甲板上浪和冲击载荷的数值模拟

Numerical Simulations for Green Water Running on Deck and Impact Loads1Lin Zhaowei, Zhu Renchuan, Miao GuopingSchool of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University Shanghai, P.R.China (200030)E-mail：gpmiao@Abstract A technique of dynamic mesh is specially introduced in a 2-D numerical wave tank to simulate the green water incident on a fixed FPSO model in head waves and oscillating vessels in beam sea conditions, respectively. Numerical results agree well with the corresponding experimental ones. It indicates the present numerical scheme and method can be used to actually simulate the phenomenon of green water on deck, and to predict and analyze the impact forces on floating structures due to green water. It is of great significance to further guide the ship design and optimization, especially for ship bow and its strength design. The mechanism may also help seaman to impose operation restrictions to avoid severe green water incidents. Keywords: Green water， numerical wave tank， dynamic mesh.1 IntroductionGreen water is one of the strongly nonlinear ship-wave interaction problems, usually occurring in harsh sea environments when an incoming wave significantly exceeds the freeboard and water runs onto the deck. Damage of deck equipments and superstructures, and deck cargo shifting will probably be caused by the huge impact forces due to the severe green water incidents. Ship capsizing may even happen due to green water on deck for high speed vessels. For running ships, green water incidents may be avoided by taking some operation measures, such as reducing the advancing speed or altering the course. For the ocean structures, such as FPSO (Floating Production, Storage and Offloading Unit), moored in a specified sea area, however, no operation measure could be adopted to reduce the possible green water on deck, and risks resulted from green water will be even severer than other vessels. Recently, some cases of bow damage and superstructures destroy of FPSO caused by server green water incidents were reported (Mac Gregor, 1999). The green water problem is arousing more attentions for research. In the past, the research of green water problem was mainly relied on the experimental means. Some limited progresses were also achieved recently with the development of numerical techniques and methods. Traditionally, probability method, combined with hydrodynamic theory, was adopted to estimate and analyze the green water related problems. The method cannot predict the amount of water and the hydrodynamic load on decks when water runs onto deck but gives only the probability of deck wetness and the pressure distribution on the deck, based on the assumption that the pressure on the deck corresponds to the static water pressure of shipped water. Subsequently, some other theories and methods based on the potential theories were developed to tackle the problem to some extends. For example, the dam breaking theory and wave overtopping theory were used to simulate the deck flow (Buchner, 1995；Greco et al, 2000), the flooded water theory and shallow water wave theory were applied to simulate the water running on deck (Mizoguchi，1989; Huang，1995; Ogawa et al，1998). Mixed Euler-Lagrange method was also adopted for parametric research for green water incident (Buchner & Cozinijn，1997；Faltinsen et al，2002). The1Support by the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20030248014)-1- strongly nonlinear phenomena, such as water jet, wave breaking, flipping and wrapping, cannot be accurately simulated in detail by the method. With the improving of the capability of high performance computer, various numerical models such as MAC，VOF，LEVEL-SET，SPH，CIP were developed and introduced to solve the problems involving free surface (Hirtand Nichols, 1981；Osher.et.al, 1988；Gingold, Monaghan, 1977；Yabe et al, 2001). Generally, those methods can provide velocity and pressure distribution of the fluid field, capture free surface deforming after a great deal of calculation. Nielsen and Mayer (2004) carried out some numerical investigation on the green water problem by using VOF method. Good results were obtained for the case of fixed FPSO in head waves, but they did not get satisfactory results for green water on a moving vessel in incident waves. It is still a difficult task to acutely simulate the green water on deck and the impact force on the deck structures. In the present paper, a 2-D numerical wave tank is used to simulate the phenomenon of wave-ship interaction and green water on deck, in which, continuity equation and Navier-Stokes equation are regarded as governing equations for modeling the fluid filed, the source technique is adopted to realize the incident wave generation and the absorption for reflected and transmitted waves, and the VOF method is used to capture the free surface deformation. Numerical simulations execute for green water incident of a fixed FPSO model in head waves and oscillating vessels in beam sea conditions, respectively. Ship oscillating motions are calculated by the potential theory. A technique of dynamic mesh is specially introduced in the numerical simulation to realize the green water occurring for an oscillating ship. Numerical results agree well with the corresponding experimental ones. It indicates the present numerical scheme and method can be used to actually simulate the phenomenon of green water on deck, and to predict and analyze the impact forces on floating structures due to green water. It is of great significance to further guide the ship design and optimization, especially for ship bow and its strength design. The mechanism may also help seaman to impose operation restrictions to avoid severe green water incidents.2. Governing equations and numerical wave tankNumerical simulations for green water phenomenon are executed in a 2-D numerical wave tank with the functions of incident wave making and the effective reflected and transmitted wave adsorption, which is similar to a 2-D physical wave tank. As shown in Fig.2 and Fig. 9, the numerical wave tank is divided into four zones: the wave making zone, the absorption zones for reflected waves and transmitted wave and the working zone, EF denotes the still water surface to divide the air region (upper side) and the water region (lower side). Supposing a right-handed Cartesian coordinate system o-xy is defined with the origin on the undisturbed free surface and oy axis pointing vertically upward, the continuity equation and Navier-Stokes equations can be written as∂ρ ∂ ( ρ u ) ∂ ( ρ v) + + =0 ∂t ∂x ∂y⎛ ∂ 2 u ∂ 2 u ⎞ ∂p ∂( ρ u) ∂( ρ u) ∂( ρ u) +u +v = µ⎜ 2 + 2 ⎟− + Sx ∂t ∂x ∂y ∂y ⎠ ∂x ⎝ ∂x⎛ ∂ v ∂ v ⎞ ∂p ∂( ρ v) ∂ ( ρ v) ∂ ( ρ v) +u +v = µ⎜ 2 + 2 ⎟− − ρ g + Sy ∂t ∂x ∂y ∂y ⎠ ∂y ⎝ ∂x2 2(1) (2) (3)where u and v denote the velocity components of the fluid particles in the x and y direction, ρ denotes the fluid density, µ the dynamic viscosity and g the gravitational acceleration. The additional momentum source terms Sx and Sy in the x and y direction are specially introduced to realize the incident wave generation and the absorption for reflected and transmitted waves (Milgram，1970; Gilbert，1978). -2- The VOF method is adopted to capture the free surface deformation, the equations can be written as：∂aq ∂t +2∂ (uaq ) ∂xq+∂ (vaq ) ∂y=0， q = 1, 2(4)(5) where aq is the volume fraction, denoting the ratio of volume of the q-th phase fluid occupying in a cell. Here q=0 and 1 denote the air phase and the water phase, respectively.∑aq =1=13. Green water of FPSO in head wavesA 2-D investigation of green water incidents on a space fixed FPSO vessel was performed by Greco (Greco, 2001) both experimentally and numerically. The experimental results are used for the further validation of the efficiency of the present method and computations. The FPSO model and wave tank dimensions are shown in Fig.1 and Fig.2, respectively. In the experiments of Greco (2001), the FPSO model is simplified as a rectangular structure with its length and height being 1.5m and 0.248m, respectively, and a small circle of radius of 0.08 m at the bow corner. As same as the physical wave tank for the experiments, the length of numerical wave tank in the present computation is also taken as 13.5m. The lengths of the wave-making zone, the reflected wave absorption zone, the working zone and the transmitted wave absorption zone are taken as 2m, 2m, 7.5m and 2m, respectively. The water depth for both the physical and numerical wave tank is 1.035m and FPSO model, with the draft of 0.198m and the freeboard of 0.05m, is set at the center of working zone.WL1 WL2 WL3 0.075m MWL 0.15m 0.2275m r=0.08m PR1 0.062mFPSOFig.1 FPSO model3.1 Mesh generationThe mesh for the numerical model is generated basically on the following considerations. To avoid dissipation for wave propagation, the number of grids in the horizontal direction should keep large enough. In the present computation, the grid size ∆x is kept about 1% of the incident wave length in the wave-making zone, the reflected wave absorption zone and the working zone. For the aim to have an effective wave adsorption, the size of the grid in the transmitted wave absorption zone increases gradually larger with a ratio of 1.2 from the rightmost boundary of the working zone. In the vertical direction,∆y is taken about 5% of the wave height near the free surface region in order to reconstruct the free surface accurately, and increases larger gradually outside the region. Additionally, more refined grids are used near the bow of the FPSO to describe accurately the nonlinear phenomena due to the wave and body interaction.-3-A Stillwaterlevel E Wave making zone Reflected wave absorption zone FPSO F L=1.5m Working zone H=1.035m D 2m m 2 7.5m 2m D=0.19 8 Transmitted wave absorption zone CBFig.2 The numerical wave tank and its function zones3.2 Wave making and absorptionWave making and absorption are realized by using the technique of Wang and Liu (2005) in the present numerical simulation. Suppose the flow velocity and pressure in the fluid field after the wave making and absorption being:(6) in which the subscripts C and I denote the values for the calculated physical variables and the incoming waves, λ=λ(x) is a smooth weighting function varying with the space position. The momentum source terms Sx and Sy may be deduced by substituting Eq. (6) to Eqs. (2) and (3), neglecting the viscosity terms and taking a linear difference discretization for the transient terms. The detailed deduction and expressions can be found in Wang and Liu (Wang and Liu, 2005) , In the present numerical scheme, the smooth weighting function is a sine function varying from 0 to 1, i.e.⎧u M = λuC + (1 − λ )u I ⎪ ⎨vM = λvC + (1 − λ )vI ⎪ p = λp + (1 − λ ) p C I ⎩ Mλ ( x) = sin(x − x1 π ) x 2 − x1 2 ，(7)in which x1 , x2 are the horizontal coordinates of the ends of the wave making or absorption zone. Suppose the horizontal coordinates being positive to the right direction in Fig.2, the velocities, pressure and the smooth weighting functions for different zones may be expressed as follows, i.e. In the wave making zone:λ u M = λu I , v M = λv I , p M = λp I ,and λ xmin = 0， xmax = 1In the reflected wave adsorption zone, the velocities can be expressed as Eq.(6) with(8)λ x =0min，λx=1max(9)And in the transmitted wave adsorption zone: (10) It is easy to understand from the above discussions that the required waves in the fluid field can be generated with the corresponding flow velocities. In the case of the generation of the incident waveu M = λuC , v M = λvC , p M = λpC ,and λ xmin = 1 λ xmax = 0ζ = ζ a cos(kx − ωt ) , its velocity components can be written as⎧u = ζ aω cos(kx − ωt ) ⎨ ⎩v = ζ aω sin(kx − ωt )(11)3.3 Solution of the problem-4- As shown in Fig. 2, the rigid wall condition is imposed on the FPSO body surface and the left side AB, the right side CD and the bottom BD of the tank. On the upper side AC, the atmospheric pressure is used as a pressure boundary condition. At the initial stage, the tank is calm (i.e. u=0 and v=0) with an initial hydrostatic pressure distribution. Laminar flow assumption is adopted in the numerical simulation of the whole fluid domain. The first order implicit difference scheme is used for the discretization of the transient terms and the upper wind difference scheme for the discretization of the convection and diffusion terms in the momentum equations. SIMPLE scheme is adopted to solve the pressure and velocity iteratively. The present computational programs are interpreted and executes on the software platform of FLUENT by the way of User Define Function (UDF).3.4 Numerical simulation and analysisIn the present computations, the wave length λ and wave height H are taken as 2m and 0.16m, respectively, and the advancing time step is 0.001 times of the wave period.0.150.1Elevation height (m)0.050-0.05-0.1 12 14 16 18 20Time(s)Fig. 3 Time history of the wave profile at the location of 3m before the FPSO bowFig.3 shows the time history of the wave profile at the location of 3m before FPSO bow in the time interval from 12s to 20s. Although in that time interval the waves reflected by the bow already reach the wave making zone, a quite stable incident wave train can still generated from the wave making zone. It indicates that the present numerical scheme is effective for long time simulation due to the application of the reflected wave absorption technique. In the numerical simulation of green water occurrence, both the wave heights on deck and impact pressure on the deck structures are obtained. Figs.4-6 show the time histories of the wave height at the three probes WL1, WL2 and WL3, located respectively at the front end of the bow, 0.075m and 0.15 m from the bow end. The time histories of wave height at the three probes are also compared with the experimentally measured results shown as dotted curves in the figures. Well agreement between the numerical results and the experimental ones can be observed in the figures. At the early stage of green water occurrence when the first incident wave peak reaches the ship bow, the wave height in front of the bow is drove up due to the wave reflection. Thus wave height, when the next wave peak coming, becomes even higher in front of the bow, causing much severer green water on deck. As observed in Figs. 4-6, the wave height at the second time of green water is much larger than that at the first time. The results shown in Figs. 4-6 are obtained for the case of a space fixed FPSO without superstructures on the bow deck. In order to get insight into the mechanism of the impact of the green water flow acting on the ship bow structures, the case of a space fixed FPSO with a vertical structure mounted on the bow deck were also conducted. The structure is mounted on the fore part of the deck, 0.2275 m from the bow (refer to Fig. 1), and-5-0.15 is high enough to protect the wave overtopping from the top. The impact pressure at the location of 12 mm high from the deck on the structure, corresponding to the location of pressure gauge on the structure in the experiments, was calculated. The time histories of the impact pressure in one wave period from both the calculation and the experiment are depicted in Fig.7 with solid and dotted curves, respectively. Two significant peaks are observed in the figure. The first peak is the result of the initial impact, and the second appears when the water front begins to run down from the wall, after having reached its maximum height. Both the results are in good agreement. Since the flow on deck and the pressure on structures are very sensitive to and strongly affected by the initial fluid flow, as indicated by Greco (2001), no exactly the same results can be obtained from the repeating of experiments and computations.-------Experiments Computations0.15Elevation height (m)0.10.050 6.5 7 7.5 Time(s) 8 8.5 9Fig.4 Time history of waves at probe WL1 located at the front end of the bow0.15------ Experiments ComputationsElevation height (m)0.10.050 6.5 7 7.5 8 8.5 9Time(s) Fig.5 Time history of waves at probe WL2 located at 0.075 from the bow end0.15Elevation height (m)------Experiments Computations0.10.050 6.5 7 7.5 8 8.5 9Time(s) Fig.6 Time history of wave at probe WL3 located at 0.15 from the bow end-6-1.2-------Experiments ComputationsPressure (k Pa)0.80.40.0 -0.50.00.51.0Time(s) Fig.7 Time history of impact pressure at probe PR1 on the mounted structure4. Green water on ships oscillating in beam wavesOscillating motions of floating bodies under the excitation of the incident waves will certainly give influence on the green water on deck and vice versa. Hence, it is necessary to count for the influence of floating body motions in the numerical simulations of green water. Numerical simulations for the case of an oscillating sectional ship model with shipping water running into its cabin were carried and the results were compared with the corresponding experimental results (Zhu and Saito, 2003).B c l f D TFig. 8 Middle sectional model of Ship 499G/T without hatchcoverFig.8 shows the middle sectional model of Ship 499G/T without hatchcover, the beam and depth of the model are 0.436m and 0.25m respectively. The hatch coaming height c is 0.032m, and the distance l between the sides of cabin and ship is 0.033m. The draught of the model is 0.19m. The physical experiments of water shipping into cabin were carried out in the 2-D tank of Hiroshima University with the water depth of 0.7m.4.1 Numerical model of ship motion and dynamic mesh techniqueTo match the experimental conditions as much as possibly, the dimensions of each functional zones and the mesh size are slightly different from those mentioned in the former section. The definition of the boundary conditions and the method to generate the meshes in the zones for wave making, reflected wave and transmitted wave adsorption are primarily the same as we have done for the numerical simulation for green water on the FPSO. A technique of moving mesh is specially introduced in the numerical simulation to realize the green water occurring for an oscillating ship. -7-Ship section Wave making zone Reflected wave absorption zoneWorking zone (Stationary grid zone)(dynamic grid zone2)1.6m1.6m11.6m2.0 m3.2 mFig.9 Tank model and zones for simulation of oscillating ship with green water occurringBesides the considerations for the mesh generation in the former section, a reasonable dynamic mesh model is setup in the working zone to simulate the ship oscillation motions. As shown in Fig. 9, a rectangular dynamic mesh domain is defined and the others are kept as static mesh domains. In the dynamic mesh domain, a circular sub-region around the body is defined as a rigid mesh region, in which the meshes are fixed to the body, moving with the body in the oscillating modes of both translation and rotation. And in the remaining sub-region of the dynamic mesh domain, the meshes move only in translational modes. The boundaries between those two sub-regions are superposed. Refined meshes are used around the moving body and in the vicinity of the region boundary to avoid numerical errors to the most extent. The advantage of such a mesh scheme is that the computational precision and efficiency are been greatly improved since different modes of motion of the body can be effectively simulated by utilizing relative motion of the meshes, and meanwhile the complex deformations of meshes around the body are avoided.4.2 Numerical simulation for green water occurrence on an oscillating shipTable 1 Motion response of the ship sectional model (wave period T=1s)Mode Sway Roll HeaveMotion ya/ζ za/ζAmplitude 0.47 0.42 0.90Phase difference -79° 101° -90°φa/kζIn the numerical simulation for green water occurrence on an oscillating ship, the ship motions, including both the motion amplitudes and phase differences, are obtained in advance by the potential flow theory and the results are validated by experiments. Those results are used to generate the body motions in the numerical fluid field. Table 1 shows the motion responses of the sectional model of a hatch coverless ship excited by a regular wave train with period of 1 second. To save the computational time, the wave information is monitored at the position before the model to obtain the exact wave phase when the activated based on the phase difference between wave and the model motions. As shown in Fig. 10, similar phenomenon can be observed from the snap shots of green water on the oscillating model in waves for both numerical simulation and experiments.-8-H=0.7m(dynamic grid zone1)Transmitted wave absorption zone(a) Visualization of computed free surface(b) Visualization of experimental free surfaceFig. 10 Snap shots of green water on a ship oscillating in waves both in numerical simulation and experiments◆Experiments600——ComputationsVolume (ml)450 300 150 0 0.0450.050.0550.060.065Wave steepness Fig.11 Comparison of shipped water volume for a ship oscillating in waves with the same period (T=1s) and various wave heightsThe shipped water volume can be obtained by integrating the water phase in the cabin. A comparison of shipped water volume for both numerical computation and experiment is made for the ship oscillating in waves with the same period (T=1s) and various wave heights. From the results shown in Fig.11, by the present computational method, not only the process of the green water shipping onto deck can be effectively simulated but also a quite alike shipped water amount with the experiments can be obtained.5. ConclusionsBy specially introducing the technique of dynamic mesh, numerical simulations have been carried out successfully for the phenomenon of green water occurrence on a fixed FPSO model in head waves and an oscillating vessel in beam seas. Numerical results agree well with the corresponding experimental ones. It indicates the present numerical scheme and method can be used to actually simulate the phenomenon of green water on deck, and to predict and analyze the impact forces on floating structures due to green water. It is of great significance to further guide the ship design and optimization, especially for ship bow and its strength design. The mechanism may also help seaman to impose operation restrictions to avoid severe green water incidents. -9- Present research mainly focuses on a 2-D dimensional problem of green water occurrence on oscillating vessel. The method can be extended to 3-D problems. And it also contributes for the further researches on strongly nonlinear problems of interaction of large amplitude wave and bodies.AcknowledgementThe present research is jointly supported by the Excellent Young Teachers Program of MOE ， the Specialized Research Fund for the Doctoral Program of Higher Education (Grant No.20030248014) and the National Natural Science Foundation of China (NSFC) (Grant No. 50379026 and 50579034) All helps are greatly acknowledged by the authors.ReferenceBuchner B. (1995), “The impact of green water on FPSO design”, OTC’95, pp 76-98, Houston. Buchner B and Cozijin J L. 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