Download Wave-induced drift of small floating objects in regular waves

Ocean Engineering 38 (2011) 712–718
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Ocean Engineering
journal homepage: www.elsevier.com/locate/oceaneng
Short Communication
Wave-induced drift of small floating objects in regular waves
Guoxing Huang, Adrian Wing-Keung Law n, Zhenhua Huang
School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798, Republic of Singapore
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 19 July 2010
Accepted 9 December 2010
Editor-in-Chief: A.I. Incecik
Available online 2 February 2011
Water waves induce a slow drift of an object floating on the water surface. In this study, we examined, by a
series of laboratory experiments, the drift motion of small rigid floating objects driven by regular waves in
deep water. Different shapes of planar objects, including square, circular and elliptical, were investigated
for two different submergences, and their drift motions in waves were determined using an infrared
motion monitoring system. The corresponding measurements enabled the quantification of the drift
characteristics with respect to the wave characteristics and object shapes. Numerical simulations based
on an existing theory were presented and comparisons between the experimental data and the
predictions by the existing theory were performed.
& 2010 Elsevier Ltd. All rights reserved.
Keywords:
Small floating objects
Regular waves
Wave-induced drift
1. Introduction
Water waves induce a slow drift of objects floating on the water
surface in the direction of wave propagation. These rigid/flexible
floating objects can vary vastly in sizes, from small biomass such as
phytoplankton, to medium size flexible oil patches (i.e. Kang and
Lee, 1995; Law, 1999; Wong and Law, 2003), to large rigid floating
ice floes in the ocean (Wadhams, 1983). Understanding of their drift
behavior is important for engineering purposes. For example, for
floating oil patches in the nearshore region, the wave-induced drift
is one of the dominant mechanisms responsible for the beaching of
oil patches. A quantitative understanding of the drift behavior is
thus necessary for the oil fate and transport modeling (Cheng et al.,
2000; Law and Huang, 2007). For offshore structures in cold
regions, drifting icebergs can be extremely hazardous. Generally,
wind and ocean currents are considered to be the primary factors
causing the iceberg drift (e.g. El-Tahan and El-Tahan, 1983).
However, for ice floes in tens-of-meters size range, Wadhams
(1983) pointed out that the wave-induced drift can be a dominating
factor, even under strong winds. Based on theoretical arguments
alone, Arikainen (1972) drew a similar conclusion that for isolated
ice floes, the wave-induced drift can be as large as the windgenerated drift and thus should not be neglected. Harms (1987)
performed laboratory measurements on the drift of ice floe models
under regular wave conditions. He obtained an empirical formula
to predict the wave-induced drift of these ice floes. Huang (2007)
studied the variations of the wave-induced surface drift in a wave
flume with time and in space. The effects of side-wall on the drift
velocity were discussed.
n
Corresponding author. Tel.: + 65 6790 5296; fax: + 65 6791 0676.
E-mail address: [email protected] (A.W.K. Law).
0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2010.12.015
In terms of analytical analysis, there are two existing methods to
examine the motion of a floating object under wave action: one is
based on the potential flow theory for large objects and the other is
based on Morison’s equation for small objects. The first method
solves the flow surrounding these large objects which scatter
waves, using the surfaces of the objects as flow boundaries. The
velocity and pressure fields around the objects can be calculated by
the potential theory. As to smaller objects, their disturbance to the
wave field can usually be neglected, and Morison’s equation can
then be used to predict the wave forces on these objects (e.g.
Sorensen, 1978). Both methods had been applied extensively to
compute the wave-induced loadings. However, they have not been
well explored in term of analyzing the time-averaged behavior of
wave-induced drifting motion for small rigid objects.
For small rigid floating objects, Rumer et al. (1979) was probably
the first to use Morison’s equation to investigate the wave-induced
drift. In their approach, the water surface is considered to be an
oscillating slope; the gravity component normal to the slope is
balanced by buoyancy, and the component tangential to the surface
slope induces the movement of the object. Based on the work of
Rumer et al. (1979), Shen and Zhong (2001) obtained analytical
solutions for two special cases where either the added mass
coefficient Cm or drag coefficient Cd vanishes for small amplitude
waves in deep-water conditions. Furthermore, the drift velocity of
different objects with realistic added mass and drag coefficients
were solved numerically. After comparing the results, Shen and
Zhong (2001) concluded that the wave-induced drift of floating
objects will decrease if the added mass coefficient and/or the drag
coefficient increase. Theoretical studies on the topic can also be
found in Marchenko (1999) and Grotmaack and Meylan (2006).
Grotmaack and Meylan (2006) compared the models of Rumer et al.
(1979) with that of Marchenko (1999). They pointed out that
Rumer et al. (1979) incorrectly used the vertical inertia instead of
G. Huang et al. / Ocean Engineering 38 (2011) 712–718
the centripetal force when deriving his equation. Grotmaack and
Meylan (2006) derived a system of equations using Hamilton’s
principle and computed the drift of small objects by a numerical
method. They showed that after a sufficiently long time, the
floating object either surfs with the wave or moves slowly relative
to the wave.
This study investigates experimentally the wave-induced drift
of three-dimensional small rigid floating objects. Different object
shapes were investigated for two submergences in order to provide
a range of wave-induced inertial force and drag force acting on the
objects. The experimental set-up and data analysis procedure are
described in Section 2, and the measurement results are presented
in Section 3. The main conclusions are summarized in Section 4.
2. Description of the experiment
2.1. Experimental setup and test preparation
The measurements were carried out in a wave flume in the
Hydraulics Laboratory, Nanyang Technological University, Singapore. The flume was 45 m long, 1.55 m wide and 1.5 m deep. The
large flume size allowed deep-water waves to be generated (d/L4
1/2). A piston type wave-maker was located at one end of the tank
to generate the desired waves. The wave generation system was
equipped with a DHI Active Wave Absorption Control System
(AWACS) to reduce reflection from the wave paddle. At the other
end of the wave tank, a wave absorbing beach was used to dissipate
the wave energy and reduce the wave reflection.
Four capacitance-type wave probes were mounted on a steel
frame, which was positioned about 6.0 m from the wave paddle. The
wave probes were capable of measuring the water level fluctuations
to the nearest 0.5 mm. The diameter of the probe wire was 0.6 cm,
thus their placement in the water did not cause any significant
modification to the wave field. A motion monitoring system
(Qualisys Track Manager) was installed to capture the trajectory
of the small moving objects. The system consisted of 2 ProReflex
infrared cameras, a laptop computer, and several markers (each is
2 cm in diameter and 5 g in weight) coated with reflective material.
The two cameras were angled at each other to cover an intersecting
span of about 3 m, viewable by both cameras (see Fig. 1). In general,
a minimum of three markers on an object were needed to determine
the motion of the moving object. When the object moved, however,
the rotation of the objects might cause one or two markers to be out
of the view of a camera. Therefore, a group of five markers was
placed on the small objects in a ‘‘X’’ pattern to ensure that at least
three markers were visible at all time during the experiments. The
images of the markers were continuously acquired by the two
cameras, and the signals were processed to give the instantaneous
position of each marker in a calibrated coordinate system. The drift
Infrared camera 1
713
behavior of the small objects can be retrieved from the recorded
object trajectory.
In the experiments, Stokes waves with target periods of 1.0 and
0.9 s were examined. The wave parameters for the drift measurements under deep water conditions are listed in Table 1. The water
depth was fixed at d¼0.8 m, thus the wave lengths were L¼1.56 m
for T¼1.0 s and 1.26 m for T¼0.9 s. The wave height H was varied
from 0.02 to 0.06 m with a 0.01 m interval. The wave steepness ka
(where k is the wave number and a the wave amplitude) thus
ranged from 0.04 to 0.15.
Polyethylene plates, with a density of 0.96 g cm 3 were used to
model the small objects. Two thicknesses, 3.0 and 4.5 cm, were
used to create two different submergences. The models used in the
experiments are shown in Fig. 2. They included the planar shape of
Square, Circle and Ellipse-I (planar aspect ratio of 3:4) and Ellipse-II
(planar aspect ratio of 1:2). All these shapes were cut out precisely
from AutoCAD generated templates. The dimensions of the plates
used in the experiments are listed in Table 2, where Lg is the length
of the longitudinal axis of the polyethylene plates (square, circle or
ellipse), Lt the length of its transverse axis, and D the uniform
thickness of plates. Since it is generally accepted that the effect of
wave diffraction is unimportant when Lg/Lo0.2 and Lt/L o0.2
(Isaacson, 1979), the longitudinal length of the plate Lg was chosen
as 0.2 m in all our experiments.
2.2. Experimental procedures
Before starting an experiment, the desirable water depth d¼0.8
m was first established in the wave flume. This was followed by the
calibration of the two infrared cameras, where they were placed
above the wave flume at a location about 7 m from the wave
generator. The water was confirmed to be sufficiently calm by
inspecting the motion of the markers for a short period of time. Any
discernible movement of the markers would suggest the presence
of a residual current. Typically a lapse of at least 15 min between
Table 1
Wave parameters in the experiments.
Expt. series
H (m)
T (s)
d (m)
L (m)
ka
d/L
DA
DB
DC
DD
DE
DF
DG
DH
DI
DJ
0.02
0.03
0.04
0.05
0.06
0.02
0.03
0.04
0.05
0.06
1.0
1.0
1.0
1.0
1.0
0.9
0.9
0.9
0.9
0.9
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
0.80
1.56
1.56
1.56
1.56
1.56
1.26
1.26
1.26
1.26
1.26
0.04
0.06
0.08
0.10
0.12
0.05
0.075
0.10
0.125
0.15
0.53
0.53
0.53
0.53
0.53
0.65
0.65
0.65
0.65
0.65
Infrared camera 2
Direction of wave
propagation
Markers
Small object
~3.0 m
Fig. 1. Schematic layout of the experimental set-up.
Wave flume
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G. Huang et al. / Ocean Engineering 38 (2011) 712–718
y
y
x
Lt = 200
Lt = 200
Lg = 200
x
L g = 200
y
y
Lt = 150
x
x
Lt = 100
Lg = 200
Lg = 200
Wave propagation
Fig. 2. Shapes of plates used in the experiments: (a) square, (b) circle, (c) Ellipse-I and (d) Ellipse-II (dimension in mm).
Table 2
Dimensions of the objects used in the experiments.
Shape series
Shape codes
Lg (mm)
Lt (mm)
D (mm)
Square
Square1
Square1a
200
200
200
200
45
30
Circle
Circle1
Circle1a
200
200
200
200
45
30
Ellipse-I
Ellipse1
Ellipse1a
200
200
150
150
45
30
Ellipse-II
Ellipse2
Ellipse2a
200
200
100
100
45
30
two subsequent runs was required. After it was confirmed that
there was no residual current, the wave generator was activated
and data recording began. During the tests, the infrared cameras
captured the images of the reflective markers at a frequency of
50 Hz. These images were processed to produce the instantaneous
displacement of the object, which were then used to reveal the drift
velocity.
Before each test, the polyethylene plate was first placed on the
still water surface with its longitudinal axis parallel to the direction
of wave propagation, which was assessed by bare-eye observations
with the sidewall of the wave flume as the reference line (a small
tolerance was allowed as it was difficult to make the longitudinal
axis strictly parallel to the wave direction). After the waves were
generated, the polyethylene plate began to drift along the wave
flume. Fig. 3 shows several snapshots of the drift of the polyethylene plate, Ellipse-II (with Lg ¼200 mm, Lt ¼100 mm and
D ¼30 mm), by water waves with a steepness of ka ¼0.08. Observations showed that the longitudinal axis of the Ellipse-II was
nearly parallel to the wave propagating direction at all time; the
same had been observed for plates of square shapes as well.
Fig. 4 shows the measurements of the horizontal displacement
of Square1 under the Expt. series of DG. In the absence of wave
reflection, three stages can be identified: (i) from 0 to 5 s, the water
was still; (ii) from 5 to 20 s, the object started moving with the
waves with a significant acceleration, but a steady state had yet to
be established; and (iii) from 20 s onward, a quasi-steady state with
an approximately constant drift velocity had been established. This
implies that the drift velocity is not a function of the initial position,
which was also pointed out by Grotmaack and Meylan (2006).
2.3. Typical movement of a polyethylene plate
Possible effects of wave reflection from the wave absorbent
beach were avoided for all runs by limiting the test duration. In the
experiments, the wave phase velocity was 1.56 m/s for T¼ 1.0 s and
1.40 m/s for T¼0.9 s. The distance from the test span to the end of
wave flume was approximate 35 m. Therefore, it would take about
45 s for a wave train to propagate a round trip between the test
section and the wave absorbent beach. The experimental duration
was typically about 80 s for all runs to monitor the motion behavior
of the small objects. As the flume was wider than the one used by
Huang (2007), the effects of secondary flow on the drift was not
significant within this duration. In the drift velocity determination,
the data during 20–45 s were used to avoid the reflection effect.
2.4. Determination of the drift velocity
The constant drift velocity in the quasi-steady state can be
computed by two approaches. One is to obtain the instantaneous
mean velocity within one wave period based on an up-crossing
method used in analyzing irregular waves. In this method, the
period-averaged mean drift velocity is a function of time, and can
be calculated by dividing the horizontal displacement between two
neighboring peaks by the wave period. An example is shown in
Fig. 5, which corresponds to the trajectory of the object shown in
Fig. 4. Choosing a quasi-steady time interval, from 25 to 45 s, the
period-averaged mean velocity calculated from the figure is
G. Huang et al. / Ocean Engineering 38 (2011) 712–718
715
t’ = 0
t’ = 10
t’ = 20
t’ = 30
t’ = 40
t’ = 50
Fig. 3. Typical movement of Ellipse2a tracked by a fixed video camera. Waves propagate from left to right. t0 ¼ t/T is the non-dimensional time.
0.8
0.025
0.6
0.02
0.2
0
0
-0.2
10
20
30
40
Time (s)
50
60
70
80
-0.4
Crest drift (m/s)
x trace (m)
0.4
0.015
0.01
0.005
0
0
-0.6
Fig. 4. Time history of horizontal movement recorded by the motion monitoring system.
0.019 m/s. The second method is to calculate the mean drift
velocity in the quasi-steady stage by determining the slope of a
best-fitting linear trend line from the horizontal trajectory. An
example is shown in Fig. 6. After line-fitting the trajectory from 25
to 45 s, a drift velocity of 0.020 m/s is obtained. The drift velocities
determined by both approaches are thus almost identical. We shall
use the first approach to calculate the drift velocity in this study.
10
20
-0.005
30
40
Time (s)
50
60
70
80
Fig. 5. Drift velocity determined by analyzing the wave crests.
3. Results and discussion
Fig. 7 shows the celerity-normalized drift velocity, ud/c where c is
the wave celerity, as a function of the wave steepness, ka. Fig. 7(a) is
for the circular and elliptical shapes, and Fig. 7(b) is for the circular
and square shapes. Also shown in Fig. 7 is the theoretical Stokes drift.
It can be observed that for these small objects, the measured drift
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G. Huang et al. / Ocean Engineering 38 (2011) 712–718
velocities were all significantly larger than Stokes drift. In addition,
the circular objects drifted slightly faster than either the square or
elliptical objects. The Ellipse-I objects, with an aspect ratio of 3:4
which is close to a circle in terms of the planar shape, drifted in a way
0.2
0.1
Time (s)
x trace (m)
0
20
25
30
35
40
45
-0.1
-0.2
50
similar to the square objects, but at a slightly larger velocity
compared to the Ellipse-II objects which had an aspect ratio of
1:2. The celerity-normalized drift velocity showed a quadratic
dependence on wave steepness ka, which is similar to the Stokes
drift. Also, the objects with 45 mm submergence (denoted by suffix
a) drifted faster than those with 30 mm submergence regardless of
the object shape. The drift behavior can be seen more clearly in Fig. 8,
where the Stokes-normalized drift velocity ud/us is plotted against
the wave steepness. The normalized drift decreased when the wave
steepness increased, and approached an asymptotic constant value
for large wave steepness.
Shen and Zhong (2001) and Grotmaack and Meylan (2006)
proposed the following equation for the drift motion of a small
floating object as follows:
y = 0.0196x - 0.7771
R2 = 0.9079
-0.3
-0.4
Fig. 6. Drift velocity determined by the linear trend line.
ð1 þ Cm Þ
dv
@Z rs Aw Cd
¼ g
þ
9Vv9ðVvÞ
dt
@x
rAb D
ð1Þ
where v is the instantaneous velocity of the small object, V the wave
orbital velocity, g the gravity acceleration, Z the water surface
displacement, rs the density of the small object, r the water
density, Aw the wetted area, and Ab the bottom surface area of the
object. In the following, we performed numerical simulations to
Fig. 7. Effect of wave steepness on celerity-normalized drift velocity for different shapes of polyethylene plates: (a) circle and Ellipse-I&II, and (b) circle and square.
G. Huang et al. / Ocean Engineering 38 (2011) 712–718
717
Fig. 8. Effect of wave steepness on Stokes-normalized drift velocity for different shapes of polyethylene plates: (a) circle and Ellipse-I&II, and (b) circle and square.
Table 3
Numerical results of drift velocity based on Eq. (1).
D/L
0.03
0.03
0.03
0.03
0.03
Aw/Ab
1.9
1.9
1.9
1.9
1.9
ka
0.04
0.06
0.08
0.10
0.12
us/c
0.0016
0.0036
0.0064
0.0100
0.0144
ud/c (calculated)
Cd ¼0 Cm ¼0
Cd ¼0.5 Cm ¼ 0
Cd ¼ 0.5 Cm ¼ 0.4
0.00198
0.0045
0.0079
0.012
0.017
0.00091
0.0020
0.0035
0.0054
0.0079
0.00037
0.00083
0.0015
0.0023
0.0034
calculate the drift velocities of the small objects in the present
study with various added mass and drag coefficients based on
Eq. (1). The small objects were all released at the wave crest. Both
the surface displacement and the wave orbital velocity were given
by the linear wave theory.
Sample numerical results are given in Table 3. It can be
concluded that: (i) the drift velocity is approximately equal to
the Stokes drift if both the added mass coefficient Cm and the drag
coefficient Cd vanish; (ii) the drift will be reduced when the added
mass coefficient and/or drag coefficient increase. It is not difficult to
infer that the predicted drift velocity for finite-size objects based on
Eq. (1) would always be smaller than the Stokes drift as both the
added mass coefficient and drag coefficient are not zero in reality.
However, our experimental data showed an opposite trend from the
predictions based on the existing theory. Hence, further research is
necessary to clarify the discrepancy.
4. Conclusions
In this study, the wave-induced drift velocity of small floating,
thin objects with various shapes and two different submergences
were investigated experimentally under deep water wave conditions. Our results show that the drift velocity of the floating object
would increase from rest to reach a quasi-steady constant magnitude within a short time. The constant drift velocity was found to
increase with the wave steepness at an approximately quadratic
rate which is similar to the Stokes drift, but the magnitude was
higher than the Stokes drift in all cases. These experimental results
differ from the predictions by the existing theory. We are currently
conducting a theoretical analysis to clarify the discrepancy.
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G. Huang et al. / Ocean Engineering 38 (2011) 712–718
Acknowledgements
This work was supported by the Ministry of Education, Singapore, through the AcRF Tier 2 Grant no. MOE2008-T2–070. The
authors would like to thank the anonymous reviewers for their
valuable comments, which improved the quality of this manuscript.
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