Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System

Transcription

1 ABS TECHNICAL PAPERS 5 Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System Jang W. Kim, Kunho Kim, Peter S. Kim and Yung S. Shin American Bureau of Shipping Houston, Texas, USA Presented at the ISOPE 5 conference held in Seoul, Korea, June 9-4, 5, and re-printed with the kind permission of the International Society of Offshore and Polar Engineers (ISOPE) ABSTRACT A numerical approach to evaluate the coupling effect on the sloshing load is presented. The motion of liquid cargo inside a LNG tank is solved by a finite-element method under the assumption of potential theory in the time domain. The nonlinear free-surface conditions are satisfied exactly on the actual free surface of the liquid cargo. The hydrodynamic load on the tank wall is decomposed into impulsive and nonimpulsive components. The integration of the impulsive pressure results in time-varying added mass for the liquid cargo motion, which is considered as part of the virtual mass when solving the ship motion. The transient motion of LNG carrier is mathematically modeled by the use of impulse response function (IRF) and wave exciting force calculated from the three-dimensional frequency-domain panel method. The calculated ship motion is compared with the experimental result to validate the numerical scheme. The sloshing impact load on the tank wall is investigated for various loading conditions and sea states KEY WORDS: Coupling Effect, Impact Pressure, LNG Carrier, Partial Filling, Sloshing INTRODUCTION Sloshing Load at Standard Filling Level Ocean going LNG vessels have been designed to operate -5 years of service life in the severe wave conditions of the North Atlantic, which is referred to as unrestricted service. While the design load of the main hull structure is governed by the external wave load and inertial load due to ship motion, the sloshing impact load due to liquid motion inside the tank governs the design load for the LNG containment system and surrounding bulkheads. LNG carriers are fully loaded when they leave the loading terminal. In case of membrane-type LNG containment system, where the pressure and temperature inside the tank is maintained near atmospheric and cryogenic, LNG cargo keeps boiling during the voyage. As a result, LNG cargo tanks can slack down to 9%~95% of tank height during the voyage in the ocean, depending on the type of insulation system and the distance and length of the trip. LNG carriers with membrane-type containment system have been operating at these high filling conditions without significant damage for the last 4 years. Recently, the growing LNG market demands new LNG carriers with cargo capacity 5%~8% larger than the existing vessels. The size and shape of the tank change accordingly. The safety of the containment system for the new design has to be re-evaluated. The impact pressure due to liquid cargo motion has been regarded as one of the most important load factors when designing new liquid cargo hold and containment system for the membrane-type LNG carriers. Sloshing model test and extensive numerical simulation have been performed to identify the critical sloshing load during the lifetime period of the LNG carrier. But still the true impact load in the full-scale LNG tank is not known yet. The model test is scaled only for the limited parameters including length and gravity (Froude scaling). Full-scale pressure is scaled up proportional to the liquid density and length dimensions (Euler scaling), neglecting the effect of gas trapping and compressibility of liquid. In other words, the dynamic similarity between model test and full-scale LNG tank is not correctly established. Dynamic similarity is also violated for the tank structure too. The model tests and calculations are usually performed for tanks with rigid boundary. However, the inner skin of the LNG tank is covered by insulation system that consists of thin metal membrane to prevent the LNG leakage, insulation material to maintain the temperature of the cargo in the cryogenic condition. Unlike the steel hull structure, most of the structural members in the LNG containment system consist of flexible materials such as plywood and polymer foams. As a result, active fluid-structure interaction is expected during the sloshing impact. As a result, the impact pressure from the sloshing analysis and model test cannot be directly used for the strength assessment of insulation system before considering the combined effect of the following phenomena: Gas trapping and cushioning effect during the impact Compressibility of liquid Fluid-structure interaction For example, the sloshing model tests performed in the /7~/5 scale model show that the maximum sloshing impact load in the existing LNG vessel exceeds the impact strength of the existing containment system by factor of to 3 if those factors are ignored. This contradicts to the successful service experience of the existing LNG fleet for the last 4 years. Numerical and experimental efforts are being made to explain the difference between model- and full-scale impact loads. ABS introduced composite scale law to consider the gas trapping and compressibility of liquid in a practical way by inspecting pattern of impact pressure time history (see, Lee et al. 4; Kim et al. 4). A preliminary study on fluid-structure interaction during the sloshing impact on the Mark III LNG containment system has also been performed (also see Kim et al. 3). A theoretical pressure profile from the slamming theory (Zhao & Faltinsen, 993) has been used to model the traveling impact load due to sloshing. It has been found that fluidstructure interaction, or hydroelasticity, and material damping play important role to the structural response of the Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System 49

2 ABS TECHNICAL PAPERS 5 containment system for the traveling impact load with thin profile and short duration. Sloshing Load at Partial Filling Condition Recently there have been growing interests in the operation of LNG carriers at partial filling conditions due to the emergence of spot market and offshore LNG terminal. LNG operators are looking into possibility of unloading partial LNG cargo onto LNG terminals of small capacity along the coast during the delivering voyage of LNG carriers. During the unloading operation in the offshore terminal, LNG vessels may have to disconnect from the receiving terminal due to weather or other unexpected circumstances in the partially loaded condition. Similar operational scenario can occur for the LNG vessels and LNG FPSO at the production site. The environmental conditions at the sites where the partial filling conditions are expected is much less severe than the North Atlantic for which the standard filling conditions are being operated. However, recent experimental and numerical study shows that even at the milder sea states, the sloshing load at the filling level near the 3% of tank height can be as high as the sloshing load at the high filling level at the North Atlantic (see e.g., Lee et al., 4; Tveitnes et al., 4). To make things worse, the sloshing pressure at this low filling level has wider impact area and longer duration. When the duration of impact load is longer than the natural period of the structure, hydroelasticity plays little role and impact load can be treated as quasi-static load, which implies more impact on the structural response of the LNG containment system and hull. While the hydroelastic response of containment system plays important role at high filling condition, two different types of fluid-structure interaction are expected when LNG tank is partially loaded. Since the wider area and the longer duration of impact pressure introduce more impulse on the hull, part of the sloshing impact is absorbed by rigid body motion of the ship. The duration of impact force at low filling level is order of ms, which can induce impulsive motion of the hull to modify the magnitude and shape of the sloshing impact pressure. However, the duration of the impact pressure is still much shorter than the duration of global ship motion. As a result, the impact force does not affect the global motion of the ship significantly. It is the non-impact dynamic pressure due to liquid motion that affects the global ship motion. As a matter of fact, it has well been known that the phase difference between the liquid motion inside a anti-roll tank reduces the roll motion of a ship significantly although the mass of liquid inside tank is only around % of the whole ship mass. The change of global ship motion will also affect the magnitude of the sloshing impact pressure. The sloshing-ship motion coupling effect on the global ship motion of LNG carrier has recently been studied by experiment and numerical calculation. To name a few, Rognebakke & Faltinsen () and Molin et al. () presented experimental and numerical study on the coupling effect between rectangular barge and liquid motion inside a rectangular tank. Gaillarde et al. (4) presented wave tank test of /5 scale LNG FPSO and LNG carrier performed by SALT project (MARIN, 3). The experimental results show significant coupling effect on the roll motion of the ship. The numerical models that have been used in these studies are based on linear potential theory (Molin et al., ; Gaillarde, 4) or nonlinear model for restricted geometry and filling level (Rognebakke & Faltinsen, ). Coupling effect in a liquid tank of more general geometry has been considered by Kim (). A Navier-Stokes solver based on SOLA-SURF scheme, ABS SLO3D, has been adopted to solve the liquid motion inside an anti-roll tank. LAMP (Large Amplitude Motion Program) has been used to solve the ship motion. Numerical results show good agreement with the wave tank test by Bai & Rhee (989). Lee et al. (5) studied the coupled motion of a LNG FPSO. They used the sloshing code ABS SLO3D coupled with a linear time-domain ship motion solver. The results are compared with the wave tank test performed by Gaillarde et al. (4). Most of the experimental and numerical work on the coupling study has been focused on global ship motion. The effect of coupling on sloshing impact load has rarely been studied. Although the roll motion decreases due to the coupling effect near the sloshing resonance frequency, roll and sway RAO increases at the higher frequency. Since the main driving motion parameter at low filling level is the transverse acceleration, concern has been raised that reduction of roll motion may not necessarily result in reduction of sloshing impact pressure. The lack of idea on how the sloshing-ship motion coupling will affect the sloshing impact load, and the relatively high impact pressure measured at low-filling model test without considering the coupling effect motivated this study. A finite-element sloshing simulation code, SLOFE, is used for the sloshing simulation after coupled with a time-domain seakeeping module. SLOFE is especially working well for the simulation of the sloshing impact pressure at low filling level because of its robust impact pressure calculation scheme (Kim et al., ). The short duration impact pressure is evaluated by solving a boundary value problem for the impulsive pressure component. In the coupling problem, the impulsive pressure component is further decomposed into terms proportional to the instantaneous acceleration of tank motion. Integrating the impact pressure, the force due to sloshing motion can be given in terms of added mass and non-impulsive sloshing-induced force. The added mass due to sloshing motion is equivalent to the infinite-frequency added mass for linear time-domain ship motion analysis. In case of sloshing motion, the added mass is time varying because the change of the liquid domain inside LNG tank is fully considered during the nonlinear sloshing simulation. The developed numerical code is first applied to the experimental work of Rognebakke & Faltinsen () for simple rectangular tank. SLOFE predicts the coupling effect in sway motion quite well. Next we validate the coupling effect in roll motion by simulating one of the wave tank tests performed by MARIN (3), where the coupling effect of more realistic LNG ship in a random sea has been studied. The coupling effect in ship motion and sloshing impact load for actual LNG ship of 45, m 3 capacity has been performed for different loading conditions, filling levels and sea states. Roll and sway motion has been investigated to see the coupling effect on ship motion. Then the sloshing impact load from coupled and uncoupled case has been compared. Correlation between moving cargo weight and the reduction factor of impact pressure has also been investigated. Coupling effect is more significant in impact pressure than global ship motion and worked favorably in most of the cases. 5 Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System

3 ABS TECHNICAL PAPERS 5 MATHEMATICAL FORMULATION Coordinate System A Cartesian coordinate system fixed to the ship is used. The x- axis directs from stern to bow, y-axis directs to the portside and z-axis directs vertically upward, see Figure. The rigid body motion of ship is defined at the ship center of gravity, ( x G, yg, zg ), as shown in Figure. where M: Mass matrix of ship without liquid cargo δm( Added mass of hull at infinite frequency C: Linear damping matrix K: Restoring matrix from hydrostatics, moorings and weight distribution without liquid cargo, R: Impulse response function f: External force on ship including wave exciting and sloshing force The retardation function, or impulse response function, R(t), can be obtained from the wave damping coefficient matrix, C w, from the frequency domain solution: () sin ωt R t = C ( ω) dω π w () ω The external force f consists of wave exciting force f w and sloshing induced force f slosh, which will be decomposed into impulsive part and non-impulsive part, f. Impulsive part of the of the sloshing force is given by instantaneous added mass due to liquid cargo, δm slosh : Figure LNG Cargo Hold and Coordinate system heave, ξ 3 yaw, ξ 6 s w a y, ξ pitch, ξ 5 roll, ξ 4 s u r g e, ξ (x G, y G, z G ) f ( t) = f w ( t) + f slosh ( t) = f () t δm ()() t && ξ t + () t w slosh Then the coupled equation of motion can be written as ( M + δm( ) + δm ) && ξ + Cξ & slosh + Kξ t (4) = R ( t τ) ξ& ( τ) dτ + f + f w Derivation of added mass matrix and exciting force due to liquid cargo motion is described in the next. f (3) Two-Dimensional Sloshing Analysis Figure Definition of Ship Motion Linear Transient Ship Motion in Time Domain Because of high nonlinearity in sloshing motion and force, the equation of motion of the ship is formulated in time domain. Assuming mild sea conditions, linear assumptions are made on the ship motion and hydrodynamic loads on the LNGC. Then the transient motion of LNG carrier is mathematically modeled by the use of impulse response function (IRF) and wave exciting force calculated from the three-dimensional frequencydomain panel method. The equation of motion for ship in time domain can be written as t & () ( M + δm( ) ) ξ& + Cξ& + Kξ = R ( t τ) ξ& ( τ) dτ + f It has been known that the maximum sloshing impact load at low filling conditions occur at the beam sea condition. We performed two dimensional sloshing calculation instead of full three dimensional simulation of the sloshing motion. The two dimensional motion in yz plane at the middle of LNG cargo tank is considered to evaluate hydrodynamic force and impact pressure. The transverse cross-sectional area of the LNG tank is modeled for the sloshing analysis. The two-dimensional motion of the tank is derived from the sway, heave, roll, pitch and yaw motion of the ship, which is given by ξ = ξ ξ3 = ξ3 ξ 4 = ξ 4. ( zct zg ) ξ 4 + ( xct xg ) ( x x ) ξ, CT G 5 ξ6, The liquid motion inside the tank is governed by translational acceleration, A(t), rotational angle, θ(t), and rotational velocity, Ω(t), of the tank. The two-dimensional motion of the tank derives those parameters: (5) Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System 5

4 ABS TECHNICAL PAPERS 5 d ξ d ξ3 = A,,, θ = ξ,. (6) 4 Ω = ξ& 4 dt dt The liquid motion inside the LNG tank is formulated within the scope of potential flow theory. The stratification due to temperature gradient and the gas in the void space of cargo hold are ignored. Assuming incompressibility of the liquid, the two dimensional velocity components, v and w, observed in the moving tank-fixed coordinate can be given by stream function ψ ( y, z, t) and velocity potential φ ( y, z, t) : ψ φ ψ φ v = = + Ωz, w = = Ωy (7) z y y z Note that the velocity potential φ alone cannot describe the liquid motion in the moving coordinate system because the velocity field is rotational due to the roll velocity of the tank, Ω(). t Following Kim et al (), stream-function formulation is used to solve the flow field inside the tank. Once the velocity field inside the tank is calculated, the velocity potential formulation is utilized to calculate the pressure field inside the liquid domain and along the wall. The governing equations and boundary conditions for the stream function and velocity potential are given in Appendix A. f = d 3d 4d. p n S p n S p n S () The added mass matrix and sloshing force in the threedimensional equation of motion, Eq. (4), can be obtained from the two-dimensional values. The relation between ξ and ξ given by Eq. (5) can be written in matrix form as ξ = Tξ where T (3) zct + zg xct xg T = xct + xg (4) The relation between sloshing force in two and three dimensions is given by T f slosh = L Tank T f slosh, (5) The pressure in the liquid is given by the Euler integral: where L Tank denotes the length of tank. φ ρ ρω p = ρ ( ψ ψ) + x x ρg x ρa x. (8) t φ Among the pressure terms, ρ and ρa x contains t hydrodynamic and internal pressure proportional to the acceleration of the tank. As described in the Appendix B, the pressure can be decomposed into impulsive pressure proportional to the tank acceleration, p, p3, p4, and the nonimpulsive pressure, p : & c ξ + p && 3ξ3 + p & 4ξ 4 + p. (9) p = p & Integrating the pressure on the tank surface, we can obtain force on the hull due to sloshing motion: f = δm & ξ + f slosh slosh where the added mass matrix, sloshing induced force, S W δmslosh = pnds ( Sym. ) f () δm slosh, and non-impulsive, are defined by pn3ds p3n3ds pn4ds p3n4ds, p4n4ds () From Eqs.(3) and (5), the following relation between added mass matrix and sloshing force in two and three dimensions can be derived: T δm slosh = L Tank T δm slosht (6) T f = L Tank T f (7) The two-dimensional added mass matrix, δmslosh, updates continuously as the shape of liquid domain changes. METHODS OF SOLUTION Ship Motion Analysis in Frequency Domain Seakeeping analysis is performed using program PRECAL, which is a frequency-domain program based on threedimensional potential theory developed by MARIN (MARIN, ). The vessel s added mass and damping coefficients, and wave exciting force in beam sea were calculated for 45 wave frequencies from to 3.6 radian/second (rad/s) with.5 rad/s intervals. A vessel speed of 75% of the design speed was considered in the seakeeping analysis. Sloshing Simulation in Time Domain A computer program SLOFE has been used to solve liquid motion inside tank. Spatial discretization of liquid domain is made by finite-element method. Quadrilateral 4-node bilinear element has been used. The finite-element mesh is updated at every time step. Uniform mesh spacing is used for the horizontal direction. In the vertical direction quadratic spacing with the finer mesh near the free surface. Figure 3 shows a mesh system for a moment when sloshing wave hits the tank 5 Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System

5 ABS TECHNICAL PAPERS 5 wall. Time-evolution of the discretized free-surface elevation and velocity potential is integrated by the 4 th -order Runge-Kutta method. More detail numerical scheme can be found in Kim et al. (). To consider the coupling effect, a time-domain seakeeping module is integrated with the SLOFE. SLOFE provides seakeeping module instantaneous added mass matrix and sloshing exciting force every time step. Seakeeping module integrates the equation of motion from the updated added mass matrix and exciting force to obtain ship and tank motion. The new tank motion is used to update the liquid motion inside the tank at the next time step. Figure 4 shows the flow chart of the coupling scheme. coupling effect. At the frequency higher than the resonance frequency, sway motion increases due to the coupling effect. Sway RAO (m/m) ω (rad/s).5 Sway RAO (m/m) Figure 3: Finite-element mesh system: 5%H filling level t=t+ t Initial Condition Update Mesh Update R * ξ &, f w Solve BVP for p, p, p p Update δm slosh, f Update 3 4, φ, ˆ ζ Solve BVP for Sloshing Module ψ Solve Eq. of Motion M tot = M + δm( ) + δm slosh & ξ = M f + f - R ξ& L) tot ( w w Update Tank Motion A, Ω Seakeeping Module Figure 4: Flow chart for coupled sloshing analysis COMPARISON WITH EXPERIMENT Rectangular Tank Mounted on Rectangular Barge Regnebakke & Faltinsen () performed an experiment to study the coupling effect between liquid motion inside a rectangular tank on a barge. The experiment has been performed in a wave flume with wave conditions and barge motion maintained in two-dimensional. Only the sway motion is allowed during the test. Two identical tanks with length 5% of the barge length have been mounted. Tests with one or two tank filled have been performed to see the effect of different cargo volume at the same filling levels. Figure 5 shows the calculated sway RAOs compared with the experimental results. Agreement is reasonably good. Both the numerical and experimental results show N-spikes near the sloshing resonance period. Sway motion decreased at the frequency lower than the sloshing resonance frequency due to Sway RAO (m/m) Sway RAO (m/m) ω (rad/s) (a) h/b =.49, L ank /L =.5 t (b) h/b =.49, L tank /L = ω (rad/s) ω (rad/s) (c) h/b =.5, L tank /L =.5 (b) h/b =.77, L tank /L =.5 Figure 5: Sway RAO of a rectangular tank mounted on a square barge : Uncoupled motion; : Coupled motion; : Experiment (Rognebakke & Faltinsen, ) LNG Carrier in Transit Condition MARIN performed series of wave tank test for a LNG carrier in transit condition (see, Gaillarde et al., 4). A /5 model equipped with two prismatic tank partially filled with fresh water has been used. The total mass of the fresh water is roughly comparable to the mass of LNG cargo that will fill all four tanks at the same filling level. The input wave is a longcrested irregular wave by JONAP spectrum for a given wave height, H s, and peak period, T p. Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System 53

6 ABS TECHNICAL PAPERS 5 One of the experimental conditions is simulated using a numerical model of an actual LNG carrier with similar configuration. The prototype ship is the same 45K LNG carrier that will be described in the next section. Minor correction of the roll radius gyration (k xx ) and roll damping coefficient has been made to match the magnitude and period of roll RAO maximum to the model. The input wave is a longcrested irregular beam sea condition. Figure 6 shows roll RAO at various filling level. Experimental and numerical results for three different filling conditions are presented. The roll RAO has been obtained from the Fourier transformation of the roll motion record divided by the Fourier transformation of incident wave. the ship. The principal particulars of the ship and No. tank are given in Table and Table, respectively. Three different filling levels, 5%H, 3%H and 5%H, have been simulated. For each filling levels, 4-tank partial loading and -tank partial loading cases are simulated. The 4-tank loading is the case in which all tanks are partially filled at the specified level, while the -tank loading is the case in which only one tank (Tank No.) is partially filled and the other tanks are fully filled. Table 3 shows the loading conditions corresponding to the test cases. Around the sloshing natural frequency, significant reduction of roll motion can be seen. Coupling effect is more significant at the higher filling level, presumably due to more moving liquid mass. However, roll motion increases due to coupling effect at frequencies higher than the sloshing natural frequency. The numerical results follow the trend well. The sharper RAO peak in the numerical results is presumably due to simple roll damping model, linear damping of % critical damping, adopted in the present seakeeping analysis. Figure 7: Midship section and profile of a 45K LNGC (a) Experiment Table : Principal Dimensions of a 45K LNGC Length O.A. (m) 83. Length B.P. (m) 7. Breadth (MLD) (m) 43.4 Depth (MLD) (m) 6. Design Draft (MLD) (m).4 Scantling Draft (MLD) (m).4 Displacement at Design Draft (ton) 59. Service Speed (knots). Roll RAO (deg/m) 6 4 Hs = 5.m, Tp =.s, Heading 9 deg %H FLVL (R) 8.6%H FLVL (R) 37.%H FLVL (R) 55.8%H FLVL (R3) Table : Cargo tank dimensions of Tank No. (membrane to membrane) Length (m) Breadth (m) Height (m) Upper chamfer height (m) 8.8 Lower chamfer height (m) 3.87 Upper chamfer angle ( deg ) 45. Lower chamfer angle ( deg ) 45. from A.P. to Tank center (m) Wave Freq (rad/s) (b) SLOFE Simulation Figure 6: Roll response of a 45K LNGC in a beam sea COUPLING ANALYSIS FOR A 45K LNGC Principal Particulars and Analysis Conditions The coupling effects on ship motion and sloshing impact pressure on a LNG carrier of 45 m 3 capacity has been performed. Figure 7 shows the midship section and profile of Table 3: Loading conditions Case Loading condition 4-tank partial (5%H, 3%H) All 3%H 4-tank partial (5%H) All 5%H -tank partial Full load Hydrostatics of 45K LNGC are shown in Table 4 for Full Load, All tank 5%H, and All tank 3%H. DGMT means the loss of transverse GM due to free surfaces All cases have an uncoupled and coupled pair for the comparison purpose. For completeness, all the test conditions are summarized in Table 5. The wave conditions for each filling level have been selected 54 Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System

7 ABS TECHNICAL PAPERS 5 based on the sloshing model test without coupling effect. The selected wave conditions correspond to maximum sloshing load measured in the model test. In the figures hereafter, UC denotes an uncoupled simulation, and CP denotes a coupled simulation. The duration of simulation is Tp. The irregular wave system has been constructed by synthesising, wave components with random phases. Table 4: Hydrostatic characteristics of 45K LNGC Load Case Full Load All 5%H 3 All 3%H 3 Draft m-bl LCG m-amid KG m-wl Trim m: Tf-Ta GML m GMT m DGMT m In this roll angle comparison, it is found that the coupling has significant effect in roll reduction (maximum 56% decrease from uncoupled case) in 4-tank loading cases while negligible effect in -tank loading cases. filling ratio (%H) roll angle (deg) Figure 8: Maximum roll angles versus filling ratios, Hs = 5m kxx 4 m kyy 4 m kzz 4 m Table 5: Test conditions for the sloshing analysis Hs (m) FL(%H) Tz(sec) Tp(sec) COUPLING EFFECTS ON SHIP MOTION In order to study the coupling effects in ship motion, roll angles and transverse accelerations are compared for the test cases. Roll Motion Figure 8 shows the maximum roll angles for Hs = 5m. In comparison of uncoupled and coupled cases, large difference can be observed in 4-tank loading cases, while very small differences is observed in -tank loading cases. This large effect of coupling is due to the relatively large inertia effect in 4-tank loading. Regarding filling levels, it is difficult to find a trend, but in general 5%H shows larger roll angles than other filling levels.. Figure 9 shows the maximum of 5%H, 3%H, and 5%H for each wave height. The roll angle increases as the sea condition becomes rough. Once again, the coupling effect in roll angles is significant in 4-tank loading cases, while almost negligible in - tank loading cases. roll angle (deg) Hs (m) Figure 9: Maximum roll angles versus wave heights: 45K LNGC Transverse Acceleration Another index for ship motion, transverse acceleration is investigated. It should be noted that the gravity term is not included in the consideration. Transverse acceleration is also an important parameter in sloshing pressure. Figure shows the maximum transverse accelerations in the simulation. Compared to roll angles, the coupling effect is relatively small even in 4-tank-filling cases. Both in -tank and 4-tank filling cases show the increase in transverse acceleration in coupled cases except 4-tank loading at 3%H. This increase in transverse acceleration is probably due to the increase of sway RAO at high frequency that we observed in Figure 5. Figure shows transverse acceleration along with wave heights. Almost linear increase in the acceleration with wave height can be observed in the figure. The differences between coupled and uncoupled cases are relatively small in 5 m and 7 m compared to 9 m wave height. The transverse acceleration generally increases slightly due to the coupling effect. Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System 55

8 ABS TECHNICAL PAPERS 5 filling ratio (%H) trans.acc. (m/s^) Figure : Maximum transverse accelerations versus filling ratios: Hs = 5 m. Table 6: Trend in sloshing load in coupling study for 45K LNGC filling ratio (%H) 4-tank and -tank loadings P UC > P CP Uncoupled P 4T > P T Coupled P T > P 4T UC=Uncoupled, CP=Coupled, 4T=4-tank loading, T=-tank loading Maximum Sloshing Load at North Atlantic, Beam sea, 3%H trans.accel (m/s^) Hs (m) Figure : Maximum transverse accelerations versus wave heights: 45K LNGC COUPLING EFFECTS ON SLOSHING IMPACT filling ratio (%H) relative load factor (P/Pmax) (a) Hs = 5 m relative load factor (P/Pmax) (b) Hs = 7 m Maximum Sloshing Load at North Atlantic, Beam sea, 3%H Following the ABS sloshing analysis procedure (ABS, 5), the maximum value of sloshing pressure averaged over meter panel has been used for the evaluation of sloshing impact on the containment system. The relative value of impact load compared to the maximum sloshing load in the North Atlantic beam sea condition is compared (see Lee et al., 4). The sloshing impact load is shown in Figure with filling ratios in the vertical axis. The figure clearly shows that the impact load is reduced by the coupling effect. As in the roll angles, the reduction is significant in 4-tank loading, up to 8% decrease from uncoupled case. The average decrease is about 6% in 4-tank loading, and about % in -tank loading. The trend in sloshing impact load is summarized in Table 6. As shown, the coupled cases show less impact load than uncoupled cases in both 4-tank and -tank loadings. For uncoupled cases, the impact load in 4-tank loading is larger than -tank loading. This can be explained by the difference in ship motion response due to loading conditions, i.e. 4-tank loading is partial loading condition and -tank loading is the full load condition (see Table 5). filling ratio (%H) Maximum Sloshing Load at North Atlantic, Beam sea, 3%H relative load factor (P/Pmax) (c) Hs = 9 m Figure : Maximum sloshing pressures versus filling ratios: 45K LNGC On the other hand, for coupled cases, the impact load in 4-tank loading is even smaller than -tank loading. In other words, the coupling effect reduces the impact load even more than what is reduced due to ship motions. Again, this coupling effect is more significant in 4-tank loading than -tank loading. 56 Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System

9 ABS TECHNICAL PAPERS 5 Figure 3 shows the relative sloshing impact loads versus wave heights. As shown, the sloshing load at mild sea states with Hs less than 7 meter is less than 4% of the North Atlantic value when coupling effect is considered. The sloshing model test at the North Atlantic condition has shown that the sloshing impact load at 3%H filling level is about two times higher than that at high filling level, when coupling effect is not considered. This means that the maximum impact load at partial filling condition when Hs is less than 7 meter is roughly comparable to the maximum sloshing load at the operational condition of the existing LNG vessel in the North Athantic. Figure 4 presents sloshing load reduction factor, defined by the ratio between sloshing impact load with and without the coupling effect, plotted against the weight ratio between moving cargo and total ship weight. The straight line represents reduction factor linearly decreasing from one when there is no moving cargo, to zero when all cargo is moving. The computed reduction factor is almost always under the straight line. Conservatively speaking, the reduction factor decreases linearly from one to zero as the weight of moving cargo increases. relative load factor (P/Pmax) Maximum Sloshing Load at North Atlantic, Beam sea, 3%H Hs (m ) Figure 3: Maximum sloshing pressures versus wave heights: 45K LNGC load reduction factor due to coupling moving_cargo/total_displacement 5%H 3%H 5%H Figure 4: Load reduction factor due to coupling vs moving cargo ratio CONCLUSIONS Sloshing analysis has been performed for a 45K LNC carrier in beam sea condition in a number of sea conditions. The coupling between the motion of liquid cargo and ship motion not only affect the ship motion but also changes the impact pressure on the containment system. Significant reduction of impact pressure has been found. The effect is partially due to the reduction of roll motion at the sloshing resonance period and coupling due to inertia of ship mass and sloshing motion. The coupling effect on the transverse acceleration is minor and slightly increases the motion. It can be concluded that current strength assessment practice of LNG containment system with the coupling effect neglected is still conservative. The sloshing load decreases roughly proportional to the weight ratio between moving cargo and whole ship when all partially filled tanks are slacked at the same filling level. Based on the present study and the previous model test results, it can be concluded that the sloshing load on the containment system at partial filling condition do not exceeds the maximum sloshing load on the existing vessel operating at high filling condition in the North Atlantic, if Hs is less than 7 meter and all LNG cargo holds are at the same filling level or fully filled. It should be noted that the present numerical analysis based on the linear seakeeping analysis for the ship motion. More careful analysis considering nonlinear ship motion is necessary for the further evaluation of coupling effect at severe sea states. REFERENCES American Bureau of Shipping (5) Guidance Notes for Sloshing and Structural Analysis of Pump Tower for Membrane-Type LNG Carriers Bai, K.J. and Rhee, K.P. (987) Roll-damping tank test, Seoul National University, Project Report Kim, J. W., Shin, Y. S. and Bai, K. J. () A Finite-Element Computation for the Sloshing Motion in LNG Tank, ISPOPE. Kim, J.W., Hwang, C.G. and Lee, H.S. (3) A Numerical Simulation of Sloshing Motion in Membrane-Type LNG Tanks with Fluid-Structure Interaction, the 8th International Conference on Numerical Ship Hydrodynamics Kim, J. W., Lee, H. and Y. Shin 4 Sloshing Impact Load and Strength Assessment of Membrane-Type LNG Containment System in Large LNG Carriers The 4 th Offshore Symposium - LNG: From Source to Market, Houston, Texas. Kim, Y. H. () A Numerical Study on Sloshing Flows Coupled with the Ship Motion; Anti-Rolling Tank Problem, Journal of Ship Research Lee, H., Kim, J.W. and Hwang, C.-G. (4) Dynamic Strength Analysis for Membrane Type LNG Containment System due to Sloshing Impact Load, Int. Conf. on Design and Operation of Gas Carriers, RINA, 4, London, UK Lee, S.J., Kim, M.H., Kim, J.W., Kim, Y.H. and Y. Shin (5) The effects of LNG-tank sloshing on the global motions of LNG-carriers, to be published. MARIN () PRECAL V 5., User s Guide. Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System 57

10 ABS TECHNICAL PAPERS 5 MARIN (3) Seakeeping tests for the JIP SALT, LNG carrier, Vol., Report No. 87- & 3-SMB. Molin, B., Remy, F., Rigaud, S. and de Jouette, Ch. () LNG- FPSO's: frequency domain, coupled analysis of support and liquid cargo motions, Proc. IMAM Conference, Rethymnon, Greece. Rognebakke, O. R. and Faltinsen, O. M., (), Effect of sloshing on ship motions, 6th International Workshop on Water Waves and Floating Bodies, Hiroshima, Japan Zhao, R. and Faltinsen, O. (993) Water entry of twodimensional bodies, J. of Fluid Mech, pp APPENDIX A. Initial Boundry Value Problem for Liquid Motion Inside the Tank The governing equations for φ and ψ are given by φ = or ψ = Ω() t in D (8) The pressure inside the liquid domain is given by the following two equivalent form of Euler integrals: p φ = + ρ t Ω ( ψ ψ) x x + g x + A x (9) p φ φ φ + = + Ω z y φ φ + g x + A x () ρ t y z On the tank wall, the following kinematic boundary conditions, stating that there is no leakage from the tank wall, are given: φ = n (, Ωz, Ωy) or ψ = on () n On the free surface the kinematic boundary condition states that normal velocity of the liquid is equal to the normal velocity of the boundary: ψˆ = = t y n ζ φ on n S F 3 The dynamic boundary condition is given by () p = p f (3) where p f is the prescribed pressure on the free surface. On the free-surface an additional boundary condition to relate φ and ψ are given: ψ φˆ ζ y on n y y = + Ω ζ S F APPENDIX B. Pressure Decomposition (4) The pressure is decomposed into regular pressure and impulsive pressure as ( y, z, t) p ( y, z, t) p ( y, z t) p = +, (5) where φ p ρ ρa x t (6) ( ) Ω p = ρ + ψ x, y, t g x x x (7) φ Since both and A x satisfy the Laplace equation, t is governed by the same equation: p = in D (8) The boundary condition for the impulsive pressure, a Neumann-type condition is imposed on the tank wall and a Dirichlet-type condition is imposed on the free surface: p = ρω& n ( z, y) ρa n on (9) n p p Ω = f + ρ g x x x + ψ on S F (3) Since the forcing term in the boundary condition on the tank wall, Eq. (9), consists of acceleration and none of the term in Eq. (3) involves the acceleration, p, can further be decomposed into p = p & ξ + p && 3ξ3 + p & 4ξ 4 + p, (3) where p, p 3 and p4 satisfy p i = in D (3) pi = ρni on n p on p (33) i = S F (34) for i =, 3, 4 and p satisfies p = in D (35) p = on n S W (36) p = p f p on S F (37) As a result, we can write loc loc loc p = p & ξ + p && 3ξ3 + p & 4ξ 4 + p, (38) where ( x, y t) p p p, = + (39) 58 Sloshing-Ship Motion Coupling Effect for the Sloshing Impact Load on the LNG Containment System