DSpace at VNU: A fully coupled ship motion and sloshing analysis in various container geometries

J Mar Sci Technol (2012) 17:139–153 DOI 10.1007/s00773-012-0157-2 ORIGINAL ARTICLE A fully coupled ship motion and sloshing analysis in various container geometries S Mitra • L V Hai • L Jing • B C Khoo Received: July 2010 / Accepted: 27 December 2011 / Published online: February 2012 Ó JASNAOE 2012 Abstract In the present study a novel modeling approach is presented to solve the combined internal sloshing and sea-keeping problem The model deals with interesting effects arising due to the coupled interaction between the sloshing in partially filled containers of several geometries and the ship motion The study is very important for the liquid cargo carrier operating in rough sea or under different environmental conditions The resulting slosh characteristics that include transient pressure variation, free surface profiles and hydrodynamic pressure over the container walls have been reported in this study In addition, the effects of coupled ship response and sloshing on ship motion parameters have also been investigated The equations of motion of fluid, considered inviscid, irrotational, and partially compressible, are expressed in terms of the pressure variable alone A finite difference-based iterative time-stepping technique is employed to advance the coupled solution in the time domain Several parameters of interest, including the container parameters, level of liquid, thrusters modeling and some important environmental factors are investigated S Mitra (&) Á B C Khoo Department of Mechanical Engineering, National University of Singapore, Engineering Drive 2, Singapore 117576, Singapore e-mail: aeromitra@gmail.com L V Hai Department of Civil Engineering, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet Street, District 10, Ho Chi Minh City, Vietnam L Jing Institute of High Performance Computing, A STAR, Singapore, Singapore Keywords Sloshing Á Ship motion Á Coupled sloshing and sea-keeping Á Fluid–structure interaction Á Finite element method Á Wave equation Introduction Sloshing is an important dynamic phenomenon in liquid storage and transportation Sloshing flow in liquid ship cargo is firstly excited by ship motion, but the sloshing flow itself affects the ship motion in return Because of the coexistence of free surfaces outside and inside a liquid cargo tank, they provide different dynamic loads on the ship Most of the published works neglect the coupled effect of internal sloshing and external hydrodynamics because of the complexity of the problems Ships with large ballast tanks and liquid bulk cargo carriers (e.g oil tankers) often have to deal with significant sloshing loads during their operations A good example is an anti-rolling tank (ART), which is equipped to reduce the amplitude of roll motion [1] The motion instability caused by deck sloshing is another example, among many others Although the sloshing dynamics are important, unfortunately there is a lack of broad consensus on the best modeling practice for such sloshing flows In conventional ship motion analyses, the effects of tank sloshing on the global motions of a ship were assumed to be negligible and hence ignored These studies are only valid when the ship size is much larger than the size of the containers and the liquid load is full Otherwise, the interaction between sloshing and ship motion becomes significant as the ratio of volume of the containers to that of the ship exceeds a critical value [19] Hence, both the sloshing phenomenon and associated ship motion behaviour should strictly be treated in a coupled manner Lee and 123 140 Choi [2] conducted experiments and numerical analysis on the sloshing problem in cargo tanks The fluid motion was predicted using a high order boundary element method, and the structure was modeled using the thin plate theory In cases of low filling depths, hydraulic jumps were obtained when the excitation frequency is approaching the resonance frequency, whereas in the cases of high filling depths, large impact pressure was obtained In another development, Kim [3] employed numerical technique to solve the coupling problem of the ship motion and sloshing flow The study was focused on the ART The three dimensional sloshing flow was simulated using the finite difference method, while the ship motion was obtained using a time domain panel method There have been several other interesting studies on the coupling analysis The recent studies can be categorized into two main approaches: the frequency-domain approach assuming linear sloshing flow, and the time-domain approach adopting nonlinear sloshing flow For example, the coupling between ship motions and sloshing undertaken by Molin et al [4] and Malenica et al [5] was based on the linear potential theory in the frequency domain According to their study, the assumption of linear ship motion appears adequate in the coupled analysis However, issues invariably arise regarding the applicability of linear sloshing flow assumption as the amplitude or intensity increases [6] To evaluate frequency and time domain approaches, Rognebakke and Faltinsen [7] investigated two dimensional experiments of a hull section containing tanks filled with different levels of water excited in sway by regular waves and the coupled effect between ship motions and sloshing inside the container Steady-state results were obtained for the sway amplitude Other simulations were also performed using a linear and nonlinear sloshing model by assuming linear external flow Good agreement between experiments and computation was reported Zhang and Suzuki [8] carried out numerical simulations of collisions between a container ship and a double-hull crude carrier The results revealed that the fluid–structure interaction of a liquid cargo-filled tank has a significant effect on the motion and the structural response of the cargo tank Kim et al [1] studied the coupling effects between the ship motion and sloshing flows by employing the impulse–response– function formulation for linear ship motion and a computational fluid dynamics (CFD) simulation for nonlinear sloshing flows Their parametric study was carried out using the panel method, which is computationally efficient, but limited to linear ship motion Recently, Lee et al [9] studied rectangular LNG tanks subject to external loadings using a CFD program The CFD simulations were verified against experimental results They concluded that the effects of viscosity and the density ratio of fluids on impact pressures are insignificant, while the compressibility of the 123 J Mar Sci Technol (2012) 17:139–153 fluid plays an appreciable role Further, Lee et al [10] analyzed the coupling and interaction between ship motion and inner-tank sloshing using a time-domain simulation scheme Wind and current, however, are not accounted for in their ship calculations The study considered environmental forces that were due to the wave loading only Experimental and numerical studies have shown that the coupling effect between liquid cargo sloshing and LNG ship motion is significant under partial filling conditions [7] As such, knowledge of the global and local pressures imposed by the sloshing liquid are deemed important The response of a LNG carrier during offloading operation is one crucial factor affecting the safety and operability of offshore LNG terminals Some of the recent studies [11–14] show that there is still no consensus on an effective modeling approach to 3D coupled nonlinear sloshing and ship motion due to the all important environmental factors In summary, most of the aforementioned studies have been largely confined to the coupling interaction problem in two dimensions and/or linear ship motion There is a dearth of works considering the three primary environmental effects, i.e combined wind, wave and current on the sloshing dynamics In operations, a vessel may carry axisymmetric and non-axisymmetric containers For liquid cargo carrier carrying non-axisymmetric container, the two dimensional/axisymmetric findings may not be applicable Further work is therefore necessary for the non-axisymmetric sloshing analysis In addition, these past studies treated the ship motion using the simple linear models, and there are no explicit analyses or benchmark solution of the sloshing inside liquid filled containers by considering combined environmental factors In the present study, the coupling effects between the vessel motions and inviscid, irrotational liquid sloshing in partially filled conditions for axisymmetric and non-axisymmetric tanks are investigated It has been reported by Bass et al [15] and Lee et al [16] that the viscous effect on sloshing motion is not so significant The purpose of this analysis is to present an integrated and yet fairly straightforward approach to the mathematical description of the effects of liquid sloshing in containers due to ship motion and vice versa The fluid domain is discretized using 8-node trilinear brick elements for rectangular container (Fig 1a) and 4-node quadrilateral element (Fig 1c) for axisymmetric case with pressure as the nodal degree of freedom In essence, the problem is defined as an initial boundary value problem by the use of the governing differential equations and boundary conditions based on the Eulerian approach, and are formulated using the Galerkin’s principle The Newmark constant-average acceleration method [17] is employed to solve the dynamic forced vibration equation in time domain The system natural frequencies and mode J Mar Sci Technol (2012) 17:139–153 141 Fig a Typical layout and nomenclature of a general liquid cargo carrier b Coordinate system of a marine vessel c A cylindrical tank with schematic mesh d A hemispherical and a cylindrical container and its nomenclature shapes are computed from the assembled matrices using MATLAB The ship and liquid-cargo motions are coupled by the kinematic and dynamic relations in that the vessel motion will excite the tank sloshing, while the sloshinginduced load will in turn influence vessel motions Numerical results obtained on coupled sloshing and ship motion is compared to available existing solutions Modeling of sloshing 2.1 Governing equations for liquid sloshing and boundary conditions The governing equation for the inviscid compressible fluid motion in terms of the excess pressure variable p(x, y, z, t) is the wave equation in three dimensions, which can explicitly be written as, o2 p o2 p o2 p o2 p þ þ ¼ ox2 oy2 oz2 c2 ot2 ¼ p€ðx; y; z; tÞ in X: c r2 pðx; y; z; tị ẳ 1ị Pressure formulation [18] has certain numerical advantages compared to the velocity potential and the displacement based formulations, as the number of unknowns per node is only one and the pressure at the structure–fluid interface is directly obtained Besides, the compressibility comes in a natural way in a pressure formulation and can be retained without incurring considerable additional computations In general, the fluid boundary is composed of three types of boundary condition These are solid–liquid interface, free surface boundary, and non-reflecting or radiating boundary The radiating boundary is not considered for sloshing flow inside a container Figure shows different container configurations The appropriate conditions at the boundaries are as follows: Solid–liquid interface boundary (BI): continuity of normal displacement at the solid–liquid interface leads to the following relation for the linearized problem, op ¼ Àqf u€n on BI on ð2Þ The normal acceleration of the interface is denoted by u€n Free surface boundary (Bf): the linearized free surface boundary condition is given by, p ¼ qf gf; ) op p€ ¼ À on Bf on g ð3aÞ ð3bÞ where f is the free surface displacement 123 142 J Mar Sci Technol (2012) 17:139–153 2.2 Finite element discretization In the present study, a finite element method is used to calculate fluid pressure in the fluid domain in the container The smaller mesh size is chosen near the free surface than that near the bottom to resolve the large motion accurately In this method, an element in the local coordinate system (n, g, f) is mapped to the prismatic element in the global coordinate system (x, y, z) For an element with ne nodes, the shape function is denoted by Nj, and the solution variable and geometry approximation can be written as follows: pðn; g; f; tÞ ¼ X Nj ðn; g; fÞ pj ðtÞ ð4Þ matrix) and the slosh term M2 (free surface mass matrix), and can be written as Z Z 1 ðMf ịij ẳ NfTi Nf j dX ỵ NfTi Nf j dC: ð8Þ c g 2.5 Force vector The evaluation of the force vector involves computation of surface integrals on the interface elements On the interface elements, the natural coordinate n has fixed values of n ẳ ặ1, with n = -1 on the left wall and n = on the right wall, j¼1 xf ¼ ne X Nj n; g; fị xjf : 5ị jẳ1 Application of the divergence theorem to the weighted residual of Eq and implementation of the boundary conditions, we have reduced to the following Galerkin formulation, ! Z o2 p o2 p o2 p T o p " N ỵ ỵ À dX ox2 oy2 oz2 c2 ot2 X ! ! Z Z op op p ỵ qf un dC þ dC ¼ 0: N"T N"T À on on g BI Bf The element natural coordinate system with its origin at the centre of the element is invoked to transform the integrals The integrals are then evaluated using two-point Gaussian quadrature in each direction The Galerkin finite element formulation finally reduces the governing equation to as follows, ẵMf fpg ỵ ẵKf f pg ẳ fFf g: 6ị The parameters Kf, Mf and Ff are the stiffness matrix, mass matrix and element load vector for a fluid element f For free vibration problem the right hand side of Eq is dropped as it represents a forcing function 2.3 Stiffness matrix The parameter Kf, the stiffness matrix for a fluid element f, is given by  Z  oNf i oNf j oNf i oNf j oNf i oNf j ỵ ỵ Kf ịij ẳ dX: 7ị ox ox oy oy oz oz X 2.4 Mass matrix The parameter Mf in Eq is the element mass matrix, which consists of the impulsive inertia M1 (domain mass 123 Bf X Fi ¼ Z NfTi q€ un dC ¼ q u€n BI y where ẵJ force ẳ ;g y;f z;g z;f Z1 Z1 NfTi jJ jforce dgdf !À1 À1 P Ni;g yi ẳ iẳ1 Ni;f yi Ni;g zi Ni;f zi 9ị ! : f 2.6 Free surface position In the linear analysis the free surface boundary conditions are satisfied on the initial undisturbed position It is necessary to compute the displacements of the surface nodes to update the free surface position at each time step In the present analysis, Eq 3a is used to deduce the free surface displacement for all rigid containers The components of the acceleration of the fluid particles are obtained from the following inviscid flow equations (Euler equations), qf v_i ỵ p;i ẳ 0; i ẳ x; y; z ð10Þ where v_ i is the acceleration of the fluid particle The velocity components of the fluid particles are evaluated from the known values of acceleration at any instant of time using simple time integration scheme given by, vt ẳ vtDt ỵ v_ t Dt; The nodal displacements are computed from the velocities in a similar manner 2.7 Shear force and overturning moment The total shear force at the base (Q) of the tank at a given time t0 is computed using the following relation ZZ ZZ Qðt0 Þ ¼ pð0; x; z; t0 Þdxdz À pðW; x; z; t0 Þdxdz; CL CR ð11Þ where p is the hydrodynamic pressure, W is the width of the container The UL and UR represents left and right wall respectively with respect to the centre line (keel) of the J Mar Sci Technol (2012) 17:139–153 143 ship Similarly, the overturning moment (M) at the bottom of the tank about x-axis is calculated from ZZ ZZ Mt0 ị ẳ zp0; x; z; t0 ịdxdz ỵ zpW; x; z; t0 ịdxdz CL À ZZ CR ðW=2 À yÞpð0; y; z; t0 Þdydz; ð12Þ CB where UB represents the bottom boundary of the tank Modeling of marine vessels A ship at sea is subject to six oscillating motions as shown in Fig 1b These motions are produced by environmental actions depending on the heading and size of the ship with respect to the waves 3.1 Coordinate systems and transformation equations As shown in Fig 1b, two coordinate systems are commonly used to describe the positions of marine vessels One of the coordinate systems is the earth-fixed inertial reference frame Measurements of the vessel’s position and orientation coordinates are described in this frame relative to a defined origin The other system is the bodyfixed reference XYZ-frame, which is fixed on the ship and moves with it In this frame, the x-axis is oriented along the ship’s longitudinal axis and is directed from stern to bow The y-axis is oriented along the ship’s transverse axis and is directed towards starboard The z-axis is oriented along the ship’s normal axis and is directed from top to bottom The transformation between different coordinate frames is accomplished by kinematics equations which treat only geometrical aspects of motion According to Fossen [19], the position and orientation vector g ¼ ½x y z / h wŠT of the moving ship is measured in the earth-fixed coordinate system, where the variables x, y and z represent the motions in surge, sway and heave, respectively, and /, h and w are the rotations about the earth-fixed reference axes, roll, pitch and yaw, respectively The linear and angular velocity vector t = [u v w p q r]T of the ship motion is measured in the body-fixed XYZ coordinate system, where the variables u, v and w are the surge, sway and heave linear velocities, respectively, and p, q and r are the roll, pitch and yaw angular velocities, respectively The relation between the two velocity vectors in earthfixed and body-fixed reference frame can be expressed as, g_ ẳ J gịt; _ t ẳ Egị g 13ị where Jgị is the transformation matrix between the earthfixed coordinate system and the body-fixed coordinate system, EðgÞ is its inverse The matrices JðgÞ and EðgÞ are given as ð14Þ ð15Þ in which c(·) = cos(·), s(·) = sin(·) and t(·) = tan(·) 123 144 J Mar Sci Technol (2012) 17:139–153 3.2 Vessel model A moving ship is modeled as a single rigid body with six degrees of freedom The components of nonlinear ship motion such as displacement, velocity and acceleration are presented in this study for low frequency (LF) and wave frequency (WF) modes [19–21] At each time step, the ship motion induces sloshing of fluid in the liquid-filled containers and, in return, these containers produce reaction forces that affect the ship motion The resulting motions of liquid sloshing inside container is applied to the moving ship in the form of external forces scon1 and scon2 ; which will be defined shortly, for WF and LF models, respectively The motion of the ship is governed by the sea state, which is defined by several parameters These parameters and their values are based on the NORSOK wind gust and JONSWAP wave spectrums that use 30 harmonic wave components The spectral density function is written as:   Hs2 944 Y Sxị ẳ 155 x exp À x ð16Þ c ; T1 T1 where the significant wave height Hs ¼ 0:5m, gamma parameter c ¼ 3:3 and "   # 0:191xT1 À p Y ẳ exp ; 17ị 2r where T1 is the average wave period, x the frequency of the spectrum and & 0:07 for x 5:24=T1 r¼ 0:09 for x [ 5:24=T1 Based on suggestions by Sørensen [21], the coupled equations of linear WF in the body-fixed XYZ-frame and nonlinear LF models that allow for the coupled interaction between containers and the moving ship are given, respectively, as, Mxịv_ w ỵ Dp xịvw ỵ GgRw ẳ swave1 þ scon1 ; ð18Þ Mv_ þ CRB ðvÞv þ CA vr ịvr ỵ DL t ỵ DNL tr ; cr ị ỵ Ggị ẳ senv2 ỵ sthr ỵ scon2 : ð19Þ Here, M is the inertia mass matrix of system; gRw is the WF motion vector in the earth-fixed frame; G is the matrix of linear generalized gravitation and buoyancy force coefficients; tr is the relative velocity vector with respect to water current; cr is the current direction relative to the vessel; CRB(v) and CA(vr) are the skew-symmetric Coriolis matrix and centripetal matrix of the rigid-body and potential induced added mass of the current load, respectively; Dp(x) is the wave radiation damping matrix; DL is the strictly positive linear damping matrix caused by the linear wave drift damping and the laminar skin friction; DNL(tr, cr) is the 123 nonlinear damping vector; scon1 and scon2 are the vectors consisting of forces and moments produced at the bases of multi-containers for WF and LF models, respectively They can be expressed as, scon1 ¼ ½ Fx Fy Fz Mxx Myy Mzz ŠWF ; 20ị scon2 ẳ ẵ Fx Fy Fz Mxx Myy Mzz ŠLF ; ð21Þ where Fx, Fy, Fz are the resultant forces from the container which are applied at the centre O of the ship in x, y and z directions, respectively, and Mxx, Myy, Mzz are the corresponding resultant moments in three directions In Eq 18, swave1 is the first order wave excitation vector, which is modified for varying vessel headings relative to the incident wave direction, and senv2 is the slowly varying environmental loads and consists of mean wind load and second-order wave drift load vector given by senv2 ẳ swind ỵ swave2 : 22ị The thruster was simply modeled by applying forces and moments in three directions to the ship model It should be noted that the ship was modeled by lumped mass The effect of the thruster can be adjusted by varying the magnitude of forces and moments to the ship model In this work more details of the mean wind load swind and second-order wave drift load vector swave2 can be found in Fossen [19], whereas detailed thruster modeling and description can be found in Sørensen [21] and Nguyen et al [22] sthr in Eq 19 is the control vector consisting of forces and moments produced by the thruster system The main parameters of the thruster system (Ka 4-70 propeller in 19A duct, P/D = 1.0, EAR = 0.7, number of blades = 4, diameter = 4, propeller max RPM = 150, rotational inertia = 5,000) can be found in Table It should be noted that the ship was modeled by one lump mass and most of ships use thrusters to maintain their position and heading Both fixed pitch (FP) and controllable pitch (CP) propellers are used for this purpose [19] The thrust force vector sthr can be expressed as: sthr ¼ Kthr uthr Rr ð23Þ where r is the number of thrusters; Kthr is thrust coefficient matrix, which is a diagonal matrix of thrust coefficients defined as: Kthr ẳ diagfK1 n1 ị; K2 ðn2 Þ; Kr ðnr Þg ð24Þ in which n (rpm) is the propeller revolution, and uthr Rr is a dynamic positioning control variable defined as: For FP propellers : uthr ẳ ẵn1 ; n2 ; ; nr T 25ị For CP propellers : uthr ẳ ½p1 ; p2 ; ; pr ŠT ð26Þ J Mar Sci Technol (2012) 17:139–153 145 Table Thruster description ‘Ka 4-70 propeller in 19A duct, P/D = 1.0, EAR = 0.7, quadrant data’ Thruster data defined with van Lammeren quadrant data = Number of blades = Diameter = EAR = A_E/A_0, blade area ratio = 0.7 Pitch/diameter ratio = 1.0 Propeller max RPM = 150 Rotational inertia of propeller, shaft and gears, seen from propeller = 5000 Thruster position is to be redefined when loading the thrusters = [-37.8, -5, 4] Propeller motor RPM max speed = 880 Max motor power (W) = 7000000 Max motor torque (Nm) = 200000 Motor time constant for 1st order model = 0.1 in which p = P/D represents the pitch ratio, P is the traveled distance per revolution and D is the propeller diameter The effects of thruster can be adjusted by varying the magnitude of forces and moments applying to the ship model, and more detailed thruster modeling can be found in Sørensen [21] and Nguyen [22] Algorithm of fully coupled liquid-container-ship program The flow chart in Fig summarizes the algorithm involved in analyzing the coupled liquid–container–ship motion Ship parameters including the dimension of the ship and thruster data are input into the program Next, the parameters of container including geometry, location on the ship and material are also given as input These parameters pertain to the initial condition of the container and ship At each subsequent time step, the environmental conditions that include the wind, wave and current data are read into the program serving as the external loads acting on the moving ship It should be noted that the program is initiated from zero external forces on the moving ship at the first time step In our time domain analysis the time step considered is 0.02 s At each time step the output from the ship program consists of the forces of the ship acting on the containers due to the external excitations caused by the wind, wave, current and thruster These forces will induce sloshing flow in the liquid-filled containers and in return, the corresponding sloshing analysis of containers yields the external forces at the base of each container that affect the ship motion at the same time step till a desired convergence is Fig Flow chart of fully coupled liquid–container–ship program achieved The value of the convergence criteria is assumed as 10-5 Resulting motions of liquid sloshing inside container are applied to the moving ship in the form of the external forces scon1 and scon2 for WF and LF models, respectively This loop of coupled interaction will continue for each incremental time step Dt until the desired total time duration T is obtained Results and discussions 5.1 Validation Comparison is carried out for the case of time domain analysis to assess the validity of our developed slosh code The slosh dynamics of the liquid in a three dimensional rectangular container is addressed in Fig 4a, which shows the transient variation of the sloshing wave amplitude at the middle cross-section The tank considered in this case is 50 m long, 20 m wide and 10 m high with water depth of 123 146 J Mar Sci Technol (2012) 17:139–153 m The exciting ground acceleration is the NS component of the 1940 El Centro earthquake and is acting along the 20 m wall The computed solution compares satisfactorily with the one by Koh et al [23], who employed a coupled BEM–FEM code Validation of our coupled slosh code is also carried out for the time domain to assess the accuracy The dimension of the hull is assumed after Kim [1, 3] as 47.5 m (length) 13.7 m (breadth) 4.0 m (draft), and rectangular tanks are placed in such a way that tank centre coincides with ship roll centre The tank considered in this case is 2.8 m in length and 13.7 m in breadth and 2.4 m in height The motion of the ship is governed by the sea state, which is defined by several parameters These parameters and their values adopted are (1) the calculated peak wave frequency 0.72 rad/s, (2) the mean wind velocity at a height of 10 m, u10 = 7.65 m/s and (3) the mean wind direction 60° These values are based on the NORSOK wind gust and JONSWAP wave spectrums derived by Hasselmann et al [24, 25] that use 30 harmonic wave components The supply ship design main parameters and the JONSWAP wave spectrum are given in Table and Fig 3, respectively The hull motion due to three dimensional external hydrodynamics with ITTC spectrum and significant wave height m is applied to a 50% filled tank The computed solution Fig 4a–c compares satisfactorily with the one by Kim et al [1], who employed an impulse response function for ship motion and the finite difference method for sloshing flow The small difference in results observed is attributed to the difference in the sea states and Table Supply ship design parameters 123 also to the present ship model considering 6° of motion The ship model, as we reckoned, is fairly robust and has been consistently used by several other researchers as found in the works of Fossen et al [19], Balchen [20] and Sørensen [21] Recently the same ship model is also used by Nguyen et al [22] for both numerical simulations and experiments to verify the proposed hybrid-controller dynamic positioning system in varying environmental conditions from calm to extreme sea states Similar coupling (weakly) strategy is adopted by Hai et al [26] using pressure based FEM code for slosh dynamics and the MCsim b4 program for ship motion calculation 5.2 Rectangular tank sloshing coupled with ship motion The typical small liquid cargo carrier varies from 4000 to 16000 cbm For the present study the marine vessel of size 80 m long, 17.4 m width and 5.6 m draft is considered Tank dimension for a rectangular tank and other associated parameters are listed in Table The rectangular containment (50 m 12 m m) is placed in such a position in the ship that the bottom of container stays 1.5 m below the centre of buoyancy of the ship where the COB is acting m from the bottom of the ship (Fig 1a) It is important to mention that the tank dimension for code validation in Sect 5.1 is smaller than the typical tank dimension for the small LNG carrier of present interest As such, for the present coupling problem, the tank dimensions are larger Vdis Displacement volume 6000 m3 m Mass 2433 Ton 80 m Loa Overall length of ship B Breadth of ship 17.4 m T À Á ssurge max À Á ssway max À Á syaw max Design draught Maximum thrust force in surge 5.6 m 0.3 106 N Maximum thrust force in sway 0.12 106 N Maximum thrust force in yaw 9.7 106 N (Awind)surge Projected wind force area, surge 330 m2 (Awind)sway Projected wind force area, sway 720 m2 (Acurrent)surge (Acurrent)sway Projected current force area, surge Projected current force area, sway 100 m2 430 m2 22 m (r)gyr_x Radius of gyration, roll (r)gyr_y Radius of gyration, pitch 6.1 m (r)gyr_z Radius of gyration, yaw 16.3 m COB Center of buoyancy m from bottom COG Centre of gravity [-2.3, 0, -5.2] Vessel parameter Vector from COH to COB [-2.3, 0, -3.0] Vessel parameter Vector from COH to centre of added mass [0, 0, 0] Vessel parameter Vector from COH to centre of damping [-2.3, 0, -5.2] J Mar Sci Technol (2012) 17:139–153 147 Fig JONSWAP wave spectrum than the tank for the validation case Furthermore, the dimension of the ART cannot be used for small liquid cargo carrier The coupling problem for small size cargo carrier has enabled us to carry out many case studies The measurement of the sloshing loadings with/without coupling effect is performed The density of the containment oil is taken as 820 kg/m3 to get more realistic behavior of liquid sloshing for a cargo carrier when it is cruises in the sea The base line sea state is described in Table A series of parametric studies on ship parameters and container systems is carried out to evaluate the influence of the various environmental effects on the ship and subsequently to the sloshing inside liquid–tank containment system and vice versa If a fluid–structure–fluid system is weakly coupled, the sloshing mechanism produces small fluid displacements In that case slosh displacement is not supposed to affect the ship motion considerably; it is then possible to solve the ship motion and the sloshing problems separately using a fixed mesh for the fluid dynamic computation To get a set of uncoupled equations we have assigned the value of scon1 and scon2 for WF and LF models in Eqs 18 and 19 are equal to zero As such the dynamic response of the coupled liquid–tank system due to the motion of the ship transporting the liquid-filled containers can be obtained by solving Eq applicable to the sloshing problem Since the internal sloshing motion inside a tank is generated by the external ship motion, the sloshing-fluidinduced hydrodynamic forces will also be a part of the forcing functions of the ship motion These forces include the internal liquid representing the partially loaded tanks, and external hydrodynamic pressures over the hull under different sea states The former can be called sloshing induced loads which is pulse impact in nature, and the latter can be referred as sea-keeping where the wave loads Fig a Slosh displacements at the middle cross-section in a 3D rectangular tank b Transient coupled ship responses (roll) for 20% filling condition c Transient coupled ship responses (roll) for 50% filling condition d Significant roll amplitudes in random waves 123 148 J Mar Sci Technol (2012) 17:139–153 Table Baseline sea state Wave Wave spectrum = JONSWAP Significant wave height = Number of frequency components in simulation = 20 Number of wave directions in simulation = Mean direction of wave = 60 p/180 Cut-off frequency for spectrum = Spreading factor for direction spectrum = Gamma value for JONSWAP spectrum = 3.3 Wind Calculated mean wind velocity = 13.79 m/s Mean wind direction (rad.) = 30 p/180 Wind spectrum = NORSOK wind spectrum Sea surface drag coefficient = 0.0026 Sea surface drag coefficient = 0.0026 NORSOK spectrum parameter = 0.468 Number of frequency components in wind gust = 20 Minimum frequency in gust realization (Hz) = 1E-4 Maximum frequency in gust realization (Hz) = 1E-1 Max variation in wind direction = p/180 Current Current velocity (m/s) = Current direction (rad) = 90 p/180 Max variation in current velocity = 0.2 Max variation in current direction = p/180 is the dominant factor Figure 5a represents the transient pressure variation at free surface of the container at critical wind, current and wave direction for coupled and uncoupled case Figure 5b, c depicts the temporal variation of roll motion for coupled and uncoupled case It is observed from Fig 5a–c that in the coupled analysis the slosh and ship motion parameters tend to be amplified on comparison to the uncoupled case All other coupled ship responses are presented in Fig 6a–e 5.2.1 Effects of significant wave height In the definition of sea state codes [27], one can observe that the sea state changing from the calm to rough sea Hence, the effects of sea states on the sloshing response of liquid-filled containers are considered by varying the significant wave heights Hw from to m at a interval of m keeping all other parameter fixed as described in Table Figure shows the effect of significant wave heights Hw on the maximum hydrodynamic pressure on the free surface of the liquid-filled rectangular container In particular the slosh height is noted to increase when the significant wave heights are increased 123 Fig a Time history of hydrodynamic pressure variation at the free surface b Time history of roll motion (coupled case) c Time history of roll motion (uncoupled case) J Mar Sci Technol (2012) 17:139–153 149 Fig a Temporal variation of pitch response (coupled case) b Temporal variation of yaw response (coupled case) c Temporal variation of surge response (coupled case) d Temporal variation of sway response (coupled case) e Temporal variation of heave response (coupled case) 5.2.2 Effects of wave, wind and current direction As far as wind and current parameters are concerned, we have seen that their contributions are small compared to waves It may be worthwhile to mention that the American Bureau of Shipping standard recommends that the effect of wind on a small size LNG carrier is negligible, and the effect of current is also not considered compared to waves The effect of coupled sloshing characteristics of the container system due to change in the mean wave direction has been studied keeping all other parameters unaltered (Table 2) Mean directions of waves acting on the ship are changed from 0° to 180° (0° is aft, 180° is stern direction) at 30° intervals Figure shows that the maximum slosh height reaches its highest value when the mean wave acts at 90° to the ship motion surge It is also clear that the value of the highest maximum slosh displacements occurs at wind direction of 90° and current direction of 90° but the 123 150 J Mar Sci Technol (2012) 17:139–153 Fig Maximum hydrodynamic pressure due to change in significant wave height and direction Fig Effects of mean wave directions on maximum hydrodynamic pressure magnitude achieved due to the change in the direction of wind and current is much smaller compared to the wave parameter 5.2.3 Base shear and overturning moment Subsequent plots from Fig 9a, b represent base shear and overturning moments acting on the container base due to coupled sloshing and ship motion at maximum wave and wind and current effect which is acting at 90o to the direction of the motion of the ship All other values remain the same as Table The base shear in sway direction and overturning moment in roll direction happens to be severe compared to the other direction of motion The data is very useful for dynamic stability analysis of the marine vessel Fig a Transient base shear variation b Temporal variation of overturning moment and overturning moment which is very useful in designing any liquid filled containment structures 5.3 Some studies on axisymmetric tank sloshing coupled with ship motion 5.2.4 Free surface profile and hydrodynamic pressure distribution 5.3.1 Effect of percentage of liquid fill in the cylindrical tank The following Fig 10a represents a snapshot of free surface profile at 25 s to show the effect of three dimensionality and random excitation acting on the tank in six displacement directions The hydrodynamic pressure variation data as shown in Fig 10b are used to find base shear The influence of level of liquid filling on global motion of the ship is studied next keeping the radius of the tank (6 m) and ship parameters constant (Table 2), and the liquid filling ratio in the tank is varied from to The liquid height and the tank height ratio are denoted by b Figure 11 123 J Mar Sci Technol (2012) 17:139–153 151 Fig 12 Effects of thruster modeling on ship response Fig 10 a Snapshot of free surface profile at 25 s b Snapshot of hydrodynamic pressure distribution along wall Fig 13 Effect of shape of containers on ship motion in roll cargo can be a useful vibration absorber for ship motion subject to external forcing frequencies of the ship that are away from the natural sloshing frequencies 5.3.2 Effect of thruster modeling Fig 11 Transient ship response corresponding to varying liquid volumes depicts the effect of liquid-to-container height ratio b on the roll motion of the ship It is seen from the representative plot that because of the liquid inertia, the amplitude of roll motion is reduced as we increase the liquid volume The conclusion can be made that in some cases, the liquid Among several factors that can affect the moving ship, the environmental settings and thruster forces are the most significant The effects of thruster modeling on the behavior of the liquid-filled containers are studied by considering two cases: with and without thruster modeling in ship calculations The incorporation of a thruster in vessel modeling is studied to investigate its effect on global motion of the ship The thrusters are placed in pods that can be rotated in any horizontal direction and they provide the ship with better maneuverability and stability Primary advantages of thrusters are electrical efficiency, better ship space usage, and lower maintenance costs Ships with thrusters not need tugs to dock, though they still require tugs to maneuver in difficult places As shown in Table 2, 123 152 the maximum thrust forces in surge, sway and yaw are (ssurge)max = 0.3 106 N, (ssway)max = 0.12 106 N and (syaw)max = 9.7 106 N, respectively In this section, the effect of having thrusters on the ship motion and the sloshing fluid-filled container are investigated In this study, the two curves presented correspond to the case when thruster is considered and when thruster is not considered All containers are assumed to be completely filled with liquid, i.e b = It can be seen from Fig 12 that there is a significant difference in the responses of ship motion between the cases for which the thruster is absent and for which the thruster is present The presence of a thruster reduces the amplitudes of motion and acceleration of the ship, thereby making the ship more stable Therefore, thrusters are recommended for marine vessels carrying liquid cargo in tanks on board Thruster modeling needs to be considered while calculating behavior of a liquid cargo carrier cruising in rough seas 5.3.3 Effect of container geometry Finally, the influence of geometry of the container is studied for a constant volume of fluid The problem geometry and nomenclature are defined in Fig 1c where Hshp is the height of the liquid in a spherical tank and Hcyl f f represents height of liquid in a cylindrical tank The geometry of the container is considered as hemispherical and cylindrical tanks The radius for both of the containers is taken as m and the liquid height for hemispherical and cylindrical containers are taken as 2.75 and m, respectively, to keep the volume of liquid constant It is seen from the results plotted in Fig 13 that there is not much variation in the ship behavior as the total weight of the liquid containment system remains the same Conclusion A weak-form Galerkin finite element model solving the fully coupled liquid sloshing analysis for liquid storage tanks of various geometries mounted on the ship and subjected to external wave loads has been developed For time-domain simulations, both ship-motion programs and inner-tank-sloshing programs are independently developed and they are coupled by a newly developed scheme Many influencing factors to the vibration response of liquid– structure–liquid coupled systems are also investigated These factors include container parameters, significant wave height, thrusters modeling, as well as the directions of waves, wind and currents acting on the ship With the increase in the liquid depth or volume in the container, noticeable reduction in the roll amplitude is observed This is due to the increased coupled frequency of the ship- 123 J Mar Sci Technol (2012) 17:139–153 container system and the consequent difference with the ship excitation frequency As the significant wave height is increasing the slosh parameters, the free surface hydrodynamic pressure changes significantly It is found from the present study that the coupled effect has a significant role in analyzing fluid–structure–fluid systems It can also be concluded that the coupled sloshing and the ship parameters are significant at the beam sea case As compared to other coupling sloshing studies, considering the effects of thrusters is rather new This is one of our contributions to the scientific community The developed computer code is useful in predicting the sloshing displacement and the hydrodynamic pressure on the walls in a small size liquid filled cargo tank of various geometries due to important environmental effects It is believed that the research presented in this paper advances the understanding of the dynamic behavior of liquid cargo carrier in axisymmetric and non-axisymmetric containers interacting with the external hydrodynamics and provides results of practical value useful to the community Acknowledgments The authors greatly acknowledge the Centre for Ships and Ocean Structures (CeSOS) at Norwegian University of Science and Technology (NTNU) for ship motion MCsim b4 program developed by Oyvind Notland Smogeli and students The first author is grateful for the financial support provided by the Centre for Offshore Research and Engineering (CORE) at the National University of Singapore This work is partially supported by a grant from Qatar National Research Foundation under NPRP-08-691-2-289 References 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Fig a Typical layout and nomenclature of a general liquid cargo carrier b Coordinate system of a marine vessel c A cylindrical tank with schematic mesh d A hemispherical and a cylindrical container. .. coupling analysis The recent studies can be categorized into two main approaches: the frequency-domain approach assuming linear sloshing flow, and the time-domain approach adopting nonlinear sloshing