
Earth Observatory of Singapore (EOS), Nanyang Technological University, 50 Nanyang Avenue, Singapore 639651
Received 12 May 2009; revised 9 October 2009; accepted 9 October 2009
Abstract
Economic and long-term social impacts of tsunami waves, as shown in the 2004 Indian Ocean tsunami, have been devastating. In addition to
the passive measures such as the tsunami early warning systems, active protective measures such as tsunami breakwaters and tsunami-resistant
buildings are also necessary. Slotted barriers are low cost structures that can be very effective in reducing the transmitted energy of long waves.
In this study, the transmission of tsunami waves, with the leading wave being modeled by a solitary wave, through slotted barriers in the form of
a row of circular cylinders is studied experimentally. The results are also analyzed by a method based on long wave approximations. It is found
that the spacing between two adjacent cylinders is one of the main factors that control the transmission of solitary waves through slotted barriers.
Reflection coefficients and the reduction of drag force on coastal structures protected by slotted barriers are also discussed in the paper.
Ó 2009 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights
reserved.
Keywords: Tsunami waves; Solitary waves; Slotted barriers; Tsunami breakwaters; Tsunami-resistant structures
1. Introduction
Tsunami waves generated by mighty underwater earthquakes/landslides, which can occur at any time, can strike in
minutes, and cause damages to coastal areas. Economic and
long-term social impacts of tsunami waves, as shown in the 2004
Indian Ocean tsunami, have been devastating (Titov et al.
(2005)). After the tsunami in 2004, tsunami early warning
systems have been implemented by many coastal nations.
However, these systems are effective only when all infrastructures and people in vulnerable coastal communities are prepared
and respond appropriately in a timely manner upon the recognition of a potentially destructive tsunami event. Some tsunami
waves, which might not be strong enough to trigger an early
warning system to issue a warning, can still cause damages to the
* Corresponding author at: School of Civil and Environmental Engineering,
Nanyang Technological University, 50 Nanyang Avenue, Singapore 639651.
E-mail address: [email protected] (Z. Huang).
harbor facilities due to ship collisions. Thus, in addition to the
passive measures such as the tsunami early warning system,
active protective measures such as breakwaters are also necessary to prevent ships from breaking mooring lines and hitting the
port facilities because of the tsunami-induced current.
Pile or slotted breakwaters are low cost breakwaters (see
Mani and Jayakumar (1995) for a cost estimation for pipe/
slotted breakwaters.) that can be very effective in reducing the
transmitted energy of long waves (see Mei et al. (1974)).
Fig. 1 shows a section of pile breakwater along Singapore
coast. A lot of research has been carried out on the interactions
of regular waves with slotted barriers in the absence of
currents (see Kakuno and Liu (1993); Isaacson et al. (1998);
Huang (2007), etc. for example). Recently, the effects of
currents on the scattering of regular waves by slotted barriers
have been examined by Huang (2006) and Huang and
Ghidaoui (2007). The authors are not aware of published work
on tsunami wave interaction with slotted barriers.
Recently, suggestions have been proposed to build tsunamiresistant structures in the tsunami vulnerable areas to serve as
1570-6443/$ - see front matter Ó 2009 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.
doi:10.1016/j.jher.2009.10.009
180
Z. Huang, Z. Yuan / Journal of Hydro-environment Research 3 (2010) 179e185
Fig. 1. A segment of pile breakwater found in a section of the Singapore coast.
tsunami shelters in the event of human evacuation. One of the
concerns in the design of tsunami-resistant builds is the drag
force acting on the structure by tsunami waves. It is anticipated that slotted barriers can be good measures to effectively
reduce the drag force on structures protected by slotted
barriers.
One of the aims of this study is to examine experimentally
the transmission and reflection of tsunami waves (solitary
waves) through pile breakwaters. Another aim of this study is
to assess the possibility of using slotted barrier to convert
some of the coastal structures into tsunami-resistant buildings.
In laboratory, the first peak of tsunami waves is normally
modeled by a solitary wave in view of the extremely long
length of such waves. The interactions between solitary waves
and slotted barriers form the basis of this study. The scale
effects are not addressed in this study. The rest of paper is
structured as follows: experimental setup and method used to
analyze data are given in Section 2; a model based on long
wave approximation is introduced in Section 3 to estimate the
transmission and reflection coefficients for solitary waves
scattered by a slotted barrier; The measured and calculated
transmission and reflection coefficients are presented in
Section 4, where the possible reduction of drag force on
coastal structures protected by slotted barriers is also estimated; finally, main conclusions drawn from this study are
summarized in Section 5.
Fig. 2. A view of the slotted barrier installed in the wave flume.
was a piston-type wave generator (HR Wallingford), which
was used to generate the solitary waves. Fig. 3 shows the
experimental setup, where the wave probe G1 was used to
measure the incident and reflected waves while the wave probe
G2 was used to measure the transmitted wave. Before studying
the interactions between solitary wave and slotted barrier, we
first confirmed the solitary wave generated by the wave-maker
by comparing it with the theoretical solitary wave described
by equation (1), which describes a solitary wave of height H
and propagating in water of depth h.
h ¼ Hsech2 k½ðx x0 Þ Cðt t0 Þ;
ð1Þ
where x0 is the location of the solitary wave peak at t ¼ t0.
The phase velocity C and the equivalent wave number k are
given by
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C ¼ gðh þ HÞ and k ¼
rffiffiffiffiffiffiffi
3H
;
4h3
ð2Þ
2. Experimental setup and data analysis
respectively.
When slotted barrier was installed in the midway of the
wave flume, part of wave energy will be transmitted to the lee
A series of experiments were conducted in a wave flume
located at the Hydraulics Laboratory, NTU, Singapore, to
study the transmission of solitary waves through pile/slotted
breakwaters consisting of an array of circular cylinders of
diameter D ¼ 3 cm. Two wave probes (UltraLab sensors,
General Acoustics) were used to measure the surface elevations at location G1 and G2. Fig. 2 shows a view of the pile
breakwater used in our experiments. The wave flume was 32 m
long and 54 cm wide. Installed at one end of the wave flume
Fig. 3. Sketch of experimental setup showing the relative locations of the
slotted barrier and the two wave probes.
Z. Huang, Z. Yuan / Journal of Hydro-environment Research 3 (2010) 179e185
side of the barrier, part of energy will be dissipated into
turbulence, and the rest of the wave energy will be reflected by
the barrier, resulting in two peaks in the surface displacement
measured by the wave probe G1. Fig. 4 shows an example of
the measured incident, reflected, and transmitted solitary
waves. As a result of the nonlinear interactions between the
barrier and the solitary waves, an enhanced undulating tail can
be observed in the reflected solitary wave.
To study the effects of water depth and incident wave
height, five wave heights varying from 4 cm to 8 cm were
examined for h ¼ 15 cm and six wave heights varying from
5 cm to 10 cm were examined for h ¼ 20cm. Three different
slotted barriers were used in the experiments, with the centerto-center distance between two adjacent cylinders (spacing)
being S ¼ 4.5 cm, S ¼ 4.2 cm, S ¼ 3.64 cm, respectively. As
the diameter of the cylinder was 3 cm, the spacing-to-diameter
ratio S/D ranged from 1.50 to 1.21 in this study.
3. A long wave approximation to transmission coefficient
Similar to regular waves, we can define the reflection (CR)
and transmission (CT) coefficients by
CR ¼
HR
HT
and CT ¼ ;
HI
HI
ð3Þ
where HR, HT and HI are the heights of reflected, transmitted
and incident solitary waves, respectively. In the following, we
will show how to estimate the transmission coefficient using
the theory developed for long waves interacting with a slotted
barrier in the presence of a uniform current (Huang and
Ghidaoui (2007)).
For the solitary wave described by equation (1), the corresponding horizontal velocity u(t) is determined by
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
gðh þ HÞ;
uðtÞ ¼
hþH
ð4Þ
Surface displacement [cm]
6
4
which is alway positive. However, the velocity u(t) can be
viewed as the sum of a mean velocity u(averaged over
a pseudo-period T ) to be defined later and a time-varying part
~uðtÞ, i.e.,
uðtÞ ¼ u þ ~uðtÞ;
which is graphically shown in Fig. 5.
After we define a pseudo-wave length L by
rffiffiffiffiffiffi
2p
h3
¼ 4p
L¼
3H
k
a pseudo-period T can be calculated by
sffiffiffi
L
h 4p
1
pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi:
T¼ ¼
C
g 3 H=hð1 þ H=hÞ
ð5Þ
ð6Þ
ð7Þ
Now, a solitary wave can be regarded as one representative
wave in a train of solitary waves with the spatial distance
between two peaks being equal to L or with the temporal
interval between two peaks being T, as shown in Fig. 6.
Now the transmission coefficient CT of the time-varying
part ~uðtÞ, with a period T and wave length L, can be estimated
by the equation (8), which was originally developed by Huang
and Ghidaoui (2007) for the scattering of long waves by
a slotted barrier in the presence of a uniform current,
CT ¼
2
pffiffiffiffiffi
:
bð1 u= ghÞ þ 2
ð8Þ
In the above equation, the equivalent energy loss coefficient
b is calculated by
b¼
f 0 uref
pffiffiffiffiffi;
2 gh
ð9Þ
with f0 being the quadratic friction coefficient and uref a reference velocity down stream of the barrier, both to be determined
experimentally (see Mei et al. (1974) for the case of long
regular waves). If we take uref ¼ gumax, with 0 < g 1 being
an empirical constant and umax the maximum of the horizontal
G1
Incident wave
181
G2
Transmitted wave
Reflected wave
2
0
-2
26
28
30
32
34
time [s]
36
38
40
Fig. 4. Solitary waves measured by probes G1 and G2 in the presence of
a slotted barrier of spacing-to-diameter ratio S/D ¼ 1.21. Target wave
height ¼ 6 cm and water depth ¼ 15 cm.
Fig. 5. Definition sketch of the velocity decomposition for a solitary wave.
Z. Huang, Z. Yuan / Journal of Hydro-environment Research 3 (2010) 179e185
182
As the incident and reflected solitary waves will interact in
a nonlinear way, it can be expected that the reflection coefficients calculated by linear wave theory will contain relatively
large error for large HI/h, which is a measure of the nonlinearity of shallow water waves. Nevertheless, an attempt is given
here to estimate the reflection coefficient based on linear wave
theory, anticipating that some useful information can be
obtained when the nonlinearity is weak. After the transmission
coefficient CT and the parameter b are obtained from equations
(16) and (14), the reflection coefficient CR can be calculated by
Fig. 6. Long wave approximation to solitary wave.
velocity of the solitary wave, the energy loss coefficient b is
formally related to the maximum velocity umax by
f umax
b ¼ pffiffiffiffiffi;
2 gh
f ¼ gf 0 ;
sech2 ðxÞdx ¼ 0:99999;
ð11Þ
0
it then follows from equations (1) and (4) that the maximum
velocity umax and the mean velocity u, averaged over the
pseudo-period T, are given by
pffiffiffiffiffi H=h
ghpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ H=h
ð12Þ
u ¼ aumax ; a ¼ 0:159:
ð13Þ
umax ¼
and
With equation (12), the equivalent energy loss coefficient
b can be expressed in terms of the transmission coefficient CT
and the relative incident wave height HI/h,
f
HT =h
f
CT HI =h
b ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi:
2 1 þ HT =h 2 1 þ CT HI =h
4. Results and discussion
4.1. Reflection and transmission coefficients
In our experiments, two water depths were examined with
D/h ¼ 1/5 and D/h ¼ 3/20, respectively. After analyzing our
experimental data, it was found that the measured reflection
and transmission coefficients are nearly independent of D/h,
which can also be seen from the long wave theory outlined in
the previous section.
Figs. 7e9 show the measured reflection and transmission
coefficients, together with those predicted by equations (16)
and (17), for spacing-to-diameter ratio S/D ¼ 1.21, 1.40, and
1.50, respectively. For a given S/D, the measured transmission
coefficient decreases with increasing HI/h, while the measured
reflection coefficient is insensitive to the change in HI/h. These
trends are similar to those found in long waves scattered by
a slotted barrier in the presence of a uniform current (Huang
and Ghidaoui (2007)).
ð14Þ
1
2f
:
bðf 2abÞ þ 2f
ð16Þ
The transmission coefficient CT can be obtained by solving
equations (14) and (16) for given f and HI/h. Since CT 1 and
b > 0, equation (16) requires that f 2ab. In fact, we can
show pfrom
equations
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi (14) and (13) that f [2ab since
a 1 þ ðCT HI =hÞ=ðCT HI =hÞ for rational values of CTHI/h.
D/h=1/5
D/h=3/20
0.8
CR and CT
ð15Þ
which, after being substituted in equation (8), leads to
CT ¼
ð17Þ
See Huang and Ghidaoui (2007) for the detailed derivation of
this equation.
From equation (13) and equation (14), the mean velocity u can
be expressed in terms of the equivalent energy dissipation
coefficient b
u
aumax
HT =h
2ab
pffiffiffiffiffi ¼ pffiffiffiffiffi ¼ apffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼
;
f
gh
gh
1 þ HT =h
pffiffiffiffiffi
bð1 þ u= ghÞ
u
2ab
pffiffiffiffiffi
; pffiffiffiffiffi ¼
:
f
gh
bð1 u= ghÞ þ 2
ð10Þ
with f being an equivalent quadratic loss coefficient to be
determined from the experiments.
Note that
Z2p
CR ¼
D/h=1/5
0.6
CT
0.4
CR
D/h=3/20
0.2
0
0
0.2
0.4
0.6
0.8
1
HI/h
Fig. 7. Comparison between the measured and predicted hydrodynamic
coefficients for S/D ¼ 1.21. Solid lines ( f ¼ 18), chains ( f ¼ 14.4), and dashed
lines ( f ¼ 21.6).
Z. Huang, Z. Yuan / Journal of Hydro-environment Research 3 (2010) 179e185
1
D/h=1/5
D/h=3/20
CR and CT
0.8
CT
D/h=1/5
D/h=3/20
0.6
0.4
CR
0.2
0
0
0.2
0.4
0.6
0.8
1
HI/h
Fig. 8. Comparison between the measured and predicted hydrodynamic
coefficients for S/D ¼ 1.40. Solid lines ( f ¼ 6), chains ( f ¼ 4.8), and dashed
lines ( f ¼ 7.2).
For a given incident wave height HI/h, the measured
transmission coefficient increases with increasing spacing-todiameter ratio S/D, while the reflection coefficient decreases
with increasing S/D. These trends agree with the following
theoretical observations: in the limit of S=D/N, theoretically
CT /1 and CR /0; in the limit of S=D/0, theoretically
CT /0 and CR /1.
The transmission coefficients are calculated by using
equations (14) and (16) for a given barrier and incident wave
conditions. Numerical experiments show that the following
values of f can produce the best fits to the measured transmission coefficients for the three barriers: f ¼ 3 for S/D ¼ 1.5,
f ¼ 6 for S/D ¼ 1.4 and f ¼ 18 for S/D ¼ 1.21. Figs. 7e9 show
the comparison between the predicted and measured transmission coefficients for all three barriers studied in the
experiments. It can be seen that the long wave approximation
can predict the transmission coefficient satisfactorily.
1
CR and C T
0.8
The predicted reflection coefficients are also shown in
Figs. 7e9. The long wave approximation can still provide
reasonable prediction of the reflection coefficients for relatively small HI/h, i.e., the nonlinearity is weak. For large HI/h,
long wave approximation over-predicts the reflection coefficients for all three barriers. This is expected that the nonlinear
interaction between the incident and reflected solitary waves
cannot be handled by long wave theory of Huang and
Ghidaoui (2007).
The sensitivity of the predicted transmission and reflection
coefficients to the variation in f is shown in Figs. 7e9 by
allowing the values of f to deviate from the best fit values
by 20%; the maximum change in either CT or CR corresponding to a 20% change in f is about 10% for all test
conditions.
4.2. Protection of coastal structures with slotted barriers
As a potential application of slotted barrier in mitigation of
tsunami hazard, we examine here the reduction of drag force
on typical coastal structures by installing a row of circular
cylinders in front of the structures, as shown in Fig. 10. It is
expected that the reduction of the tsunami wave height by the
slotted barrier can significantly reduce the draft force on the
structures under consideration.
After a solitary wave passing the slotted barrier, the
turbulence intensity in the flow may be enhanced, to a certain
degree, by flow separations. The horizontal velocities
measured at different elevations are shown in Fig. 11 for
several selected instants. After a solitary wave passing through
a slotted barrier, the horizontal velocity is more or less
uniform on a vertical plane, just like the velocity distribution
without any barrier interacting with solitary waves. We may
assume that the wake effects can be ignored and the drag
coefficient of the coastal structure under consideration can be
assumed to be the same as that without the barrier.
For solitary waves considered here, the drag force acting on
a coastal structure,FD, dominates. Let CD be the drag coefficient, the drag force FD can be calculated by
1
2
FD ðtÞ ¼ CD rAf uðtÞ ;
2
D/h=1/5
CT
D/h=3/20
D/h=1/5
D/h=3/20
0.6
0.4
0.2
0
CR
0
0.2
0.4
0.6
0.8
1
HI/h
Fig. 9. Comparison between the measured and predicted hydrodynamic
coefficients for S/D ¼ 1.50. Solid lines ( f ¼ 3), chains ( f ¼ 2.4), and dashed
lines ( f ¼ 3.6).
183
Fig. 10. Coastal structures protected by slotted barriers.
ð18Þ
Z. Huang, Z. Yuan / Journal of Hydro-environment Research 3 (2010) 179e185
184
by using the calculated CT (from equation (16)) in equation
(19). It can be seen that the slotted barriers can significantly
reduce the drag force on the protected structures, especially
when the spacing between the two adjacent cylinders is small.
For S/D ¼ 1.21, about 60% of the drag force can be reduced by
the barrier. Therefore, it is possible to convert some existing
coastal structures into tsunami-resistant shelters by protecting
them with well-designed slotted barriers. To provide effective
protection for the coastal structures, the slotted barrier must be
higher than the tsunami wave height. For huge tsunami waves,
say, of hight as large as 10 m, other innovative measures
should be explored from practical point of view.
Distance from bottom [m]
0.2
0.15
t=
0.6s
0.8s
0.1
0.9s
1.0s
0.05
1.1s
1.2s
1.4s
0
0
0.2
0.4
Velocity [m/s]
0.6
5. Conclusion remarks
Fig. 11. Vertical distribution of the horizontal velocities measured in the lee
side of the barrier and at several instants within a period. S/D ¼ 1.21, h ¼ 0.2m
and HI ¼ 0.1m. In the figure t ¼ 0 is chosen such that the solitary wave peak
passes the probe at t ¼ T/2.
where Af is the frontal area. The ratio of the maximum drag
force on the structure protected by a slotted barrier to that
without barrier is a function of HI/h and the transmission
coefficient CT,
2
u
½FD with barrier
¼ 2 T with barrier
R FD ¼
½FD without barrier ½uI without barrier
ð19Þ
2 2
CT HI =h
HI =h
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = pffiffiffiffiffiffiffiffiffiffiffi ;
1þCT HI =h
1þHI =h
where operator [.] means taking the magnitude of its argument,uT is the velocity for the transmitted solitary wave, and uI
for the incident solitary wave. RFD is a drag force reduction
factor, which is a measure of drag force reduction by slotted
barriers.
Based on equation (19), the estimated reduction factors of
the drag force on structures for all data collected in our
experiments are shown in Fig. 12, where symbols were
obtained by using the measured CT in equation (19) and lines
1
D/h=0.20, S/D=1.21
D/h=0.15,S/D=1.21
D/h=0.20, S/D=1.40
Acknowledgments
This is Earth Observatory of Singapore (EOS) Contribution
No. 3. The reported work was funded partially by the Ministry
of Education, Singapore, through Project NTU-SUG 3/07, and
partially by Earth Observatory of Singapore, Nanyang Technological University (NTU), Singapore. Mr. Jun Wang,
a former exchange postgraduate student from Hohai University, China, and Mr. Lip-Yung Koh, a former FYP student at
NTU, are acknowledged for their initial involvement in the
experiments. Preliminary experimental results of this study
were presented at the Second International Symposium for
Shallow Flows, December, 2008, Hong Kong.
D/h=0.15,S/D=1.40
0.6
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D
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