Experimental Study on A Sloshing Mitigation Concept Using Floating Layers of Solid Foam Elements

A sloshing mitigation concept taking advantage of floating layers of solid foam elements is proposed in the present study. Physical experiments are carried out in a liquid tank to investigate the hydrodynamic mechanism of this concept. Effects of the foam-layer thickness, excitation amplitude, and excitation frequency on the sloshing properties are analyzed in detail. It is found that the floating layers of solid foam elements do not evidently affect the fundamental natural sloshing frequency of the liquid tank evidently among the considered cases. At the resonant condition, the maximum wave height and dynamic pressure are greatly reduced as the foam-layer thickness increases. Higher-order pressure components on the tank side gradually vanish with the increase of the foam-layer thickness. Cases with different excitation amplitudes are also analyzed. The phenomenon is observed when the wave breaking in the tank can be suppressed by solid foam elements.


Introduction
A great proportion of the natural gas reserve is located in the deep to ultra-deep offshore at present. The concept of the floating liquefied natural gas (FLNG) platform has been considered as a promising technology for the offshore gas exploitation. During the operation, the liquefied natural gas (LNG) is stored in several huge tanks of the FLNG platform. For around 20-25 years of offshore service, the FLNG platform can experience any complex sea state with any fill in tanks. The liquid sloshing in tanks can be easily excited, which can endanger the safe operation of the platform structure in certain situations. Thus, it is of great significance to design the effective sloshing mitigation techniques for the structural safety of FLNG tanks.
For a general liquid tank, various sloshing mitigation techniques have been investigated, among which the bafflebased techniques (including baffles, screens, bulkheads, and so on) are mostly considered. Typical studies in the recent decade are briefly summarized as follows. Faltinsen et al. (2011) studied the steady-state liquid sloshing in a rectangular tank with a slat-type screen in the middle. Akyildiz (2012) presented effects of baffle height on free surface el-evation and pressure in a 2D rectangular tank by means of volume of fluid technique. Jung et al. (2012) studied effects of the vertical baffle installed on the tank bottom on the liquid sloshing and discussed the vortex effects from the baffle tip. Song et al. (2012) studied the hydrodynamic characteristic of horizontal and vertical baffles in a 2D rectangular tank. Wang et al. (2013) analytically studied the sloshing response of liquid in a rigid cylindrical container with multiple annual rigid baffles. Hasheminejad et al. (2014) studied the transient sloshing in a horizontal circular container with three common baffle configurations. Jin et al. (2014) conducted experiments on a rectangular liquid tank with an inner submerged horizontal perforated plate. Lu et al. (2015) simulated the sloshing in a rectangular tank with two horizontal baffles using both viscous and potential-flow numerical methods. Wei et al. (2015) considered anti-sloshing effects of the vertical screen structure on the bottom of the tank. Goudarzi and Danesh (2016) simulated the liquid response in a rectangular tank with bottom baffles subjected to the seismic excitations, using the commercial software ANSYS CFX. Wang et al. (2016) studied effects of Tshaped baffles on liquid sloshing in the horizontal elliptical tanks using a semi-analytical scaled boundary finite-element method (SBFEM). Cho and Kim (2016) investigated dual vertical porous baffles for obvious sloshing suppression. Kumar and Sinhamahapatra (2016) analytically studied dynamic characteristics of fluid within a tank with the vertical perforated baffle and screens of different solidity ratios. Hwang et al. (2016) studied and demonstrated the effects of elastic baffles on the sloshing phenomena through experimental and numerical methods. Xue et al. (2017) carried out the experimental study on the hydrodynamic characteristics of different anti-sloshing baffles in a square tank. Sanapala et al. (2018) investigated the parametric liquid sloshing in a horizontally baffled rectangular container. Kim et al. (2018) numerically studied the liquid tank fitted with horizontal baffles on two sides to suppress the impact pressure of sloshing by using an air-trapping mechanism. Khayyer et al. (2018) developed an enhanced fully-Lagrangian meshfree computational method (named enhanced ISPH-SPH solver) for incompressible fluid-elastic structure interactions, with which sloshing flows in tanks with elastic baffles were simulated. Falahaty et al. (2018) utilized a stress point integration in particle-based modeling of fluid-structure interactions, where violent sloshing in a tank with a hanging elastic baffle was modelled. As comprehensively reviewed by Gotoh and Khayyer (2018), readers can find more sloshing modelling cases using particle methods.
The above baffle-based techniques still require further verification for the special case of LNG tanks. Because the internal surface of an LNG tank is fully covered by thin invar membranes to prevent gas leakage at an extremely low temperature, installing baffles through the membrane surface may affect mechanical properties of the membrane and bring safety risks. Consequently, alternative sloshing mitigation techniques have kept emerging in the recent decade. For example, Hernández and Santamarina (2012) studied effects of dynamic baffles in the liquid tank on the real-time adjustment of the hydrodynamic characteristics and sloshing mitigation. Koh et al. (2013) utilized the improved Consistent Particle Method (CPM) to probe the effect of a constraint floating baffle (CFB) in sloshing mitigation. Zhang (2015) proposed a design of LNG ships with various wedged tanks from an overall perspective, aiming to reduce the resultant sloshing dynamics from all tanks acting on the ship. Arai et al. (2014) and Yu et al. (2017) installed two floating plates in the tank to dynamically change the natural sloshing frequencies. Hosseini et al. (2017) used suspended annular baffles to reduce the floating roof motion and presented the efficiency of this technique in experiments. At present, innovative designs are still encouraged to further lower the cost and construction complexity of the sloshing mitigation system.
The present study is inspired by Sauret et al. (2015) which experimentally studied the effect on sloshing of 'liquid bubble foams' on the top of a liquid tank. From the ex-perimental observation, it was found that only a few layers of bubbles can be sufficient to significantly damp the liquid oscillations, while the bubbles close to the walls have a significant impact on the dissipation of energy. However, for a practical LNG tank, it may not be easy to controllably generate the desired size and amount of LNG bubble foams. Thus, this study proposes an alternative idea by using 'solid foam elements' instead of 'liquid bubble foams'. The solid foam elements can also be considered as fragments of the floating anti-sloshing blanket introduced by Samsung Heavy Industry (SHI) and BASF Company Ltd (Kim et al., 2011), although the present idea has aborted linkages among the fragments and applied multiple layers. This design is expected to inherit advantages of the above two sloshing mitigation strategies, so that both the sloshing violence and boil-off rate of the liquid cargo can be reduced with a much simpler technical operation.
In this study, the sloshing mitigation effects of floating layers of solid foam elements in rectangular tanks are investigated. Physical experiments are carried out at the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. Section 2 describes the experimental set-ups. Section 3 illustrates the experimental results. Effects of the foam-layer thickness, excitation amplitude, and excitation frequency on the sloshing properties are analyzed in detail. Finally, conclusions are drawn in Section 4.

Experimental set-up
Physical experiments are conducted in the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The key experimental set-up is shown in Fig. 1a. A liquid tank is fixed on an oscillation test platform produced by Beijing Aerospace Hill Testing Technology Co. LTD of China. The platform works as a low frequency electric vibration system and it vibrates according to the displacement equation , where x e is the displacement along the tank length, A is the vibration amplitude and f is the vibration frequency. A simplified rectangular tank made of plexiglass is used. The internal dimensions of the tank are 0.5 m in length, 0.08 m in width and 0.4 m in height. The thickness of the tank wall is 0.02 m, so that the liquid tank can be considered as rigid. The tank is filled with fresh water dyed by the rhodamine stain. Spherical foam elements with the diameter of 0.005 m and density of 37 kg/m 3 are uniformly distributed on the free liquid surface in the tank. The foam element samples are shown in Fig. 1b. Material properties of the foam layers are given in Table 1. During the experiments, the wave profiles are recorded by a video camera (100 frames per second), and six pressure sensors on the tank side (as in Fig. 2) are used to record histories of the dynamic pressure. A Cartesian coordinate system is set, with its origin at the bottom left corner of the tank. The x-axis points to the right, while the z-axis points vertically upward. We use P, P1, P2, P3, P4 and P5 to denote the pressure sensor located at 0.35 m, 0.20 m, 0.15 m, 0.10 m, 0.07 m and 0.04 m above the tank bottom, respectively.

Results and discussions
3.1 Identification of natural sloshing frequencies Natural sloshing frequencies of the liquid tank with foam elements must be first identified. From the linear potential-flow theory, it is well known that the natural sloshing frequencies of a 2D rectangular tank can be expressed as: where L is the tank length, H is the initial liquid depth in the tank, g is the gravitational acceleration, and w n represents the natural sloshing frequency of the n-th mode. The lowest natural sloshing frequency w 1 that is of the most interest in practice is mainly considered in this study. In the first series of experiments, the liquid has the depth of H=0.2 m, the excitation amplitude is A=0.002 m and three thicknesses of the foam layer (i.e. T/H=0, T/H=0.1 and T/H=0.2) are considered. The liquid tank is excited at seven different frequencies of w=0.88w 1 , 0.92w 1 , 0.96w 1 , 1.00w 1 , 1.04w 1 , 1.08w 1 , and 1.12w 1 , respectively. ρ Fig. 3 shows the entire variation process of dynamic pressure histories at the pressure sensor P1 and P2, for the case of w=w 1 . It can be seen that there exists an initialization stage when the excitation amplitude gradually increases to the target amplitude, and a cooling down stage before the machinery is switched off. For the present oscillation test, the steady stage maintains during t/T e =100 to 180. The steady stage is mainly analyzed in subsequent studies. In the remaining manuscript, T e is used to denote the excitation period of the oscillation test platform, =1025 kg/m 3 is the density of water, and g=9.81 m/s 2 is the gravitational acceleration. Fig. 4 compares the nondimensionalized wave height (H s -H)/H along the left tank wall for three different foamlayer thicknesses T/H=0, 0.1 and 0.2, where H s denotes the maximum free-surface height above the tank bottom at the steady sloshing state. It can be seen that, for each condition of the foam-layer thickness, the largest wave height occurs at w=1.00w 1 , corresponding to the resonant condition. The implementation of the foam elements does not affect the   fundamental natural sloshing frequency of the liquid tank evidently. The value w 1 predicted by the linear potential-flow theory can be considered the fundamental natural sloshing frequency of a liquid tank with the foam elements in subsequent experiments. Through comparison, it is also clear that two thicknesses of the foam layers can both reduce the sloshing wave height effectively. As the thickness of the foam layers increases from T/H=0 to 0.2, the wave height in the tank decreases from 0.31H to 0.06H under the resonance condition, as an evidence of the sloshing mitigation. Fig. 5 shows the snapshots of sloshing waves in a steady state when the free-surface elevation reaches its maximum at the left tank wall, for the case of T/H=0, T/H=0.1 and T/H=0.2. The effect of the foam elements on suppressing the maximum wave height can be identified visually. It is also noticed that some foam particles can inevitably attach to the tank walls in the experiments, which may reduce the sloshing mitigation effect of the foam elements to some extent.

Effects of foam-layer thickness on sloshing dynamics
In this subsection, the effects of the foam-layer thickness on the wave profile and hydrodynamic pressure on the tank side are investigated. All experiments are conducted at w = 1.00w 1 to produce the resonance scenario. Fig. 6 shows that the effect of the foam-layer thickness on the sloshing mitigation, where H fs denotes the free-surface height along the tank wall in the steady state, and H 0s is the value of H fs in the case of T=0. We use Δ to indicate the difference between the free-surface height and the initial liquid depth, i.e. ΔH fs = H fs -H, and ΔH 0s = H 0s -H. It can be seen that the wave height decreases sharply when the foam-layer thickness increases from T=0 to T=0.1H. Compared with the case without the foam layers, the wave height can be reduced by 70% at T=0.1H. As the foam-layer thickness further increases from T=0.1H to T=0.2H, the reduction rate of the wave height becomes lower. The physical mechanism of the sloshing mitigation using the foam layers can be understood as follows. During the free-surface motion, the elastic foam layers can absorb the energy of the liquid impulse. The energy can spread within the foam layers and be further dissipated through the particle-to-particle interactions and the friction between the foam layer and side walls. A thicker foam layer has more particle-to-particle interaction chances and larger interface of the foam layer and side walls, which can result in a more effective energy dissipation. With considering the economy in practice, the thickness of the foam elements should be as small as possible, as long as the sloshing mitigation effect can be guaranteed. For the present situation, a suitable choice of the foam-layer thickness can be T=0.1H. This will be further tested in the tanks with different liquid depths.
To verify the repeatability of the experiment, each case has been tested twice under the same condition. Figs. 7a and 7b take for example two cases of T/H=0.035 and T/H=0.04, respectively. For each case, the dynamic pressure histories   NING De-zhi et al. China Ocean Eng., 2019, Vol. 33, No. 1, P. 34-43 37 from two repetitive tests are compared. It is clear that the obtained results are almost unchangeable when a repetitive test is conducted. Here, it is noted that the lower half of pressure histories (with p<0) at P1 does not have practical meaning. This is because when the pressure sensor P1 is expected to be exposed in the air, the sensor is actually disturbed by liquid drops or wet foam elements attached on it.
For the pressure sensor P2 that is always within the liquid, the pressure signals have excellent repeatability properties. Then, the effects of foam-layer thickness on the dynamic pressure in the fluid domain are investigated. Fig. 8 shows the time histories of the dynamic pressure detected at P1 and P2 during the steady state for all thickness cases. In each subfigure of Fig. 8, the recorded pressure histories on the sloshing tank side with different foam-layer thicknesses are compared. It is evident that the pressure amplitude decreases as the foam-layer thickness increases.
In Fig. 9, an FFT analysis is carried out on the pressure histories at P2. For the case without the foam layers, the wave components at w 1 , 2w 1 and 3w 1 can be observed. To analyze the pressure components in detail, Fig. 10 presents the variation of the pressure amplitudes corresponding to the three frequencies, as the foam-layer thickness increases. As the foam-layer thickness increases, the pressure component at the dominant frequency w 1 reduces significantly, and the pressure amplitudes of higher-order components corresponding to 2w 1 and 3w 1 gradually vanish. This indicates that the hydrodynamic nonlinearity reduces with increasing the foam-layer thickness, together with a decrease of the wave height. The trend of the w 1 pressure component is similar to

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NING De-zhi et al. China Ocean Eng., 2019, Vol. 33, No. 1, P. 34-43 that of the wave height in Fig. 6. Fig. 11 further shows the vertical distribution of the pressure amplitude for different foam-layer thicknesses. It is found that the largest amplitude of the measured pressure occurs at P1. Below P1, the pressure amplitude declines gradually as the liquid depth increases. At each pressure sensor, the pressure amplitude decreases as the foam layer becomes thicker.

Large excitation amplitude cases
In this subsection, we further amplify the excitation amplitude from A/H=0.01 to A/H=0.04, where A denotes the excitation amplitude. The excitation frequency is under the resonant condition w=w 1 . Fig. 12 depicts the wave height variation along the tank wall as the excitation amplitude increases. Three foam-layer thicknesses of T/H=0, T/H=0.1 and T/H=0.2 are considered. It can be seen that, for the cases of three foam-layer thicknesses, the wave height in the tank increases with the excitation amplitude almost linearly. The growth rate of the wave height is generally smaller for the case with a thicker foam layer. Moreover, when the tank is excited by smaller excitation amplitude, the wave height difference between two cases of T/H=0.1 and T/H=0.2 is small. However, as the excitation amplitude A increases, the foam layer with T/H=0.2 produces a more effective sloshing mitigation effect than that in the case of T/H=0.1.
At this preliminary test stage of the sloshing mitigation concept, only mild sloshing cases are considered, so that sloshing waves are not allowed to impact the tank roof. Thus, a pressure sensor P (as in Fig. 2) is installed on the top of the tank side instead of the tank roof. Among these experimental tests, only for the excitation amplitude of A/H=0.04, the sensor at P has non-zero signals. For the remaining cases, the liquid surface cannot reach the sensor P. Fig.13 shows the dy-     NING De-zhi et al. China Ocean Eng., 2019, Vol. 33, No. 1, P. 34-43 namic pressure at P for A/H=0.04, where the pressure history presents strong nonlinear characteristics. The sloshing snapshots of the liquid domain for A/H=0.04 are shown in Fig. 14.  From Fig. 14a, it can be seen that the liquid motion in a tank without the foam elements is so violent that the wave profile breaks near the top corner of the tank. From Figs. 14b and 14c when the foam elements are deployed on the free surface, the wave breaking is evidently suppressed and the maximum wave height is greatly reduced. It is also noticed that the maximum wave height is not right on the tank wall but at a nearby location (marked by a black circle in Figs. 14b and 14c) when the foam elements are used in this large amplitude case.
Similar to Fig. 10, the amplitudes of the pressure components corresponding to w 1 , 2w 1 and 3w 1 are shown in Fig. 15.   Fig. 14. Typical snapshots of sloshing waves for A/H=0.04, w=1.00w 1 .
Three foam-layer thicknesses are compared. It is found that for all three cases, the amplitude of each pressure component increases with the excitation amplitude. As the excitation amplitude increases, higher-order pressure components of 2w 1 and 3w 1 become more evident. An interesting phenomenon can be found for the case without the foam elements. As the excitation amplitude increases, the amplitude of the 3w 1 component becomes larger than that of the 2w 1 component, showing a strong nonlinear property.

Effects of fill depth
In this subsection, two more fill depths of H=0.1 m and H=0.15 m are further discussed. The liquid tank is excited under the resonant condition (calculated by Eq. (1)) with the amplitude A/H=0.02.
Figs. 16-18 are for H=0.1 m and Fig. 19 is for H=0.15 m. Fig. 16 shows the effects of the foam-layer thickness on the sloshing wave height. Fig. 17 compares the amplitude spectra of the dynamic pressure at P4 for the cases with different foam-layer thicknesses. Fig. 18 gives in detail the amplitudes of the pressure components corresponding to w=w 1 , 2w 1 and 3w 1 . These results are consistent with observations of the high fill depth case in the previous subsections. As the foam-layer thickness increases from T/H=0 to T/H=0.1, the sloshing wave height in the tank can be reduced by 64%. As the foam-layer thickness increases, each pressure component reduces its amplitude evidently. Fig. 19 shows the effects of the foam-layer thickness on sloshing wave height for the case of H=0.15 m. From T/H=0 to T/H=0.1, the sloshing wave height can be reduced by 77%. Based on the observations in Figs. 6, 16 and 19 for three liquid depths, it is expected that T=0.1H can be a suitable choice of the foam-layer thickness that can effectively reduce the sloshing violence in the liquid tanks.   NING De-zhi et al. China Ocean Eng., 2019, Vol. 33, No. 1, P. 34-43 41 4 Conclusions A sloshing mitigation concept taking advantage of the floating layers of the solid foam elements is proposed in this study. Physical experiments are conducted in a rectangular tank to investigate the corresponding hydrodynamic properties. Floating layers of the solid foam elements made up of EPS are deployed on the free surface. The tank fixed on the experimental platform is excitated harmonically along the length direction of the liquid tank. Both the wave profile and dynamic pressure of the sloshing liquid are recorded. Effects of the foam elements on the sloshing properties are analyzed in detail.
It is found that the floating layers of the solid foam elements do not affect the fundamental natural sloshing frequency of the liquid tank evidently among the considered cases. Under the resonant condition, the foam elements can efficiently damp the free surface oscillations and reduce the pressure amplitude on the tank side. Higher-order frequency components of the dynamic pressure on the tank side gradually vanish as the foam-layer thickness increases. To be economical in practice, the foam-layer thickness should be as small as possible, as long as the sloshing mitigation effect can be guaranteed. For the present situation, a suitable choice of the foam-layer thickness can be one-tenth the mean liquid depth. The above findings are applicable to both high and low liquid fill depth cases.
Cases with different excitation amplitudes are also analyzed. Both the wave height and pressure amplitude in the tank increase with the excitation amplitude. The maximum wave height and dynamic pressure are greatly reduced as the foam-layer thickness increases. This phenomenon is observed when the wave breaking in a tank can be suppressed by the foam elements.
It should be noted that this study has only considered the mild sloshing cases, when the complex wave impacts on the tank roof are not involved. In future experiments, the mitigation effects of the foam elements on violent sloshing waves will be further investigated. A more systematic study on the mechanism of this sloshing mitigation concept is still required before achieving a practical application.