China Ocean Eng., 2021, Vol. 35, No. 2, P. 301–307
DOI: https://doi.org/10.1007/s13344-021-0027-y, ISSN 0890-5487
http://www.chinaoceanengin.cn/ E-mail: coe@nhri.cn
Experimental Studies on Sloshing Mitigation Using Dual Perforated Floating Plates in A Rectangular Tank
YU Yue-mina, b, *
a College of Civil Engineering and Architecture, Hainan University, Haikou 570228, China
b State Key Laboratory of Marine Resource Utilization in South China Sea, Hainan University, Haikou 570228,
China
Received July 25, 2020; revised December 19, 2020; accepted January 28, 2021
©2021 Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature
Abstract
The performance of dual perforated floating plates in a rectangular tank is investigated based on the model tests
under different external excitations for different filling rates. It is found that dual perforated floating plates in the tank
can remarkably mitigate violent resonant sloshing responses compared with the clean tank, especially when the
external excitation frequency is in the vicinity of the first-order resonant frequency. Next, the parametric studies
based on different filling rates and external excitation amplitudes are performed for the first-order resonant
frequencies. The presence of dual perforated floating plates seldom shifts the sloshing natural frequencies. Further,
dual perforated floating plates change the sloshing modes from the standing-wave mode in the clean tank to the Utube mode, which can arise from the sloshing reduction to some extent.
Key words: suppressing sloshing, experiment, dual perforated floating plates, the lowest natural frequency, sloshing
mode
Citation: Yu, Y. M., 2021. Experimental studies on sloshing mitigation using dual perforated floating plates in a rectangular tank. China Ocean
Eng., 35(2): 301–307, doi: https://doi.org/10.1007/s13344-021-0027-y
1 Introduction
With growing demand for the deep-sea natural gas,
Floating Liquefied Natural Gas (FLNG) units as a new type
of ocean structure has been paid increasing attention to.
During offloading and exploration operations, partially
filling rates in the tank are inevitable, and the liquid sloshing will occur under wave excitations. Liquid sloshing is a
physical phenomenon in a tank for partially filling rates under external excitations. When the external excitation frequency is close to the lowest resonant frequency of the free
surface, tremendous impact loads due to sloshing may lead
to the structural failure or even damage.
Owing to significant wave energy dissipation, the perforated structures are commonly utilized in suppressing
sloshing problem. The sloshing flow through the holes is effectively attenuated but this increase is partially offset by
the decrease in the baffle weight (Dodge, 2000). In recent
decades, a great deal of efforts which refer to the liquid
sloshing in tanks with the horizontal or vertical perforated
plates have been made analytically, numerically and experimentally by many researchers. Jin et al. (2014) explored experimentally the effects of the horizontal plate with perfor-
ated slots on liquid sloshing in a rectangular tank. The solidity ratio can affect the wave run-up, while the submergence
depth of the inner plate changed the resonant frequency.
Cho et al. (2017) investigated the sloshing problem in a
swaying rectangular tank with the perforated horizontal
baffles for centered and sided locations by means of the analytic solution, the numerical approach and the model test respectively. Based on the boundary element method, Zang et
al. (2019b) studied the effects of the perforated baffles on
sloshing mitigation in the tanks with axisymmetric geometries under lateral excitation. The lowest natural frequencies
decreased as permeability of the perforated baffle decreased
for both the top and bottom-mounted vertical perforated
baffle. Furthermore, Zang et al. (2019a) used the boundary
element method to predict sloshing characteristics in a twodimensional rectangular tank with various perforated
baffles, which were fixed at the tank. As the solidity ratios
of the perforated vertical baffle increased, the rectangular
tank was gradually divided into two relatively independent
compartments, and thus the lowest natural frequencies became smaller. Moreover, the change of baffle length cannot
significantly alter the smallest natural frequencies. On the
Foundation item: This study is financially supported by Hainan Provincial Natural Science Foundation of China (Grant No. 519MS026) and Scientific
Research Foundation of Hainan University (Grant No. KYQD(ZR)1878).
*Corresponding author. E-mail: yuyuemin@hainanu.edu.cn
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YU Yue-min China Ocean Eng., 2021, Vol. 35, No. 2, P. 301–307
basis of analytical, numerical and experimental approaches,
Poguluri and Cho (2019) investigated the performance of a
vertical slotted screen fixed in the center of a rectangular
tank on suppressing sloshing. The lower the submergence
depths of the vertical slotted screen is, the higher the wave
run-up along the side-wall will be. Based on Moving
Particle Semi-implicit method, Bellezi et al. (2019) performed the numerical investigation to determine the optimal geometry of perforated vertical bulkheads in order to mitigate sloshing in a rectangular tank under pitching excitations for different filling rates. As the submerged solidity ratios of the vertical bulkhead decreased, the resonant frequencies increased monotonically. Xue et al. (2013) studied
experimentally and numerically liquid sloshing in a rectangular tank with an orifice bulkhead. The orifice can reduce
the surface area, so that it mitigates the impact pressure acted on the vertical bulkhead.
Many other analytical, experimental and numerical studies have also been performed with regard to the sloshing reduction by dual perforated plates. Nasar and Sannasiraj
(2019) investigated experimentally the effects of dual perforated plates on sloshing mitigation in a tank under a sequence of excitation frequencies for different solidity ratios.
The tank was mounted inside the barge, which was subjected to regular beam waves. On the basis of the scaled
boundary finite element method, Ye et al. (2018) investigated sloshing characteristics in a cylindrical tank with a
coaxial dual perforated structure. A modal expansion technique was employed by Love and Tait (2010), which accounted for the effects of higher sloshing modes in the
tuned liquid damper tanks with one, two or three vertical
screens. Wave elevations, sloshing forces exerted on the
side walls and the vertical screens were compared between
the modelling results and the experimental data. Next, to determine the suitability of different fluid models for sloshing
responses inside a tuned liquid damper with dual perforated
vertical screens, Love and Tait (2013) considered three fluid models, which were the shallow water wave model, the
small depth and the intermediate depth multimodal models
respectively. It indicated the suitability of three fluid models related to the Ursell parameter. Cho and Kim (2016) investigated the effects of dual perforated vertical plates on
the sloshing mitigation in a rectangular tank both analytically and experimentally, which were fixed to the tank symmetrically with respect to the vertical-axis. There was a
trade-off between wave elevations and forces on plates. Yu
et al. (2019) analyzed experimentally the effects of multiple
vertical slotted screens on sloshing reduction in a rectangular tank under a series of excitation frequencies including
the higher natural frequencies. The peaks of the wave elevation decreased as the number of screens increased for a wide
range of excitation frequencies. Moreover, the results of the
wavelet transform showed that the amplitudes for different
frequency components were significantly mitigated in the
tank with vertical slotted screens.
The above-mentioned investigation clearly indicates that
there exist very limited literatures on suppressing sloshing
in the tanks with dual perforated floating plates. In order to
systematically study the effects of perforated floating plates
on sloshing mitigation in a rectangular tank, a sequence of
model tests is carried out on the biaxial shake table. The organization of this paper is presented as follows. The details
of the experimental setup and test conditions are shown in
Section 2. In Section 3, experimental results and discussion
will be illustrated with the suppressing effects, parametric
study and sloshing modes. Finally, the main conclusions of
this work are stated in Section 4.
2 Model tests
2.1 Experimental setup
The experimental test rig is set up to identify the range
of parameters for conducting the sloshing dynamics studies
on storage tank. The biaxial shake table facility available at
Structure Laboratory of Hainan University is deployed for
mounting the partially filled storage tank. The shake table is
capable of supplying a continuous velocity of 1 m/s with a
peak input amplitude of 500 mm, over an operating frequency range from 0 Hz to 50 Hz. It is 3 m×3 m in size and
Fig. 1. MTS biaxial shake table experimental facilities showing (a) bare table and (b) overall view of the partially filled rectangular tank with dual perforated floating plates mounted on the shake table.
YU Yue-min China Ocean Eng., 2021, Vol. 35, No. 2, P. 301–307
can accommodate structures weighing up to 10 tons. This is
a unique facility with an excitation capability of ±1.1 g in
horizontal direction. Fig. 1a shows the experimental setup
of the shake table loaded with the instrumented tank. The
built-in control laws enable imposition of harmonic excitations under displacement or acceleration control modes.
The inner dimensions of the rectangular tank are 90
cm×30 cm×91 cm (length L×width W×height H), and it is
made of transparent plexiglass with 2 cm thickness. The
perforated floating plates were made of the paulownia wood
with the dry density of about 230−300 kg/m3 in order to ensure the draft and they were waterproofed by the varnish.
The layout consists of a cluster of circular holes distributed
along 3 rows and 9 columns, and the spacing and diameter
of equally distributed circular holes are both 1.5 cm. The
diameter, quantity and distribution of the circle holes are dependent on the sizes of the floating plate and the inner dimensions of the tank. In addition, the submerged solid area
of the floating plate is minimized as much as possible to ensure that there is enough flux through the circle holes. Furthermore, dual perforated floating plates (with the size of
2.7 cm×21 cm) are installed at L/6 and 5L/6 of the tank by
four two-fold tracks, which can ensure the smooth motion
of perforated floating plates along the two-fold tracks. The
capacitance-type wave probe running at a sampling frequency of 1 kHz is calibrated and 1.7 cm apart from the side
wall, which is used for recording the wave elevations. During the experiment, the flow visuals are captured through
the high-definition video camera. The video camera, focused to minimize perspective errors in the images, is
placed in front of the tank, as shown in Fig. 1b.
2.2 Test conditions
To understand the dynamics of liquid sloshing in a rectangular tank, a systematic experimental program was designed and implemented with the help of a shake table for
various conditions. The dynamics of liquid sloshing under
the influence of resonant excitations is studied. The forced
surge excitation of the tank is taken as a horizontal harmonic displacement for the base motion as:
s = Asin (2π f t) ,
(1)
where, A is the external excitation amplitude, f refers to the
fundamental sloshing frequency, and t is time.
A series of test cases have been considered with different vibration amplitudes and filling rates. For each case,
periodic vibration test of the tank is conducted and times
series of the free surface elevations are recorded by the
wave probes. Table 1 summarizes the principal experimental conditions in the model tests. Excitation frequencies used
in the model tests are based on the analytical sloshing natural frequency of the clean tank as well as additional excitation frequencies in the vicinity of the smallest natural frequency.
Furthermore, experiments on the test rig with clean tank
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Table 1 Experimental conditions
Excitation amplitude
A(m)
0.002
0.004
0.006
Filling rate R
(%)
30
50
70
30
50
70
30
50
70
Excitation frequency f (Hz)
0.6−1.1a, 0.76, 0.77, 0.78, 0.79
0.6−1.1a, 0.84, 0.86, 0.87, 0.88
0.6−1.1a, 0.89, 0.91
0.6−1.1a, 0.76, 0.77, 0.78, 0.79
0.6−1.1a, 0.84, 0.86, 0.87, 0.88
0.6−1.1a, 0.87, 0.88, 0.89, 0.91
0.6−1.1a, 0.76, 0.77, 0.78, 0.79
0.6−1.1a, 0.83, 0.84, 0.86, 0.87
0.6−1.1a, 0.86, 0.87, 0.88, 0.89,
0.91
aThe increment is 0.05 Hz.
are also carried out by removing the perforated floating
plates. The free surface elevations and the natural frequency can also be identified from the wave probe measurements, although in this case they can be analytically calculated.
3 Experimental results and discussions
The sloshing eigenfrequency for the clean tank can be
obtained as (Wu, 2007):
1 √
fn =
gkn tanh(kn d),
(2)
2π
where fn is the n-th order sloshing eigenfrequency and kn is
the n-th order wave number. The gravitational acceleration
is g and the water depth is d .
The most effective frequencies, in response to the external excitation, correspond to the lowest natural frequencies, because of the fact that they always include most of the
total energy of liquid sloshing (Gedikli and Ergüven, 1999).
A series of test cases have been conducted with different excitation frequencies every other 0.05 Hz from 0.6 Hz to
1.1 Hz in order to obtain the response-frequency, which correspond to maximal response-amplitude. Then the maximal
wave runup along the side wall is observed and measured to
estimate the lowest natural frequency after the excitation
frequency is densified every other 0.01 Hz in the vicinity of
the response-frequency. Table 2 presents the comparison of
the first-order natural frequencies obtained from analytical
and experimental results in the clean tank for different test
cases. The experimental result for the clean tank is always
Table 2 Sloshing eigenfrequencies
Filling rate
R (%)
30
50
70
The first-order natural frequency f1 (Hz)
Excitation
amplitude A(m) Analytical results
Experimental results
0.002
0.80
0.80
0.004
0.80
0.80
0.006
0.80
0.79
0.002
0.89
0.87
0.004
0.89
0.86
0.006
0.89
0.85
0.002
0.92
0.90
0.004
0.92
0.89
0.006
0.92
0.88
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YU Yue-min China Ocean Eng., 2021, Vol. 35, No. 2, P. 301–307
slightly smaller than the analytical counterpart due to some
discrepancies between analytical and experimental cases.
Moreover, the solid floating plates can change the first-order resonant frequency to smaller one (Yu et al., 2017),
while this phenomenon is not evident in the cases of dual
perforated floating plates.
3.1 Effects on sloshing mitigation for the tank with dual
perforated floating plates
To explore the effects of dual perforated floating plates
on sloshing mitigation, both the results of the clean tank and
those of the tank with dual perforated floating plates for
three different filling rates (R=30%, 50%, 70%) are plotted
in Figs. 2−4. The vertical-axis is the maximum wave run-up
measured 1.7 cm away from the side wall. Furthermore, the
percentage of sloshing reduction is calculated as follows:
P=
η0max − η1max
η0max
× 100%,
(3)
where P is the percentage of mitigation in sloshing response. The maximum wave run-up inside the clean tank is
η0max and the maximum wave run-up inside the tank with
dual perforated floating plates is η1max .
In Fig. 2, the maximum wave run-up is shown for comparison between the clean tank and the tank with dual perforated floating plates for the 30% filling rate. The horizontal-axis is the external excitation frequency (ranging from
0.6 Hz to 1.1 Hz). The blue circles refer to the results of the
clean tank and the red dots represent the results of the tank
with dual perforated floating plates. From Fig. 2a, for the
2 mm excitation amplitude, the percentage of mitigation is
as high as nearly 86% in the first-order natural frequency.
As shown in Fig. 2b, for the excitation amplitude of 4 mm,
the percentage of mitigation is in the order of 82%. When
the excitation amplitude is 6 mm, as shown in Fig. 2c, the
percentage of mitigation is about 82%. The plates and their
opening structure increase viscous damping (Saghi et al.,
2020) for the low filling rate 30% and part of the sloshing
Fig. 2. Frequency response plots of maximum run-up along the side wall for 30% filling rate.
Fig. 3. Frequency response plots of maximum run-up along the side wall for 50% filling rate.
Fig. 4. Frequency response plots of maximum run-up along the side wall for 70% filling rate.
YU Yue-min China Ocean Eng., 2021, Vol. 35, No. 2, P. 301–307
energy is converted into the mechanical energy of dual perforated floating plates, which have positive effects on suppressing sloshing.
As the filling rate increases from R=30% to R=50%,
Fig. 3 shows the maximum wave run-up as functions of the
external excitation frequency (ranging from 0.6 Hz to
1.1 Hz). As shown in Fig. 3a, for the 2 mm excitation amplitude, the percentage of mitigation is in the order of 83% in
the first-order natural frequency. From Fig. 3b, for the 4 mm
excitation amplitude, the percentage of mitigation is about
84%. When the excitation amplitude is 6 mm, as shown in
Fig. 3c, the percentage of mitigation is observed to be about
80%. One part of fluid can move through the opening structure and another part flows through the lower part of dual
floating plates for the 50% filling rate. Partial sloshing energy is converted into the mechanical energy of dual perforated floating plates.
Fig. 4 shows the maximum wave run-up as the function
of the external excitation frequency (ranging from 0.6Hz to
1.1 Hz) for the 70% filling rate. From Fig. 4a, for the 2 mm
excitation amplitude, the percentage of mitigation is as high
as nearly 85% in the first-order natural frequency. As shown
in Fig. 4b, for the 4 mm excitation amplitude, the percentage of mitigation is in the order of 74%. When the excitation amplitude is 6 mm, as shown in Fig. 4c, the percentage
of mitigation is about 67%. Most of fluid moves through the
lower part of the floating plate for the 70% filling rate. Energy conversion plays a critical role in sloshing mitigation.
Therefore, it is concluded that dual perforated floating
plates in the rectangular tank can effectively suppress the
305
sloshing runup for three different filling rates under three
different excitation amplitudes, especially when the external excitation frequency is close to the first-order natural frequency of the free surface.
3.2 Parametric study of the tank with dual perforated floating plates
The solid floating plates appreciably shift the lowest natural frequencies to smaller frequency region, as discussed in
Yu et al. (2017) and Arai et al. (2013). The presence of perforated floating plates little shifts the first-order resonant
frequencies. As compared with the solid floating plate although the perforated floating plate can provide higher viscous damping, the perforated plate is lighter in weight and
the fluid can move through the opening structure.
Fig. 5 shows the first-order natural frequency as functions of the filling rate R. Two different cases have been
considered, which are the clean tank and the tank with dual
perforated floating plates. The smallest natural frequency increases as the filling rate R increases, although the rate of
increase is decreasing. As the filling rate R increases from
30% to 70%, the sloshing tank switches from an intermedium-water condition to a deep-water condition. It can be seen
that the lowest resonant frequencies are all insensitive to the
presence of the perforated floating plates for different filling
rates except for the filling rate 30%.
Fig. 6 shows the lowest natural frequency as functions
of the excitation amplitude A, and the results for the clean
tank and the tank with dual perforated floating plates are put
together. It can be demonstrated that when the filling rate R
is fixed, the lowest natural frequency decreases as the excit-
Fig. 5. First-order natural frequency as functions of the filling rate R.
Fig. 6. First-order natural frequency as functions of the excitation amplitude A.
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ation amplitude A increases, and the response is nearly linear. In addition, the dual perforated floating plates shift the
first-order natural frequencies to smaller frequency region
for the filling rate 30% under three different excitation amplitudes. This may be because viscous damping plays a major role in resonant frequencies for the low filling rate and
the opening structure can provide higher viscous damping.
By comparing Fig. 6 with Fig. 5, it is observed that the
change of smallest resonant frequency by the excitation
amplitude A is less significant than that by the filling rate R.
Therefore, for sloshing tank with dual perforated floating
plates, the filling rate R is still the main frequency influence
parameter while the excitation amplitude A acts as the secondary influence parameter.
3.3 Sloshing modal properties of the tank with dual perforated floating plates
Kobayashi et al. (2006) and Kobayashi and Koyama
(2010) studied the sloshing characteristics in a rectangular
tank with a bulkhead, which divided the free surface into
two sections. The sloshing mode is separated into the separated-compartment mode and the U-tube mode.
Fig. 7 shows the snapshots of wave surface profiles captured in the model tests for R=70% and A=6 mm under resonant frequencies. From Fig. 7a, it can be observed that the
liquid motion in the clean tank is so violent that the free surface can reach the tank ceiling at t=2T/4 and t=T under the
smallest natural frequency for the 70% filling rate. As
shown in Fig. 7a, the standing-wave mode can be observed
clearly. Moreover, the effects of dual perforated floating
plates on sloshing mitigation can be visually identified, as
can be seen from Fig. 7b. It is also noticed that dual perforated floating plates change the sloshing modes from the
standing-wave mode in the clean tank to the U-tube mode.
The free surface in the rectangular tank is divided into three
sub-regions by dual perforated floating plates. Most fluid
runs back and forth through the lower part of dual perforated floating plates from one sub-region to another. A lot of
sloshing energy is dissipated in this process, which is
mainly converted into the mechanical energy of dual perforated floating plates. Fig. 8 shows time history of wave elevations 1.7 cm apart from the side wall in the tank with dual
perforated floating plates for R=70% and A=6 mm during
resonant sloshing. The process is mainly divided into two
stages, one of which is the transient stage and the other is
the stable stage.
4 Conclusions
In this study, a sequence of experiments for sloshing responses in a rectangular tank with dual perforated floating
plates is conducted for three different filling rates under different external excitations. To assess the effects of dual perforated floating plates on sloshing attenuation, the comparisons are made for both the results of the clean tank and those
of the tank with dual perforated floating plates. The conclusions from the present experimental studies are as follows.
(1) Compared with the clean tank case, for the filling
rate of 30%, the percentage of sloshing mitigation is
between 82% and 86% in the smallest natural frequency under three different excitation amplitudes in the tank with
dual perforated floating plates. For the 50% filling rate, the
percentage of mitigation is about 80%−84%. When the
filling rate increases to 70%, the percentage of mitigation is
between 67% and 85%.
(2) The first-order resonant frequency increases as the
filling rate increases, although the rate of increase is decreasing. The smallest natural frequencies are all insensitive to the presence of the dual perforated floating plates.
Fig. 7. Experimental wave elevation for R=70% and A=6 mm at different time instants during resonant sloshing.
YU Yue-min China Ocean Eng., 2021, Vol. 35, No. 2, P. 301–307
Fig. 8. Time history of wave elevations 1.7 cm away from the side wall in
the tank with dual perforated floating plates for R=70% and A=6 mm during resonant sloshing.
Furthermore, the lowest resonant frequency decreases as the
excitation amplitude increases, and the response is nearly
linear.
(3) The standing-wave mode can be observed clearly in
the clean tank, while dual perforated floating plates change
the sloshing modes from the standing-wave mode to the Utube mode. Partial sloshing energy is converted into the
mechanical energy of dual perforated floating plates and the
opening structure increases viscous damping as well.
Acknowledgments
The author would like to thank Mr. ZHANG Cong, Mr.
LIN Yu-Qi and Mr. TAO Jun from Hainan University, for
their assistance during the model tests.
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