Numerical study on structural response of anti-sloshing baffles of different configurations in a sloshing tank considering hydroelasticity

Ocean Engineering 188 (2019) 106290

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Numerical study on structural response of anti-sloshing baffles of different configurations in a sloshing tank considering hydroelasticity Hao Qin a, b, Lin Mu a, b, *, Wenyong Tang c, Zhe Hu d a

College of Marine Science and Technology, China University of Geosciences, Wuhan, 430074, China Shenzhen Research Institute, China University of Geosciences, Shenzhen, 518057, China c State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China d Key Laboratory of Ships and Ocean Engineering of Fujian Province, Jimei University, Xiamen, 361021, China b

A R T I C L E I N F O

A B S T R A C T

Keywords: Structural response Hydroelastic effect Anti-sloshing baffle Baffle configuration Structural resonance

With the enlargement of liquid containers, unexpected deformation and damage of the containers might happen. As a commonly used way to suppress intense sloshing, anti-sloshing baffles usually sustain liquid loads with a large wetted area. Therefore, it is necessary to investigate the structural response of the anti-sloshing baffles taking hydroelasticity into account to guarantee the structural safety. This paper presents a numerical study on the structural response of horizontal and vertical baffles of different configurations in a sloshing tank considering hydroelasticity. Two validations on the effectiveness of the numerical methods and convergence tests are con­ ducted. Influences of baffle height and length of the horizontal and vertical baffles on the structural response are examined through a series of simulations, from which amplitudes, frequencies of different vibration components and energy ratio of the wetted natural vibration caused by hydroelasticity are obtained and analyzed. Addi­ tionally, possible structural resonances of the horizontal and vertical baffles caused by the periodic sloshing motion and hydroelastic effects are shown, indicating the necessity of considering hydroelasticity. Meaningful conclusions are drawn on the characteristics of structural response, the effects of baffle configuration and the occurrence of structural resonance of the baffle from a structural safety point of view.

1. Introduction

including the interaction between liquid and walls of the tank and the interaction between liquid and anti-sloshing baffles in the tank. In particular, sloshing suppression using anti-sloshing baffles and struc­ tural responses of the tank structures attract attentions increasingly from a structural safety point of view. In order to guarantee operational safety of liquid tanks, anti-sloshing baffles are commonly used to suppress violent liquid sloshing. Mecha­ nisms of anti-sloshing baffles in suppressing sloshing and natural fre­ quency of liquid with baffles were studied. For example, Celebi and Akyildiz (2002) investigated the nonlinear liquid sloshing inside a partially filled rectangular tank, revealing that flow over a vertical baffle produced a shear layer and energy was dissipated by viscous action especially in shallow water. Akyildiz (2012) further investigated the liquid sloshing in a moving partially filled rectangular tank with a ver­ tical baffle, showing that the suppression was because of the hydrody­ namic damping of baffle including the blockage effects and viscosity of baffle walls. It was claimed that limitations of their works lied in the ignorance of turbulence and the three-dimensional effects. Cho et al.

Liquid sloshing in partially filled tanks is widely observed during the operation of ships and offshore structures on the sea, which might lead to stability loss in maneuvering of floaters and large deformation or even damage to containers, causing serious safety threats (Ibrahim, 2005; Faltinsen and Timokha, 2009). Therefore, comprehensive un­ derstandings on sloshing phenomena, sloshing loads and their influences on structures are of vital importance in assessing structural safety in the design and check stage of liquid cargo carriers. Faltinsen (1978) firstly proposed a Boundary Element Method (BEM) to study mechanisms of liquid sloshing, which inspired many researchers to devote efforts on the investigations of the free surface evolution (Frandsen, 2004; Liu and Lin, 2008; Shao et al., 2012), the slamming loads (Akyildiz and Ünal, 2005; Lee et al., 2007; Colagrossi et al., 2010) and the flow-motion coupling of floaters (Rognebakke and Faltinsen, 2003; Kim et al., 2007; Mitra et al., 2012) caused by liquid sloshing. In recent decades, some researchers focused on the fluid-structure interactions in liquid sloshing problems,

* Corresponding author. Shenzhen Research Institute, China University of Geosciences, Shenzhen, 518057, China. E-mail address: [email protected] (L. Mu). https://doi.org/10.1016/j.oceaneng.2019.106290 Received 31 March 2019; Received in revised form 30 June 2019; Accepted 6 August 2019 Available online 13 August 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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(2005) performed numerical analysis of the resonance characteristics of liquid sloshing in a baffled tank, through which sloshing frequency response with respect to the number, location and opening width of baffle were parametrically investigated. However, linearized potential flow theory was applied and nonlinearity behaviors were not consid­ ered. Ebrahimian et al. (2013) presented a developed successive method to determine sloshing natural frequencies for multi-baffled axisym­ metric containers with arbitrary geometries, confirming that the baffle position on sloshing natural frequencies was more important than the baffle size. In their study, multi-baffles were only represented by ring baffles. Chu et al. (2018) studied the sloshing phenomenon in a rect­ angular water tank with multiple bottom-mounted baffles, indicating that the natural frequency of the tank can be altered by the multiple baffles and be related to the effective water depth above the baffles, and yet only vertical baffles were considered. Performances of anti-sloshing baffles highly depend on their con­ figurations, including the layouts and sizes of the baffles. Therefore, great attentions were paid on anti-sloshing baffles with different con­ figurations in sloshing suppression. For example, Maleki and Ziyaeifar (2008) analyzed the damping potential of baffles in circular-cylindrical storage tanks, showing that horizontal ring baffles were more effective in reducing the sloshing oscillations compared with vertical blade baf­ fles. However, their theoretical model was limited to the assumption that baffles were not uncovered during the sloshing. Belakroum et al. (2010) predicted the damping effect of baffles on sloshing, in which the horizontal baffle, vertical baffle at the bottom of tank and vertical baffle at the free surface were introduced. It was concluded that mounting a baffle vertically just at the free surface was effective on the attenuation, but the effects of baffle sizes were not considered. Xue and Lin (2011) and Xue et al. (2012, 2013, 2017) carried out a series of experimental and numerical studies on the influences of anti-sloshing baffles with different configurations thoroughly, including ring baffles, horizontal baffles, vertical baffles and perforated vertical baffles. The performances of baffles with different configurations in suppressing violent sloshing were examined, and sloshing loads on the tank walls were discussed. Their studies highlighted the better selected layouts and sizes of anti-sloshing baffles, which were valuable from a structural design point of view. Jung et al. (2012) studied the effect of the vertical baffle height on the liquid sloshing in a rectangular tank, which showed the changes of vortex shedding and the mean maximum pressures with baffle height. In their study, commercial software FLUENT was applied without considering the thickness of the vertical baffle. Hasheminejad et al. (2014) developed rigorous mathematical models for transient sloshing under sufficiently small free surface elevation in baffled circular cylin­ drical vessels, in which effectiveness of baffle configuration on sup­ pression of the induced destabilizing lateral forces were examined. Nayak and Biswal (2015) conducted a series of experiments in a rigid rectangular tank, in which bottom-mounted vertical baffles, surface-piercing wall-mounted vertical baffles, and bottom-mounted submerged-blocks were tested. Their results showed that the surface-piercing wall mounted baffles were the most effective, but no comment on the performance of horizontal baffles was given. Shao et al. (2015) investigated different baffles on mitigating liquid sloshing in a rectangular tank, and found that the mitigation performance depended on the shape, structure and location of the baffles. However, the hori­ zontal baffle was only fixed on one-side of the wall rather than two-sides, which would underestimate the performance of suppression. Wang et al. (2016) applied a semi-analytical scaled boundary finite-element method to study the effects of the T-shaped baffle on liquid sloshing in horizontal elliptical tanks. The effects of liquid fill level, baffled arrangement and length of those baffles upon the sloshing frequencies, the associated sloshing mode shapes and sloshing wave height were investigated. Kim et al. (2018) introduced a concept of moving baffles with a spring system as one of the attempts for reducing sloshing impact, and tested the performance in suppression experimentally. However the filling ratio and the stiffness of springs were not deeply studied. Cho et al. (2017)

applied a matched eigenfunction expansion method to obtain the analytical solution for the sloshing with porous horizontal baffle. It was found that the porous baffles installed on container walls can more significantly reduce the violent sloshing than those installed at the tank center, and yet no comparison was conducted between porous baffles and non-porous baffles. Wang et al. (2017) investigated the natural frequencies, the associated mode shapes, and sloshing forces of liquid sloshing in a rigid circular/elliptical toroidal tank with various baffles. Horizontal bottom-mounted, surface-piercing ring baffles as well as their combination form, bottom-mounted, surface-piercing ring baffles as well as their combination, and free surface-touching baffle were included. Sanapala et al. (2018) studied the sloshing dynamics in a partially filled rectangular container as well as the optimal position of the baffle and its width systematically. It was claimed that the optimal baffle position and width was effective in controlling the total response from resultant slosh wave amplitude, but not when imposed with pure vertical seismic excitation. Wang et al. (2019) developed an analytical technology to solve the dynamic response of the liquid in the circular cylindrical container with multiple rigid baffles, in which the effects of baffles’ parameters and the pitching excitation frequency on the liquid responses were examined. When the pitching excitation frequency was relatively large, resultant force and moment were limited on the angular acceleration and the Stokes-Joukowski potential. Nasar and Sannasiraj (2019) carried out experimental tests to explore effectiveness of porous baffles with different arrangements. It was found that porous baffle placed at 1/2, 1/3 and 2/3 configurations of the tank length acted better than solid submerged baffle placed at 1/2 of the tank length. Kargbo et al. (2019) studied layered liquid sloshing in a rectangular tank con­ taining two density stratified immiscible fluids. Various input parame­ ters as well as the effect of various baffle configurations and a submerged block of various sizes and positions on the layered sloshing interfacial wave were considered. Yu et al. (2019) investigated the screen-affected sloshing experimentally by varying solidity ratio, slot size, location and slat screen number, indicating that sloshing reduction didn’t always apparently increase as screen number increased under the screen posi­ tions specified in experiments. On the structural response of tank structures, vibration of structures caused by liquid sloshing loads and its influence on the liquid pressures are investigated in existing literatures. For a long time, analytical de­ ductions have been used for calculation of the structural response in liquid sloshing problems, which were usually limited by non-viscous and linearized potential assumptions. For example, Amabili (2001) studied the vibrations of circular plates resting on a sloshing liquid free surface using a fully coupled Rayleigh-Ritz method, in which it was stated that when the natural frequency of plate modes was close to those of sloshing modes of the free surface, very large coupling between the two families of modes was expected. Kim and Lee (2005) developed an analytical approach to investigate the vibration characteristics of the sloshing and bulging modes for a liquid-filled rigid circular cylindrical storage tank with an elastic annular plate in contact with sloshing surface of liquid. It was found that effects of liquid level and radius ratio on the sloshing and bulging frequency were great. Askari et al. (2013) proposed a theoretical method to investigate free vibrations of circular plates immersed in fluids and presented a series of experimental tests to validate the model, in which bulging and sloshing natural frequencies and mode shapes were obtained by solving the coupled governing equations simulta­ neously. However, in these above studies, the structural responses were limited in the frequency domain, and liquid sloshing loads on the plates were not checked. Graczyk et al. (2007) addressed the long-term extreme pressure in the membrane LNG tank and structural response of the Mark III containment system, in which importance of accounting for the steel flexibility in calculating sloshing response was shown. The method was based on the idea of Response Amplitude Operator (RAO) for linear systems and the influence of fluid-structure coupling was neglected. Biswal and Bhattacharyya (2010) studied the dynamic interaction between the liquid and elastic tank-baffle system to evaluate 2

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researched. In addition, there is a lack of analysis on the vibration characteristics and vibration components of the response of metal baf­ fles under sloshing loads. Significance of hydroelasticity is rarely examined on the structural response of anti-sloshing baffles, although it was emphasized by Amabili (2001) that large coupling between liquid and structure arises when their natural frequencies are close to each other, and by Faltinsen and Timokha (2009) that hydroelasticity matters when duration of local loads is comparable with the natural period of structures. In present paper, numerical study on structural response of antisloshing baffles of different configurations considering hydroelasticity is conducted using a strong coupled fluid-structure interaction (FSI) scheme, which is based on the monolithically coupled FSI equation initially proposed by Hu et al. (2016). In order to verify the effectiveness of numerical methods, a validation on the hydroelastic response is conducted with the case of a dam-breaking flow slamming an elastic vertical wall, the results of which are compared with the results by Sun et al. (2015) and Zhang and Wan (2018). Convergence and validation on sloshing loads are further conducted in a surging tank, the results of which are compared with the numerical results by Jung et al. (2012) and experimental data by Kang and Lee (2005). Simulations on liquid sloshing with anti-sloshing baffles of different configurations consid­ ering hydroelaticity are further carried out in two surging tanks with horizontal and vertical baffles respectively, in which effects of baffle height and length on the structural response of the baffles are examined. Analysis on vibration characteristics and vibration components of structural response of the baffles are performed, through which the significance of wetted natural vibration caused by hydroelasticity is revealed. Additionally, possible structural resonances of the horizontal and vertical baffles are shown when frequency of the forced excitation caused by sloshing motion is close to the wetted natural vibration fre­ quency caused by hydroelasticity. This present paper aims to numeri­ cally study the structural response of anti-sloshing baffles of different configurations considering hydroelasticity. Using a direct strong coupled FSI scheme, the possible structural resonance induced by the substantially reduced wetted natural vibration frequency due to hydroelastic effects is also examined. Although liquid sloshing with baffles has been carefully studied in a wide range of literatures, contri­ butions were mainly focused on the influence of rigid baffles without considering the hydrodynamic-elasticity interaction between the slosh­ ing liquid and the elastic baffle. In reality, even the metal tank structures with relatively large stiffness are usually elastic rather than rigid, particularly for engineering applications with full scale or experimental tests with large scale. For example, large size bulkheads of liquid tanks are undouble elastic structures in liquid cargo carriers, the deformation of which induced by the sloshing loads should not be neglected. Actu­ ally, it was found by Lugni et al. (2014) that pressures on elastic tank structures subjected to sloshing were with a much larger amplitude compared with pressures on rigid tank structures, and it was claimed by Zhang and Wan (2018) that deformations of elastic tank structures would finally affect the safety assessment of structures. As such, in­ vestigations on structural response of tank structures considering hydroelasticity would benefit the community in both understanding of the hydrodynamic-elasticity interaction mechanism and the practical engineering applications. As far as the authors concern, this present paper might be innovative in structural response prediction and analysis of metal anti-sloshing baffles considering hydroelastic effects. The importance of consider­ ation of hydroelasticity in structural response prediction is emphasized, drawing attention from a structural safety point of view. Discussion of the influences of baffle configurations on baffle responses might be of reference value in the preliminary structural design or safety assessment stages. This might contribute to the ocean engineering community from a structural response perspective similar to Lugni et al. (2014) and Zhang and Wan (2018), who brought out the issue of the hydrodynamic-elasticity interaction between sloshing liquid and elastic

the coupled system response, and observed that both the liquid and structural responses could be controlled by using the baffle. However, the research was focused on the coupled system frequency, while free surface elevation and sloshing loads were not included. Noorian et al. (2012) studied the interaction of sloshing and structural vibrations of baffled tanks, and confirmed that the variation of sloshing and structural dynamics frequencies versus baffle flexibility was a function of the di­ rection of both sloshing and structural vibration mode shapes. In the study, BEM method was used for modeling of liquid behavior, as such it was restricted by linearity of free surface motions. Lee et al. (2015) presented a procedure for the numerical computation of nonlinear impact responses in FLNG cargo tank structures under sloshing loads using ANSYS/LS-DYNA and focused on the effect of the spatial and temporal patterns of the load profile and the tank shell thickness. And yet, the study aimed to establish a procedure for structure verification rather than focusing on the sloshing event itself. With the growth of computation capability and the advancement of computational dynamics, numerical methods of coupled fluid-structure interaction play a more and more important role in prediction of structural response in liquid sloshing problems. For example, Idelsohn et al. (2008) applied a Particle Finite Element Method (PFEM) for the simulation of the interaction between an elastic baffle and free surface flows in a sloshing tank and calculation of structural response. However, the baffle was made of rubber prone to large deformation, which was quite different from the metal anti-sloshing baffles in tanks of liquid cargo carriers. Eswaran et al. (2009) carried out numerical simulations of sloshing waves in baffled and un-baffled tanks, the results of which were compared with available experimental data to demonstrate the reduction of sloshing effects. The model was developed within com­ mercial software ADINA, in which nonlinearity of free surfaces was not fully considered. Liao and Hu (2013) and Paik and Carrica (2014) simulated the interaction between liquid sloshing flow and an elastic baffle in sloshing tanks by coupling Finite Difference Method (FDM) and Finite Element Method (FEM). Similar to Idelsohn et al. (2008), the baffles in their studies were made of rubber, and components of struc­ tural responses were not analyzed. Lugni et al. (2014) analyzed the hydroelastic effects during the evolution of a flip-through event upon a vertical aluminum wall in a sloshing tank numerically and experimen­ tally, showing that the hydroelastic response of the vertical wall was divided into quasi-static regime, fully hydroelastic regime and free vi­ bration regime. However, experimental data were needed for their developed model, posing limitations of its practical usage. Hwang et al. (2016) simulated sloshing flows in rolling tanks with elastic baffles using a modified particle-based fluid-structure interaction solver, which also applied a rubber baffle under large deformation without consideration of the actual property of anti-sloshing baffles in liquid cargo carriers. Fourey et al. (2017) proposed an efficient coupling between Smoothed Particle Hydrodynamics (SPH) and FEM dedicated to violent fluid-structure interaction, in which a case of liquid column acting on an elastic beam was simulated as the benchmark test. And yet, in their study the liquid on the baffle was hydrostatic, and thus the effects of sloshing motions were not included. Zhang and Wan (2018) applied a fully Lagrangian method by coupling the Moving Particle Semi-Implicit (MPS) method and the FEM to study the liquid sloshing with elastic structures considering and the structural response. Their study analyzed hydroelasticity similar to Lugni et al. (2014), but in both of the studies only wall structures were examined. Although plenty of research concerning the liquid sloshing in baffled tank has been carried out, most of them were focused on the suppression performance of anti-sloshing baffles on liquid free surface elevation. Since the anti-sloshing baffles also sustain sloshing loads similar to the tank walls, the structural response of the baffles should be checked to guarantee structural safety. However, existing studies mainly concen­ trated on the structural response of elastic tank walls, as well as the rubber-made baffles prone to large deformation, structural response of metal baffles concerning strong hydroelastic effects hasn’t been deeply 3

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tank walls. Also, the report of structural resonance when the forced vi­ bration frequency is close to the wetted natural vibration frequency using a directly strong coupled FSI approach might be new, which might draw designers’ attention to avoid resonance between the wetted nat­ ural vibration and the leading excitation in practice. Consequently, the structural safety threat caused by the lack of consideration of hydro­ elasticity, which was emphasized by many literatures such as Amabili (2001) and Faltinsen and Timokha (2009), might be reduced.

Boundary condition at the fluid-structure interface is defined by the velocity and force continuity of the liquid domain and structure domain, written as:

Z

∂ θFf

��

The baffle is supposed to meet the thin plate assumption and cylin­ drical bending assumption, which can be simplified and discretized into two-dimensional Euler beams (Timoshenko and Woinowsky-Krieger, 1959) using Finite Element Method. From Eq. (4), the structural dy­ namic equation can be written as:

�

(3)

∂t þ r⋅ θuFf ¼ 0

where M is the global matrix of mass, C is the global matrix of damping, K is the global matrix of stiffness, F is the nodal force vector. At time t þ Δt, Eq. (11) is solved using Houbolt scheme with time step Δt:

� � � 2M=Δt2 þ 11C=ð6ΔtÞ þ K wtþΔt ¼ FtþΔt þ 5M=Δt2 þ �3C=Δt wt � 4M=Δt2 þ 3C=ð2ΔtÞ wt

where u, P, ρf , f f , ν , Ff and θ are the velocity, pressure, density, body force, kinematic viscosity of liquid, transportation volume of liquid and the ratio of the volume occupied by liquid (Lin, 2007). Note that tur­ bulence model is not applied in this paper, since turbulence is not the focus of the paper and there are different opinions on whether or which turbulence model should be used (Godderidge et al., 2009; Thiagarajan et al., 2011; Liu et al., 2016). The structures in the tank are assumed to be linear elastic with no material failure happening. Governing equation of the structure is given by the structure momentum conservation equation: s 2

2

s

ρ ∂ w = ∂t ¼ r⋅σ þ f

s

(4)

8 > tþΔt < Fnode j ¼ tþΔt > : Fnode

(5)

uτ ¼ 0

(8)

jþ1

_ tþΔt j). ðwÞ node

node_ jþ1.

where p, p0 and μ are the liquid pressure, air pressure and the dynamic viscosity coefficient. un and uτ are the normal component and the tangential component of velocity vector on the boundary respectively. Boundary condition at the solid wall is given as: (7)

(13)

�. � 2 Δy ptþΔt þ ptþΔt i;j i;j 1 �. � tþΔt tþΔt 2 ¼ Δy pi;jþ1 þ pi;j

(15)

Here, utþΔt iþ1=2;j is the liquid velocity at the right boundary of the grid (i,

(6)

un ¼ 0

(12)

2Δt

where Bcouple is conversion matrix function formed in coupling calcula­ tion. On the FSI interface sketched in Fig. 1, Eq. (9) and Eq. (10) can be discretized as: �. � _ tþΔt utþΔt 2 (14) _ tþΔt node j þ ðwÞ node jþ1 iþ1=2;j ¼ ðwÞ

Boundary condition at the free surface is given as:

p0

� � þ M=Δt2 þ C=ð3ΔtÞ wt

liquid velocity utþΔt couple at time step t þ Δt can be written as: � � tþΔt PtþΔt couple ¼ Bcouple ucouple

2.2. Boundary conditions

p þ 2μ∂un =∂n ¼

Δt

SIMPLE method (Patankar and Spalding, 1972) is used to solve the liquid governing equations. From Eqs. (1) and (2), the SIMPLE-based pressure Poisson equation can be obtained with a two-step projection method. Therefore, the relation between liquid pressure PtþΔt couple and

where w, ρs , f s and σs are the displacement, density, body force and first Piola-Kirchhoff stress tensor of the structure.

∂un = ∂τ þ ∂uτ =∂n ¼ 0

(11)

€ þ Cw_ þ Kw ¼ F Mw

(1) (2)

(10)

TÞ ¼ 0

2.3. Fluid-structure coupling scheme

A self-developed numerical sloshing tank is built in two-dimensional to carry out simulations of liquid sloshing, which solves the NavierStokes (N-S) equations and reconstructs the free surface with a Vol­ ume of Fluid (VOF) method. Due to the limitation of the solver, effects of bubble and entrapped air are excluded. The liquid in the tank is assumed to be viscous, Newtonian and incompressible. Governing equations of liquid includes the continuity equation, the momentum conservation equation and the volume transportation equation:

�

ðσs ns

where nf , ns and T are the unit outward normal vector of the liquid boundary, the unit outward normal vector of the structure boundary, and the normal traction force on the interface.

2.1. Governing equations

∂u = ∂t þ uru ¼ rP ρf þ f f þ ν r2 u

Z

σf n f þ

2. Numerical methods

r ⋅ ðθuÞ ¼ 0

(9)

u ¼ ∂w=∂t

j

_ tþΔt and ðwÞ node

ptþΔt i;j 1

jþ1

are velocities of the structure at node_ j and

, ptþΔt and ptþΔt i;j i;jþ1 are the liquid pressure at the center of the

tþΔt tþΔt grid (i, j-1), (i, j) and (i, jþ1). Fnode j and Fnode

jþ1

are the liquid forces

acting on node_ j and node_ jþ1. Eq. (14) and (15) can be further written into their matrix form: _ tþΔt utþΔt couple ¼ Acouple 1 w

4

(16)

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Fig. 1. Sketch of the fluid-structure interface. Note that superscript t þ Δt is omitted in figure.

FtþΔt ¼ Acouple 2 PtþΔt couple

Fig. 2 gives the flow chart of the fluid-structure coupling scheme, which is explained as:

(17)

tþΔt tþΔt where utþΔt are the liquid velocity, liquid pressure couple , Pcouple , and F

(1) Before FSI calculation begins, meshing of liquids and structures, motion of surge tank and initial simulation parameters are set in module ‘Set Initial Condition’. (2) At each time step of FSI calculation, tentative velocity of liquid is obtained in module ‘Calculate Temp Velocity’ using the two-step projection following the SIMPLE method. (3) Assembling of the matrices in Eq. (18) and solving of coupled FSI Eq. (19) using Houbolt scheme are then proceeded in module ‘Solve Coupled FSI Equation’. (4) When the coupled FSI equation is solved, structural response

and structure nodal force respectively. Acouple 1 and Acouple 2 are con­ version matrices generated in coupling calculation. Combining Eq. (13), (16) and (17), the structure nodal force can be written as: � (18) FtþΔt ¼ Acouple 2 Bcouple Acouple 1 w_ tþΔt

Substituting Eq. (18) into Eq. (12), the coupled FSI equation at time step t þ Δt can be finally obtained as:

� � � � � 2M=Δt2 þ 11C=ð6ΔtÞ þ K wtþΔt ¼ Acouple 2 Bcouple Acouple� 1 w_ tþΔt þ 5M=Δt2 �þ 3C=Δt �wt � 2 t Δt þ M=Δt2 þ C=ð3ΔtÞ wt 4M=Δt þ 3C=ð2ΔtÞ w

2Δt

(19)

including structural velocity is calculated, while liquid velocity at FSI interface is obtained through Eq. (16) in module ‘Get Struc­ ture Response’.

Here, Eq. (19) is a monolithically coupled FSI equation initially proposed by Hu et al. (2016), where detailed explanation and formation from Eq. (13)–(18) can be found as reference.

Fig. 2. Flow chart of the fluid-structure coupling scheme. 5

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(5) Using the obtained velocity at FSI interface, liquid pressure is corrected through SOR iteration of Eq. (13) in module ‘Correct Liquid Pressure’ until it meets the convergence criterion. (6) After Liquid pressure converges, real velocity of liquid is updated using the corrected liquid pressure to replace the tentative ve­ locity in module ‘Update Liquid Velocity’, and then the liquid free surface is reconstructed with VOF method. (7) So far, the FSI calculation for one time step is finished. Updating the time to the next step, calculation goes back to module ‘Calculate Temp velocity’ again and loops till the whole simula­ tion is over.

simulation times of the three groups are kept the same as 3.5s. Param­ eters of the water dam and the vertical wall are listed in Table 1. Fig. 3 shows the snapshots of the dam-breaking flow slamming the vertical wall, from which it is seen that the water dam collapses to the floor, reaches the wall, climbs along the wall and rolls back down to the floor. Significant interaction between the hydrodynamic water and the elastic wall happens during the slamming process, leading to the phe­ nomenon of hydroelasticity. Fig. 4 depicts the comparisons of dis­ placements at upper wall end and pressures at monitor PG among the three groups of grid sizes and time steps, showing that Group 2 is enough for the convergence of grid size and time step. As such, displacements at upper wall end and pressures at monitor PG among present results, re­ sults by Sun et al. (2015) and results by Zhang and Wan (2018) are further compared in Fig. 5, indicating that the present numerical methods is effective and accurate in the simulation of FSI considering hydroelasticity. Note that there still are some discrepancies among the present results, results by Sun et al. (2015) and results by Zhang and Wan (2018), which might be caused by the difference between FDM used in present method and MPS used in existing literatures. The small structural displacement assumption applied in present method would certainly contribute to the tolerable difference as well, which was excluded in their methods. It should be mentioned that the wetted natural vibration frequency of the wall in the first-period is seen as 1.0 Hz approximately, which is much smaller than the dry natural vibration frequency of the wall calculated as 1.7 Hz. This phenomenon reveals the strong added mass effect caused by hydroelasticity, since the elastic wall has to ‘carry’ the water in attach with it to move. As such, sharp reduction of the struc­ tural vibration frequency in the FSI process is expected considering hydroelasticity. Additionally, several examples of present method in dealing with hydroelastic response of the structures can further be found in Hu et al. (2016) and Qin et al. (2017). In these examples, a case of free oscillation of a vertical baffle (or bulkhead) in a liquid tank was intro­ duced in Section 5.3 of Hu et al. (2016). The case compares the hydroelastic response of baffles between using present numerical methods and using simple theoretical deductions under very similar conditions, that is, the conditions of baffled tank and small displacement assumption. The case might be very suitable in validating the hydroe­ lastic response in present paper, however, it has been contained in our previous published paper Hu et al. (2016). As such, the case is not included in present paper and it is suggested that readers who are interested in the case could refer to Hu et al. (2016).

Advantages of the developed FSI scheme lie in that it is a SIMPLEbased monolithically strong coupled scheme, that is, the structure and liquid are solved simultaneously at each time step. By replacing the liquid boundary velocity with the structural velocity at the FSI interface, the pressure Poisson equation in SIMPLE method is modified to form the monolithically strong coupled FSI equation Eq. (19), which was proven effective in FSI simulations with strong added mass effect (Hu et al., 2016) and transient slamming issues (Qin et al., 2017). Therefore, it is believed that the developed FSI scheme based on SIMPLE method has certain advantages in dealing with hydroelasticity problems such as the interaction between sloshing liquid and elastic baffle in this study. Limitations of the developed FSI scheme is that the structures are simplified as two-dimensional Euler beams without considering the ef­ fects of structure thickness, and response of the structure is limited within elastic range and no plastic behavior and structural failure is introduced in the scheme. Also, turbulence model and threedimensional effects are not included in the scheme. These limitations can be eliminated by further improving the FSI scheme with more complicated structures, turbulence model and three-dimensional development. 3. Validations on numerical methods 3.1. Validation on hydroelastic response In order to show the effectiveness of the present numerical methods in simulation of FSI events, a validation on the hydroelastic response of an elastic vertical wall slammed by a dam-breaking flow is carried out. The validation case is designed similar to the ones by Sun et al. (2015) and Zhang and Wan (2018) using the MPS-FEM method, in which a 0.1m � 0.2 m-size water dam breaks and hits a 0.304 m-high vertical wall, placed 0.352 m away from the water dam. The vertical wall is fixed at the lower end and free at the upper end. A pressure gauge PG is placed on the wall, 0.02 m from its lower end. Three groups of grid sizes and time steps are applied to test convergence first. Grid sizes are selected as 0.010 m � 0.010 m (Group 1), 0.005 m � 0.005 m (Group 2) and 0.003 m � 0.003 m (Group 3), while time steps are selected as 0.00020s (Group 1), 0.00010s (Group 2) and 0.00006s (Group 3). Under these selections of grid sizes and time steps, the maximum Courant number is about 0.05≪1, much smaller than the Courant number requirement for unknown fluid fields (John and Anderson, 1995). The element lengths of the vertical wall are kept the same with the grid lengths of the fluid in the three groups. Whole

3.2. Validation on liquid sloshing In order to show the effectiveness of present methods in simulation of liquid sloshing events, a convergence test and a validation on violent liquid sloshing containing impact phenomena are carried out. The validation is designed similar to the ones by Jung et al. (2012) which conducted numerical simulation using software FLUENT, and by Kang and Lee (2005) which conducted experimental tests in the ship designing of DSME Co., Ltd. The tank is with a length (L) of 0.80 m, a height (H) of 0.50 m and a liquid-depth (Hw) of 0.35 m. Two pressure gauges PG1 and PG2 are installed at the left side wall, 0.00525 m from the bottom, and at the top wall, 0.01 m from the left top corner to monitor the sloshing-induced liquid pressure. Three groups of grid sizes and time steps are applied to test convergence of grid and time, which are 0.010 m � 0.010 m (Group 1), 0.005 m � 0.005 m (Group 2) and 0.003 m � 0.003 m (Group 3). The whole simulation time is 16s, while time steps are selected as 0.0010s (Group 1), 0.0005s (Group 2) and 0.0003s (Group 3). The maximum Courant number is about 0.15≪1, which meets the Courant number criterion for unknown fluid fields (John and Anderson, 1995). The other parameters of the liquid are the same with the ones listed in the left column of Table 1. The sloshing tank is subject to pure surge motion following the si­

Table 1 Simulation parameters. Water dam

Vertical wall

Liquid kinematical viscosity Liquid density Acceleration of gravity

1.145 � 10 6 m2/ s 1000 kg/m3 9.81 m/s2

Elastic modulus Poisson’s ratio Wall density

Air pressure

101263.4 Pa

Wall thickness

0.2 GPa 0.3 7860 kg/ m3 0.006 m

6

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Fig. 3. Snapshots of a dam-breaking flow slamming a vertical wall.

Fig. 4. Comparisons of (a) wall displacements and (b) liquid pressures among three groups of grid sizes and time steps.

Fig. 5. Comparisons of (a) wall displacements and (b) liquid pressures among present results, results from Sun et al. (2015) and Zhang and Wan (2018).

nusoidal function: X ¼ Asurge sinðωsurge tÞ

PG2 using the three groups of grids and times, from which it is seen that the dependence on grid sizes and corresponding time steps is relatively low and Group 1 or Group 2 is enough for the accuracy of the simulation. Therefore, in the rest part of the paper, grid size of Group 2 is selected to balance the accuracy and efficiency of the simulations. Fig. 6 also gives the comparisons of liquid pressures at PG1 and PG2 among results from present method, the numerical results by Jung et al. (2012) and the experimental data by Kang and Lee (2005), indicating satisfactory agreement. Though there are some discrepancies at the sharp peak values of pressures at PG1 where strong slamming happens and some errors are allowed, the results from present method is much closer to the experimental data by Kang and Lee (2005) compared to the results by Jung et al. (2012). Therefore, present method is well validated in the

(20)

where X, Asurge and ωsurge are the distance, amplitude and circular frequency of surge motion. According to Jung et al. (2012), parameters of surge motion are selected as Asurge ¼ 0:02m (0.025L) and ωsurge ¼ 5:82rad=s respectively, the latter of which is of the same value with the 1st order natural frequency of the liquid in the tank. Fig. 6 shows the snapshots of the liquid sloshing phenomena in one period with velocity field presented using black arrows, from which it is seen that violent free surface evolution and slamming on the tank’s top wall occur due to the resonance between the surge motion and the liquid in the tank. Fig. 7 plots the comparisons of liquid pressures at PG1 and 7

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Fig. 6. Snapshots of liquid sloshing in an unbaffled tank.

Fig. 7. Comparisons of pressure time histories at (a) PG1 and (b) PG2.

8

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Fig. 8. Sketches and mesh plots of the sloshing tanks with horizontal and vertical baffles.

used to capture the high vibration frequency of the structural responses of the baffles. Parameters of the liquid and the steel baffle are listed in Table 2. Same to the simulation in Section 3.2, the tanks are subject to pure surge motion in the horizontal directions according to Eq. (20), with parameters selected as Asurge ¼ 0:02m (0.025L) and ωsurge ¼ 5:82rad=s respectively.

Table 2 Simulation parameters. Liquid

Steel Baffles 6

2

Liquid kinematical viscosity Liquid density Acceleration of gravity

1.145 � 10 m / s 1000 kg/m3 9.81 m/s2

Elastic modulus Poisson’s ratio Baffle density

Air pressure

101263.4 Pa

Baffle thickness

210 GPa 0.3 7860 kg/ m3 0.0005 m

4.2. Structural response of horizontal and vertical baffles of different configurations Structural response of horizontal baffles of different baffle height Hb and baffle length Lb is investigated firstly. As shown in Fig. 8a, two groups of horizontal baffle configurations are considered respectively. In Horizontal-Baffle Group 1 (shorted as HBG 1), non-dimensional baffle length Lb/Hw is kept as 0.6, while non-dimensional baffle height Hb/Hw is selected as 0.6, 0.8, 0.9, 1.0, 1.1 and 1.2 respectively. In HorizontalBaffle Group 2 (shorted as HBG 2), non-dimensional baffle length Lb/ Hw is selected as 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8 respectively, while nondimensional baffle height Hb/Hw is kept as 0.9. Structural response of the vertical baffle of different baffle height Hb and baffle length Lb is investigated secondly. As shown in Fig. 8b, two groups of vertical baffle configurations are considered respectively. In Vertical-Baffle Group 1 (shorted as VBG 1), non-dimensional baffle length Lb/Hw is kept as 0.6, while non-dimensional baffle height Hb/Hw is selected as 0.7, 0.8, 0.9, 1.0, 1.1 and 1.2 respectively. In VerticalBaffle Group 2 (shorted as VBG 2), non-dimensional baffle length Lb/ Hw is selected as 0.4, 0.6, 0.8, 1.0, 1.2 and 1.4 respectively, while nondimensional baffle height Hb/Hw is kept as 1.0 which means that the vertical baffle is mounted at the bottom of the tank.

simulation of liquid sloshing problems and is able to give prediction on sloshing phenomena and sloshing loads effectively. 4. Structural response of anti-sloshing baffles of different configurations 4.1. Numerical set-ups Structural response of anti-sloshing baffles of different configura­ tions is studied in this section in two tanks, one of which is with two horizontal baffles placed on the side walls and the other of which is with a vertical baffle placed above the mid-center of the bottom, illustrated in Fig. 8a and Fig. 8b. The length of tank is 0.8 m (L), the height of tank is 0.5 m (H) and the liquid-depth is 0.35 m (Hw), which are the same to the ones in Section 3.2. The horizontal baffles are fixed Hb from the bottom with a length of Lb, while the vertical baffle is fixed Hb from bottom to its upper end with a length of Lb. Baffle height Hb and length Lb of both the horizontal baffle and the vertical baffle are selected with different values to set up different baffle configurations in the tanks. For the horizontal baffles, the outside ends are fixed supported and the inside ends are simple supported. For the vertical baffle, the lower end is fixed sup­ ported and the upper end is simple supported. Grid size of 0.005 m � 0.005 m (Group 2) selected from Section 3.2 is applied in the liquid domain, while the finite element length is chosen as 0.005 m to divide the baffles, shown in Fig. 8c and d. The whole simulation time is 16s, while a much smaller time step of 0.00001s is

4.2.1. Horizontal baffles of different height Defining non-dimensional displacement as Db/Asurge, where Db and Asurge are the displacement of the baffle and surge amplitude of the tank, structural responses of the horizontal baffles with the different baffle heights in HBG 1 are plotted in Fig. 9. With the increase of baffle height, structural response of the horizontal baffle changes significantly. It is seen that response of the horizontal baffle mainly shows a 9

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Fig. 9. Non-dimensional displacements at baffle midpoint in HBG 1.

superposed vibration consists of the forced vibration caused by periodic surge motion, the wetted natural vibration affected by hydroelasticity, and the high (2nd and 3rd mainly) order harmonic vibration caused by nonlinearity of free surface. Before the horizontal baffle height reaches the liquid-depth when the baffle is fully immersed in liquid, the forced vibration dominates the structural response, while with the increase of Hb/Hw the wetted natural vibration becomes more significant, shown in Fig. 9a–d. In the meanwhile, for the response of immersed horizontal baffles, asymmetry between crest and trough in one forced vibration period appears and increases as Hb/Hw ascends, indicating that there is a global deformation in the vertical upward direction caused by extru­ sion of the sloshing liquid. After the horizontal baffle leaves the liquid and forms a gap between the baffle and the free surface, sloshinginduced liquid slamming on the baffle happens, leading to a mutation of structural response shown in Fig. 9e and f. On one aspect, baffle response shows transient characteristics with a much larger displace­ ment peak rather than the relative ‘smooth’ histories when immersed in the liquid. On the other aspect, the forced vibration frequency remains stable, yet the wetted natural vibration frequency changes sensitively due to the continuously changing wetted area and the nonlinear liquid slamming excitation. By using a fast Fourier transform (FFT) method on all the cycles of displacements, frequency and amplitude of the structural response are extracted, plotted in Fig. 10. As expected, the largest amplitude peak occurs at about 0.93 Hz (f1), corresponding to 5.82 rad/s of the circular

frequency of surge motion. From Fig. 10a–d, it is seen that there are two smaller peaks at 1.86 Hz (f2 ¼ 2f1) and 2.79 Hz (f3 ¼ 3f1) corresponding to the 2nd and 3rd order harmonic vibration frequencies respectively, and one small peak varies from 5.80 Hz to 8.47 Hz corresponding to the wetted natural vibration frequency. Due to the added mass effect, the wetted natural vibration frequency is much smaller than the dry natural vibration frequency of the horizontal baffle in the air, which is about 45.56 Hz. From Fig. 10e and f, it is seen that when the horizontal baffle is placed above the still liquid surface, the high order harmonic vibration amplitudes are enlarged, while many small peaks representing wetted natural vibrations can be observed from 7.93 Hz to 45.56 Hz, due to the frequently changing liquid-baffle interface in the sloshing-induced slamming events. Non-dimensional amplitudes of different vibration components changing with Hb/Hw are illustrated in Fig. 11a, while energy ratio of wetted natural vibration component against forced vibration component changing with Hb/Hw is displayed in Fig. 11b. The energy ratio is calculated by Ahydroelastic2 =A12 where A1 and Ahydroelastic are the amplitudes of the forced vibration and the wetted natural vibration. It is seen that with the increase of Hb/Hw between 0.6 and 1.0, the amplitude of forced vibration decreases sharply and the amplitude of wetted nat­ ural vibration increases slightly, indicating that the decrease of the forced vibration component is the main reason for the increase of energy ratio of wetted natural vibration component against forced vibration component. The amplitudes of the high order harmonic vibrations are 10

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Fig. 10. Corresponding FFTs of non-dimensional displacements in HBG 1.

Fig. 11. Plots of (a) non-dimensional amplitudes of vibrations and (b) energy ratios of the wetted natural vibration component against forced vibration component in HBG 1.

rather small, and there is a cross-over point of their Amplitude-Hb/Hw curves. When Hb/Hw is selected as 1.1 and 1.2, amplitudes of the forced vibrations are enlarged compared to the ones of the baffle placed near free surface because of the transient impact. The wetted natural vibra­ tion is changing on both the frequency and the amplitude due to the frequently changing liquid-baffle interface, and amplitudes of high order harmonic vibrations are greatly enlarged due to the strong

nonlinearity of liquid slamming. 4.2.2. Horizontal baffles of different length Similarly in HBG 2, structural responses of the horizontal baffles with the different baffle lengths are plotted in Fig. 12. It is seen that when the horizontal baffle is fixed at the height of Hb/Hw ¼ 0.9, response of the baffle varies with the change of baffle length obviously. As the 11

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Fig. 12. Non-dimensional displacements at baffle midpoint in HBG 2.

horizontal baffle length increases, structural response of the baffle in­ creases sharply, including both the forced vibration caused by periodic surge motion and the wetted natural vibration affected by hydro­ elasticity. Asymmetry between crest and trough in one forced vibration period is observed even more clearly than the one in HBG 1, which grows with the increase of horizontal baffle length Lb/Hw. When Lb/Hw is larger than 0.6, the trough reaches to a positive value, indicating that the global deformation caused by the difference of pressures beneath and upon the baffle is larger than the amplitude of the forced vibration, shown in Fig. 12d–f. Additionally, longer baffle reduces the structural rigidity and increases the natural vibration frequency, and thus the wetted natural vibration caused by hydroelasticity is with less period­ icity but larger amplitude. The FFT plots showing frequency and amplitude of the structural response are given in Fig. 13. Similarly to the results in HBG 1, the largest amplitude peak occurring at about 0.93 Hz (f1) representing the forced vibration component, and smaller peaks at 1.86 Hz (f2 ¼ 2f1) and 2.79 Hz (f3 ¼ 3f1) representing the 2nd and 3rd order harmonic vibra­ tion components can be observed. Besides, there appears another peak representing the wetted natural vibration component, the corresponding frequency of which varies from 3.17 Hz to 36.14 Hz, much smaller than the dry natural frequency of the horizontal baffle. Apparently, the change of horizontal baffle length leads to much larger variation of the wetted natural frequency compared to the change of horizontal baffle height, which is caused by the change of rigidity and thus the dry natural

vibration frequency of the baffle. Non-dimensional amplitudes of different vibration components changing with Lb/Hw are illustrated in Fig. 14a, indicating that the amplitudes of the forced vibration component, the wetted natural vi­ bration component and the high order harmonic vibration components increase with Lb/Hw due to the decrease of baffle rigidity. In the meanwhile, energy ratio of wetted natural vibration component against forced vibration component is plotted in Fig. 14b, showing that the energy is transmitting into the wetted natural vibration component from the forced vibration component with the increase of Lb/Hw. In other words, the increase of the horizontal baffle length leads to larger in­ fluence of hydroelasticity. Being different from the reason of the in­ crease of energy ratio in HBG 1, the increase of energy ratio here is caused by the quick growth of the wetted natural vibration component. In addition, amplitude of the 2nd order harmonic vibration increases much severely than the 3rd order harmonic vibration, which might because of the weaker free surface nonlinearity in the interaction be­ tween liquid and the longer horizontal baffle. 4.2.3. Vertical baffles of different height Fig. 15 gives the structural responses of the vertical baffles repre­ sented by non-dimensional displacements at baffle midpoint with the different baffle heights in VBG 1. It is seen that similar to the horizontal baffle cases, response of the vertical baffle also shows a superposed vi­ bration consists of the forced vibration caused by the periodic surge 12

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Fig. 13. Corresponding FFTs of non-dimensional displacements in HBG 2.

Fig. 14. Plots of (a) non-dimensional amplitudes of vibrations and (b) energy ratios of the wetted natural vibration component against forced vibration component in HBG 2.

motion, the wetted natural vibration affected by the hydroelasticity, and the 3rd order harmonic vibration caused by the nonlinearity of free surface. The structural response of the baffle decreases with the increase of Hb/Hw from 0.7 to 0.9 due to the enlarged gap between the lower end of baffle and the tank bottom, and then increases with the increase of Hb/Hw from 0.9 to 1.1 due to the strong overtopping of liquid across the upper end of baffle, depicted in Fig. 15a–e. When Hb/Hw is further increased from 1.1 to 1.2, the reduced interaction area might be the

reason of the decrease of structural response since quite a part of the baffle is exposed in the air, shown in Fig. 15e and f. In the meanwhile, the wetted natural vibration becomes much more significant as the vertical baffle is closer to the free surface of the liquid, indicating that the wetted natural vibration might be excited by the evolution of free surface when liquid overtopping across the upper end of baffle. Note that the wetted natural vibration can be easily observed in Fig. 15d, and is almost invisible in Fig. 15a. 13

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Fig. 15. Non-dimensional displacements at baffle midpoint in VBG 1.

Frequency and amplitude of the structural response are extracted and plotted in Fig. 16 using a FFT method. As expected, the largest amplitude peak occurs at about 0.93 Hz (f1) corresponding to the forced vibration component, while one peak at 2.79 Hz (f3 ¼ 3f1) can be seen corresponding to the 3rd order harmonic vibration component. Compared to these two peaks, the peak representing the wetted natural vibration component is relatively smaller, and is much observable in Fig. 16c and d, which matches with the description before. The wetted natural vibration frequency ranges from 6.45 Hz to 15.14 Hz approxi­ mately as Hb/Hw increases from 0.7 to 1.2 due to the change of added mass effect, including the change of wetted area and the liquid acceleration. Non-dimensional amplitudes of different vibration components changing with Hb/Hw are illustrated in Fig. 17a, showing that the am­ plitudes of forced vibration and the wetted natural vibration reach their maximums with a non-dimensional baffle height of 1.1 and 1.0 respec­ tively. Amplitude of the 2nd order harmonic vibration generally reaches the maximum with a non-dimensional baffle height of 0.9, and becomes insignificant with a non-dimensional baffle height over 1.0. Energy ratio of wetted natural vibration component against forced vibration component is plotted in Fig. 17b, from which it is seen that the energy ratio has a peak value with a non-dimensional baffle height of 1.0. Here, the maximum energy ratio is mainly induced by the increase of the wetted natural vibration component, revealing that hydroelastic effects

are most significant when the upper end of baffle is placed near the free surface. 4.2.4. Vertical baffles of different length Similarly in VBG 2, structural responses of the vertical baffles with the different baffle lengths are plotted in Fig. 18. It is seen that with the increase of the vertical baffle length, the structural response of the baffle increases rapidly since longer baffle leads to smaller rigidity. Although the superposition characteristics can be found through observation of the displacement time histories, the high order harmonic vibration and the wetted natural vibration components are not obvious, except for the case of Fig. 18c where significant superposition of vibration components is seen. The structural response is superposed by the forced vibration, the 3rd order harmonic vibration and the wetted natural vibration in most cases. Interestingly, in the case of Fig. 18c, the superposition is special because the frequencies of the 3rd order harmonic vibration and the wetted natural vibration are very close, which produces large vi­ bration from the storaged vibrational energy and thus leads to the structural resonance. Therefore, structural response dominated by the forced vibration is severely influenced by the resonance between the 3rd order harmonic vibration and the wetted natural vibration, showing significant superposition characteristics. The FFT plots showing frequency and amplitude of the structural response are given in Fig. 19, which displays a main peak at about 14

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Fig. 16. Corresponding FFTs of non-dimensional displacements in VBG 1.

Fig. 17. Plots of (a) non-dimensional amplitudes of vibrations and (b) energy ratios of the wetted natural vibration component against forced vibration component in VBG 1.

0.93 Hz (f1) representing the forced vibration component, and a smaller peak at 2.79 Hz (f3 ¼ 3f1) representing the 3rd order harmonic vibration component. Frequency of the wetted natural vibration varies from 2.25 Hz to 13.73 Hz due to the different wetted area, while amplitude of the wetted natural vibration component changes greatly with the in­ crease of Lb/Hw and reaches the maximum at 2.74 Hz. From Fig. 19c, it is seen that the resonance between the 3rd order harmonic vibration and

the wetted natural vibration leads to a highly-raised superposed peak, which is much larger than all the components of the 3rd order harmonic vibration and wetted natural vibration in other cases where structural resonance doesn’t happen. The amplitudes of different vibration components are illustrated in Fig. 20a, from which it is seen that the amplitude of forced vibration increases with Lb/Hw due to the increase of baffle flexibility, in the 15

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Fig. 18. Non-dimensional displacements at baffle midpoint in VBG 2.

meanwhile, the amplitudes of 3rd order harmonic vibration and wetted natural vibration reach the maximums at Lb/Hw ¼ 0.8 due to the structural resonance. Fig. 20b gives the energy ratio of wetted natural vibration component against forced vibration component, showing a sharp peak when resonance between the 3rd order vibration and the wetted natural vibration happen. The greatly enlarged energy ratio at Lb/Hw ¼ 0.8 reveals the energy superposition between different vibra­ tion components, which should be paid attention to from a structural point of view.

determined not only by the suppression degree but also by the frequency relation of different components. On another aspect, the parameter change of the baffles actually affects the liquid-baffle interaction in three aspects, the acting way of liquid loads, the significance of added mass effect, and the change of inherent structural property. The acting way of liquid loads can be categorized into a mild long-process way and a vi­ olent transient way, the latter of which includes obvious slamming phenomenon and leads to completely different structural response pattern. The significance of added mass effect affects the wetted natural vibration frequency in hydrodynamic-elasticity interaction, while the inherent structural property not only affects the wetted natural vibration frequency but also the vibration amplitude. For the horizontal baffles, the height change of horizontal baffles would influence the acting way of liquid loads and the significance of added mass effect. Fully uncovered baffles placed upon the free surface lead to slamming, which certainly change the acting way of liquid loads from a milder way to a violent way. As the horizontal baffles are placed higher, the added mass of liquid around the baffles reduces but the structural mass remains the same, and thus the added mass effect is less significant. The length change of horizontal baffles would influence the significance of added mass effect and the inherent structural property, since longer baffles lead to larger wetted area which would cause greater added mass, and smaller structural rigidity. It is seen that the responses between baffles fully covered in liquid and uncovered baffles

4.2.5. Discussions on physical mechanisms From the observations of displacement time histories, corresponding FFTs and energy ratios of the wetted natural vibration component against forced vibration component, some physical mechanisms can be analyzed. Generally, responses of the baffles are composed of compo­ nents of the forced vibration caused by the sloshing motion, high order harmonic vibration caused by nonlinearity of free surface and the wetted natural vibration affected by hydroelasticity respectively. It needs to mention that, hydroelasticity influences the wetted natural vibration in many ways. Firstly, the hydrodynamic loads, especially the loads from nonlinear free surface evolutions, would excite the wetted natural vibration. Secondly, the added mass effect would influence frequency of the wetted natural vibration in the hydrodynamic-elasticity interaction. Lastly, amplitude of the wetted natural vibration is 16

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more attenuation of the free surface, without influencing the liquidbaffle interaction mechanism. For the vertical baffles, the height change of the vertical baffle would influence the acting way of liquid loads and the significance of added mass effect. As the vertical baffle is located higher, the added mass re­ duces especially when the baffle is exposed in the air. Also when the baffle upper end is close to the free surface, liquid overtopping behavior would contribute to the excitation of wetted natural vibration, although no violent slamming happens. The length change of the vertical baffle would influence the significance of added mass effect and the structural property, which is similar to the horizontal baffles. In addition, when the baffle upper end is close to the free surface, acting way of liquid loads changes as well since liquid overtopping behavior happens, which is similar to the change of vertical baffle height. It is seen that responses of baffles with different height and length share similar tendency, indi­ cating slamming plays an insignificant role. The forced vibration am­ plitudes of the baffles are governed by the long-term hydrodynamic loads from the periodic sloshing motion. The wetted natural vibration frequencies are governed by the structural rigidity and wetted area when baffle length and height change. A special case is found when frequency of the wetted natural vibration is close to the high order harmonic vibration. Under structural resonance conditions, the storaged vibrational energy is released to a large extent through the greatly enlarged displacement, which would strongly influence the velocity and pressure of the liquid around the baffle. As such, hydroelasticity, or in other words the interaction between the hydrodynamic liquid and the elastic baffle, should be carefully considered in the liquid-baffle inter­ action process. In fact, the structural resonance happening in this Fig. 18c is not dangerous enough since the wetted natural vibration resonant with the 3rd order harmonic vibration, the latter of which contains merely a small part of energy. However, this phenomenon shows the real possi­ bility of structural resonance of the anti-sloshing baffle in a tank. Because of the hydroelastic effects, frequency of the wetted natural vi­ bration is greatly decreased to a level that might be close to the fre­ quency of tank’s motion. Therefore, under certain conditions, resonance between the forced vibration and the wetted natural vibration might occur as well. When structural resonance at the leading excitation fre­ quency happens, the storaged vibrational energy of the baffle would be greatly released, causing sharp amplification of the baffle response (displacement, velocity, etc.). The amplified baffle response would in turn influences the liquid domain through the strong-coupling between liquid and baffle, leading to sharply enlarged liquid pressure (sloshing loads). Note that when this structural resonance happen, the liquid domain is usually affected in a quite large area, and even the sloshing loads acting on the side walls of the tank are influenced to a great extent. Analysis of structural resonance at the leading excitation frequency will be shown in Section 4.3.

Fig. 19. Corresponding FFTs of non-dimensional displacements in VBG 2.

withstanding slamming are quite different in pattern. The forced vi­ bration amplitudes of the fully covered baffles are governed by the hy­ drodynamic loads induced by the periodic sloshing motion, which is a relatively long-term process as the free surface is suppressed by the baffles and no drastic change of wetted area. The wetted natural vi­ bration frequencies, on another aspect, are mainly affected by the in­ crease of added mass as the baffles approach the free surface, which is easy to understand since the liquid upon baffles becomes thinner. However, when the baffles are placed above the free surface, the reduction of free surface suppression leads to larger hydrodynamic loads, while the violent change of wetted area leads to various liquid added mass. As such, slamming happens and responses of the baffles are dominated by the nonlinear slamming loads in amplitude and the continuously changing added mass in frequency. The change of length affects the horizontal baffles in a much milder way. Placed near the free surface, the baffles attenuate the free surface greatly even when the baffle length is relatively small, since the liquid energy is mostly focused on the free surface evolutions. Therefore, longer baffle only leads to

Fig. 20. Plots of (a) non-dimensional amplitudes of vibrations and (b) energy ratios of the wetted natural vibration component against forced vibration component in VBG 2. 17

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4.3. Analysis on hydroelasticity: possible structural resonance at the leading excitation frequency In Section 4.2, structural responses of horizontal and vertical baffles are studied, showing that the baffle response is superposed by the forced vibration induced by periodic surge motion, the wetted natural vibra­ tion affected by hydroelasticity, and the high order harmonic vibration caused by nonlinearity of free surface. The influence of baffle height and baffle length on the baffle response are investigated, from the results of which a case including the resonance between the 3rd order harmonic vibration and the wetted natural vibration is found, depicted in Figs. 18c and 19c. Caused by the resonance, the amplitude of the superposed vibration at 2.74 Hz is greatly enlarged, although the resonance is not induced by the superposition containing the forced vibration, which is usually the main component of structural response. Therefore, one may wonder is there a possibility of the resonance between the forced vibration and the wetted natural vibration, and what would be the phenomenon if the resonance happens. In this section, a research focused on the resonance between the forced vibration and the wetted natural vibration is carried out to solve the question and give an intuitive insight on the structural resonance of the anti-sloshing baffles happening in liquid sloshing events. Main dimensions of the tanks, governing equation of surge motion and simulation parameters of liquid and baffle remain the same with the ones in Section 4.1. Two study cases on the structural resonance of the baffles are considered, including a horizontal baffle case and a vertical baffle case, with parameters of baffle configuration and tank’s surge motion listed as follows. Case 1: Horizontal baffle, Hb/Hw ¼ 0.9, Lb/Hw ¼ 0.8, Asurge ¼ 0.001 m, ωsurge ¼ 19.92 rad/s. Case 2: Vertical baffle, Hb/Hw ¼ 0.8, Lb/Hw ¼ 0.8, Asurge ¼ 0.001 m, ωsurge ¼ 17.22 rad/s. Note that being different from the simulations in Section 4.2, the surge amplitude selected here is a small value, so that when severe structural resonance happens the baffles are still within linear elastic range. In addition, in order to meet the structural resonance require­ ment, the surge frequency of the tank is selected according to the wetted natural vibration frequency of the baffles, which are 3.17 Hz (19.92 rad/ s) for the horizontal baffle and 2.74 Hz (17.22 rad/s) for the vertical baffle under the simulations conditions in this research.

Fig. 22. Comparison of (a) sloshing forces on the elastic and rigid horizontal baffles and (b) corresponding FFT in a structural resonance case.

affected by hydroelasticity. The identical frequency of the forced vi­ bration and the wetted natural vibration leads to the superposition of these two vibration components and produces the large amplitude vi­ bration due to the storage of vibrational energy, or in other words, in­ duces the structural resonance of the baffle. Being different from the structural resonance depicted in Figs. 18c and 19c, the structural reso­ nance here is caused by the leading component of vibrations in the baffle response, therefore, the storage vibrational energy is greatly excited and transmitted into the kinetic energy of the horizontal baffle, causing the large amplitude vibration. The non-dimensional amplitude extracted from FFT plot is about 9.16, almost hundred times of the largest nondimensional amplitude in Figs. 9 and 12, indicating that the structural resonance is extremely dangerous from a structural point of view. Note that limited by the linear elastic assumption of the structure in the FSI solver, the simulation in Case 1 is conducted with small surge amplitude, Asurge ¼ 0.001 m. Under conditions when the surge amplitude is large, structure failure of the anti-sloshing baffle is certainly expected. Defined as Ftotal/(ρgHwLb) where Ftotal is the total force integrated along the baffle, comparisons of non-dimensional total forces on the horizontal baffle and the corresponding FFT plots with/without considering hydroelasticity are illustrated in Fig. 22. It is seen that the oscillation periods of the total forces are identical, corresponding to the resonant frequency at about 3.17 Hz, however, the oscillation ampli­ tudes differ greatly. Non-dimensional amplitudes of the total forces on the horizontal baffle with/without considering hydroelasticity are about 0.161 and 0.024 respectively, indicating that when structural resonance happens the force acting on the baffle is significantly increased. Fig. 23 gives the comparison of non-dimensional total forces on the left-side wall of the tank and corresponding FFT plots considering elastic/rigid horizontal baffles. Note that only the hydrodynamic parts of the forces are shown in the figure. It is seen that although the tank wall is rigid, the hydrodynamic-elasticity interaction happening at the elastic baffle would greatly affects the liquid field under structural resonance condition in a wide range and leads to amplification of total force on the tank wall. Therefore, structural resonance of the baffle not only leads to significantly increased force on the baffle itself, but also causes greatly enlarged hydrodynamic force on the tank wall, even it

4.3.1. Structural resonance of the horizontal baffle Fig. 21 gives the structural response and corresponding FFT plot of the horizontal baffle in its non-dimensional form in simulation Case 1. It is seen that compared to the ones in Figs. 9 and 12, the non-dimensional displacement in Case 1 is greatly enlarged. In the meanwhile, the FFT plot of the non-dimensional displacement shows only one high peak at about 3.17 Hz, which is not only the frequency of the forced vibration caused by surge motion, but also the wetted natural vibration frequency

Fig. 21. Plots of (a) displacement at the horizontal baffle midpoint and (b) corresponding FFT in a structural resonance case. 18

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Fig. 23. Comparison of (a) sloshing forces on the rigid left-side wall considering elastic and rigid horizontal baffles and (b) corresponding FFT in a structural resonance case.

Fig. 24. Plots of (a) displacement at the vertical baffle midpoint and (b) corresponding FFT in a structural resonance case.

happens at the baffle rather than the tank wall. It is important that this phenomenon should not be ignored from a structural safety point of view. From both the structural response of the baffle and the liquid forces acting on the baffle, it is found that structural resonance induced by the identical frequency of the forced vibration and the wetted natural vi­ bration can cause greatly enlarged amplitudes of displacement and forces. Note that the dry natural vibration frequency of the baffle with selected parameters is about 23.15 Hz, which is much larger than the wetted natural frequency 3.17 Hz and is large enough to rule out the possibility to meet with the frequency of tank’s surge motion. Therefore, it is reasonable to conclude that the added mass of hydroelasticity leads to the reduction of the wetted natural frequency of the baffle, and thus leads to a higher possibility of structural resonance when the reduced natural vibration frequency is close to the excitation frequency of the

Fig. 26. Comparison of (a) sloshing forces on the rigid left-side wall consid­ ering elastic and rigid vertical baffles and (b) corresponding FFT in a structural resonance case.

Fig. 25. Comparison of (a) sloshing forces on the elastic and rigid vertical baffles and (b) corresponding FFT in a structural resonance case. 19

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tank’s surge motion.

sloshing baffles of different configurations are studied numerically through a self-developed FSI solver, which solve the N-S equations in the liquid domain and the monolithically coupled FSI equation on the fluidstructure interface. Validation on the hydroelastic response using pre­ sent numerical methods is conducted with the case of a dam-breaking flow slamming an elastic vertical wall, from which satisfactory agree­ ments of displacement and pressure among present results, results by Sun et al. (2015) and results by Zhang and Wan (2018) are reached. Convergence and validation on sloshing loads are conducted in a surging tank, the results of which agree well with the numerical results by Jung et al. (2012) and experimental data by Kang and Lee (2005). Investigation on the structural response of anti-sloshing baffles of different configurations considering hydroelaticity is further carried out in two surging tanks with horizontal and vertical baffles. Effects of baffle height and length on the structural response of the baffles are examined in a series of simulations. In the meanwhile, vibration components of structural response are analyzed and significance of the wetted natural vibration affected by hydroelasticity is revealed. Phenomena of struc­ tural resonance of baffles are obtained, drawing attention from a structural safety point of view. From the investigation, important con­ clusions can be drawn as follows.

4.3.2. Structural resonance of the vertical baffle For the simulation Case 2, structural response and corresponding FFT plot of the vertical baffle in its non-dimensional form in Fig. 24, from which it is seen that similar to Case 1, the non-dimensional displacement significantly increased due to the structural resonance of the baffle. One high peak in the FFT plot is observed at about 2.74 Hz, which is the frequency of the forced vibration caused by surge motion and the wetted natural vibration frequency affected by hydroelasticity. On one aspect, compared to the non-dimensional displacement of the horizontal baffle in Fig. 21 in Case 1, the non-dimensional displacement of the vertical baffle in Case 2 is even larger, which might be caused by the motion direction of the tank and more significant hydroelastic effects. On the other aspect, the structural resonant frequency of the vertical baffle is lower than the one of the horizontal baffle, which in turn proves that hydroelasticity, or the added mass effect in particular, is more obvious. It should be mentioned since the frequency of tank’s surge motion is usually low, thus the vertical baffle with lower wetted natural vibration frequency might be more prone to structural resonance than the hori­ zontal baffle. Comparisons of non-dimensional total forces on the vertical baffle and the corresponding FFT plots with/without considering hydro­ elasticity are illustrated in Fig. 25, from which the non-dimensional amplitudes of the total forces on the vertical baffle with/without considering hydroelasticity are 0.294 and 0.022 approximately. Note that corresponding to the larger amplitude of displacement, amplitude of total force on the vertical baffle is larger than the one on the hori­ zontal baffle as well since hydroelasticity acts in a mutual way between liquid and baffle. Fig. 26 gives the comparison of non-dimensional total forces on the left-side wall of the tank and corresponding FFT plots considering elastic/rigid vertical baffles. Note that only the hydrodynamic parts of the forces are shown in the figure. Similar to the horizontal baffle case, the total force on the tank wall is greatly amplified by the increased liquid pressure induced from the structural resonance happening at the elastic baffle compared with the force in the rigid baffle case. However, from the comparison between Figs. 23 and 26, it is observed that larger amplification occurs in the horizontal baffle case than the vertical baffle case. This might because that the elastic horizontal baffles are closer to the tank wall, as such the hydrodynamic-elasticity interaction would affect greater the liquid field near the tank wall. Additionally, the block effect of the vertical baffle would suppress the translation of the lower part liquid in the tank, which might contribute to the distinction as well. Nevertheless, structural resonance of the baffle should be avoided even if the force amplification in the vertical baffle case is smaller. In conclusion, the simulation results in Case 1 and Case 2 show that structural resonance at the frequency of the leading vibration compo­ nent happens when the frequency of forced vibration caused by the surge motion is close to the frequency of wetted natural vibration. The occurrence of structural resonance of the baffle leads to greatly enlarged amplitudes of structural response and liquid forces as a result of the maximal excitation of the storaged vibrational energy. In addition, the significantly reduced wetted natural vibration frequency caused by hydroelastic effects surely leads to higher possibility of structural reso­ nance of the anti-sloshing baffle. Since the main dimensions of sloshing tanks in real size is much larger than the ones of the model tanks in this paper, the structural resonant frequency might be even lower, indicating that structural resonance of the baffle in real size might be more likely to occur. Therefore, attention should be paid on the structural response and the possible structural resonance of anti-sloshing baffles from a struc­ tural safety point of view.

(1) Structural response of the horizontal and vertical baffles shows a vibration superposed by the forced vibration caused by periodic surge motion, the wetted natural vibration affected by hydro­ elasticity, and the high (2nd and 3rd mainly) order harmonic vibration caused by nonlinearity of free surface. (2) With the increase of baffle height before the horizontal baffle reaches the still free surface, amplitudes of the forced vibration and wetted natural vibration of the horizontal baffle decreases, while the wetted natural vibration frequency increases due to weaker added mass effects. When the horizontal baffle is placed above the still free surface, severe slamming caused by sloshing liquid happens and causes a sharp increase of the baffle response. (3) With the increase of baffle length of the horizontal baffle, am­ plitudes of the forced vibration and wetted natural vibration in­ creases, while the wetted natural vibration frequency decreases rapidly due to the decrease of structural rigidity. Compared with the increase of forced vibration, the wetted natural vibration increases even higher, leading to the quick increase of energy ratio of the wetted natural vibration against the forced vibration. (4) Amplitudes of the forced vibration and wetted natural of the vertical baffle reach the maximum when the vertical baffle height is close the still free surface height, as well as the energy ratio of the wetted natural vibration against the forced vibration. With the increase of vertical baffle height, frequency of the wetted natural vibration increases due to the reduction of wetted area. (5) Amplitude of the forced vibration of the vertical baffle increases with the increase of vertical baffle length, while amplitude of the wetted natural vibration reaches the maximum when its fre­ quency is close to the frequency of the 3rd order harmonic vi­ bration. The resonance between the two vibration components leads to a greatly enlarged energy ratio of the wetted natural vibration against the forced vibration, showing that consider­ ation of hydroelasticity is necessary. (6) Hydroelasticity, or the added mass effect in particular, usually leads to substantial reduction of natural vibration frequency of the structure, making it possible to be close to the excitation frequency of the tank’s motion. When the frequency of the wetted natural vibration affected by hydroelasticity approaches the fre­ quency of the forced vibration caused by the surge motion, severe structural resonance happens due to the excitation of the storaged vibrational energy. Under this condition, amplitudes of displacement and forces are sharply increased to an extreme extent, which should be carefully paid attention to from a struc­ tural safety point of view.

5. Conclusions In this paper, structural responses of horizontal and vertical anti20

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(7) Structural resonance at the elastic baffle not only affects the total force on the baffle, but also amplifies the total force on the tank wall. When structural resonance happens at the leading excita­ tion frequency, the greatly enlarged baffle response would strongly influences the liquid pressure in a rather large area, leading to the enlarged force even on the tank wall. This phe­ nomenon further emphasizes the important of considering hydroelasticity and the possible structural resonance happening in the tank.

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To reaffirm the investigation purpose, this paper aims to study the structural response of anti-sloshing baffles of different configurations considering hydroelasticity through numerical simulations. The possible structural resonance induced by the substantially reduced wetted nat­ ural vibration frequency due to hydroelastic effects is also examined using a direct strong coupled FSI scheme. The results of the influences of baffle configurations on baffle responses and the report of structural resonance considering hydroelasticity might be of reference value in the preliminary structural design or safety assessment stages. Limitations of the paper lie in the ignorance of the baffle thickness, structural plas­ ticity, turbulence model and the three-dimensional effects, which are suggested to be investigated in the future. In addition, this present paper only considers simple baffles in two-dimensional, and more complex baffles such as ring baffles, perforated baffles, porous baffles, etc. should be further taken into account for practical engineering purpose. Defor­ mation of elastic tank wall should be considered for the hydroelasticity problems as well since wall structures might bear even larger sloshing loads especially when slamming happens. Acknowledgement This work was supported by the National Key Research and Devel­ opment Program of China (Grant No. 2017YFC1404700), the Discipline Layout Project for Basic Research of Shenzhen Science and Technology Innovation Committee (Grant No. 20170418), the Guangdong Special Fund Program for Marine Economy Development (Grant No. GDME2018E001). The authors would like to thank the anonymous reviewers for their meaningful comments. References Akyildiz, H., Ünal, E., 2005. Experimental investigation of pressure distribution on a rectangular tank due to the liquid sloshing. Ocean. Eng. 32 (11–12), 1503–1516. Akyildiz, H., 2012. A numerical study of the effects of the vertical baffle on liquid sloshing in two-dimensional rectangular tank. J. Sound Vib. 331 (1), 41–52. Amabili, M., 2001. Vibrations of circular plates resting on a sloshing liquid: solution of the fully coupled problem. J. Sound Vib. 245 (2), 261–283. Askari, E., Jeong, K.H., Amabili, M., 2013. Hydroelastic vibration of circular plates immersed in a liquid-filled container with free surface. J. Sound Vib. 332 (12), 3064–3085. Belakroum, R., Kadja, M., Mai, T.H., Maalouf, C., 2010. An efficient passive technique for reducing sloshing in rectangular tanks partially filled with liquid. Mech. Res. Commun. 37 (3), 341–346. Biswal, K.C., Bhattacharyya, S.K., 2010. Dynamic response of structure coupled with liquid sloshing in a laminated composite cylindrical tank with baffle. Finite Elem. Anal. Des. 46 (11), 966–981. Colagrossi, A., Lugni, C., Brocchini, M., 2010. A study of violent sloshing wave impacts using an improved SPH method. J. Hydraul. Res. 48 (S1), 94–104. Celebi, M.S., Akyildiz, H., 2002. Nonlinear modeling of liquid sloshing in a moving rectangular tank. Ocean. Eng. 29 (12), 1527–1553. Cho, J.R., Lee, H.W., Ha, S.Y., 2005. Finite element analysis of resonant sloshing response in 2-D baffled tank. J. Sound Vib. 288 (4–5), 829–845. Chu, C., Wu, Y., Wu, T., Wang, C., 2018. Slosh-induced hydrodynamic force in a water tank with multiple baffles. Ocean. Eng. 167, 282–292. Cho, I.H., Choi, J.S., Kim, M.H., 2017. Sloshing reduction in a swaying rectangular tank by a horizontal porous baffle. Ocean. Eng. 138, 23–34. Ebrahimian, M., Noorian, M.A., Haddadpour, H., 2013. A successive boundary element model for investigation of sloshing frequencies in axisymmetric multi baffled containers. Eng. Anal. Bound. Elem. 37 (2), 383–392. Eswaran, M., Saha, U.K., Maity, D., 2009. Effect of baffles on a partially filled cubic tank: numerical simulation and experimental validation. Comput. Struct. 87 (3–4), 198–205.

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