Analytical Study of Transient Coupling between Vessel Motion and Liquid Sloshing in Multiple Tanks

The transient coupling between the vessel motion and liquid sloshing in multiple tanks is investigated. External disturbance factors (e.g., spring constraint or force field) that might affect the oscillation characters of the coupling system are not involved so that the vessel motion is only excited by the liquid sloshing in tanks. The analytical solution for this coupling problem has been derived based on the potential flow theory, which converts the problem to a linear system of ordinary differential equations. The approach to determine natural frequencies of the coupling system is also given. The vessel with one or more rectangular tanks is considered for cases studies. Effects of factors, such as vessel mass, number of tanks, tank configuration and free-surface deformation on the vessel motion, liquid sloshing, and mechanical-energy components of the system are studied systematically. DOI: 10.1061/(ASCE)EM.1943-7889.0001085. © 2016 American Society of Civil Engineers. Author keywords: Sloshing; Vessel motion; Fluid-structure interaction; Free surface waves; Coupling of sloshing and seakeeping. Introduction The liquefied natural gas (LNG) carrier is a vessel designed for transoceanic LNG transportation. After several decades of development, new demands are being raised in the LNG shipping industry. For example, the world’s first offshore floating liquefied natural gas (FLNG) facility (i.e., Prelude) is coming soon. During the offloading operation from the FLNG facility to the LNG carrier, the liquid sloshing inside the LNG carrier could have nonneglectable effects on vessel motions (Zhao et al. 2011). Meanwhile, the vessel motions would further excite the liquid sloshing in return. Thus, the interaction between the liquid sloshing and vessel motions forms a coupling problem. What makes the coupling more complicated is that the vessel motion, itself, is also coupled with a complex external environment, such as ocean waves, at the same time. Moreover, each vessel may have multiple liquid tanks so that the liquid motion in each tank could have an interaction with the vessel motion. At present, simulations on this coupling problem have emerged in the literature, such as in Rognebakke and Faltinsen (2003), Molin et al. (2002), Malenica et al. (2003), Newman (2005), Kim et al. (2007), Mitra et al. (2012) and Zhao et al. (2014). However, because of the complexity of the problem, a systematic understanding of the complete coupling system is still far behind. The complexity is mainly from the inclusion of too many influencing factors (i.e., the external environment, vessel motion, and liquid sloshing in any of the tanks). Thus, to have a deep understanding of this complete problem, it is important to firstly isolate the influencing factors and make clear the coupling mechanism between different pairs of them. The present study would focus on the coupling pair between the liquid sloshing and vessel motions. Studies on the coupling of vessel motion and liquid sloshing in a single tank can be found in some literature. Cooker (1994) has introduced a typical model, which is a single-tank vessel suspended as a bifilar pendulum. Initially, the vessel with still water is dragged away from the equilibrium position before it is released from rest. During the swing motion, the vessel remains horizontal. The swing amplitude, liquid depth, and wave amplitude are all assumed to be small so that the vertical vessel displacement is neglected, and the sloshing could be analyzed using the linear shallow-water theory. It is found that the presence of the sloshing fluid could evidently change the natural oscillation frequency of the suspended system. In recent years, Cooker’s model returns to peoples’ concern. Alemi Ardakani and Bridges (2010) extended Cooker’s theory by including the nonlinear terms in shallow water equations, although the vessel motion is still restricted in a linear sense. They developed a numerical algorithm to solve the coupling problem. Their nonlinear results have shown distinctions from the linear ones in the near resonance situation. Yu (2015) applied the linear finite-depth water theory to Cooker’s model. It is found that the shallow-water theory, which assumes the pressure acting on tank walls to be hydrostatic, does not give satisfactory results for cases of shorter suspension length and larger water depth where nonhydrostatic pressure becomes significant. Then, Herczynski and Weidman (2012) performed a related experiment to measure the horizontal oscillation of a vessel solely driven by the liquid sloshing inside. The linear finite-depth solutions are validated by the experiment. Alemi Ardakani et al. (2012) further used a weakly-nonlinear finite-depth water theory for Cooker’s model. The resonance, where the natural frequency of the system coincides with one of the fluid modes, was identified. Later, Turner and Bridges (2013) investigated the subtle energy transfer from liquid sloshing to vessel motion through the nonlinearity. Turner et al. (2015) further allowed the pivoting of the vessel with a sloshing tank. Another model in dealing with the vessel-liquid-coupling problem is the tuned liquid damper (TLD) (a type of liquid tank installed on the top of high buildings to control the wind-induced oscillation). The TLD model is simplified as a liquid vessel constrained by springs and allowed to move freely only in the horizontal direction, which is essentially a mass-spring-damper system. The TLD and Cooker’s bifilar pendulum have equivalent governing Dept. of Mechanical Engineering, Univ. College London, Torrington Place, London WC1E 7JE, U.K. E-mail: [email protected] Note. This manuscript was submitted on September 17, 2015; approved on January 7, 2016; published online on March 2, 2016. Discussion period open until August 2, 2016; separate discussions must be submitted for individual papers. This paper is part of the Journal of Engineering Mechanics, © ASCE, ISSN 0733-9399. © ASCE 04016034-1 J. Eng. Mech. J. Eng. Mech., 2016, 142(7): -1--1 D ow nl oa de d fr om a sc el ib ra ry .o rg b y U ni ve rs ity C ol le ge L on do n on 0 1/ 26 /1 7. C op yr ig ht A SC E . F or p er so na l u se o nl y; a ll ri gh ts r es er ve d. equations after linearization but different characteristics in the nonlinear mathematical form. Ikeda and Nakagawa (1997) harmonically excited the TLD model from rest by an electromagnetic exciter. The free surface is weakly-nonlinear. Stability analyses for this nonlinear vibration were performed. Frandsen (2005) studied a similar vibration system. A finite different solver, based on fullynonlinear potential flow theory, is developed for the flow simulation. The coupling of sloshing and vessel vibration in an external time-varying force field is investigated. Gavrilyuk et al. (2012) derived linear modal equations to study the dynamics of a mechanical system carrying the liquid tank of the tapered conical geometry. Harmonic external forces were also applied. The previous review suggests that the literature has mainly considered single-tank vessels restrained by suspensions or springs or even placed in time-varying force fields. Studies on the purely free motion of a multiple-tank vessel seem to be rare. Purely, in this case, indicates that external factors (e.g., restoring forces from springs or suspension structures) are not involved. Because either the suspended structure or spring-mass system has an oscillation nature, their existence could totally change oscillation property of a purely coupling system. For example, resonance phenomena may occur if the motion frequency of the vessel with empty tanks coincides with certain natural sloshing frequencies. On the vessel with multiple tanks, two related works are Weidman (2005) and Turner et al. (2013). In their study, the Cooker’s experiment with a multi-tank rectangular vessel is examined. Turner et al. (2013) also gave the characteristic equations for the natural frequencies of the coupling problem. However, like most of the aforementioned literature, the problem with the purely free motion is not concerned, neither. To help fill this gap, the present work would investigate the purely coupling between vessel motions and liquid sloshing in multiple tanks. The vessel is only excited by the liquid sloshing in tanks without external factors that could disturb the oscillation characters of the coupling system. The vessel motion is initially driven by the free-surface deformation in tanks. Based on the potential flow theory, the author would derive the analytical solution of the coupling problem and introduce an approach to determine natural frequencies of the system. Dynamic properties of the coupling system with a single sloshing tank are firstly investigated. Then, multiple sloshing tanks are further involved. The effects of factors, such as vessel mass, number of tanks, and tank configurations, are studied. The present work is a follow-up study of the previous research (e.g., Zhang et al. 2015; Zhang 2015b) on liquid sloshing caused by prescribed excitations. Problem Descriptions Consider a vessel with N liquid tanks, as shown in Fig. 1. The tanks are numbered from Tank-1 to Tank-N in sequence. The fluid domain, free surface, and wetted wall in Tank-k (k 1⁄4 1 toN) are denoted by Vk, SF, and S k B, respectively. To analyze the liquid sloshing in Tank-k, a tank-fixed coordinate system Ok − xkykzk is set, with the origin Ok at the center of the mean free surface and Okzk axis pointing upward. The earth-fixed coordinate system Oo − xoyozo is also set, which coincides with the initial position of O − xyz. The vessel is undergoing a free motion in the Ooxo direction on the horizontal ground without friction. The motion of the vessel is determined by the Newton’s Second Law mv̇c 1⁄4 XN k1⁄41 Fk ð1Þ wherem = vessel mass without liquid; v̇c 1⁄4 fv̇1; 0,0g 1⁄4 fx c; 0,0g = transversal acceleration of the vessel; and Fk 1⁄4 fF1; 0; 0g = sloshing-induced force from domain Vk. A dot over a variable represents a time derivative. Hereafter, the superscript k indicates that the variable is from the fluid domain in Tank-k. The value of Fk is from the integration of the pressure over the wetted tank wall SB. The pressure could be calculated based on the potential flow theory, which assumes the fluid to be inviscid, incompressible, and flow-irrotational. For Tank-k, a scalar velocity potential φkðxk; yk; zk; tÞ, whose gradient represents the fluid velocity, is introduced. For convenience sake, the superscript k of variables in Tank-k is omitted before calculating hydrodynamic forces. The small-amplitude assumption is further adopted, which requires the amplitude of the liquid and vessel motions to be small relative to the characteristic tank dimension. Then, the velocity potential φ could be determined from the linearized boundary value problem ∇2φ 1⁄4 0; in V̄ ð2Þ ∂φ ∂n 1⁄4 v1ðtÞn1; on S̄B ð3Þ ∂φ ∂z 1⁄4 ∂η ∂t ; on S̄F ð4Þ η 1⁄4 − 1 g ∂φ ∂t ; on S̄F ð5Þ φðx; y; z; tÞ 1⁄4 0; ηðx; y; tÞ 1⁄4 η̄ðx; yÞ; for t ≤ 0 ð6Þ where n fn1; n2; n3g = unit normal vector pointing out of the fluid domain; ηðx; y; tÞ = free-surface elevation; and g = gravitational acceleration constant. The mean fluid domain, mean wet tank surface, and mean free surface are denoted by V̄, S̄B, and S̄F, respectively. The initial conditions in Eq. (6) give the free surface an initial profile. The fluid pressure p is obtained from the linearized Bernoulli’s equation p 1⁄4 −ρ ∂φ ∂t − ρgzþ CðtÞ ð7Þ where ρ = fluid density; and CðtÞ = spatial-independent function. By redefining φ appropriately, CðtÞ could be set as zero without affecting the velocity field. Theoretical Analyses General Solutions In this section, the solution of the linearized vessel-liquid-coupling problem is to be derived. The procedure for the liquid sloshing Fig. 1. Sketch of vessel with multiple sloshing tanks © ASCE 04016034-2 J. Eng. Mech. J. Eng. Mech., 2016, 142(7): -1--1 D ow nl oa de d fr om a sc el ib ra ry .o rg b y U ni ve rs ity C ol le ge L on do n on 0 1/ 26 /1 7. C op yr ig ht A SC E . F or p er so na l u se o nl y; a ll ri gh ts r es er ve d. analysis is similar to that in Zhang (2015a). Firstly, the vessel motion effect is separated from the velocity potential φðx; y; z; tÞ 1⁄4 v1ðtÞxþ φðx; y; z; tÞ ð8Þ Substituting Eq. (8) into Eqs. (2)–(5) leads to the boundary value problem of φ ∇2φ 1⁄4 0; in V̄ ð9Þ ∂φ ∂n 1⁄4 0; on S̄B ð10Þ ∂φ ∂z 1⁄4 ∂η ∂t ; on S̄F ð11Þ φ̇þ gη 1⁄4 −v̇1x; on S̄F ð12Þ This is a nonhomogeneous problem whose solution can be written in a summation form φðx; y; z; tÞ 1⁄4 X∞ i1⁄41 ξiðtÞφ̄iðx; y; zÞ ð13Þ where φ̄i = natural modes of the sloshing liquid; and ξi = corresponding time-dependent coefficients. The natural modes φ̄i indicate nontrivial solutions of the steady sloshing state. All φ̄i form an orthogonal set. Each φ̄i corresponds to a natural frequency ωi, which satisfies the relationship ∂φ̄i=∂z 1⁄4 ðωi =gÞφ̄i. Because of the conservation of the liquid volume, the following equations are satisfied Z