Experimental Studies on Sloshing Mitigation Using Dual Perforated Floating Plates in A Rectangular Tank

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 U-tube mode, which can arise from the sloshing reduction to some extent.


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 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.

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 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/m 3 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.

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: 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 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.

Experimental results and discussions
The sloshing eigenfrequency for the clean tank can be obtained as (Wu, 2007): where is the n-th order sloshing eigenfrequency and is the n-th order wave number. The gravitational acceleration is and the water depth is . 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  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: where is the percentage of mitigation in sloshing response. The maximum wave run-up inside the clean tank is and the maximum wave run-up inside the tank with dual perforated floating plates is . 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   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 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- YU Yue-min China Ocean Eng., 2021, Vol. 35, No. 2, P. 301-307 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.

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. 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.