numerical study of 3-d liquid sloshing in an elastic tank

Journal of Ship Research, Vol. 00, No. 0, Month 2019, pp. 1–11http://dx.doi.org/10.5957/JOSR.09180082

Numerical Study of 3-D Liquid Sloshing in an Elastic Tank byMPS-FEM Coupled Method

Xiang Chen, Youlin Zhang, and Decheng Wan

State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean andCivil Engineering, Shanghai Jiao Tong University, CollaborativeInnovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, China

The sloshing phenomenon in a partially loaded oil tanker or liquid natural gas ship is atypical fluid–structure interaction problem involvingmulti-physics, violent free-surfaceflow, and nonlinear structural response. In the past decades, the complex phe-nomenon has been commonly investigated without consideration of the hydro-elasticbehaviors of the bulkheads. In our previous work, the liquid sloshing phenomenon in atwo-dimensional (2-D) elastic tank was numerically studied. However, the bulkheadsof the tank will deform within the three-dimensional (3-D) space in reality. So, it isnecessary to study the 3-D sloshing problem in an elastic tank. In this article, a hybridapproach is developed within the Lagrangian system. The moving particle semi-implicit (MPS) method is used to simulate the evolution of 3-D flow with a violent freesurface, and the finite element method (FEM) is used for the numerical analysis ofstructural response due to the impact loads of the sloshing flow. To couple the MPSmethod and the FEMmethod, an interpolation scheme based on the kernel function ofthe particle method is proposed for the communication on the isomerous interfacebetween the fluid and structure domains. The reliability of force and deformationinterpolation modules is validated by two tests. Then, the sloshing phenomenon in a3-D elastic tank is numerically investigated and compared against the previouspublished 2-D results. By varying the Young’smodulus of the tankwalls, characteristicsregarding the evolutions of free surface, variation of impact pressures, and dynamicresponses of the structures are presented.

1. Introduction

To support the transportation demands of natural resources, moreandmore vessels, such as the very large crude carriers and the liquidnatural gas carriers, are manufactured. For these huge structures,risks such as local deformation or even damage of cargo-containmentsystems resulting from sloshing phenomena subsequently increase.Therefore, it is necessary to take the elasticity of the tank walls intoaccount in the research of sloshing phenomena (Dias & Ghidaglia2018). However, the phenomena involving the vibrations of the tankwalls are complex. In the process of sloshing wave interacting withelastic bulkheads, the fragmentation, splash, and fusion of water are

observed. Meanwhile, the structures vibrate nonlinearly under theimpact loads resulting from the sloshing wave. These phenomenaare hard to simulate realistically by the traditional mesh-basedmethods.

Comparatively, the meshless methods are predominant in sim-ulating these phenomena because they are flexible in dealing withstructural deformation and tracking of the violent free surface, and itis unnecessary to cope with the nonlinear convective term in themomentum equation. The moving particle semi-implicit (MPS)method, which is a representative particle method for in-compressible flow, was introduced into the simulation of fluid–structure interaction (FSI) problems by coupling with the finiteelement method (FEM) method in recent years. Till now, severaltwo-dimensional (2-D) FSI problems have been numerically in-vestigated by the MPS-FEM coupled method. For example, thebenchmark of dam-break flow impacting onto an elastic obstacle is

Manuscript received by SNAME headquarters November 7, 2018; acceptedDecember 17, 2018.

Corresponding author: Decheng Wan, [email protected]

MONTH 2019 JOURNAL OF SHIP RESEARCH 10022-4502/19/nnnnnn-0001$00.00/0

http://dx.doi.org/10.5957/JOSR.09180082

mailto:[email protected]

simulated by Mitsume et al. (2014a, 2014b). Hwang et al. (2016)used the MPS method to investigate the sloshing phenomenon inpartially filled rectangular tanks with elastic baffles. Zha et al.(2017) developed an improved MPS method to solve the hydro-elastic response of a wedge entering calm water. According to theseresults, theMPS-FEM coupled method shows great prospect in thesimulation of FSI problems. In these aforementioned 2-D simu-lations, the beam element is used for the calculation of structuralresponse, and the boundary particles of fluid domain coincide withthe structural nodes of structure domain on the interface. However,the beam element is not applicable for the three-dimensional (3-D)structure analysis any more. And it is a time-consuming task ofsimulation, whereas the consistent boundary is used on the interface.To address the practical 3-D FSI problems, further extension of theMPS-FEM coupled method should be conducted.

In our previous studies (Zhang et al. 2016; Zhang&Wan 2018),an in-house solver based on the MPS-FEM coupled method isdeveloped and used to simulate the sloshing flow interacting withelastic bulkheads in 2-D space. Continuing along the path ofprevious studies, we devote ourselves to extending the MPS-FEMcoupled method for the 3-D sloshing problem in the present ar-ticle. The thin-plate element which has four nodes within anelement is adopted to predict the structural response of the 3-Dtank. A kernel function–based interpolation (KFBI) technique isproposed for the data communication crossing the interface be-tween the fluid and the structure domains. With the benefit of theproposed technique, the spaces of the fluid domain and structuraldomain can be dispersed by particles and elements with differentspacing sizes. Based on the extended MPS-FEM method, thecomplex liquid sloshing phenomena in a 3-D elastic tank canbe simulated. To numerically investigate the influences of thestructural elasticity on the sloshing phenomena, the evolutions ofthe free surface together with the impact pressures acting on thelateral walls of the elastic tank are compared against those of therigid tank.

2. Numerical approach

In this article, an in-house solver MPSFEM-SJTU (MovingParticle Semi-implicit and Finite Element Method by Shanghai JiaoTong University) is developed for FSI problems with a violent freesurface. The solver mainly consists of three modules, the fluiddomain simulation module based on the meshless MPS method, thestructural domain calculation module based on the FEM method,and the interpolation module for data transformation between thefluid and structure domains. Here, the theories concerning the threemodules are introduced, respectively.

2.1. MPS method for fluid domain

The MPS method is originally proposed for incompressibleviscous fluid, and its governing equations are constructed in theLagrangian system as follows:

∇ �V¼ 0 (1)

DVDt

¼�1ρ∇Pþ ν∇2Vþ g (2)

where V, t, ρ, P, ν, and g represent the velocity vector, time, liquiddensity, pressure, kinematic viscosity, and the gravity accelerationvector, respectively.

The interaction between particles is described through a kernelfunction W(r) in MPS. In this article, a modified kernel functionpresented by Zhang et al. (2014) is used.

W ðrÞ¼8<:

re0:85rþ 0:15re

� 1 0≤ r < re

0 re ≤ r(3)

where r is the distance between particles and re is the radius of theinteraction area. The gradient model is re¼ 2.1l0, whereas re¼ 4.0l0is used for the Laplacian model. l0 is the initial distance betweentwo adjacent particles.

In the Lagrangian approach, particle trajectories are computedby integrating in time the material derivative of velocity obtainedfrom the governing equations. To calculate the material derivative ofvelocity by the MPS method, the differential operators of gradient,divergence, and Laplacian included in the momentum equationshould be expressed by the particle interaction models which aredefined as follows:

Æ∇ϕæi ¼dimn0

Xj�i

ϕj þϕi��rj � ri��2�rj � ri

� �W��rj � ri��; (4)

Æ∇ �Φæi ¼dimn0

Xj�i

�Φj �Φi

� � �rj � ri�

��rj � ri��2 W

��rj � ri��; (5)

Æ∇2ϕæi ¼2 dimn0λ

Xj�i

�ϕj �ϕi

� �W��rj � ri��; (6)

whereϕ is an arbitrary scalar function,V is an arbitrary vector, dim is thenumber of space dimensions, n0 is the initial particle number density forincompressible flow, and λ is a parameter defined as follows:

λ¼

Pj�i

W��rj � ri

�� rj � ri��2

Pj�i

W��rj � ri

�� ; (7)

which is introduced to keep the variance increase equal to that of theanalytical solution (Koshizuka & Oka 1996).

The incompressible condition of the MPS method is representedby keeping the particle number density constant. In each time-step,there are two stages: first, the temporal velocity of particles iscalculated based on viscous and gravitational forces, and particlesare moved according to the temporal velocity; second, pressure isimplicitly calculated by solving the pressure Poisson equation(PPE), and the velocity and position of particles are updatedaccording to the obtained pressure. The PPE in the present MPSsolver is defined as follows:

Æ∇2Pnþ 1æi ¼ð1� γÞ ρΔt

∇ �Vpi � γ

ρΔt2

Ænpæi � n0

n0; (8)

where γ is a parameter with the value between 0 and 1. The range of:01≤ γ≤ :05 is better according to numerical experiments con-ducted by Lee et al. (2010). In this article, γ¼ :01 is adopted for allsimulations.

2 MONTH 2019 JOURNAL OF SHIP RESEARCH

For the MPS method, the pressure of the fluid domain is closelyaffected by the accuracy of free-surface detection. In presentsolver, we use a free-surface detection method by Zhang et al.(2014).

ÆFæi ¼dimn0

Xj�i

1��ri � rj��ri � rj

�W�rij�; (9)

where the vector function F represents the asymmetry of ar-rangements of neighbor particles. A particle satisfying

ÆjFjæi > :9jFj0 (10)

is considered as a free-surface particle, where jFj0is the initial valueof jFj for the surface particle.

2.2. FEM method for structural domain

In the present study, the FEM method is used to solve the de-formation of structure which is governed by the equations expressedas follows:

M€yþC _yþKy¼FðtÞ; (11)

C¼ α1Mþ α2K; (12)

where M, C, and K are the mass matrix, the Rayleigh dampingmatrix, and the stiffness matrix of the structure, respectively. F isthe external force vector acting on the structure, and varies with

Fig. 3 Schematic diagram of the displacement interpolation

Fig. 4 Schematic diagram of the tank and arrangement of measuringpoints (Unit: m). (A) liquid tank. (B) measuring pointsFig. 2 Schematic diagram of the force interpolation

Fig. 1 Interface between the fluid and structure domains

MONTH 2019 JOURNAL OF SHIP RESEARCH 3

computational time. y is the displacement vector of the structure. α1and α2 are coefficients which are related with natural frequenciesand damping ratios of structure.

To solve the structural dynamic equation, another two groupfunctions should be supplemented to set up a closed-form equationsystem. Here, Taylor’s expansions of velocity and displacementdeveloped by Newmark (1959) are used:

_ytþΔt ¼ _yt þð1� γÞ€ytΔtþ γ€ytþΔtΔt; 0< γ < 1; (13)

ytþΔt ¼ yt þ _ytΔtþ1� 2β

2€ytΔt

2 þ β€ytþΔtΔt2; 0 < β < 1; (14)

where β and γ are important parameters in the Newmark method,and selected as β ¼ .25, γ ¼ .5 for all simulations in the presentarticle. The nodal displacements at t ¼ t þ Δt can be solved by thefollowing formulae (Hsiao et al. 1999):

�K ytþΔt ¼ �FtþΔt; (15)

�K¼Kþ a0Mþ a1C; (16)

�FtþΔt ¼Ft þMða0yt þ a2 _yt þ a3€ytÞþCða1yt þ a4 _yt þ a5€ytÞ;(17)

a0 ¼ 1βΔt2

; a1 ¼ γβΔt

; a2 ¼ 1βΔt

; a3 ¼ 12β

� 1; a4 ¼ γβ� 1;

a5 ¼Δt2

�γβ� 2

�; a6 ¼Δtð1� γÞ; a7 ¼ γΔt;

(18)

where �K and �F are the so-called effective stiffness matrix andeffective force vector, respectively. Finally, the accelerations andvelocities corresponding to the next time-step are updated asfollows:

€ytþΔt ¼ a0�ytþΔt � yt

�� a2 _yt � a3€yt; (19)

_ytþΔt ¼ _yt þ a6€yt þ a7€ytþΔt: (20)

2.3. Data interpolation on the fluid–structure interface

For the simulation of 3-D FSI problems based on the afore-mentioned MPS-FEM coupled method, the space of the fluid do-main will be dispersed by particles, whereas the space of thestructural domain will be dispersed by grids. In general, the fineparticles should be arranged within the fluid domain to keep asatisfactory precision for the fluid analysis. By contrast, the muchcoarser grids could be accurate enough for the structure analysis,which indicates that the fluid particles are usually not coincidedwith the structural nodes on the interface between the fluid andstructure domain, as shown as F1Fig. 1. Hence, the isomerous in-terface between the two domains may result in the challenge of dataexchange in the process of FSI simulation. In the present study, theKFBI technique is proposed to apply the external force carried bythe fluid particles onto the structural nodes and update the posi-tions of boundary particles corresponding to the displacements ofstructural nodes.

The schematic diagram of the KFBI technique for the forcetransformation from the fluid domain to the structural boundary isshown in F2Fig. 2. In the KFBI technique, the boundary particle of thefluid domain will be denoted as the neighbor particle of the structurenode when the distance between the particle and the node is smallerthan the effect radius rei of interpolation. The weighted value of thefluid force of the neighbor particleW(|ri� rn|) is calculated based onequation (3). Then, the equivalent nodal force Fn corresponding tothe node n is obtained by the summation of force componentsregarding the neighbor particles.

Fn ¼PiPi � l2cell �W ðjri � rnjÞP

iW ðjri � rnjÞ (21)

where Pi is the pressure of the boundary particle obtained from thefluid domain and lcell is the initial element size.

The schematic diagram of the technique for the deformation ofthe fluid-structure interface is shown in F3Fig. 3. The fluid boundaryconsisting of particles will deform according to the deformation ofthe structural boundary. The deflection value of the boundaryparticle wm can be obtained by the interpolation based on the kernelfunctions W(|ri � rm|) and the nodal displacement δi.

Table 1 Parameters for fluid and structural analysis

Parameters Values

Fluid Fluid density (kg/m3) 998Kinematic viscosity (m2/sec) 1� 10�6

Gravitational acceleration (m/sec2) 9.81Particle spacing (m) .005

.007.01

Total number of particles 49,633108,180229,816

Fluid time step size (s) 1� 10�4

Structure Structure density (kg/m3) 1800Young’s modulus (MPa) 40Elements per lateral wall 600Damping coefficients α1 .0128Damping coefficients α2 5.01�10�7

Structural time step size (s) 1� 10�4

Fig. 5 Comparison of the structural vibrations at the measuring point B


wm ¼Piδi �W ðjri � rmjÞPiW ðjri � rmjÞ (22)

In the present study, the rectangular thin-plate element, which hasfour nodes within an element, is used to predict the structural re-sponse. According to the Kirchhoff–Love plate theory (Love 1888),the nodes will move along the initial normal direction of the plateboundary.

3. Results and discussions

The sloshing problem in a liquid tank has long been a researchhotspot because of its effects on the safety of liquid-carrying ships.However, the research on this problem is of great scientific sig-nificance because the sloshing problem is often accompanied by aslamming phenomenonwhich can be characterized by the nonlinearimpact loads of fluids and the violent evolution of free surfaces.

In our previous studies (Yang et al. 2015; Chen et al. 2017; Wenet al. 2017), the liquid sloshing phenomenon in a rigid tank wasnumerically studied by the MPS method, and sufficient reliabilityhas been presented by the comparison between our numerical re-sults and the published experimental results. In recent years, thesloshing phenomenon in a 2-D elastic tank is numerically simulated

in consideration that the elasticity of tank wall plays an importantrole in the practical sloshing phenomenon. However, the actualstructure and constraints of the liquid tank are all with the 3-Dfeatures. It is necessary to carry out 3-D numerical simulationresearch, and investigate the influence of structural elasticity on theevolution of sloshing wave, structural dynamic response, impactloads of fluids, and other physical phenomena.

3.1. Numerical setup

In this article, the 3-D liquid sloshing in an elastic tank is nu-merically studied. The numerical model is the same as the ex-perimental facility given by Souto-Iglesias et al. (2015). Similargeometric and computational parameter values to those regardingour previous 2-D studies are used, but with the dimension of .12 malong the Z direction in this study, as shown in F4Fig. 4. On the leftwall of the elastic tank, six displacement sensors are mounted at thepointsA (0, .05, 0),B (0, .08, 0),C (0, .1, 0),D (0, .15, 0),E (0, .2, 0),and F (0, .25, 0). A pressure sensor is mounted at the point P (0,.093, 0), and a wave height probeW (.005, 0, 0) is set near the elasticwall.

The 3-D tank is forced to roll harmoniously around the axisO-O0

which is the symmetry axis of the floor. The roll motion of the tankis governed by the following equation:

θ¼ αθ0 sin�2πTt

�; (23)

where θ0 is the amplitude of the roll motion and is selected as 4°, andT is the rotation period and is set as 1.6312 seconds (3.857 rad/sec).In the initial stage, the roll motion is buffered by the coefficient α inequation (23), which is defined as follows:

Fig. 6 Comparison of the impact pressures at the measuring point P

Fig. 7 Comparison of the wave heights at the location W

Fig. 8 Pressure distribution on the particle model

Fig. 9 Force of the element node at the bottom center point


α¼

8><>:

:5

�1þ tanh

�2πTt� π

��t < T

1:0 t≥ T

: (24)

To study the influence of elasticity of the bulkhead on thesloshing phenomenon, simulation of sloshing in a rigid tank is alsocarried out in this article.

3.2. Convergence study of particle resolutions

In this section, the convergence of spacing resolution for 3-D FSIof liquid sloshing is investigated first. The fluid computationaldomain is dispersed by particles with the initial distances of .01,.007, and .005 m. Accordingly, the total numbers of particles are49,633, 108,180, and 229,816. To calculate the structural responsesof the elastic walls due to the sloshing impact loads, the lateral tankwalls of the elastic tank are dispersed by thin-plate elements with aspacing size of .01 m; and the number of the elements of the twoelastic lateral walls of the liquid tank is 600. Moreover, the Young’smodulus of the structural material is set as 40 MPa, and theRayleigh’s damping has been taken into account for the structuralanalysis. The detailed calculation parameters of both the fluid andthe structure domains are shown inT1 Table 1.

F5 Figure 5 shows the time histories of structural vibrations atmeasuring point B of three different spacing resolutions. Accordingto the figure, the amplitudes and trends of the structural vibrationsfor the three cases are in agreement with each other, although thestructural response presents strongly nonlinear characteristics.

Also, the impact pressures regarding the measuring point P of thethree particle models are comparatively presented inF6 Fig. 6. Similarcharacteristics including the fluctuation amplitudes of the curvesand the form of the pressure signals regarding the three cases arealso observed, which indicate that the three particle models are allsuitable for the investigation of sloshing in an elastic tank. How-ever, it is difficult to get a more realistic evolution of the water

surface when a lower particle resolution is used in numericalsimulations. As shown by the wave height histories regarding themeasuring point W in F7Fig. 7, the difference between the particlemodel with the initial particle spacing size of .01 m and the othertwo models is obvious. To obtain reliable structural responses,considering accurate fluid impact loads, realistic water surface, andnumerical efficiency, the particle model with a spacing size of.007 m is used to investigate the influence of the structural elasticityon the sloshing phenomenon in the following simulations.

3.3. Validation of interpolation modules

To validate the interpolation accuracy of the interface datatransformation modules including force and deformation, two testsare carried out. The calculated parameters are listed in Table 1. Theparticle spacing is .007 m and the element size is .01 m. The fluidforce transformation from the fluid domain to the structural domainis first investigated. In F8Fig. 8, the triangular pressure on the fluidparticles which is similar to the distribution of hydrostatic pressurecan be calculated as follows:

PðxÞ¼ 10; 000� 20; 000x: (25)

Fig. 10 Force distribution on the element model (t ¼ .025 seconds)

Fig. 11 Deformation distribution on the element model

Fig. 12 Displacement of the particle model at the geometric centerpoint

Fig. 13 Displacement of the particle model (t ¼ .025 seconds)


F9 Figure 9 shows the force time history of the element node atthe bottom center point. At the instant t ¼ .025 seconds, theforce distribution on the element model is shown inF10 Fig. 10. Thecalculated maximum .996 N approximates to the theoretical value1.0 N.

Finally, the accuracy of the deformation transformation on theinterface between the fluid and structure domains is studied. Thedeformation distribution on the element model is shown inF11 Fig. 11and can be calculated as follows:

DðxyÞ¼�:01f ðxÞf ðyÞsinð20πtÞ; (26)

f ðxÞ¼ 8x� 16x2; (27)

f ðyÞ¼ 1� 25009

y2: (28)

Equations (26)–(28) can ensure that the four edges are fixed andthe max displacement occurs at the geometric center point. With thehelp of the interface data interpolation module, the boundaryparticles will be moved. The displacements of the particle modelbetween the present and theoretical results at the geometric centerpoint are compared in F12Fig. 12. The variation tendencies and dis-placement amplitudes of both the results are almost the same. Inaddition, the deformations of the particle and element models arecoincident with each other at t ¼ .025 seconds, which is shown in

F13Fig. 13.

Fig. 14 Snapshots of the water surface in the 3-D tanks at four instants(A) rigid tank, (B) elastic tankE¼ 80MPa, (C) elastic tankE¼ 40MPa, and

(D) elastic tank E ¼ 20 MPa

Fig. 14 (Continued )


3.4. Influences of structural elasticity on free surface

It is well known that severe impact events will occur in theprocess of sloshing in a liquid tank while the motion frequency ofthe tank is close to its natural frequency. In this section, the phe-nomenon induced by the impact event will be investigated quali-tatively.F14 Figure 14 shows the evolutions of the wave surface in theliquid tank and the deformations of the elastic lateral wall before andafter the impact event.

In Fig. 14A, the liquid tank is rigid. It can be observed thatoverturning of the wave surface occurs at the instant t1, and thesloshing wave impacts onto the lateral wall at the instant t2. Then,the jet flow along the lateral is formed after the impact event, and thewater front will climb up to the corner of the roof wall. At the instantt4, the sloshing wave travels to the other side of the tank.

By contrast, the evolution of a sloshing wave in the elastic tank isobviously different from that in the rigid tank, as shown in Figs.14B–D. At the instant when the impact event occurs, the elasticlateral wall deforms with the 3-D ellipsoidal form at the impactregion. The fluid pressure of this region is significantly lower than

Fig. 15 The maximum deformation of the 3-D elastic wall from a localview (A) E ¼ 80 MPa, (B) E ¼ 40 MPa, and (C) E ¼ 20 MPa

Fig. 16 Snapshots of the water surface in the 2-D elastic tank E ¼ 20MPa (Zhang & Wan 2018)

Fig. 17 Time history of water level at the measuring point W (A) rigidtank, (B) elastic tank E ¼ 80 MPa, (C) elastic tank E ¼ 40 MPa, and (D)

elastic tank E ¼ 20 MPa


that in the rigid tank because the kinetic energy of fluid particles isconverted into the potential energy of the deformed tank wall whichgives rise to the buffer effect on the impact phenomenon. At theinstant t3, an angle between the direction of jet flow and the lateralwall of the tank is formed during the climbing-up of the fluidparticles along the deformed wall, and the angle will increase withthe decrease in the bulkhead’s elasticity. Moreover, the fluid par-ticles will be bounced off the lateral wall when a very flexiblematerial is used for the bulkhead, as shown in Fig. 14D. After that,the sloshing wave transfers to the other side of the elastic tank, andthe second wave with a small crest is formed near the lateral wallduring the shape recovering of the elastic tank.

In addition, we compared the 3-D free surfaces against the 2-Dresults which were published as shown in Fig. 16. It can beobserved that the present shapes of sloshing wave are quitesimilar to those of the 2-D results at all instants of the impactevent, which indicates that the 3-D coupled method is dependablein solving such complex FSI problems. However, there are also

obvious difference between the 3-D and the 2-D results. Com-paring Figs. 14D2 and 16B, the maximal deformation of lateralwall presents at the location near the impact region of the 3-D tankwhereas that arises at the middle of the wall in the 2-D tank. Thedeformation pattern of the 3-D tank shows localized propertieswhereas that of the 2-D tank shows the global characteristics,which iscaused by the different boundary constraints used for the two kinds ofsimulations. In the 3-D simulations, the fixed constraint is used for allthe four edges of the elastic lateral walls, which limits the deformationof structure far from the impact region. In contract, the bulkheads ofthe tank are simplified as beams in the 2-D simulations, and bothsides of the beam are fixed, which indicates that the deformation ofthe local area can be shared by the whole beam immediately.

F15Figure 15 shows the maximum deformation of the 3-D elasticwall from a local view. Obviously, the deformation increases withthe decrease of the Young’s modulus. The deformation of the 3-Dsimulation is similar to an ellipsoidal shape, which is different fromthat of 2-D calculation. This difference will result in the reactiveforce acting on the fluid being different. When the elastic wallrestores, the direction of force acting on the bottom fluid is towardthe upper right in the 2-D case. However, the direction of restoringforce acting on the fluid is toward the center of elliptical sphere.Therefore, the height of jet flow from Fig. 14D3 is smaller than thatin F16Fig. 16C.

To further investigate the influence of structure elasticity on theevolution of the water surface, the time histories of wave height atthe measuring pointW are presented in F17Fig. 17. By comparing withthe wave height regarding the rigid tank, the amplitudes of the waveheight decreases from .5m (rigid tank) to .24m (elastic tankwith theYoung’s modulus E¼ 20MPa) as more flexible material is used for

Fig. 18 Vibration of the lateral wall at the measuring point B (A) elastictank E ¼ 80 MPa, (B) elastic tank E ¼ 40 MPa, and (C) elastic tank

E ¼ 20 MPa

Fig. 19 Vibration of the middle point of the lateral wall simulated by 2-DMPS–FEM method (Zhang & Wan 2018)

Fig. 20 Structural vibrations of the measuring points on the left wall ofthe tank with the Young’s modulus E ¼ 40 MPa


the tank. However, the wave heights in the elastic tanks present theoscillation characteristic as shown in Fig. 17D, which indicates thatthe vibration of tank walls intensified the variation of sloshingwaves.

3.5. Influences of structural elasticity on displacementresponses

F18Figure 18 shows the time histories of structural vibrations at themeasuring point B where the impact event occurs. It can be noticedthat the tank wall oscillates with a similar pattern except the am-plitudes. The vibration pattern within one period consists of twoparts. During the impact stage, the lateral wall oscillates with largeamplitudes and long periods. After the impact stage, the wall os-cillates with small amplitudes and short periods, which is caused bythe structural elastic restoring force. The pattern of the structuralresponse is similar to that regarding the 2-D simulation results fromour earlier studies, but with a different character after the impactstage. Attenuation of structural vibration is observed after theimpact event for the 3-D simulation, whereas this phenomenon doesnot occur in the 2-D simulation, as shown in F19Fig. 19. The reasons forthis difference are that the damping regarding the structural shapeand constraint for the 3-D structure analysis are greater than that inthe 2-D calculation.

F20Figure 20 shows the time histories of structural displacementsconcerning the measuring points A–E arranged on the lateral wall

Fig. 21 Timehistories of pressure at themeasuringpointP (A) rigid tank,(B) elastic tank E ¼ 80 MPa, (C) elastic tank E ¼ 40 MPa, and (D) elastic

tank E ¼ 20 MPa

Fig. 22 Enlarged signals of time histories of pressure (A) rigid tank, (B)elastic tank E ¼ 80 MPa, (C) elastic tank E ¼ 40 MPa, and (D) elastic

tank E ¼ 20 MPa


along the vertical direction. The oscillation amplitude of themeasuring point B is much larger than that of other points awayfrom it, which is proof that the elastic wall deformswith 3-D feature.

3.6. Influences of structural elasticity on impact pressures

The elasticity of tank walls can also give rise to the difference of theimpact loads acting on the lateral walls between the elastic and rigidtanks. As shown inF21 Fig. 21, the pressure time histories correspondingto a rigid tank and elastic tank are measured. Figure 21A shows thecalculated and experimental pressure time histories at point P. Thenumerical pressure shows a good congruency with the experimentaldata. There are two pressure peaks in one period. The well-knowncharacter of the impact events, “church roof shape,” can be observed inboth results. For the pressure in the tank with elastic lateral walls, theroof shape of the impact pressure signal shows much different featurescompared with that in the rigid tank. For instance, the peaks of theimpact pressure regarding the tank with the Young’s modulus of 20MPa are less than 2000 Pa, which are obviously smaller than those ofthe rigid tank. Furthermore, the pressure curves present obvious os-cillation for the elastic tank, as shown inF22 Figs. 22B–D.According to theenlarged signals of pressureswithin one cycle of tank’s rollmotion, thepressure curves oscillate with lower frequencies and larger amplitudeswith the decrease of the structural Young’s modulus.

4. Conclusions

In this article, theMPS–FEMhybridmethod,which gains the benefitsof the meshlessMPSmethod in simulating the free-surface flow and thereliability of the FEMmethod in solving structural dynamic response, isdeveloped for the FSI problems. To realize the data transformation onthe interface between the fluid and the structure domains, a KFBItechnique is proposed. By applying the hybrid method to investigate theinfluence of structural elasticity on the liquid sloshing phenomenon in a3-D elastic tank, the following conclusions can be derived:

1) The reliability and availability of the MPS-FEM coupledmethod and interpolation technologies have been validatedby the convergence study of particle resolutions and somenumerical tests.

2) The evolution of the free surface in the tank is affected bythe variation of the structural Young’s modulus. For in-stance, the water surface presents a much more complex 3-D character and the lower amplitude of the sloshing wave isrecorded at the measuring point near the wall with thedecrease in structural stiffness of the lateral wall.

3) Different structural deformations between the 3-D and 2-Dsimulations are observed. According to the 3-D results, themaximal deformation of the lateral wall presents at theimpact region, and the amplitude of the structural vibrationdecays after the impact event. For the structural responseconcerning 2-D simulation, the maximal deformation of thetank wall exists at the middle of the wall, and the atten-uation of structural vibration is not observed.

4) The impact phenomenon induced by the sloshing wave isbuffered by the elastic deformation of the lateral wall. Theamplitudes of the impact pressure decrease with structuralstiffness.

Acknowledgments

This work is supported by the National Natural Science Foun-dation of China (51879159, 51490675, 11432009, 51579145),Chang Jiang Scholars Program (T2014099), Shanghai ExcellentAcademic Leaders Program (17XD1402300), Program for Pro-fessor of Special Appointment (Eastern Scholar) at Shanghai In-stitutions of Higher Learning (2013022), Innovative Special Projectof Numerical Tank of Ministry of Industry and Information Tech-nology of China (2016-23/09), and the Lloyd’s Register Foundationfor doctoral students, to which the authors are most grateful.

References

CHEN, X., ZHANG, Y. L., ANDWAN, D. C. 2017 Effects of the arrangement ofvertical baffles on liquid sloshing by MPS method, Proceedings, 27th In-ternational Ocean and Polar Engineering Conference, June 25–30, SanFrancisco, California.

DIAS, F. AND GHIDAGLIA, J. M. 2018 Slamming: Recent progress in the evalu-ation of impact pressures, Annual Review of Fluid Mechanics, 50(1), 243–273.

HSIAO, K. M., LIN, J. Y., ANDLIN, W. Y. 1999 A consistent co-rotational finiteelement formulation for geometrically nonlinear dynamic analysis of 3-Dbeams, Computer Methods in Applied Mechanics and Engineering, 169, 1–18.

HWANG, S. C., PARK, J. C., GOTOH, H., KHAYYER, A., AND KANG, K. J. 2016Numerical simulations of sloshing flows with elastic baffles by using aparticle-based fluid–structure interaction analysis method, Ocean Engineer-ing, 118, 227–241.

KOSHIZUKA, S. AND OKA, Y. 1996 Moving particle semi-implicit method forfragmentation of incompressible fluid, Nuclear Science and Engineering,123, 421–434.

LEE, B. H., PARK, J. C., KIM, M. H., JUNG, S. J., RYU, M. C., AND KIM,Y. S. 2010 Numerical simulation of impact loads using a particle method,Ocean Engineering, 37, 164–173.

LOVE, A. E. H. 1888 On the small free vibrations and deformations of elasticshells, Philosophical Transactions of the Royal Society (London), 17,491–549.

MITSUME, N., YOSHIMURA, S., MUROTANI, K., AND YAMADA, T. 2014a MPS-FEM partitioned coupling approach for fluid-structure interaction with freesurface flow, International Journal of Computational Methods, 11(4),4157–4160.

MITSUME, N., YOSHIMURA, S., MUROTANI, K., ANDYAMADA, T. 2014b ImprovedMPS-FE fluid-structure interaction coupled method with MPS polygon wallboundary model, Computer Modelling in Engineering Sciences, 101(4),229–247.

NEWMARK, N. M. 1959 A method of computation for structural dynamics,Journal of the Engineering Mechanics Division, 85(3), 67–94.

SOUTO-IGLESIAS, A., BULIAN, G., AND BOTIA-VERA, E. 2015 A set of canonicalproblems in sloshing. Part 2: Influence of tank width on impact pressurestatistics in regular forced angular motion,Ocean Engineering, 105, 136–159.

WEN, X., CHEN, X., ANDWAN, D. C. 2017 MPS simulation of sloshing flowsin a tuned liquid damper, Proceedings, 27th International Ocean and PolarEngineering Conference, June 25–30, San Francisco, California.

YANG, Y. Q., TANG, Z. Y., AND WAN, D. C. 2015 Numerical study on liquidsloshing in horizontal baffled tank by MPS method, Chinese Journal ofHydrodynamics, 30, 101–107.

ZHANG, Y. L., CHEN, X., ANDWAN, D. C. 2016 MPS-FEM coupled method forthe comparison study of liquid sloshing flows interacting with rigid and elasticbaffles, Applied Mathematics and Mechanics, 37(12), 1359–1377.

ZHANG, Y. X., WAN, D. C., AND HINO, T. 2014 Comparative study of MPSmethod and level-set method for sloshing flows, Journal of hydrodynamics,26(4), 577–585.

ZHANG, Y. L. ANDWAN, D. C. 2018 MPS-FEM coupled method for sloshingflows in an elastic tank, Ocean Engineering, 152, 416–427.

ZHA, R. S., PENG, H., AND QIU, W. 2017 Solving 2D coupled water entryproblem by an improved MPS method, Proceedings, 32nd Interna-tional Workshop on Water Waves and Floating Bodies, April 23–26, Dalian,China.


numerical study of 3-d liquid sloshing in an elastic tank

Documents