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Experimental and numerical studies on sloshing in a membrane-type LNG tank with two ﬂoating plates

MARK

⁎

Yue-Min Yua,b, , Ning Maa,b, She-Ming Fanc, Xie-Chong Gua,b a b c

State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, 800 Dongchuan Rd., Shanghai, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration (CISSE), Shanghai Jiao Tong University, China Shanghai Key Laboratory of Ship Engineering, Marine Design & Research Institute of China, Shanghai, China

A R T I C L E I N F O

A BS T RAC T

Keywords: Sloshing Suppressing device Model test Wave runup Impact pressure CFD

Sloshing with two ﬂoating plates in a membrane-type liqueﬁed natural gas (LNG) tank is experimentally studied at three diﬀerent ﬁlling rates under purely harmonic roll excitation. Our primary objective is to determine the performance of the suppressing device and of corresponding mechanisms. It is concluded that the suppressing device can eﬀectively damp the wave runup along the longitudinal bulkhead and the impact pressure acting on the bulkhead under the mild excitation amplitudes and, especially at the top of the tank. Furthermore, a numerical simulation based on a computational ﬂuid dynamics (CFD) program is introduced to conduct an additional investigation. The accuracy of the numerical simulation is veriﬁed against experimental results by comparing the wave elevation and heave motion of two ﬂoating plates relative to the tank. The wave runup along the vertical side wall and the velocity ﬁeld in the tank with ﬂoating plates are numerically evaluated and discussed as a supplement to obtain more dissipative mechanisms.

1. Introduction Membrane tanks are often used as storage tanks in Liquid Natural Gas (LNG) ships. In transit, sloshing (Faltinsen and Timokha, 2009) can occur as a result of six-degree-of-freedom ship motion from waves. When the excitation frequency approaches the lowest natural frequency of the free surface, sloshing becomes most violent and signiﬁcantly compromises structural safety. Therefore, sloshing suppressing devices are often considered to guarantee ship safety in realtime applications. Extensive investigations have been performed on how such devices suppress sloshing. Most of these studies have been carried out based on model tests. Anai et al. (2010) presented a new suppressing device that used membrane material with mooring ropes. The membrane material ﬂoated near the free surface with its central component connected to the vessel base by mooring ropes. Then, an anti-sloshing ﬂoating blanket attached free surfaces was proposed by Kim et al. (2011). The authors’ experimental results showed that the new device can signiﬁcantly decrease eﬀects of pressure acted on bulkheads. Recently, on the basis of the principle that beer is not as easy to shake as water, Sauret et al. (2015) proposed the placement of foams on free liquid surfaces to suppress sloshing. Wei et al. (2015) experimentally studied free surface

elevations and impact pressures of a rectangular tank with a centralized slat-screen under shallow water under high amplitude harmonic excitation. Numerical simulations are comparatively less costly and more eﬃcient than experiments in sloshing research, and especially when examining full-scale tanks. The velocity ﬁeld in a tank with suppressing devices is very complex. Jung et al. (2012) investigated the liquid sloshing eﬀects of a vertical baﬄe positioned at the centre of the lower wall of a tank. Liquid sloshing is dampened as the baﬄe height increases. Then, using the sub-domain partition method, Wang et al. (2013) semi-analytically investigated liquid sloshing response in a rigid cylindrical tank with multiple annular rigid baﬄes. Using the improved Consistent Particle Method (CPM), Koh et al. (2013) numerically simulated sloshing using a constrained ﬂoating baﬄe. The numerical algorithm was partially veriﬁed through their experimental results. Based on potential ﬂow theory, Molin and Remy (2013), who applied the multi-modal approach and represented the screen as a porous boundary, numerically studied sloshing in a rectangular tank with vertical perforated screens at mid length in the context of tuned liquid dampers (TLDs). Numerical hydrodynamic factors agreed well with experimental ones. Furthermore, Molin and Remy (2015) proposed an extension of the discharge formula of the numerical model and

⁎ Correspondence to: School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, A706, Mulan Building, 800 Dong Chuan Road, Minhang, Shanghai 200240, China. E-mail address: [email protected] (Y.-M. Yu).

http://dx.doi.org/10.1016/j.oceaneng.2016.11.029 Received 6 June 2016; Received in revised form 18 November 2016; Accepted 20 November 2016 Available online 25 November 2016 0029-8018/ © 2016 Elsevier Ltd. All rights reserved.

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accounted for an inertia term. To validate the numerical model, complementary experiments with vertical slots or circular holes were conducted. Recently, Goudarzi and Danesh (2016) numerically investigated suppressing eﬀects of vertical baﬄes within a full-scale tank under real earthquake excitation and examined the validity of the analytical method. Arai et al. (2013) proposed a simple anti-sloshing ﬂoating device that moved in only a single direction. Installed at the still water level, the tank was equally divided into two or three subtanks. The authors investigated, through model tests and numerical simulations, the resulting sloshing suppression eﬀectiveness and further optimized the shape of the ﬂoating plate. Only the wave runup at the vertical bulkhead was measured and discussed, and little is known of impact pressure levels. A ﬂoating device is positioned close to the side wall on the basis of the principle of ﬂoating breakwater (Koutandos et al., 2005). Both wave elevation and impact pressure are measured and analysed in the present study. Furthermore, the velocity ﬁeld in the tank is assessed via the numerical simulation. The outline of this paper is organized as follows. Experimental procedures and results are described in Section2. In Section3, the numerical simulation with two ﬂoating plates is veriﬁed based on our experimental results. Further, the velocity ﬁeld in the tank with two ﬂoating plates is predicted and analysed through a validated numerical simulation. The study's conclusions are drawn in Section4.

Fig. 1. Dimensions of the clean tank and the positions of pressure sensors (dimension: mm).

2. Performance test A series of model tests are conducted to investigate the free surface elevation and impact pressure under three diﬀerent ﬁlling rates in a rectangular tank with two ﬂoating plates, where a nearly two-dimensional ﬂow is forced by moving the tank through external excitation. The free surface elevation and impact pressure are investigated. The experimental system used in this paper mainly includes the sloshing platform, data acquisition system and tank.

Fig. 2. Positions and geometric parameters of the plates (dimension: mm).

frequency f and amplitude θ0: θ = θ0 sin(2πft ). The lowest natural frequencies of three ﬁlling depths in the tank without plates 1

πng

πnh

( fn = 2π L tanh( L ) ) were 0.71 Hz, 0.81 Hz and 0.87 Hz, respectively. The experiments were run at frequencies of 0.4–2.1 Hz (encompassing the smallest natural frequencies) and at amplitudes of 0.6° and 1.0°. The speciﬁc test conditions used are shown in Table 1. Model tests were performed at three ﬁlling depths under purely harmonic roll excitation and the centre of the rolling motion was located at the centre of the tank base. Each group test time was set to 120 corresponding excitation cycles, guaranteeing that the wave surface reached a steady state over a long time period. For the measurement of impact pressure levels, membrane piezo-resistive sensors were used and dynamic pressure data were acquired. The ranges of transducers were valued at 40 kPa and sampling frequencies employed 20 kHz in the model tests.

2.1. Experimental setup Experiments were performed using the Moog hexapod at the Shanghai Key Laboratory of Ship Engineering in Marine Design & Research Institute of China. A regular tank was constructed with transparent acrylic plates set on a moving MOOG platform capable of generating controlled motions within six degrees of freedom. During the experiments the model tank was subjected to purely harmonic roll motion. The internal dimensions of the clean tank are as follows: length 0.9 m, width 0.225 m and height 0.6 m. These are the same dimensions as those used in the model tests reported in Arai et al. (2013). The top of the tank is closed. The origin of the tank ﬁxed coordinate system is located in the middle of the tank base. The vertical z-axis points upward and the y-axis is parallel to the side wall. The angular motion around the y-axis is denoted as a rolling motion. Nine pressure sensors are spaced equally along the longitudinal bulkhead of the tank in positions shown in Fig. 1. The two ﬂoating plates are exactly the same with rectangular cross-sections and a speciﬁc gravity that is less than that of water. The plates are 0.03 m in width and 0.15 m in height with 0.075 m drafts. The plates are positioned 0.15 m from the side wall, where the free surface is divided into the following three sub-regions: the left, central and right compartments. Each ﬂoating plate is supported by two full extension stainless steel slides that include three interlinking slide proﬁles (cabinet, intermediate, and drawer members). Therefore, the two ﬂoating plates can only move vertically within the tank (see Fig. 2).

2.3. Impact pressure Impact pressure presents discreteness and randomness, with the former denoting that the excitation signal is sinusoidal, while the measured pressure signal is not necessarily sinusoidal and with the latter denoting that it is impossible to predict the impact force precisely and that amplitudes must therefore be described using probabilistic distributions. This section mainly compares impact pressure levels along longitudinal bulkheads with and without ﬂoating plates at three ﬁlling depths at the lowest natural frequencies for the clean tank. Table 1 Experimental conditions.

2.2. Experimental conditions

Filling rate R (%)

Excitation amplitude A (°)

Excitation frequency f (Hz)

30 50 70

0.6, 1.0 0.6, 1.0 0.6, 1.0

0.4～2.1a, 0.71 0.4～2.1a, 0.81 0.4～2.1a, 0.87

a

The tank was subjected to the forced harmonic roll motion of 218

The interval is 0.1 Hz.

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Fig. 3. Comparisons between pressure thresholds with and without plates. (R=30%, A=0.6°, f=0.71 Hz).

Fig. 4. Comparisons between pressure thresholds with and without plates. (R=50%, A=0.6°, f=0.81 Hz).

Fig. 5. Comparisons between pressure thresholds with and without plates. (R=70%, A=0.6°, f=0.87 Hz).

Fig. 6. Comparisons between pressure thresholds with and without plates. (R=30%, A=1.0°, f=0.71 Hz).

Fig. 7. Comparisons between pressure thresholds with and without plates. (R=50%, A=1.0°, f=0.81 Hz).

impact pressure are compared with and without ﬂoating plates. Furthermore, the standard deviation is superimposed onto the average of the pressure peaks, which is based on the availability of 120 excitation cycles. Hp is the location of the pressure sensor above the

Figs. 3–8 show the spatial distribution of impact pressure thresholds along the right walls of the clean and suppressing tanks. In each excitation cycle, the maximum impact pressure is referred to as a peak. In these ﬁgures, the maximum Pmax and average peak Pave of the

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Fig. 8. Comparisons between pressure thresholds with and without plates. (R=70%, A=1.0°, f=0.87 Hz).

Fig. 9. Flow evolution during half of the excitation period. (R=30%, A=1.0°, f=0.71 Hz). (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

2.4. Wave elevation

tank bottom, and d is the distance between two adjacent sensors (see Fig. 1). As seen in Figs. 3–8 for three ﬁlling depths, compared to the clean tank both the maximum and average peak of the impact pressure along the vertical bulkhead are signiﬁcantly reduced. These results indicate that ﬂoating plates can eﬀectively decrease sloshing impact loads at close to the lowest clean tank resonant frequency under the mild excitation amplitudes, and especially around the top of the tank. In the clean tank, increasing the ﬁlling depth causes the most severe slamming position to move from a near static water level to the top of the tank, and this tends to cause local damage and then result in a loss of compartment structure utility. With ﬂoating plates installed in the tank, pressure thresholds of the maximum and average peak above the static water level are approximately equal to 0, except in the case of R=70% and A=1.0° (see Fig. 8). This means that the wave elevation motion is remarkably restrained due to the damping performance of the ﬂoating plates.

Depending on the ﬁlling rate, sloshing will produce diﬀerent ﬂow phenomena, and the tank height is also a vital parameter aﬀecting the impact pressure on the top of the tank. The impact pressure reveals resulting local structural stresses. The wave elevation along the tank wall reveals the resulting global loads acting on the tank, which are important in determining the interaction between sloshing and ship motions (Wei et al., 2015). A snapshot of the wave elevation framework reveals an intuitive expression for analysing sloshing phenomena. Figs. 9–11 show the wave elevation through selected frames for half of the excitation period of the suppressing tank for case A=1.0° and ﬁlling rates of 30–70%, where the excitation frequency is equal to the lowest natural frequency of the clean tank. The corresponding roll angle of the tank (θ), which is non-dimensional with respect to the motion amplitude (θ0), is shown in the time history of the tank motion and is reported in the bottom

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Fig. 10. Flow evolution during half of the excitation period. (R=50%, A=1.0°, f=0.81 Hz). (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

(1) ∇⋅→ u =0 → where u is the ﬂow velocity. This equation is in fact a statement of the

panel. T refers to the excitation period. The red squares denote the tank position at the time of the above images. Fig. 9 shows that the tank begins to rotate clockwise (frame 1#). Subsequently, the wave elevation in the right compartment begins to drop while the wave elevation in the left compartment gradually increases like the water column in the U-shaped tube (see frames 2#−6#). Furthermore, a standing wave occurs in the central compartment due to a block of the two ﬂoating plates. According to previous studies, ﬂuid motion is referred to as the U-tube mode (Kobayashi et al., 2006). The impact pressure is considerably reduced under the Utube mode, and the heave motion of two ﬂoating plates along with the wave elevation increases sloshing energy dissipation to some extent. When the ﬁlling rate is increased, the U-tube mode still occurs as is shown in Figs. 10–11. However, a progressive wave is generated on the central compartment. Then, overtopping can be observed (as shown in frame 4# of Fig. 11), and overtopping ﬂuid splashes onto the top of side tank, potentially placing greater impact pressure on the position of pressure sensor number PP9 (see Fig. 8).

conservation of volume. For the special case of incompressible ﬂow, the momentum equation is:

⎯→ ⎯ ∂→ u 1 +→ u ⋅∇→ u = − ∇p + νΔ→ u + f ∂t ρ

(2)

In the above equation, ρ is density, p is static pressure, ν is ⎯→ ⎯ kinematic viscosity and f denotes body acceleration (per unit mass). The dynamic mesh is a feature of ANSYS FLUENT that updates mesh of the area surrounding a moving or rotating object. The six degree of freedom solver uses an object's forces and moments to compute the translational or rotational motion of an object's centre of gravity. The governing equation solving the translation of an object's centre of gravity in the inertial coordinate system is as follows:

⎯→ ⎯ → 1 xG̈ = ∑ FG (3) m → where ẍG is the translational acceleration of the centre of gravity and ⎯→ ⎯ FG is the force vector of the object. Object rotation is computed using body coordinates:

3. Numerical veriﬁcation and prediction To investigate ﬂow characteristics in the tank with two ﬂoating plates, a numerical simulation was conducted using the ANSYS FLUENT program (ANSYS FLUENT 14.5, 2012a, 2012b). Due to its application of the volume of ﬂuid method (VOF), dynamic mesh model and six degree of freedom (6DOF) solver, this program has often been preferred in sloshing studies involving ﬂoating plates (Yu et al., 2015). For an incompressible ﬂuid, the simpliﬁed mass continuity equation is:

→ → →⎞ ⎛ ⎯→ ⎯ θB̈ = L−1 ⎜∑ MB − θḂ × LθḂ ⎟ ⎝ ⎠

(4)

→ where θB̈ is the rotational acceleration of the object, L is the inertia → ⎯→ ⎯ tensor, MB is the moment vector of the object and θḂ is the rotational 221

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Fig. 11. Flow evolution during half of the excitation period. (R=70%, A=1.0°, f=0.87 Hz). (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

velocity vector of the object. ANSYS FLUENT uses a ﬁnite volume method to discretize and solve the Navier-Stokes equation. Then, the VOF (Volume of Fluid) model is applied to track the free surface. The PISO (Pressure Implicit with the Splitting of Operators) algorithm is chosen for the pressure-based solver. The tank body is given for a wall condition. For the calculations, time sizes range between 0.001 s and 0.005 s depending on the sloshing intensity. Macro works by the UDF (User-Deﬁned Function) send linear and angular velocities of a body, and ANSYS FLUENT then uses rigid body motions to update the mesh position on the dynamic zone during the next time step. The 6DOF (Six Degree of Freedom) solver computes external forces and moments for an object, and these forces are computed by a numerical integration of pressure and shear stress over the object's surfaces. This technique, along with the use of dynamic meshes, can be readily applied to sloshing ﬂow interactions and to two ﬂoating plates. The mesh around two ﬂoating plates is purely triangular, and therefore smoothing and remeshing methods are applicable and can be used together in this simulation, updating interior meshes to prevent the development of negative cell errors.

Table 2 Reynolds numbers for various excitation frequencies. Filling rate R (%)

Excitation amplitude A (°)

Reynolds number Re

30

0.6 1.0 0.6 1.0 0.6 1.0

2035～5087 3459～8648 3391～8478 5765～14413 4748～11869 8171～20178

50 70

Table 3 Description of computational meshes. Mesh

Minimum size (mm)

Maximum size (mm)

Element number (N)

M1 M2 M3

4.6 2.5 2.1

9.4 6.8 4.9

24538 48380 99256

Table 4 Convergent analysis of various mesh resolutions.

3.1. Validation of the water proﬁle

Test case

Excitation amplitude A (°)

Excitation frequency f (Hz)

Maximum runup (mm)

Relative error (%)

Case 1

0.6

0.71

M1: M2: M3: M1: M2: M3:

2.05 2.85

Case 2

1.0

0.71

21.128 21.293 20.702 32.770 33.609 32.775

In reference to the deﬁnition of the Reynolds number (Chen and Nokes, 2005) under surge excitation, here the Reynolds number under u h roll excitation is deﬁned as Re= 0ν , where the maximum roll velocity is 1

u 0 = 2 Lθ0 ω . The range of Reynolds numbers under two excitation amplitudes is shown in Table 2 for three ﬁlling rates. Convergence tests were ﬁrst conducted for the present numerical simulation when the ﬁlling rate was equal to 30%. In two cases, three runs with diﬀerent mesh resolutions were performed, namely, M1, M2

−0.02 2.54

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Fig. 12. Comparisons between water proﬁles. (R=30%, A=1.0°, f=0.71 Hz).

numerical simulation can be attributed to the intermediate member of full extension slides in the experiment, which increases the weight of the ﬂoating plate (see Fig. 14). As is shown in these ﬁgures, the numerical results are in good agreement with the model test measurements, which conﬁrm the validity of the numerical simulation.

and M3. Mesh generation information is listed in Table 3. The numerical results of the maximum runup 0.01 m away from the left side wall with various mesh resolutions are presented in Table 4. It is evident that the maximum runup values for two cases are in good agreement. The maximum relative error for ﬁne mesh M3 is less than 3% (Lu et al., 2015). These comparisons show that sloshing solutions predicted through the present numerical simulation are not sensitive to the mesh resolution. Furthermore, to ensure numerical accuracy and save computational time, the M2 is adopted as the baseline for the numerical simulation conducted in this study. The numerically simulated water proﬁle is presented in Figs. 12–14 at three ﬁlling depths for half of the excitation period. The simulated results agree quite well with the experimental results in terms of wave elevations and the locations of the two ﬂoating plates. When the ﬁlling rate is equal to 70%, slight diﬀerences between the model test and

3.2. Wave runup analysis For the ﬁlling rate R=30%, the wave runup was numerically monitored 0.01 m away from the vertical side wall of the tank in addition to the heave of the centre of gravity of the plate relative to the tank. The maximal wave runup and maximal heave of the plate are plotted against excitation frequency in Figs. 15 and 16 for two diﬀerent excitation amplitudes. Furthermore, the longitudinal coordinate of the blue circles represents the maximal wave runup while the longitudinal 223

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Fig. 13. Comparisons between water proﬁles. (R=50%, A=1.0°, f=0.81 Hz).

(2008) experimentally investigated the sloshing ﬂuid ﬂow in a rectangular tank with an vertical baﬄe mounted on top. A vortex was generated at the edge of the plate and a swirl ﬂow formed in the liquid. Furthermore, the presence of a vortex is likely to generate local security risks, and so it is critical to know how vortexes form and disappear in practical engineering. The velocity ﬁelds within the tank for varying ﬁlling depths are presented in Figs. 17a, 18a and 19a. For a more detailed account of these ﬂow characteristics enlarged local views are shown in Figs. 17– 19. A vortex appears at the edge of the right plate (in Figs. 17b, 18b and 19b), while a ﬂow occurs along the bottom of the left plate from the central compartment to the left compartment (in Figs. 17c, 18c and 19c). A similar vortex and ﬂow on the opposite side of the tank are observed during the next half of the excitation period. During this motion period, a water wave in the central compartment strikes the

coordinate of the red triangles expresses the maximal heave of the plate. The results indicate that resonant frequencies shift to a lower frequency of close to 0.6 Hz after adding two ﬂoating plates compared to the clean tank. In addition, similar changes occur between the maximal heave of the plate and the maximal wave runup while the maximal wave runup value is slightly larger than the maximal heave of the plate at the same excitation frequency. These results are similar to those of Arai et al. (2013). 3.3. Velocity ﬁeld prediction Damping eﬀects on sloshing in a tank with an upper centralized baﬄe were studied by Kobayashi et al. (2006). The authors categorized ﬂow patterns into three types: vortex, combined single swirl and vortex, and twin swirls patterns. Using a ﬂow visualization, Onuma et al. 224

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Fig. 14. Comparisons between water proﬁles. (R=70%, A=1.0°, f=0.87 Hz).

Fig. 16. Maximal wave runup and plate heave against forcing frequency. (R=30%, A=1.0°). (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

Fig. 15. Maximal wave runup and plate heave against forcing frequency. (R=30%, A=0.6°). (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.)

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Fig. 17. Velocity ﬁeld at the 7# time point. (R=30%, A=1.0°, f=0.71 Hz).

Fig. 18. Velocity ﬁeld at the 7# time point. (R=50%, A=1.0°, f=0.81 Hz).

ﬂoating plate, and a water column builds within compartment. The height of the water column is dependent on the water column's velocity (Eswaran et al., 2011). Furthermore, vortex occurrence and disappearance signiﬁcantly aﬀect the velocity ﬁeld within the tank.

(1) Our experimental results show that the suppressing device not only reduces wave runup along the longitudinal bulkhead but also decreases impact loads acting on the bulkhead at the mild excitation amplitudes. Especially at the 70% ﬁlling rate, impact pressure on the top of a tank can be greatly reduced. (2) When the excitation frequency approaches the lowest natural frequency of a clean tank, the U-tube mode of the ﬂuid motion occurs in a tank with two ﬂoating plates. The U-tube mode plays a signiﬁcant role in decreasing sloshing. (3) The simulated water proﬁles and movements of the ﬂoating plates agree well with our experimental results. More importantly, resonant frequencies shift to a lower frequency in the tank with two ﬂoating plates in comparison to those of the clean tank. The computed results show that the vortex near the edge of the ﬂoating plate, which appears and disappears in intervals, has key eﬀects on the velocity ﬁeld.

4. Conclusions The results of model tests indicate that adding two ﬂoating plates can be highly eﬀective at dampening sloshing under three diﬀerent ﬁlling rates under prescribed harmonic rolling excitation. The results of our model tests are used to calibrate and verify the accuracy of the numerical simulation. The maximal wave runup and maximal heave of the plates are then monitored and discussed through a numerical simulation of the 30% ﬁlling rate. Furthermore, the velocity ﬁeld in the tank with two ﬂoating plates is predicted and analysed at three ﬁlling depths subjected to harmonic rolling excitation. Our main conclusions are as follows:

The model tests are performed in the nearly two-dimensional 226

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Fig. 19. Velocity ﬁeld at the 7# time point. (R=70%, A=1.0°, f=0.87 Hz).

membrane-type tank with two ﬂoating plates under the mild excitation amplitudes. Thereafter, three-dimensional experimental and numerical studies should be conducted before initiating actual applications.

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Acknowledgments The authors would like to express deep gratitude to the Shanghai Key Laboratory of Ship Engineering technicians Jun Nie, Su-Jun Yang, Yong-Shun Wu, Guo-Zhao He and Da-Jian Wang for their indispensible help in arranging the experiments. The ﬁnancial funding from the National Natural Science Foundation of China (Grant no. 51279105) and the Knowledge-based Ship-Design Hyper-Integrated Platform (KSHIP) 2nd Term of Ministry of Education and Finance, P.R. China (Gran no. GKZY010004) is greatly appreciated. References Anai, Y., Ando, T., Watanabe, N., Murakami, C., Tanaka, Y., 2010. Development of a new reduction device of sloshing load in tank. In: Proceedings of the 20th International Oﬀshore and Polar Engineering Conference. Beijing, China, June 20–25, 2010 ANSYS FLUENT 14.5, 2012a. ANSYS FLUENT User's Guide. ANSYS Inc., Canonsburg, PA, USA. ANSYS FLUENT 14.5, 2012b. ANSYS FLUENT Theory Guide. ANSYS Inc., Canonsburg, PA, USA. Arai, M., Suzuki, R., Ando, T., Kishimoto, N., 2013. Performance study of an antisloshing ﬂoating device for membrane-type LNG tanks. In: Proceedings of the 15th International Congress of the International Maritime Association of the Mediterranean. A Coruna, Spain, October 14–17, 2013. Chen, B.F., Nokes, R., 2005. Time-independent ﬁnite diﬀerence analysis of fully nonlinear and viscous ﬂuid sloshing in a rectangular tank. J. Comput. Phys. 209, 47–81. Eswaran, M., Singh, A., Saha, U.K., 2011. Experimental measurement of the surface velocity ﬁeld in an externally induced sloshing tank. Proc. Inst. Mech. Eng. Part M: J. Eng. Marit. Environ. 225 (2), 133–148.

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