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. W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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) W. Wang et al. Ocean Engineering 146 (2017) 434–456 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) W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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) W. Wang et al. Ocean Engineering 146 (2017) 434–456 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. 443 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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) W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 446 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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) W. Wang et al. Ocean Engineering 146 (2017) 434–456 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) 448 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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 W. Wang et al. Ocean Engineering 146 (2017) 434–456 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. 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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. 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