Sloshing of liquid in partially liquid filled toroidal tank with various baffles under lateral excitation

Sloshing of liquid in partially liquid filled toroidal tank with various baffles under lateral excitation

Ocean Engineering 146 (2017) 434–456 Contents lists available at ScienceDirect Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng ...

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Ocean Engineering 146 (2017) 434–456

Contents lists available at ScienceDirect

Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Sloshing of liquid in partially liquid filled toroidal tank with various baffles under lateral excitation Wenyuan Wang a, b, *, Yun Peng a, b, Qi Zhang a, b, Li Ren c, Ying Jiang d a

School of Hydraulic Engineering, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian, 116024, China State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian, 116024, China School of Mechanical and Power Engineering, Dalian Ocean University, Dalian, 116023, China d Mobilities and Urban Policy Lab, IDEC, Hiroshima University, Hiroshima, 739-8514, Japan b c

A R T I C L E I N F O

A B S T R A C T

Keywords: Toroidal tank Scaled boundary finite element method Sloshing Baffles Sub-domains Sloshing frequencies Sloshing force

A technique to study effect of various baffles on liquid oscillations in partially filled rigid toroidal tanks is first developed by extending the semi-analytical scaled boundary finite element method (SBFEM), which utilizes the advantages of both the boundary element (BEM) and finite element methods (FEM). As found in the BEM, only the boundary is discretized to reduce the space by one, and no fundamental solution is needed unlike the BEM. The calculated liquid domain is divided into several simple sub-domains so that the liquid velocity potential in each liquid sub-domain becomes the class C1 with continuity boundary conditions. Based on the linear potential theory and weighted residual method, the semi-analytical solutions of the liquid velocity potential corresponding to each sub-domain are obtained by means of SBFEM, where the geometry of each sub-domain is transformed into scaled boundary coordinates, including the radial and circumferential coordinates by using a scaling centre, and the finite-element approximation of the circumferential coordinate yields the analytical equation in the radial coordinate. By discretizing the flow boundaries, the integral equation governed on the boundary is formulated into a general matrix eigenvalue problem. Based on the eigenvalue problem and multimodal method, an efficient methodology is adopted to computer the sloshing masses and sloshing force. Accuracy, simple and fast numerical computations are observed by the convergence study, and excellent agreements have been achieved in the comparison of results obtained by the proposed approach with other methods. Meanwhile, several baffle configurations are considered including the horizontal bottom-mounted and surface-piercing ring baffles as well as their combination form, bottom-mounted and surface-piercing ring baffles as well as their combination, and free surface-touching baffles. The effects of baffled arrangement, the ratio b=a of elliptical cross section, liquid fill level, and baffles' length upon the sloshing frequencies, the associated sloshing mode shapes and sloshing forces are investigated in detail and some conclusions are outlined. The results show that the present method allows for the simulation of complex 3D sloshing phenomena using a relative small number of degrees of freedom while the mesh consists of two-dimensional elements only.

1. Introduction Sloshing is a widespread physical problem and has the oscillation behavior of fluid in a partially filled tank, which is subjected to the external excitation. The understanding of the complex hydrodynamic characteristic of sloshing has far-reaching significance in the fields of engineering and technologies disciplines and has great interest concerning economy, environment and safety, which can often be found in moving vehicles such as liquid bulk cargo carriers (e.g., oil tankers, ships, trucks, railroad cars), aircrafts, spacecrafts, and rockets as well as in

storage containers, dams, nuclear vessels, and reactors undergoing seismically excitation (Ibrahim, 2005). Ibrahim (2005), and Faltinsen and Timokha (2009) have carried out extensive reviews on the applications and physics of sloshing problems. If the sloshing frequencies are sufficiently near to the structures' natural frequencies, the waves of resonant sloshing may induce amount hydrodynamic loads on the system of tank walls and associated support structures, which can reduce the fatigue life of the system and even cause failure. In order to avoid failure of structure system due to the undesirable dynamic behaviors, the hydrodynamic loads of sloshing should also be

* Corresponding author. School of Hydraulic Engineering, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian, 116024, China. E-mail address: [email protected] (W. Wang). https://doi.org/10.1016/j.oceaneng.2017.09.032 Received 10 June 2017; Received in revised form 12 August 2017; Accepted 24 September 2017 0029-8018/© 2017 Elsevier Ltd. All rights reserved.

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Fig. 1. Schematic of a toroidal tank. (a) model, and (b) cross section.

various types of baffles on the sloshing behavior of containers by using experimental, theoretical, and numerical approaches. For examples, Dodge (1971), Strandberg (1978), Younes et al. (2007), Panigrahy et al. (2009), Goudarzi et al. (2010), Hosseini and Farshadmanesh (2011), Hosseinzadeh et al. (2014), Akyldz et al. (2013), Turner et al. (2013), Wei et al. (2015) and Xue et al. (2013, 2017) investigated the parametric effects of different baffles including vertical, horizontal, annular, slat-screens, ring-type, flexible and perforated configuration concerning about the baffle shapes, dimensions, arrangements and numbers on the sloshing characteristics in the 2D, 3D and cylindrical containers. Evans and McIver (1987), Watson and Evans (1991), Gavrilyuk et al. (2006, 2007), Chantasiriwan (2009), Hasheminejad and Aghabeigi (2009), Wang et al. (2012a, 2012b), Goudarzi and Danesh (2016) and Cho and Kim (2016), Cho et al. (2017) represented the analytical or semi-analytical approaches to obtain the natural frequencies, mode shapes and the sloshing response of liquid sloshing in the partially fluid-filled the 2D, 3D and cylindrical container with horizontal, vertical, annular and porous baffles and internal bodies for different submergence depths, lengths, location and configuration, respectively. Hasheminejad and Mohammadi (2011), Hasheminejad, et al. (2014), Hasheminejad and Aghabeigi (2011, 2012), and Hasheminejad and Soleimani (2017) introduced the analytical solutions to study the free or transient liquid sloshing characteristics in the two-dimensional (2D) horizontal circular or elliptical cylindrical filled half-full or an arbitrary depth tanks with various baffles such as bottom-mounted and surface-piercing vertical baffles as well as the horizontal side baffles with arbitrary extension placed at the free liquid surface. In addition to these two methods, many useful numerical schemes have been developed for the sloshing problems with different baffles due to the great benefits of computers. Armenio and Rocca (1996) presented the numerical test to analyze the effect of a vertical internal baffle in a rectangular container on the liquid sloshing, and it has been observed that the presence of this baffle brought out a strong reduction of the sloshing response in the whole range of roll frequencies. Wang et al. (2010) adopted the cell-centered pressure-based algorithm along with the level set technique to evaluate the potential of baffles including the orientation, and the number of the fluid sloshing motion in a two-dimensional (2D) rectangular tank. Zheng et al. (2013) used FLUENT software to simulate liquid sloshing in tanks installing various kinds of baffles including the circular baffle, the conventional baffle, and the staggered baffle undergoing the constant braking excitation. Zhou et al. (2014) used multi-modal method of Lukovsky-Miles variational to model the nonlinear liquid sloshing in a cylindrical tank with annular baffles. Cho et al. (2002, 2005), Khalifa et al. (2007), Belakroum et al. (2010), Biswal and Bhattacharyya (2010), and Nayak and Biswal (2016) conducted the finite element method (FEM) to study the effects of parametric baffle including number, location and inner-hole diameter as well as liquid fill height on the natural frequencies and corresponding modes as well as resonance sloshing response in the containers. Meanwhile, Cho and Lee (2004) and Biswal et al. (2006) introduced a nonlinear finite element method (FEM) for simulating the large amplitude slosh in the two-dimensional baffled container undergoing the horizontal based on the fully nonlinear potential flow theory.

Fig. 2. The SBFEM coordinate system and the SBFEM mesh of a cubic tank. (a) SBFEM coordinate system with the typical elements on the part of the boundary (The surface that is not like closed), (b) The SBFEM mesh of a cubic tank (The surface that is like closed), and (c) The SBFEM mesh of a cubic tank having baffle with two sub-domains.

restrained. Baffles installed in the tanks have been effectively chosen as an internal components to suppress liquid sloshing, and consequently reduce the sloshing forces in most of the practical engineering problems. The issue of using baffles to suppress sloshing behavior can go back to late 50s when Abramson (1969) analyzed the influence of baffles on the sloshing in continuers of space vehicles filled with fuel. Since then, a mount of valuable studies have been conducted to study the effects of 435

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Fig. 3. An approximate solution for the sloshing frequencies in a toroidal container obtained by an annular tank analogy.

2003), Firouz-Abadi et al. (2008), Sygulski (2011), Noorian et al. (2012), Ebrahimian et al. (2013, 2014, 2015), Kolaei et al. (2015, 2017) presented the boundary element method (BEM) to estimate the effects of baffles' position, number and dimension fluid height on the natural frequencies, the corresponding mode shapes, and hydrodynamics coefficients in the 2D, 3D and axisymmetric baffled containers. Wang et al. (2016a, 2016b) conducted the scaled boundary finite element method (SBFEM) to study the free liquid sloshing in the horizontal elliptical tanks with various T-shaped or Y-shaped baffles such as bottom-mounted and surface-piercing baffles, and to analyze the transient liquid sloshing resonance in the cylindrical containers with different multiple baffles including floating ring, wall-mounted and floating circular baffles. In general, it can be observed from the literature as mentioned above that location and size of the baffles have great influence on the hydrodynamic damping, upper mounted baffles were more acceptable for a chargeable container, baffles being close to the liquid free surface brought out bigger values of the damping ratio, the vertical baffle should be preferred in comparison with the horizontally baffle or without baffle in the containers, and the ring baffles brought out more effeteness as compared to the common horizontal and vertical baffles, and so on. The above reviews clearly further indicate that the liquid sloshing in different container with various types of baffles such as the horizontal, vertical and longitudinal baffles, ring baffles, porous baffles and T- and Yshape baffles, flexible baffles, blocks, the circular baffle, and the staggered baffle have been investigated by using the experimental, analytical, and numerical approach. However, the geometries of the baffled tanks as mentioned above are most confined to the common configures such as 2D/3D rectangular tanks, horizontal cylindrical containers with the circular or elliptical section, axisymmetric cylinder or sphere. Toroidal tanks are widely applied to the pressurized fluids in medical hyperbaric

Table 1 Comparison of the calculated sloshing natural frequencies in a toroidal tank with those of literature (Frequencies (Hz)). Methods

f1

f2

f3

f4

f5

f6

Approximate solution Experiment SBFEM ð996Þ SBBEM ð2284Þ SBBEM ð4100Þ

0.5740

1.5395

2.1688

2.6576

3.0665

3.4273

0.5733 0.5732 0.5732 0.5732

1.5374 1.5374 1.5373 1.5374

2.1648 2.1661 2.1656 2.1654

2.6524 2.6562 2.6531 2.6523

3.0624 3.0749 3.0641 3.0617

3.4232 3.4579 3.4302 3.4235

Celebi and Akyildiz (2002), Akyildiz and Ünal (2006), Eswaran et al. (2009), Akyildiz (2012), and Jung et al. (2012) brought out the volume of fluid (VOF) method to investigate the baffle's height, arrangements as well as the liquid height affecting the linear or nonlinear liquid sloshing phenomenon in partially filled rectangular or 3D containers with baffles. Koh et al. (2013) and Shao et al. (2015) adopted an improved consistent particle method (CPM) and an improved smoothed particle hydrodynamics (SPH) to investigate the effect of various baffles including horizontal baffles, vertical middle baffles, porous baffles, T-shape baffles and floating baffle on fluid sloshing in the containers. Liu and Lin (2009), and Xue and Lin adopted (2011) performed a three-dimensional fluid flow model based on the virtual boundary force (VBF) method to determinate the internal effects of ring baffle's parameters including different arrangements and shapes on the liquid sloshing in a three-dimensional rectangular tank. Wu et al. (2012) employed a time-independent finite difference technique combining fictitious cell technique to analyze the natural frequencies and damping effects of the viscous liquid sloshing in the two-dimensional (2D) containers with baffles such as flat plate, bottom-mounted and surface-piercing types. Gedikli and Erguven (1999,

Fig. 4. Model and mesh of SBFEM for the toroidal tank. (a) Showing four sub-domains without the liquid free surface, (b) Showing mesh of four sub-domains without the liquid free surface, and (c) Showing mesh of four sub-domains with the liquid free surface. 436

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Fig. 5. The mode shapes on the total liquid free surface. (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode.

Fig. 6. The mode shapes along the x axis. (a) first mode, (b) second mode, (c) third mode, (d) fourth mode, (e) fifth mode, and (f) sixth mode.

sloshing in the toroidal tanks with baffles. The primary purpose of the current paper is to perform a simple and semi-analytical method named scaled boundary finite element (SBFEM) based on linear potential theory to fill this gap. The SBFEM firstly has been applied to the structural mechanics such as analysis of soil-structure interaction, which was introduced (1997) and described (2003) by Wolf and Song. A new coordinate system in SBFEM containing two local co-ordinates such as the circumferential local and radial coordinates has been established. Only the boundary in the research domain should be discretized and an

chambers, nuclear fusion reactors, diver's oxygen tanks, rocket fuel tanks, liquid petroleum gas (LPG) in motor vehicles, circumferential reinforcement for submarines, satellite antenna support structures, and protective devices for nuclear fuel containers and so on (Xu and Redekop, 2006). Meserole and Fortini (1987) have experimentally and analytically studied the fluid sloshing dynamics of circular toroidal tanks with/without baffles. However, the analyzed model was based on the equivalent mechanical models instead of the annular cylindrical tank. As far as authors know, there has not been any other report to simulate the liquid 437

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Fig. 7. The mode shapes on middle surface of the cubic tank.

discretize the whole body like the finite element method (FEM), so SBFEM has the benefits of both the FEM and the BEM without adopting the shortcomings and it has also its own attractive properties. For example, a significant advantage of SBFEM is that singularities at cracks, corners and bimaterial interfaces are analytically represented by its solutions. Therefore, accuracy can be achieved directly without fine crack-tip meshes or singular elements as required by the FEM. This

analytical solution is used within the body in the radial co-ordinate. Meanwhile, the above reviews showed that the finite element method (FEM) and the boundary element method (BEM) are most frequently for solving the liquid sloshing with baffles. However, in the FEM the entire volume of the domain is discretized by 3D elements and BEM may cannot find the fundamental solution. While the SBFEM does not require the fundamental solution as the boundary element method (BEM) or 438

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Fig. 8. The various horizontal baffles in the toroidal tank with circular cross section. (a) wall-mounted ring baffle on inner wall named inner baffle, (b) wall-mounted ring baffle on outer wall named outer baffle, (c) the combination of inner and outer baffles, and (d) floating ring baffle.

2. Theoretical description

method can also meet the infinity of the boundary condition automatically. The increasing popularity of the SBFEM has been applied to many fields, such as fracture mechanics (Song and Wolf, 2002; Li et al., 2015; Ooi et al., 2016; Chen et al., 2017), dam-water-foundation interaction (Li, 2009; Xu et al., 2016; Chen et al., 2017), soil-structure interaction (Li et al., 2013; Syed and Maheshwari, 2015; Lin et al., 2016; Bazyar and Song, 2017), magneto-electro-elastic plate analysis (Liu et al., 2016), acoustical problems (Lehmann et al., 2006), heat transfer (Bazyar and Talebi, 2015a,b; Lu et al., 2017), seepage flows (Bazyar and Grai, 2012; Bazyar and Talebi, 2015a,b), sloshing problems (Wang et al., 2016a, 2016b), electromagnetics (Liu et al., 2012; Liu and Lin, 2012), potential flow (Li et al., 2006; Tao et al., 2009; Lin and Liu, 2012), elastic waveguides problems (Gravenkamp et al., 2015), to conduct isogeometric analysis (Gravenkamp et al., 2017), and automatic image-based stress analysis (Saputra et al., 2017), and so on. This paper is outlined as follows. In section 2, the theoretical formulation of the sloshing problem with baffled toroidal tank is addressed. In Section 3, based on the linear potential theory, the formulation of the scaled boundary finite element method in SBFEM coordinate for the problem is summarized, and a zoning method is presented to model arbitrary arrangement of baffles in various baffled tanks. The procedure to solve the SBFEM equations for liquid sloshing dynamics with baffles and formulation of a general matrix eigenvalue problem and sloshing motion equation are presented, and sloshing frequencies, modes and sloshing forces are also obtained. In Section 4, convergence tests are given to demonstrate the accuracy and efficiency of the proposed numerical model. In section 5, several baffle configurations are considered including the horizontal bottom-mounted and surface-piercing ring baffles as well as their combination form, bottom-mounted and surfacepiercing ring baffles as well as their combination, and free surfacetouching baffle. The effects of liquid fill level, the ratio b=a of elliptical cross section, baffled arrangement and length of those baffles upon the sloshing frequencies, the associated sloshing mode shapes and sloshing forces are investigated in details. Finally, conclusions are stated in Section 6.

Fig. 1 shows a toroidal tank with the circular section and rigid wall (where r ¼ 0:167m, R ¼ 0:312m and h is denoted the height of liquid) filled with an incompressible and inviscid ideal fluid undergoing the lateral ground motion along the x direction, which may be equipped with various horizontal and vertical rigid baffles whose thickness has been neglected, as discussed in detail at a later stage in this paper. The free surface of the fluid is orthogonal to the z axis and the free surface waves of liquid movement are based on the linear theory with small-amplitude sloshing of the liquid. Based on the above assumptions and definitions, the potential Φðx; y; z; tÞ of liquid motion should satisfy the Laplace equation:

∂2 Φ ∂2 Φ ∂2 Φ þ 2 þ 2 ¼0 ∂x2 ∂y ∂z

in Ω

(1)

with boundaries at the rigid wall of tank

∂Φ _ b ¼ Xð e ⋅nÞ ∂n

(2)

and at the free liquid surface

∂2 Φ ∂Φ ¼0 þg ∂t2 ∂z

(3)

where X_ ¼ dX=dt, n is unit vector of the outward normal direction of any point, and b e in the x direction is the unit vector. The potential Φ can be dividing into two parts: the potential Φs with sloshing motion and the potential ΦU with uniform motion

Φ ¼ Φs þ Φu and

439

(4)

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Fig. 9. Model and mesh of SBFEM for the toroidal tank with horizontal wall-mounted ring baffle on outer wall of container having L1 =L ¼ 0:5. (a) The model without the liquid free surface, (b) Showing eight sub-domains without the liquid free surface, (c) Showing mesh of eight sub-domains without the liquid free surface, (d) Showing mesh of eight sub-domains with the liquid free surface, and (e) Showing mesh of each sub-domain.

_ x Φu ¼ XðtÞb

3. Analysis of sloshing problem using scaled boundary finite element method

(5)

It can be easily seen that Φu satisfies the Laplace Eq. (1) along with the boundary conditions (2) and (3). Thus, the other potential Φs also should meet the Laplace equation

∇2 Φs ¼ 0 in

Ω

In order to better solve the transient sloshing problem (Eqs. (6)–(8)), an eigenvalue problem subjected to the zero external excitation with XðtÞ ¼ 0 and Φu ¼ 0) has been first taken consideration. In this case, the sloshing potential Φs can be expressed the form under the homogeneous

(6)

boundary condition (8) with ∂∂tΦ2 u ¼ 0 2

and the corresponding boundary conditions

∂Φs ¼0 ∂n 2

at the wall of tank

Φs ¼ φs ðx; y; zÞeiωt

(7)

(9)

Substituting Eq. (9) into Eqs. (6)–(8) yields 2

∂ Φs ∂Φs ∂ Φu ¼ 2 þg ∂t 2 ∂z ∂t

at the free liquid surface

∇2 φs ¼ 0

(8)

∂φs ¼0 ∂n

440

in Ω at the wall of tank

(10) (11)

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Fig. 11. The different baffles including outer baffle, inner baffle, combination baffle and floating ring baffle.

Table 4 The first eleventh sloshing frequencies (Hz) of the case with wall-mounted ring baffle on the outer wall of tank. Fig. 10. Quarter model of the mesh of SBFEM for the toroidal tank with horizontal wallmounted ring baffle on outer wall of container having L1 =L ¼ 0:5.

Table 2 The nodes of mesh for different cases with different length of the baffle.

L1 =L1 ¼ 0:25 L1 =L1 ¼ 0:50 L1 =L1 ¼ 0:75

Outer baffle

Inner baffle

Combination

Floating baffle

3720 3912 4140

3720 3912 4104

3912

3656 3848 4040

Table 3 The coordinate of scaling centre of each sub-domain. x

y

z

1 2 3 4 5 6 7 8

0.3 0.3 0.3 0.3 0.312 0.312 0.312 0.312

0.3 0.3 0.3 0.3 0.312 0.312 0.312 0.312

0.085 0.085 0.085 0.085 0.04175 0.04175 0.04175 0.04175

at the free liquid surface

L1 =L ¼ 0:00

L1 =L1 ¼ 0:25

L1 =L ¼ 0:50

L1 =L ¼ 0:75

1 2 3 4 5 6 7 8 9 10 11

0.5733 0.5733 1.0138 1.0138 1.3183 1.3183 1.4664 1.5374 1.5374 1.5452 1.5453

0.5695 0.5695 0.9930 0.9932 1.2799 1.2799 1.4230 1.4976 1.4976 1.4978 1.4995

0.5555 0.5555 0.9362 0.9385 1.2050 1.2050 1.3395 1.4293 1.4295 1.4353 1.4353

0.5327 0.5327 0.8854 0.8868 1.1615 1.1615 1.2931 1.4015 1.4018 1.4185 1.4185

Table 5 The first eleventh sloshing frequencies (Hz) of the case with wall-mounted ring baffle on the inner wall of tank.

Sub-domain

∂φs ω2  φs ¼ 0 ∂y g

Mode

(12)

In order to use the SBFEM to solve Eqs. (10)–(12) for one domain of the three-dimensional free liquid sloshing problem, one can introduce the named scaled boundary coordinate system (Song and Wolf, 1997). For convenience, the abbreviated form φ represents φs . As can be seen from Fig. 2, the scaling centre Oðx0 ; y0 ; z0 Þ is chosen in a solved domain from which the entire boundary must be visible, which can always divide the complex domain into several sub-domains with their own scaling centres, as shown in Fig. 2(c) such as the SBFEM mesh of a cube having baffle with two sub-domains. The normalized radial coordinate ξ is a scaling factor, which is in the range of 0 ¼ ξ0  ξ  ξ1 ¼ 1 (ξ0 presents the scaling centre and ξ1 is corresponding to the boundary S). The circumferential directions η and ζ (two local curvilinear coordinates) can be defined, which are similar to the traditional discretization finite element method (FEM) using a doubly-curved surface element, and the eight-nodes isoparametric element has been used in this paper, as shown in Fig. 2, The relationship between the scaling coordinate system ðξ; η; ζÞ

Modes

L1 =L ¼ 0:00

L1 =L1 ¼ 0:25

L1 =L ¼ 0:50

L1 =L ¼ 0:75

1 2 3 4 5 6 7 8 9 10 11

0.5733 0.5733 1.0138 1.0138 1.3183 1.3183 1.4664 1.5374 1.5374 1.5452 1.5453

0.5731 0.5731 1.0136 1.0137 1.3187 1.3187 1.4304 1.5049 1.5049 1.5459 1.5473

0.5715 0.5715 1.0093 1.0097 1.3152 1.3152 1.3430 1.4277 1.4277 1.5438 1.5451

0.5645 0.5645 0.9898 0.9901 1.2924 1.2939 1.2939 1.3897 1.3897 1.5263 1.5278

Table 6 The first eleventh sloshing frequencies (Hz) of the case with floating ring baffle.

441

Modes

L1 =L ¼ 0:00

L1 =L1 ¼ 0:25

L1 =L ¼ 0:50

L1 =L ¼ 0:75

1 2 3 4 5 6 7 8 9 10 11

0.5733 0.5733 1.0138 1.0138 1.3183 1.3183 1.4664 1.5374 1.5374 1.5452 1.5453

0.5724 0.5724 1.0103 1.0104 1.3137 1.3137 1.4656 1.5360 1.5360 1.5410 1.5412

0.5698 0.5698 0.9998 1.0003 1.3003 1.3003 1.4604 1.5292 1.5299 1.5299 1.5307

0.5630 0.5630 0.9769 0.9773 1.2716 1.2716 1.4337 1.5028 1.5028 1.5033 1.5035

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Fig. 12. The mode shapes along the x axis for the horizontal baffles. (a) first mode, and (b) second mode.

0

and Cartesian coordinate system ðx; y; zÞ can be obtained (Song and Wolf, 1997), in which ðb x0; b y 0; b z 0 Þ is the scaling centre, ðx; y; zÞ represents the point of boundary, and ðb x; b y; b z Þ is corresponding to the any point of domain.

x ½J ¼ @ xη xζ

y yη yζ

1 z zη A zζ

(16)

thus

bx ¼ bx 0 þ ξxðη; ζÞ by ¼ by 0 þ ξyðη; ζÞ bz ¼ bz 0 þ ξzðη; ζÞ

      jJj ¼ x yη zζ  yζ zη  y xη zζ  xζ zη þ z xη yζ  xζ yη

(13)

(17)

The operator ∇ in the SBFEM coordinate system can be expressed as

in which

∇ ¼ b1 ðη; ζÞ

xðη; ζÞ ¼ Nðη; ζÞx yðη; ζÞ ¼ Nðη; ζÞy zðη; ζÞ ¼ Nðη; ζÞz

  ∂ 1 ∂ ∂ þ b2 ðη; ζÞ þ b3 ðη; ζÞ ∂ξ ξ ∂η ∂ζ

(18)

(14) where

where Nðη; ζÞ is the shape function. Then, one can obtain the spatial derivatives between the Cartesian and SBFEM coordinate systems

0 1 y z  yζ zη 1 @ η ζ xζ zη  xη zζ A; b1 ðη; ζÞ ¼ jJj x y  x y

8 9 ∂ > > > > > > > ∂b x> > > > > > = < ∂ >

0 1 yz  yη z 1 @ η xη z  yη x A b3 ðη; ζÞ ¼ jJj y x  x y

0 y z  yζ zη 1 @ η ζ xζ zη  xη zζ ¼ ∂by > > jJj x y  x y > > > > η ζ ζ η > > > > > > ∂ > > : ; ∂bz

8 ∂ 9 > > > > ∂ξ > > > > > 1> > > yζ z  zζ y yzη  yη z > <1 ∂ > = zζ x  xζ z xη z  zη x A > ξ ∂η > > xζ y  yζ x yη x  xη y > > > > > > > > > 1 ∂ > > : ; ξ ∂ζ

η ζ

η

(15)

ζ η

0 1 y z  zζ y 1 @ ζ zζ x  xζ z A; b1 ðη; ζÞ ¼ jJj xζ y  yζ x

(19)

η

Based on the weighted residual technique, the governing Eq. (10) and the corresponding boundary conditions Eqs. (11) and (12) can be transformed as

where xη , yη as well as zη and xζ , yζ as well as zζ are the partial derivative with respect to the variations η and ζ, respectively, and J is the Jacobian matrix defined as

∫ V ð∇wÞT ∇φdV  ∫ Ωw w

ω2 φdΩ  ∫ Ωb wvdΩ ¼ 0 g

(20)

where V is the solved domain, Ωw and Ωb are the boundaries of the liquid free surface and the tank wall, and v is velocity normal to the boundary. The vectors φðξ; η; ζÞ and the weighting function vector wðξ; η; ζÞ can also be written in the form of shape function Nðη; ζÞ

φðξ; η; ζÞ ¼ Nðη; ζÞφðξÞ

(21)

wðξ; η; ζÞ ¼ Nðη; ζÞwðξÞ ¼ ðwðξÞÞT ðNðη; ζÞÞT

(22)

The space infinitesimal volume can be obtained

  ∂x   ∂ξ   ∂x   ! !  dV ¼ d ξ ⋅ d ! η dζ ¼  ∂η   ∂x   ∂ζ Fig. 13. The lateral acceleration. 442

∂y ∂ξ ∂y ∂η ∂y ∂ζ

 ∂z   ∂ξ   ∂z  dξdηdζ ¼ ξ2 jJjdξdηdζ ∂η   ∂z   ∂ζ

(23)

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where Ωw and Ωb represent the boundaries of liquid free surface and tank wall, and

and the infinitesimal area is as follow

 ! ! !  i j k      ∂x ∂y ∂z   2 dΩ ¼ ξ  ∂η ∂η ∂η dηdζ      ∂x ∂y ∂z    ∂ζ ∂ζ ∂ζ n 2  2  2 o12 ¼ ξ2 yη zζ  yζ zη þ xζ zη  xη zζ þ xη yζ  xζ yη dηdζ

B1 ¼ b1 ðη; ζÞNðη; ζÞ

(27)

B2 ¼ b2 ðη; ζÞNðη; ζÞη þ b3 ðη; ζÞNðη; ζÞζ

(28)

The first part of Eq. (26) can be written as

 T T T I1 ¼ ∫ ξ2 wðξÞξ E0 φðξÞξ þ ξwðξÞξ ET1 φðξÞ þ ξwðξÞ E1 φðξÞξ

(24)

ξ

 þ wðξÞT E2 φðξÞ jJjdξdηdζ

For the sake of convenience, Eq. (24) is abbreviated as 2

dΩ ¼ ξ kb1 ðη; ζÞk2 jJjdηdζ

(25) where

Substituting Eqs. (18), (21), (22), (23) and (25) into Eq. (20), one can obtain

∫ V ð∇wÞT ∇ϕdV  ∫ Ωw w Tω

∫ wðξ1 Þ Nðη; ζÞ T

Ωw

2

g

ω2 ϕdV  ∫ Ωb wvb dV ¼ ∫ g V

(29)



Nðη; ζÞφðξ1 Þξ21 kb1 ðη; ζÞk2 jJjdηdζ

E0 ¼ ∫ Ω B1 ðη; ζÞT B1 ðη; ζÞjJjdηdζ

T  ! 1 1 B1 φðξÞξ þ B2 φðξÞ dVþ B1 wðξÞξ þ B2 wðξÞ ξ ξ þ ∫ wðξ1 Þ Nðη; ζÞ T

Ωb

T

(30)

(26)

νξ21 kb1 ðη; ζÞk2 jJjdηdζ

Fig. 14. Time histories of sloshing force of the toroidal tanks for various length L1 of the horizontal baffles. (a) wall-mounted ring baffle on outer wall of tank (outer baffle), (b) wallmounted ring baffle on inner wall of tank (inner baffle), (c) floating ring baffle, and (d) different baffle with L1 =L ¼ 0:5.

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Fig. 15. Time histories of sloshing force of the toroidal tanks for various input time with L1 =L ¼ 0:5. (a) wall-mounted ring baffle on outer wall of tank (outer baffle), (b) wall-mounted ring baffle on inner wall of tank (inner baffle), (c) floating ring baffle and (d) the combination of inner and outer baffles.

E1 ¼ ∫ Ω B2 ðη; ζÞ B1 ðη; ζÞjJjdηdζ T

(31)

E2 ¼ ∫ Ω B2 ðη; ζÞT B2 ðη; ζÞjJjdηdζ

(32)

I2 ¼

ω2 T wðξ1 Þ M0 φðξ1 Þ g

(36)

The third part of Eq. (26) can be written as

I3 ¼ ∫ wðξ1 ÞT Nðη; ζÞT νξ21 kb1 ðη; ζÞk2 jJjdηdζ

Adopting integration by parts to the terms including δφðξÞ;ξ in the form of ξ in Eq. (29) leads to

Substituting Eqs. (33)–(35) into Eq. (26) yields

ξ e   ξ  I1 ¼ wðξÞT E0 φðξÞξ ξ2   ∫ ξei wðξÞT E0 2ξφðξÞξ þ ξ2 φðξÞξξ dξ ξi   ξ1 ξ T T T þ wðξÞ E1 φðξÞξξ  ∫ ξ10 wðξÞ ET1 φðξÞ þ ξφðξÞξ dξ 0   ξ1 ξ T T þ ∫ ξ0 wðξÞ E1 φðξÞ þ ξφðξÞξ dξ þ ∫ ξ10 wðξÞT E1 ξφðξÞξ dξ

∫ V ð∇wÞT ∇ϕdV  ∫ Ωw w

ω2 ϕdV  ∫ Ωb wvb dV g

 ξ ¼ wðξ1 ÞT E0 φðξ1 Þξ ξ21  wðξ0 ÞT E0 φðξ0 Þξ ξ20  ∫ ξ10 wðξÞT E0 2ξφðξÞξ  þ ξ2 φðξÞξξ dξ  wðξ1 ÞT ET1 φðξ1 Þξ1  wðξ0 ÞT ET1 φðξ0 Þξ0   ξ ξ  ∫ ξ10 wðξÞT ET1 φðξÞ þ ξφðξÞξ dξ þ ∫ ξ10 wðξÞT E1 ξφðξÞξ dξ

ξ

þ ∫ ξ10 wðξÞT E2 φðξÞdξ

 ξ T T T ¼ wðξ1 Þ E0 φðξ1 Þξ ξ21  wðξ0 Þ E0 φðξ0 Þξ ξ20  ∫ ξ10 wðξÞ E0 2ξφðξÞξ  þ ξ2 φðξÞξξ dξ  wðξ1 ÞT ET1 φðξ1 Þξ1  wðξ0 ÞT ET1 φðξ0 Þξ0   ξ ξ  ∫ ξ10 wðξÞT ET1 φðξÞ þ ξφðξÞξ dξ þ ∫ ξ10 wðξÞT E1 ξφðξÞξ dξ

ξ

þ ∫ ξ10 wðξÞT E2 φðξÞdξ þ

ω2 wðξ1 ÞT M0 φðξ1 Þ g

þ ∫ wðξ1 ÞT Nðη; ζÞT νξ21 kb1 ðη; ζÞk2 jJjdηdζ Ωb

ξ

þ ∫ ξ10 wðξÞT E2 φðξÞdξ

¼0 (33)

(38)

The second part of Eq. (26) can be expressed as

I2 ¼ ∫ wðξ1 ÞT Nðη; ζÞT Ωw

ω2 Nðη; ζÞφðξ1 Þξ21 kb1 ðη; ζÞk2 jJjdηdζ g

The following equations can be obtained if Eq. (38) is satisfied for all sets of the weighting functions wðξÞ

(34)

Introducing

M0 ¼ ∫ Nðη; ζÞ Ωw

T

Nðη; ζÞφðξ1 Þξ21 kb1 ðη; ζÞk2 jJjdηdζ

(37)

Ωb

E0 φðξ0 Þξ ξ2i þ ET1 φðξ0 Þξ ξ0 ¼ ∫ Ω Nðη; ζÞvkb1 k2 jJjξ20 dηdζ

(39)

E0 φðξ1 Þξ ξ2e þ ET1 φðξ1 Þξ ξ1 ¼ ∫ Ω Nðη; ζÞvkb1 k2 jJjξ21 dηdζ

(40)

(35) E0 φðξ1 Þξ ξ21 þ ET1 φðξ1 Þξ ξ1 ¼

Eq. (34) becomes

ω2 φðξ1 ÞM0 g

    E0 ξ2 φðξÞξξ þ ξ 2E0  E1 þ ET1 φðξÞξ þ ET1  E2 φðξÞ ¼ 0 444

(41) (42)

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Fig. 16. The mode shapes on the two side surfaces of different baffles.

tank wall and liquid free surface, respectively. The derivations as mentioned above are only for on element of SBFEM in the shadow color element in Fig. 2(a). In order to model the total space, an assemblage for the entire boundary should be processed as in the conventional finite element method. For example, in Fig. 2(c), the cubic tank has a floating surface-piercing baffle, which can be divided into two sub-domains by

Eq. (42) represents the fundamental governing equation of SBFEM about the three-dimensional liquid sloshing problem, which is a standard homogeneous second-order ordinary differential equation. Eqs. (39)–(41) are the interior and exterior boundary conditions in SBFEM. Among them, Eq. (39) will vanish due to that it becomes a scaling center ðξ ¼ 0Þ. Eqs. (40) and (41) satisfy the exterior boundary conditions at the 445

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Fig. 16 (continued).

as that of models in Fig. 2(c), in which the sub-domains can be obtained by dividing the domain using the baffles and the extending virtual surface (another principle is that the entire boundary of a sub-domain must be visible. Therefore, the sub-domain can also be obtained by dividing the domain using the virtual surface). A dual variable is introduced to solve Eq. (42), which is defined as

extending the baffle to the bottom of the container with the baffle's and virtual surface. And the surface of the baffle should be discretized two times (the element with color red nodes), while the virtual surface should be discretized one times to connecting the two sub-domains, which conform the continuities of velocities and velocity potentials of the two sub-domain. The processing in the example discussed below is the same

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Fig. 16 (continued).

Table 7 The first eleventh sloshing frequencies (Hz) in an elliptical toroidal tank with wall-mounted ring baffle on outer wall of tank. Modes

b=a ¼ 0:50

b=a ¼ 0:75

b=a ¼ 0:90

1 2 3 4 5 6 7 8 9 10 11

0.4045 0.4045 0.7096 0.7131 0.9350 0.9350 1.0632 1.1279 1.1288 1.1480 1.1480

0.4888 0.4888 0.8409 0.8438 1.0951 1.0951 1.2305 1.3106 1.3109 1.3247 1.3247

0.5305 0.5305 0.9015 0.9041 1.1658 1.1658 1.3013 1.3878 1.3880 1.3971 1.3971

with

qðξÞ ¼ E0 φðξÞξ ξ2 þ ET1 φðξÞξ ξ

Fig. 17. The various horizontal baffles in the toroidal tank with elliptical cross section.

(44)

Then, Eq. (42) becomes the state equation

 XðξÞ ¼

ξþ1=2 φðξÞ ξ1=2 qðξÞ



ξXðξÞ;ξ ¼¼ ½ZXðξÞ (43) where

447

(45)

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Fig. 19. Time histories of sloshing force of the toroidal tanks with different filled height and L1 =L ¼ 0:5.

Fig. 18. Time histories of sloshing force of the elliptical toroidal tanks with the length of baffle L1 =L ¼ 0:5.

QðξÞ ¼ Φ21 ξλi c1

Table 8 The first eleventh sloshing frequencies (Hz) in a toroidal tank with wall-mounted ring baffle on the outer wall of tank filled different liquid height. Mode

h=r ¼ 0:75

h=r ¼ 1:00

h=r ¼ 1:25

1 2 3 4 5 6 7 8 9 10 11

0.6223 0.6223 1.0181 1.0183 1.2929 1.2929 1.4164 1.5165 1.5165 1.5212 1.5213

0.5733 0.5733 1.0138 1.0138 1.3183 1.3183 1.4664 1.5374 1.5374 1.5452 1.5453

0.4759 0.4759 0.8311 0.8326 1.0865 1.0865 1.2474 1.2985 1.2999 1.3283 1.3283



T 0:5I  E1 0 E1 Z¼ T E1 E1 E þ E2 0 1

E1 0 T E1 E 0 1  0:5I

Therefore, considering the boundary condition ðξ ¼ 1Þ leads to the equilibrium equation through Eqs. (40) and (41)

Kφðξ ¼ 1Þ ¼ Qðξ ¼ 1Þ

K ¼ Φ21 Φ1 11

(46)

Φ¼

Φ11 Φ21

Φ12 Φ22

φb ¼ K1 bb Ka φa

(49)

XðξÞ ¼

ξþ1=2 φðξÞ ξ1=2 qðξÞ



 ! ξ Φ11 ξ½λ c1 þ Φ12 ξ½λ c2   ξþ1=2 Φ21 ξ½λ c1 þ Φ22 ξ½λ c2

(50)

where the parameters c1 and c2 are integration constants. A finite solution existing at the scaling centre leads to c2 ¼ 0. The solution of Eq. (50) can further be obtained

φðξÞ ¼ Φ11 ξλi c1

 ¼

 0 0

(55)

(56) (57)

K0 ¼ Kaa  Kab K1 bb Kba

(58)

1 M0 ¼ M0 g

(59)

The eigenfrequencies and eigenvectors φa can be solved through the eigenvalue Eq. (47). Then, φb can be calculated by using Eq. (56). Meanwhile, the eigenvectors of the interior points on the wall can be obtained from Eq. (51). In order to solve the transient sloshing problems, an additional function φ* ðx; y; zÞ is introduced and then the weak state equation in the variational form of Eqs. (1)–(3) by using Green's theorem can be obtained similar as the Reference (Patkas and Karamanos, 2007; Karamanos et al., 2009).

þ1=2

¼

φa φb

 K0  ω2 M0 φa ¼ 0



(48)

Thus, one can achieve the solution of Eq. (45)





nodes. For convenience, the abbreviated form φa and φb represent φa ðξ ¼ 1Þ and φb ðξ ¼ 1Þ. One can employ the condensation from Eq. (55) to eliminate the vectors φb at the boundary of tank wall such as

and the corresponding eigenvector matrix is partitioned in the form



0 0

2

dλi c



ω2 M0 Kab  Kbb 0 g

Qa ðξ ¼ 1Þ ¼ wg M0 φa ðξ ¼ 1Þ and Qb ðξ ¼ 1Þ ¼ 0 are the velocities of the

(47)

⌈  λi c

Kaa Kba

where φa and φb represent the velocity potentials of the nodes on boundaries of the liquid free surface and the tank wall, respectively.

in which λi andλi are eigenvalues and are arranged to the diagonal matrices

Λ¼

(54)

Eq. (53) can be split into the following form



Eq. (45) can induce the eigenvalue problem such as



(53)

where



½ZΦ ¼ ΦΛ

(52)

(51)

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Fig. 20. The various vertical baffles in the toroidal tank with circular cross section. (a) bottom-mounted baffle, (b) surface-piercing baffle, and (c) combination with bottom-mounted and surface-piercing baffles.

  1 ∂2 Φs 1 ∂2 Φ U * ∫ ∇Φs ∇φ* dV þ ∫ 2 φ* dΩ ¼  ∫ φ dΩ g Ωw ∂t g Ωw ∂t 2 V

  ∂ΦU ∂Φu ∂Φs ðbe ⋅nÞdΩ  ρ∫ Ωb þ ðbe ⋅nÞdΩ ∂t ∂t Ωb ∂t   ∂Φu ∂Φs þ ðbe ⋅nÞdΩ ¼ ML X€  ρ∫ Ωb ∂t ∂t

F ¼ Fu þ Fs ¼ ρ∫

(60)

where Ωw is corresponding to boundary of the free liquid surface. Using the orthogonality of the natural sloshing eigenvectors, the velocity potential Φs ðx; y; z; tÞ and φ* ðx; y; zÞ are formed as (Patkas and Karamanos, 2007).

Φs ðx; y; z; tÞ ¼

∞ X

Y_ n ðtÞφn ðx; y; zÞ

where Ωb is corresponding to boundary of tank wall and ML represents the mass of total liquid. The vector Fs can be performed as

(61)

m¼1;2;3⋯

Fs ¼ ρ∫

Ωb

φ* ðx; y; zÞ ¼

∞ X

bn φn ðx; y; zÞ

X ∂Φs ðbe ⋅nÞdΩ ¼  F n Y€n ∂t n

1 1 ∫ ψ xdΩ ¼ φTn ∫ NT xdΩ g Ωw n g Ωw

ðn ¼ 1; 2; 3⋯Þ

(68)

Introducing the variables

 an ¼

 Mn Yn Pn

(69)

The motion of liquid sloshing about Eq. (63) becomes (Patkas and Karamanos, 2007).

where ωn are the eigenvalues of the free liquid sloshing solved by using SBFEM, and

Pn ¼

Mnc a€n  ML X€

n¼1;2;3⋯

(63)

ðn ¼ 1; 2; 3⋯Þ

∞ X

F ¼ Fs þ FU ¼ 

where the dot is the derivation about time, bn and Y_ n ðtÞ represent the constants and generalized parameters of coordinates, respectively, and φn ðx; y; zÞ ðn ¼ 1; 2; 3⋯Þ are eigenvectors solved by using SBFEM. Substituting Eqs. (61) and (62) into Eq. (60) leads to (Patkas and Karamanos, 2007).

1 1 1 M n ¼ ∫ ψ2n dΩ ¼ φTn ∫ NT NdΩφn ¼ φTn M0 φn g Ωw g Ωw g

(67)

Thus, the total force becomes

(62)

m¼1;2;3⋯

M n Y€n þ ω2n M n Yn ¼ Pn X€ ðn ¼ 1; 2; 3⋯Þ

(66)

€ b a€n þ ω2n an ¼  X ðtÞ

ðn ¼ 1; 2; 3⋯Þ

(70)

(64) 4. Validation of SBFEM model This section provides an example to verify the proposed SBFEM for the determination of the sloshing natural frequencies. As illustrated in Fig. 1, a typical toroidal tank without baffle is provided as test cases to investigate the lowest six sloshing natural frequencies which bring out the sloshing masses, and the results are compared with those of an

(65)

The total force Fðx; y; z; tÞ on the tank can be split into two parts about the rigid uniform body force Fu and sloshing hydrodynamic force Fs 449

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Fig. 21. Model and mesh of SBFEM for the toroidal tank with vertical surface-piercing ring baffle having L1 =L ¼ 0:5. (a) The model without the liquid free surface, (b) Showing twelve subdomains without the liquid free surface, (c) Showing mesh of twelve sub-domains without the liquid free surface, (d) Showing mesh of twelve sub-domains with the liquid free surface, and (e) Showing mesh of each sub-domain.

considering the flow separation by the baffle, a cubic tank with surfacepiercing baffle has been taken for example. As shown in Fig. 2(c), the entire domain is divided into two sub-domains in the SBFEM with 1386 nodes. Considering the symmetry of this model, only the first fifth mode shapes on the left baffle surface and the virtual surface (which are corresponding to the middle surface of the cubic tank) are shown in Fig. 7. It is can be seen from this figure that the present results by SBFEM are in good agreement with those from FEM using the business software named ANSYS (which should discretize the entire domain), and the flow can be separated by the baffle, especially for the firth and third modes. Meanwhile, the more obvious flow separation by the baffle can be validated in the following numerical examples.

approximate solution as illustrated in Fig. 3 with an annular tank analogy (in which R0 and Ri are the exterior and interior radiuses of the annual cylindrical tank, respectively, and ha is the liquid height, which are defined in Eqs. (71)–(73)) and the experimental technique (Meserole and Fortini, 1987) in Table 1. The filled liquid height is taken as H ¼ r, as shown in Fig. 1. As shown in Fig. 4, the total domain is divided into four sub-domains in the SBFEM, and three meshes including 996, 2284 and 4100 nodes with 8-node element are used to illustrate the convergence of the method. It is can be observed form Table 1 that the present results by SBFEM are in good agreement with those from experimental technique, the present method has more higher accuracy than the approximate solution, and can achieve excellent convergence. Meanwhile, the mode shapes (the lowest six sloshing modes which bring out the sloshing masses) along the x axis and on the total liquid free surface observed in the oscillation are illustrated schematically in Figs. 5 and 6, which are normalized by the maximum value of the point for the case of the x axis. From those figures, the results are very similar to those in the literature (Meserole and Fortini, 1987).

Ri ¼ R  2r

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h=2rð1  h=2rÞ

(71)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h=2rð1  h=2rÞ

(72)

Ro ¼ R þ 2r " ha ¼ r=2

1

5. Numerical results and discussion In this section, a parametric study is carried out to investigation effects on the normalized sloshing frequencies, the associated mode shapes and sloshing forces in partially filled rigid tanks with various baffles including liquid fill level, the ratio of the major axis and the minor axis of the elliptical cross section, baffled arrangement and length of those baffles. 5.1. Effect of the horizontal length of the various baffles

#

cos ð1  h=rÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ð1  h=rÞ 2 h=2rð1  h=2rÞ

In this section, the effects of the length of the baffles in the toroidal tank with circular cross section on the first eleventh sloshing frequencies, the associated partial mode shapes and sloshing force are investigated, and four horizontal baffles such as wall-mounted ring baffle on inner wall

(73)

Meanwhile, in order to validate the proposed method accuracy 450

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Ocean Engineering 146 (2017) 434–456 Table 10 The nodes of mesh for different cases with different baffled length. bottom-mounted baffle

surface-piercing baffle

L1 =L1 ¼ 0:25 L1 =L1 ¼ 0:50

3644 4204

3644 4204

L1 =L1 ¼ 0:75

4764

4764

combination

4092 (twenty subdomains)

Table 11 The first eleventh sloshing frequencies (Hz) of the case with bottom-mounted baffle.

of tank named inner baffle, wall-mounted ring baffle on outer wall of tank named outer baffle, the combination of inner and outer baffles, and floating ring baffle are considered, in which all the baffles are located in the middle height of the tank, the filled liquid height is h ¼ r, L1 is the horizontal length of baffle, the length L is defined as the distance of the baffle extending to the tank wall and the cross section of those baffles are as shown in Fig. 8. The calculated domain of all modes is divided into eight sub-domains in the SBFEM, and the case with outer baffle ðL1 =L ¼ 0:5Þ is taken for example where there is 3912 nodes on the boundaries and the two faces of the ring baffle has been discretized, as illustrated in Figs. 9 and 10. In Fig. 10, the first and fifth sub-domains (which is the quarter model) taken for example to illustrate the scaling centre and discretization on the boundary of each sub-domain for the

Table 9 The coordinate of scaling centre of each sub-domain. x

y

z

1 2 3 4 5 6 7 8 9 10 11 12

0.23 0.23 0.23 0.23 0.2 0.2 0.2 0.2 0.24 0.24 0.24 0.24

0.23 0.23 0.23 0.23 0.2 0.2 0.2 0.2 0.24 0.24 0.24 0.24

0.1 0.1 0.1 0.1 0.04175 0.04175 0.04175 0.04175 0.04175 0.04175 0.04175 0.04175

L1 =L ¼ 0:00

L1 =L1 ¼ 0:25

L1 =L ¼ 0:50

L1 =L ¼ 0:75

1 2 3 4 5 6 7 8 9 10 11

0.5733 0.5733 1.0138 1.0138 1.3183 1.3183 1.4664 1.5374 1.5374 1.5452 1.5453

0.5736 0.5736 1.0124 1.0133 1.3169 1.3169 1.4551 1.5284 1.5284 1.5442 1.5451

0.5707 0.5707 1.0039 1.0039 1.3071 1.3071 1.3985 1.4842 1.4842 1.5376 1.5380

0.5658 0.5658 0.9870 0.9871 1.2533 1.2897 1.2897 1.3735 1.3735 1.5257 1.5266

SBFEM. Meanwhile, the mesh nodes of various cases with different length of the baffle are list in Table 2, the coordinate of scaling centre of each sub-domain is also list in Table 3, and the different baffles including outer baffle, inner baffle, combination baffle and floating ring baffle are illustrated in Fig. 11. Tables 4–6 give the first eleventh sloshing frequencies (Hz) of the toroidal tank with inner, outer, floating ring baffle, and floating ring baffles having different length ðL1 =L ¼ 0:00; 0:25; 0:5; 0:75Þ, and Fig. 12 illustrates the mode shapes of the four cases with various baffles ðL1 =L ¼ 0:5Þ mentioned above along the x axis of the first two sloshing modes which will bring out the sloshing forces. As can be seen from those tables and figures, the same values of the natural frequencies appear in pair because of the symmetry of the toroidal tanks about the x and y coordinates, increasing the horizontal baffle's length yields the systemic natural frequencies to decrease, and the horizontal baffles being close to the exterior wall of the tank have the most possibility to reduce the natural frequencies. The location of the baffles has small effect on the free-surface profiles of the second mode but strongly alters the free surface of the first mode, especially when the baffles move near to the exterior wall of the tank. The time histories of lateral sloshing forces subjected to the turning maneuver excitation (as shown in Fig. 13) for the above four baffle with different length ðL1 =L ¼ 0:00; 0:25; 0:5; 0:75Þ are as illustrated in Fig. 14. From this figure, extending the length of wall-mounted ring baffle on outer wall of tank carries out the greater suppression of the sloshing force, while increasing the lengths of the other three kinds of baffle has very small effect on the lateral sloshing force. Fig. 15 investigates the effects of the input time ðt0 ¼ 0:1; 0:5; 1:0; 2:0Þ on the sloshing force for the different baffles. It can be observed that the input time has sensitive to the lateral sloshing force for all case with different baffles, and a shorter rise time will bring out higher value of the sloshing force, which is due to the higher rate of the input's change. Meanwhile, the lowest eight sloshing mode shapes along the x axis and on the two side surfaces of the different baffle are illustrated schematically in Fig. 16. It can be seen from this figure, the modes shape between the two side surfaces have a distinct difference and sometimes the maximum values will appear the edge of the baffles especially for the outer baffles, which means that flow separation due to the baffles is obvious. The same value of the mode shapes (only the direction of modes of shapes is different) appearing in pair verifies the results of Tables 4–6. Meanwhile, the outer baffle compared to the inner and combination baffles is an efficient means for controlling the sloshing behavior of partially filled toroidal tanks.

Fig. 22. Quarter model of the mesh of SBFEM for the toroidal tank with vertical surfacepiercing ring baffles of container having L1 =L ¼ 0:5.

Mode

Modes

451

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Table 12 The first eleventh sloshing frequencies (Hz) of the case with surface-piercing and combination baffles. Modes

1 2 3 4 5 6 7 8 9 10 11

surface-piercing baffle

combination baffle

L1 =L ¼ 0:00

L1 =L1 ¼ 0:25

L1 =L ¼ 0:50

L1 =L ¼ 0:75

L1 =L ¼ 0:50

0.5733 0.5733 1.0138 1.0138 1.3183 1.3183 1.4664 1.5374 1.5374 1.5452 1.5453

0.5727 0.5727 1.0104 1.0105 1.3121 1.3121 1.4228 1.4966 1.4966 1.5381 1.5391

0.5708 0.5708 1.0001 1.0002 1.2958 1.2958 1.3171 1.4028 1.4028 1.5224 1.5233

0.5754 0.5754 0.9976 1.0049 1.2173 1.2970 1.2970 1.3227 1.3227 1.5242 1.5305

0.5746 0.5746 1.0115 1.0139 1.3151 1.3151 1.4159 1.4922 1.4922 1.5413 1.5444

Fig. 23. The mode shapes along the x axis for the vertical baffles. (a) first mode, and (b) second mode.

Fig. 24. Time histories of sloshing force of the toroidal tanks for various lengthL1 of the vertical baffles.(a) bottom-mounted baffle, and (b) surface-piercing and combination baffles.

entails an increase in the sloshing frequencies. Fig. 18 shows the variations of the lateral sloshing forces with time against the ratio b=a with the horizontal length of baffle L1 =L ¼ 0:5. From the figure, as the ratio b=aincreases, the sinusoidal variation of the sloshing force has been transformed to an intense oscillatory nature, and the bigger the ratio b=a is, the bigger frequencies have. Therefore, the ratio b=a has significant effect on sloshing frequencies and the lateral sloshing forces.

5.2. Effect of the ratio of the major and minor semiaxes The elliptical cross section of a toroidal tank with wall-mounted ring baffle on outer wall of tank (installed in the middle of the water) is illustrated in Fig. 17, in which a and b are the major and minor semiaxes, the filled liquid height h ¼ b, Lis the length extending the baffle to the wall of the tank, and L1 is the length of the baffle. In this model of SBFEM, there are eight sub-domains with 3912 nodes for each case. Table 7 gives first eleventh sloshing frequencies with different ratio b=a ¼ 0:50; 0:75; 0:90 and L1 =L ¼ 0:5. It can be seen from the table, a higher effect of the baffle the ratio b=a is observed, and increasing the ratio b=a

5.3. Effect of fill levels Table 8 list the first eleventh sloshing frequencies (Hz) of the toroidal 452

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illustrated in Fig. 21. Similar to Section 5.1, the quarter model including the scaling centre and meshes of SBFEM is illustrated in Fig. 22, and the coordinates of the scaling centres are list in Table 9. Meanwhile, the mesh nodes of various cases with different baffled length are list in Table 10. Tables 11 and 12 list the first eleventh sloshing frequencies (Hz) of the toroidal tank with those baffles having different length ðL1 =L ¼ 0:00; 0:25; 0:5; 0:75Þ, and Fig. 23 shows the mode shapes of the three cases with various baffles ðL1 =L ¼ 0:5Þ mentioned above along the x axis of the first two sloshing modes which will bring out the sloshing forces. It can be observed from those tables and figures that a shorter bottom-mounted baffle leads to negligible influence on the sloshing frequencies, irrespective of the baffle being close to the liquid free surface, and the results generally show lower magnitudes of sloshing frequencies with increasing baffle length for the bottom-mounted baffle. As to the surface-piercing baffle, the length of baffle brings out the fluctuation of first two sloshing frequencies, while the values of the other higher sloshing frequencies monotonic descent with an increase of the baffle length. The location of the vertical baffles has small influence on the free-surface profiles of the first mode but strongly alters the free surface of the second mode for the case of surface-piercing ring baffle. The time histories of lateral sloshing forces subjected to the turning maneuver excitation for the above three baffle with different length ðL1 =L ¼ 0:00; 0:25; 0:5; 0:75Þ are as illustrated in Fig. 24. The results demonstrated that increasing the length of the bottom-mounted vertical baffle initially has very small influence on the sloshing force, while this effect becomes notable as the baffle moves to the liquid free surface. Extending the surface-piercing vertical baffle has a clearly higher influence on the sloshing force, and an optimal baffle length could be used to suppress the sloshing force. In generally, the surface-piercing vertical baffle is overall more effective than the bottom-mounted and combination baffles with the same length.

Table 13 The first eleventh sloshing frequencies (Hz) of the case with free surface-touching baffle. Modes

L1 =L ¼ 0:00

L1 =L1 ¼ 0:25

L1 =L ¼ 0:50

1 2 3 4 5 6 7 8 9 10 11

0.5733 0.5733 1.0138 1.0138 1.3183 1.3183 1.4664 1.5374 1.5374 1.5452 1.5453

0.6603 0.6603 1.1679 1.1679 1.5253 1.5253 1.7915 1.7917 1.8179 1.8801 1.8801

0.7946 0.7946 1.3613 1.3614 1.7307 1.7307 1.9913 1.9915 2.1960 2.1960 2.3365

5.5. Effect of the length of free surface-touching baffle Table 13 presents the first eleventh sloshing frequencies (Hz) of the toroidal tank with free surface-touching baffle having different length ðL1 =L ¼ 0:00; 0:25; 0:5Þ, in which L ¼ r ¼ h, as shown in Fig. 25. Meanwhile, Fig. 21 gives the 2D liquid free surface contour plots of the first six mode shapes with different baffle including wall-mounted horizontal ring baffle on outer wall of tank, bottom-mounted vertical baffle, surfacepiercing baffle, and free surface-touching baffle. From Table 13 and Fig. 26, it can be found that an increase of the length of free surfacetouching baffle causes the rapid increase of the sloshing frequencies, and the baffle touching the liquid free surface brings out apparent effect on the 2D liquid free surface contour plots. The time histories of lateral sloshing forces subjected to the lateral excitation for the free surfacetouching baffle with different length ðL1 =L ¼ 0:00; 0:25; 0:5Þ are as depicted in Fig. 27. From the figure, the length of the free surfacetouching baffle has great influence on the sloshing force, and an appropriate length can have the effectiveness to suppress the sloshing force. In general, free surface-touching baffle has higher influence that those of the other baffles. The baffle being near to the outer wall can more effectively suppress the liquid sloshing of toroid tanks.

Fig. 25. The free surface-touching baffle in the toroidal tank with circular cross section.

tank with wall-mounted ring baffle on outer wall of tank filled liquid with different height ðh=r ¼ 0:75; 1:00; 1:25Þ as shown in Fig. 7(b), in which the horizontal length of baffle is L1 =L ¼ 0:5, and the time histories of lateral sloshing forces with different fill levels ðh=r ¼ 0:75; 1:00; 1:25Þ are also as illustrated in Fig. 19. In this model of SBFEM, there are eight sub-domains with 3912 nodes for each case. It can be found that all the sloshing frequencies and sloshing force increase with an increase of fill levels due to greater wetted area, and when the fill height increase, the sinusoidal variation of the sloshing force becomes more obvious. Therefore, the effectiveness of the filled levels on the sloshing frequencies and sloshing force is bigger.

5.4. Effect of the vertical length of the various baffles 6. Conclusion In this section, the effects of the vertical length of the baffles in the toroidal tank on the first eleventh sloshing frequencies, the associated partial mode shapes and sloshing force are investigated, and three vertical baffles such as bottom-mounted and surface-piercing ring baffles as well as their combination form are considered, in which all the baffles are located in the vertical central axis of the cross section of the toroidal tanks, the filled liquid height is h ¼ r and L ¼ h, L1 is the vertical length of baffle as shown in Fig. 20. The calculated domain of all modes is divided into twelve sub-domains in the SBFEM, and the case with surfacepiercing baffle ðL1 =L ¼ 0:5Þ is taken for example where there is 4140 nodes and the two faces of the ring baffle has been discretized, as

In this paper, a semi-analytical technique named scaled boundary finite element method (SBFEM) has been developed to obtain the natural frequencies, the associated mode shapes, and sloshing forces of liquid sloshing in a rigid circular/elliptical toroidal tank with various baffles containing the horizontal bottom-mounted and surface-piercing ring baffles as well as their combination form, bottom-mounted and surfacepiercing ring baffles as well as their combination, and free surfacetouching baffle. By adopting zone technique, the liquid domain is divided into several liquid sub-domains so that the velocity potential in each sub-domain becomes class C1 due to the continuity boundary 453

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Fig. 26. The 2D liquid free surface contour plots of the first six mode shapes with different baffles.

dimensional problem. Then, the solution for the SBFEM equations is derived in detail, and the eigenvalue problems for free liquid sloshing with SBFEM are also obtained. Using multimodal method, an efficient methodology is adopted to calculate the sloshing masses and sloshing forces. The results of numerical examples show that the proposed method

conditions. The analytical solutions of the liquid velocity potential in each liquid sub-domain based on the linear potential theory and weighted residual technique are obtained by using scaled boundary finite element method, and only the boundary of the domain should be discretized, which transforms the three-dimensional problem into the two454

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free surface brings out apparent effect on the 2D liquid free surface contour plots. The length of the free surface-touching baffle has great influence on the sloshing force, and an appropriate length can have the effectiveness to suppress the sloshing force. Those results may present a valuable tool for the practical design engineering. Acknowledgments This research was supported by Grants 51779037, 51279026 and 51309049 from National Natural Science Foundation of China, for which the authors are grateful. References Abramson, H.N., 1969. Slosh Suppression. NASA Technical Report Sp-8031. Akyildiz, H., Ünal, N.E., 2006. Sloshing in a three-dimensional rectangular tank: numerical simulation and experimental validation. Ocean. Eng. 33, 2135–2149. 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. Akyldz, H., Erdem Unal, N., Aksoy, H., 2013. An experimental investigation of the effects of the ring baffles on liquid sloshing in a rigid cylindrical tank. Ocean. Eng. 59, 190–197. Armenio, V., Rocca, M.L., 1996. On the analysis of sloshing of water in rectangular containers: numerical study and experimental validation. Ocean. Eng. 23, 705–739. Bazyar, M.H., Grai, A., 2012. A practical and efficient numerical scheme for the analysis of steady state unconfined seepage flows. Int. J. Numer. Anal. Methods Geomech. 36, 1793–1812. Bazyar, M.H., Talebi, A., 2015a. A scaled boundary finite-element solution to nonhomogeneous anisotropic heat conduction problems. Appl. Math. Model. 39 (23), 7583–7599. Bazyar, M.H., Talebi, A., 2015b. Transient seepage analysis in zoned anisotropic soils based on the scaled boundary finite-element method. Int. J. Numer. Anal. Methods Geomech. 39 (1), 1–22. Bazyar, M.H., Song, C., 2017. Analysis of transient wave scattering and its applications to site response analysis using the scaled boundary finite-element method. Soil Dyn. Earthq. Eng. 98, 191–205. 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. Biswal, K.C., Bhattacharyya, S.K., Sinha, P.K., 2006. Nonlinear sloshing in partially liquid filled containers with baffles. Int. J. Numer. Methods Eng. 68, 317–337. Celebi, M.S., Akyildiz, H., 2002. Nonlinear modeling of liquid sloshing in a moving rectangular tank. Ocean. Eng. 29, 1527–1553. Chantasiriwan, S., 2009. Modal analysis of free vibration of liquid in rigid container by the method of fundamental solutions. Eng. Anal. Bound. Elem. 33, 726–730. Chen, K., Zou, D., Kong, X., 2017. A nonlinear approach for the three-dimensional polyhedron scaled boundary finite element method and its verification using Koyna gravity dam. Soil Dyn. Earthq. Eng. 96, 1–12. Cho, I.H., Kim, M.H., 2016. Effect of dual vertical porous baffles on sloshing reduction in a swaying rectangular tank. Ocean. Eng. 126, 364–373. Cho, I.H., Choi, J.S., Kim, M.H., 2017. Sloshing reduction in a swaying rectangular tank by an horizontal porous baffle. Ocean. Eng. 138, 23–34. 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. Cho, J.R., Lee, H.W., 2004. Numerical study on liquid sloshing in baffled tank by nonlinear finite element method. Comput. Methods Appl. Mech. Eng. 193, 2581–2598. Cho, J.R., Lee, H.W., Kim, K.W., 2002. Free vibration analysis of baffled liquid-storage tanks by the structural-acoustic finite element formulation. J. Sound Vib. 258, 847–866. Dodge, F.T., 1971. Engineering Study of Flexible Baffles for Slosh Suppression. NASA Contractor Reports CR-1880. Ebrahimian, M., Noorian, M.A., Haddadpour, H., 2014. Equivalent mechanical model of liquid sloshing in multi-baffled containers. Eng. Anal. Bound. Elem. 47 (1), 82–95. 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. Ebrahimian, M., Noorian, M.A., Haddadpour, H., 2015. Free vibration sloshing analysis in axisymmetric baffled containers under low-gravity condition. Microgravity Sci. Technol. 27 (2), 97–106. 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, 198–205. Evans, D.V., McIver, P., 1987. Resonant frequencies in a container with a vertical baffle. J. Fluid Mech. 175, 295–307. Faltinsen, O.M., Timokha, A.N., 2009. Sloshing. Cambridge University Press, Cambridge. Firouz-Abadi, R.D., Haddadpour, H., Noorian, M.A., Ghasemi, M., 2008. A 3D BEM model for liquid sloshing in baffled tanks. Int. J. Numer. Methods Eng. 76, 1419–1433.

Fig. 27. Time histories of sloshing force of the toroidal tanks for various length L1 of the surface-touching baffles.

can achieve excellent accuracy and convergence, and have higher efficiency compared to the approximate solution and experimental technique. The influences of the wide range parameters including liquid fill level, baffled arrangement and length of those baffles in detail and the results in terms of the sloshing frequencies, sloshing mode shapes and sloshing force are presented. Increasing the horizontal baffle's length yields the systemic natural frequencies to decrease, and the horizontal baffles being close to the exterior wall of the tank have the most possibility to reduce the natural frequencies. The location of the baffles has small effect on the free-surface profiles of the second mode but strongly alters the free surface of the first mode, especially when the baffles move near to the exterior wall of the tank. Extending the length of wallmounted ring baffle on outer wall of tank carries out the greater suppression of the sloshing force, while increasing the lengths of the other three kinds of baffle has very small effect on the lateral sloshing force. The input time has sensitive to the lateral sloshing force for all case with different baffles, and a shorter rise time will bring out higher value of the sloshing force. A higher effect of the baffle the ratio b=a of elliptical cross section of the toroidal tank is observed, and increasing the ratio b=a entails an increase in the sloshing frequencies. As the ratio b=a increases, the sinusoidal variation of the sloshing force has been transformed to an intense oscillatory nature, and the bigger the ratio b=a is, the bigger sloshing forces have. Therefore, the ratio b=a has significant effect on sloshing frequencies and the lateral sloshing forces. All the sloshing frequencies and sloshing force increase with an increase of fill levels due to greater wetted area, and the effectiveness of the filled levels on the sloshing frequencies and sloshing force is bigger. A shorter bottommounted baffle leads to negligible influence on the sloshing frequencies, irrespective of the baffle being close to the liquid free surface, and the results generally show lower magnitudes of sloshing frequencies with increasing baffle length for the bottom-mounted baffle. As to the surface-piercing baffle, the length of baffle brings out the fluctuation of first two sloshing frequencies, while the values of the other higher sloshing frequencies monotonic descent with an increase of the baffle length. The location of the vertical baffles has small influence on the freesurface profiles of the first mode but strongly alters the free surface of the second mode for the case of surface-piercing ring baffle. Increasing the length of the bottom-mounted vertical baffle initially has very small influence on the sloshing force, while this effect becomes notable as the baffle moves to the liquid free surface. Extending the surface-piercing vertical baffle has a clearly higher influence on the sloshing force, and an optimal baffle length could be used to suppress the sloshing force. In generally, the surface-piercing vertical baffle is overall more effective than the bottom-mounted and combination baffles with the same length. An increase of the length of free surface-touching baffle causes the rapid increase of the sloshing frequencies, and the baffle touching the liquid 455

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