J Mai- Sci Technol (2014) .19:302-313 DOI 10.1007/S00773-013-Ö249-7
O R I G I N A L A R T I C L E
Green water loading on a floating structure with degree
of freedom effects
Xizeng Z h a o • Zhouteng Y e • Y i n g n a n F u
Received: 19 August 2013/Accepted: 22 December 20i3/Publislied online: 10 January 2014 © JASNAOE 2014
Abstract The aim o f the present woric is to investigate whether the degree o f freedom ( D O F ) o f a floating body has a notable effect on the m a x i m u m impact pressure due to green water on deck. The analysis is carried out f o r a box-shaped floating structure w i t h a deckhouse, using experimental and numerical means to model the green water load. Green water on deck and impact o n the deck-house is generated by the impingement o f a focusing wave group on a floating structure. Computations are performed using a twodimensional constrained interpolation p r o f i l e -based model solving the Navier-Stokes ( N - S ) equations w i t h free surface boundary condition to deal w i t h nonlinear water-structure interactions. The free surface is captured by a v o l u m e o f fluid ( V O F ) - t y p e tangent o f hyperbola f o r interface capturing/slope w e i g h t i n g ( T H I N C / S W ) , w h i c h is more accurate than the original T H I N G scheme. The ver-ifications o f the simulation through a series o f model-to-model comparisons are performed i n a two-dimensional glass-wall wave tank. Experimental water surface eleva-tions, body motions and impact pressure are compared satisfactorily w i t h the computed results f o r d i f f e r e n t DOFs cases. A s a result, the peak impact pressure due to green water decreases rapidly w i t h the increasing D O F .
K e y w o r d s Green water • C I P method • T H I N G scheme • Degree o f freedom • Floating structure • V O F method
X. Zhao (13) • Z, Ye • Y. Fu
Ocean College, Zhejiang University, Hangzhou 310058, China e-mail: xizengzhao@zju.edu.cn
X, Zhao
State Key Laboratory of Satellite Ocean Environment Dynamics, The Second Institute of Oceanography, Hangzhou 310012, China
1 Introduction
The green water phenomenon occurs when an i n c o m i n g wave significantly exceeds the freeboard and wet the deck. Green water loading may cause extensive structural dam-ages to naval ships as w e l l as floating offshore units [1], and is o f considerable concern to the stability and surviv-ability o f ships and offshore platforms. Therefore, there is a great need f o r both scientists and engineers to accurately estimate the loads related w i t h green water impact.
The green water problems have been studied experi-mentally and numerically. Cox and Ortega [ 2 j p e r f o r m e d a small-scale laboratory experiment to quantify a transient wave overtopping a horizontal deck and a fixed deck, w i t h the attentions focused on the free surface and velocity iTieasurements. Ariyarathne et al. [ 3 j studied the green water impact pressure due to p l u n g i n g breaking waves i m p i n g i n g on a s i m p l i f i e d , three-dimensional model structure i n the laboratory and to relate the impact pressure w i t h the measured velocity as w e l l as v o i d f r a c t i o n on the deck. Chang et al. [4] investigate the evolution and m a x i -m u -m velocities o f green water flow on a 3 D structure i n the laboratory using the bubble image velocimetry technique. However, laboratory tests are l i m i t e d by high costs and technical limitations o f the experimental facilities. T o overcome these limitations, computational modeling o f the green water loadings has been w i d e l y used.
The most coinmonly used model is w i t h i n the f r a m e -w o r k o f the potential flo-w theory. This k i n d of models, l i k e the finite element method ( F E M ) or the boundary element method ( B E M ) , is only available f o r the simple cases such as linear problems. Greco et al. [ 5 j and Faltinsen et al. [ 6 j investigated the water on deck f o r a fixed barge-shaped structure experimentally and numerically. N u m e r i c a l l y , a boundary element method ( B E M ) f o r unsteady nonlinear
J Mar Sci Teciinol (2014) 19:302-313 303
free surface flow was developed f o r the analysis of water-on-deck phenomena. It is shown that potential flow mod-eling suffices to give a robust estimate o f green water loads until large breaking phenomena. However, many nonlinear phenomena are related w i t h green water loadings such as slamming, violent impact force, wave breaking, water-air m i x i n g and associated turbulence. Obviously, the B E M -type method is not able to capture these nonlinear features [ 7 ] . Therefore, numerical analysis based on Navier-Stokes ( N - S ) equations is required to describe the evolution o f the fluid-body interactions.
A second and more recent alternative is the use o f the mesh-free methods l i k e the smoothed particle hydrody-namics (SPH) method [ 8 ] ; or the m o v i n g particle semi-i m p l semi-i c semi-i t (MPS) method [9, 10]. Sueyoshsemi-i et al. [11] ssemi-imu- simu-lated the nonfinear motions o f a floating body induced by the water on deck using a m o v i n g particle semi-lagrange (MPS) model. Shibata and Koshizuka [12] simulated the three-dimensional shipping water on a stationary deck i n head seas using the M P S method. Shibata et al. [13] developed a three-dimensional numerical model using the MPS method f o r a ship m o t i o n w i t h a f o r w a r d speed under h i g h wave height conditions where water o n deck occurs. Rudman and Cleary [14] applied the SPH method to sim-ulate the impact o f a breaking rogue wave on a moored semi-submersible tension leg p l a t f o r m , w i t h the attentions focused o n the wave impact angle and m o o r i n g line pre-tension o n the subsequent p l a t f o r m motions. However, the mesh-free methods are l i m i t e d by their osciUations i n the pressure field and l o w computational efficiency. As a consequence, i m p r o v e d versions and promising results are currently available [15, 16].
A t h i r d alternative numerical approach is the mesh-based methods solving the N - S equations w i t h free surface modeling, such as the volume o f fluid ( V O F ) approach [17]. The advantages o f the V O F method are its mass conservation and easy to implement. M a n y i m p r o v e d V O F - t y p e schemes have been proposed such as P L I C - V O F [ 1 8 ] , T H I N G [ 1 9 ] , T H I N G / W L I C [20] ( W L I G : weighed line interface calculation) and T H I N G / S W scheme [21]. I n this paper, the T H I N G / S W scheme is combined w i t h the GIP-based model to treat the dis-torted f r e e surface. Kleefsman et al. [22] investigated some aspects o f water impact and green water loading using an enhanced VOF-based model. Yamasaki et al. [23] proposed a GIP-based finite difference simulation o f green water impact on fixed and m o v i n g bodies. M a i n attentions were focused on the fixed body, and no v a l i -dations were presented f o r the fioating body cases. H u and Kashiwagi [24] investigated a floating body using a GIP-based Gartesian grid approach f o r predicting hydrodynamic loads associated w i t h strongly nonlinear s h i p -wave interactions. Regular -waves were analyzed f o r the
i n c o m i n g wave conditions, and benchmark laboratory tests were also shown f o r validation. Zhao and H u [25] studied nonlinear interactions between extreme waves and a floating body using an enhanced two-dimensional GIP-based model. They have paid attentions to the green water loadings and the t w o degrees o f f r e e d o m body motions w i t h the sway m o t i o n fixed, and computations were compared w i t h the experimental measurements w i t h a good agreement. Recently, L i a o and H u [26] simulated the fluid-structure interactions using a partitioned approach by coupling the finite element method to the GIP-based model. I n this regard, the GIP-based model is capable o f solving such complex problems i n c l u d i n g distorted free surface and large amplitude body motions w i t h reasonable accuracy when compared to experimen-tal data.
The green water problems have been studied both experimentally and numerically f o r decades; above studies to date have been covered to free surface elevations, body morions, impact forces, and probability o f deck wetness and so on. There exist f e w models that are capable o f predicting impact pressure w i t h acceptable accuracy, the understanding o f impact pressure is still poor due to the complex and violent process o f green water on deck. The studies on the effect o f degree o f f r e e d o m on green water impact force are seldom f o u n d .
The objective o f the present study is to invesrigate the D O F effects on the impact force o f the green water as extreme waves i n the v i c i n i t y o f a floating structure. A n improved version o f the GIP-based model is proposed to study the interaction o f green waters w i t h a floating structure b y introducing the T H I N C / S W scheme f o r the f r e e surface treatment. Fuithermore, the model is used to simulate a focused wave group f o r the green water loadings, w h i c h may contribute to reduce the effect o f single frequency o f regular waves and obtain green water on deck more easily. The paper is organized i n the f o l l o w i n g manner. The GIP-based numerical model is presented briefly i n Sect. 2. Section 3 describes the physical model tests performed f o r model validation, i n c l u d i n g wave elevation, impact pressure and body response measurements under focused wave groups. Section 4 is devoted to the representative numerical exam-ples, comparing w i t h experimental results. F i n a l l y , some conclusions are drawn.
2 A C I P - b a s e d numerical model
Computation o f w a v e - b o d y interactions is performed using a two-dimensional wave tank w i t h a flat bottom as shown i n F i g . I . A piston-type wavemaker located at ;i- = 0 is used to generate the incident wayes. A floating body is placed at x — 7.0 m away f r o m the wavemaker. M o r e
304 J Mai- Sci Teclinol (2014) 19:302-313
details can be f o u n d i n the early w o r k [25] f o r reference. The fluid is treated as an incompressible, viscous fluid. The governing equations are as f o l l o w s :
V • (7 = 0 dt + {u • V ) S - - V p + - V ^ i 7 P P (1) (2)
where u and t are the velocity vector and time, respectively; p is the density, is the viscosity, F is the external force, i n c l u d i n g gravitational force.
The fluidstructure interaction is treated as a m u l t i -phase problem that includes water, air and solid body. A stationary Cartesian grid that covers the w h o l e computation domain is chosen. A volume f r a c t i o n field (j),,, (in = 1 , 2 , and 3 indicate water, air and solid, respecdvely) is adopted to represent and track the phase interface. The total v o l u m e f u n c t i o n f o r the water and structure is solved using the f o l l o w i n g advection equation.
# 1 3
8/ •-|-t7-V(/)i3 = 0 (3)
Here, (/)i3 = - f (pj. The density and viscosity o f the solid phase are set to be the same as those o f the l i q u i d phase to ensure numerical stability. Computations show that such treatment could speed the velocity-pressure coupling and save the C P U time. The v o l u m e f u n c t i o n f o r the solid (pj is determined by a Lagrangian method i n w h i c h a r i g i d body is assumed [ 2 7 ] . The position o f the water is calculated by = C ^ B - ips, where the position o f the Hquid and solid phase (/)i3 is captured by a free surface/interface capturing method. The volume f u n c t i o n f o r air cpj is then determined by (/)2 = TO — — (fis-A f t e r all v o l u m e functions have been calculated, the physical property A, such as the density p and viscosity /«, are calculated by the f o l l o w i n g f o r m u l a .
3
E
« 1 = 1
(4)
F o l l o w i n g Zhao and H u [25], the governing equations are discretized using a high-order finite difference method
on a Cartesian g r i d system. A fractional step scheme is used to solve the N - S equations. The intermediate velocity is computed by the C I P method [28] first; then a pressure is obtained by solving the Poisson equation derived by enforcing the continuity constraint; and the final velocity is updated by simple algebraic operations.
T o model the body motions, the wave-body interaction is coupled using the f r a c t i o n a l area v o l u m e obstacle rep-resentation ( F A V O R ) method developed i n i t i a l l y by H i r t [29]. The effect o f a m o v i n g sofid body on the flow is included by imposing the velocity field o f the solid body into the flow at the solid edge. The f o l l o w i n g equation is introduced to update the local i n f o r m a t i o n o f the fluid domain covered b y the body.
Ü = (t>m + (i-(t>,yi
(5)Here, is the local velocity o f the solid body and Ü denotes the flow velocity obtained f r o m the fluid flow solver. The sohd phase tp^ is obtained using a v i r t u a l par-ticle method [ 2 4 ] . Ub, the local velocity of the solid body, is tracked by a Lagrange method. B y integrating the pressure on the body surface, the hydrodynamic forces acting on the body are first calculated. W i t h Newton's L a w , the body motion , accelerations, velocities are calculated. M o r e details can be f o u n d i n previous references [24, 2 5 ] .
A n accurate interface capturing scheme, the T H I N C / S W scheme (Xiao et al. [21]) is used to calculate the free surface. The T H I N C / S W scheme is also a V O F - t y p e method. I n the T H I N C / S W method, a variable steepness parameter is adopted instead o f the constant steepness parameter that used i n the original T H I N G scheme. This variable parameter helps to maintain the thickness o f the j u m p transition layer [ 2 1 ] . I n this procedure, m u l t i
-dimensional computations can be performed by a direc-tional splitting method.
The one-dimensional advection equation f o r a density f u n c t i o n é is written i n conservation f o r m as f o l l o w s :
8/ + V{u(l>) = du
'dx (6)
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Equation 6 is discretized by a finite volume method. For a k n o w n velocity u", integrating E q . 6 over a computational cell [ A - , _ I / 2 , A - , + 1 / 2 ] and a t i m e interval [/", results i n :
+
^ " + ^ ( « ' • - 1 / 2 - g / + l / 2 )
(7)
" + 1
where Ax,- = Xi+1/2 - A V I / 2 > = r
fi„ {ii(p),±.Ld' is the flux across the cell boundary
(;r = A-,±i/2), and (p =i;,k*f (t^{x,t)dx is the cell-averaged density f u n c t i o n defined at the cell center
{x = Xi). The fluxes are calculated by a semi-Lagrangian
method. Similar to the CIP method, the profile o f (p inside an u p w i n d computation c e l l is approximated by an interpolation f u n c t i o n . Instead o f using a p o l y n o m i a l i n the CIP scheme, the T H I N G scheme uses a hyperbolic tangent f u n c t i o n to avoid numerical smearing and oscillation at the interface. Since 0 < x < 1 > and the variation o f x across the f r e e surface is step-like, a piecewise m o d i f i e d hyperbolic tangent f u n c t i o n is used to approximate the profile inside a computation cell, w h i c h is displayed as f o l l o w s :
A - ' •y tanh (8)
where a, y, 5, flare parameters to be specified. Parameters a and yare used to avoid interface smearing, which are given by:
^ • + 1 i f </';+i > 4>i~] ^ , _ ] otherwise y
1 i f > (/.,.„ 1 — I otherwise (9) Parameter S is used to deteimine the m i d d l e point o f the hyperbolic tangent f u n c t i o n , and is calculated by s o l v i n g the f o l l o w i n g equation:
1
(10)
Parameter P is used to control the sharpness o f the variation o f the color f u n c t i o n . I n the original T H I N G scheme, a constant — 3.5 is usually used w h i c h may result i n r u f f i i n g the interface w h i c h aligns nearly i n the direction o f the velocity. Therefore, a refined T H I N G scheme, the T H I N G / S W , by determining adaptively according to the orientation o f the interface was proposed by X i a o et al. [ 2 1 ] .
In a two-dimensional case, parameters could be determined by the f o l l o w i n g equations:
^ , = 2 . 3 - f 0.01
= 23\uy \ +0M (11)
where n = (n_^, 11^) is the unit n o r m vector o f the interface. A f t e r Xi{x) is determined, the flux g,- at the cell boundary can be calculated. I n F i g . 2, g,+i/2 f o r Hj+1/2 > 0 is i n d i -cated by the dashed area. A f t e r all o f the fluxes across the cell boundaries have been computed, the cell-integrated value at the new time step can be obtained by E q . 7. This cell-integrated value is used to determine the f r e e surface position. Therefore, mass conservation is automatically satisfied f o r the l i q u i d or water part.
A validation test, k n o w n as Zalesak's problem [ 3 0 ] , has been p e r f o n n e d to check the T H I N G / S W scheme. This test is one o f the most popular scalar advection tests. Gases w i t h d i f f e r e n t g r i d sizes have been earned out f o r para-metric study. A velocity field is given by u = (y - 0.5, 0.5 — ,\-) w i t h A f = 271/628. I n general, one r e v o l u t i o n is completed i n 628 time steps. The numerical error is defined and estimated b y :
306 J Mar Sci Teclinol (2014) 19:302-313
Table 1 Numerical errors for Zalesalc's test problem
Grid 50 X 50 100 X 100 200 X 200 500 X 500 number THING 1.38 X 10"' 9.11 X 10"^ 4.95 X 10"^ 2.04 X 10"^ THING/ 1.22 X 10"' 5.16 X 10"' 2.58 X 10"- 1.01 X 10"-SW EiTor = (12)
Here,
<f>'jj
is the exact solution o f The result o f numerical eiTor is summarized i n Table 1. I n addition, the shape distortion after one rotation is evaluated as shown i n Fig. 3 w i t h the contours o f 0.05, 0.5 and 0.95. The dashed contour line indicates the exact shape and the solid contour line shows the numerical results. I t can be seen f r o m F i g . 3 and Table 1 that a finer g r i d produces better shape reten-tion, and the numerical eiTor o f the T H I N C / S W scheme is lower than that o f the original T H I N C scheme. Comparing w i t h the original T H I N C scheme, the T H I N C / S W scheme has a better shape after one rotation. Therefore, the T H I N C / S W scheme is chosen and used i n this study.3 Model tests
Details of the model tests were described i n the early w o r k by Zhao and H u [ 2 5 ] . However, f o r the completeness, a b r i e f description is p r o v i d e d as f o l l o w s .
The physical experiments have been performed i n a t w o -dimensional glass-wall wave tank i n the Research Institute for A p p l i e d Mechanics ( R I A M ) o f Kyushu University, Japan. I t was p e r f o r m e d by the author when he was w o r k i n g as a research f e l l o w i n R I A M , Kyushu University. The wave tank is 18-m long, 0.30-m wide, and filled w i t h
tap water to a depth o f h = 0.4 m. A wedge-type wave-maker is located at one end o f the tank to generate specific waves. Another wedge-type absorbing wavemaker is placed at the opposite end to damp incident waves.
Figure 4 displays the experimental setup i n detail. A photo o f the floating body is shown i n F i g . 5a, w h i l e Fig. 5b shows a photograph o f the caiTiage and guide r a i l . A simple box-shaped geometry w i t h a rectangular deckhouse is used. The floating body is 0.5-m long, 0.29-m w i d e and 0.123-m high. The w i d t h o f model (0.29 m ) is nearly as same as the w i d t h o f wave flume (0.3 m ) . Then, the 3 D experimental test can be approx-imately regarded as a 2 D case. The main model geo-metrical and hydrostatic parameters are summarized i n Table 2. This box-type floating body is connected w i t h a heaving r o d through a rotational j o i n t . T h e rotational j o i n t is placed i n i t i a l l y at the stiU free surface. The heaving r o d is set i n and moves smoothly between the shder mechanisms installed i n a carnage on the guide rails. The body is free to move i n heave and r o l l . The sway m o t i o n is restrained by a spring, connecting the carriage and the guide rail. The mass o f the floating body, heaving rod and carriage is — 14.5 k g , 1112 — 0.276 k g and in^ = 2.13 k g , respectively. The heave m o t i o n is shared by the heave r o d and the floating body. The horizontal sway m o t i o n is shared b y the heave rod, the carriage and the floating body. The r o l l m o t i o n is shared only b y the floating body. The corresponding mass f o r heave, r o l l and sway m o t i o n is {iiii + 1112), (m^) and (nil + ' " 2 + ' " a ) , respectively. The spring constant is added to the C F D code by imposing a horizontal drag f o r c e ƒ to the body by the equation ƒ = —kx (k = N/ m, the spring constant and x is the distance away f r o m the balance posifion). The allowed body motions are measured by potentiometers. The prescribed wave parameters are checked w i t h wave probes located along
Fig. 3 Zalesak's problem after one rotation (solid for simulation and dashed for theory and plotted are the contours of 0.05, 0.5 and 0.95): THING (upper) and THING/SW (down): a grid: 50 x 50; b grid: 100 X 100; c grid: 200 x 200; d grid: 500 x 500
(a) (b) (c) (d)
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Fig. 4 E.xpeiimental .setup
Unit: m
/ / / / / / / / /
the wave tanlv. The wave probes are placed at x = 3.0, 5 . 1 , 7.0, 8.9 and 11.0 m away f r o m the wavemaker, respectively. The sampling rate f o r the wave probes i n these measurements is 100 H z , w h i l e a h i g h sampling frequency of 1,000 H z is chosen f o r the body m o t i o n and pressure measurements. A pressure gauge is placed o n the deckhouse at a height o f 0.01 m above the deck to record the water-on-deck impact pressure, as shown i n F i g . 6.
The laboratory measurements have been used to inves-tigate the 2 - D O F body motions under extreme wave con-ditions by Zhao and H u [ 2 5 ] . I n this study, temporal wave elevations, body position and green water impact pressure measurements are used to validate the numerical model o f dealing w i t h the 3 - D O F body response to a transient wave groups.
4 Results and discussions
4.1 W a v e generation
I n this subsection, waves are generated numerically using a piston-type wave generator. A c c o r d i n g to the linear
wavemaker theory, the incident velocity can be calculated by the f o l l o w i n g equation f o r a specified regular wave.
CO
r ( c o ) ' 9/ (13)
Where )/ is the expected wave elevation, and co is the wave frequency; x is the wavemaker potions; T((o) is the transfer f u n c t i o n f o r a piston-type wavemaker and can be calculated by the f o l l o w i n g equation.
2(cosh2/t/? - 1) T{w) =
2kh + sinh 2kh (14)
where h is the water depth and k is the wavenumber cor-responding to CO.
T o obtain green water easily, the focusing wave theory is used, and the target transient wave elevation is repre-sented as
A'f
>Kx,t)=^i-li{x,t) 1 = 1
A'f
= J2 «/ cos {ki {x - Ap) - 2nfi { t ~ t , ) ) (15)
1 = 1
and the wave elevation at the wavemaker position, x — 0, should be
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Fig. 5 a Photograph of the body model; b photograph of the carriage and guide rail
Table 2 Main parameters of the body model
Item Value (m)
Length Breadth Draft
Gyration radius
Center of gravity (from the bottom)
0.5 0.29 0.10 0.1535 0.0796 Nf Nf '1(0, t ) ^ Y . '?'(0' ^) = £ '^os {k (0 1 = 1 7 = 1 Xp) - « , ( / - r p ) ) (16) The con-esponding input velocities at the wavemaker boundary can be given by
Nf Nf
(17)
OJ,
i=l 1 = 1 T{wi) dt
where ; is the ith component wave. Ap and tp are f o c a l position and time, respectively. A'f is the number o f the wave components and is chosen to be 29 to approximate a continuous spectrum. The components, are equispaced across a bandwidth o f df and centered at frequency fc-W a v e conditions are described as f o l l o w s . A computational domain o f 14.5 x 1.4 m is discretized using a variable grid. For the focused wave group, the component wave frequency ranges f r o m 0.6 to 1.6 w i t h ƒ ; — 0.83 s as the peak frequency and r^, = 1.2 s as the peak period. The waves are focused at t = 20.0 s and at x = 7.0 m away f r o m the generator. The amplitudes o f the i n d i v i d u a l wave components, <7,-, are calculated based on the Joint N o r t h Sea Wave Project ( J O N S W A P ) type spectra.
The time histories o f the wave elevations (without the body) along the tank are presented and compared w i t h the experimental data. Focusing wave amplitudes, A — 0.07 m , has been chosen, and the input positions are adjusted to focus the wave energy at the target location.
Pressure sensor
0.01m
Fig. 6 Details of the pressure transducer
The time series o f the input velocity and the wavemaker position is shown i n F i g . 7. I t should be pointed out that the wavemaker position is useless i n the present model because only the input velocity at the wavemaker boundary is used here. Comparison o f the time series o f the wave elevations between computation and laboratory experiment is dis-played i n F i g . 8. Here, both the original T H I N C scheme and the T H I N C / S W scheme are used f o r the free surface treatment. Comparing these numerical results, i t is d i f f i c u l t to tell the difference between them because only small difference c o u l d be noticed at the end o f the simulation. Notice that the computational results coincide pretty w e l l w i t h the experimental data, and the wave elevation reaches its largest value at the focusing position (x/h = 17.5), w i t h a symmetric distribution o f wave p r o f i l e about the focusing time. I t can be also noticed that the crest is twice times o f the trough. Then the green water occurs easily, w h i c h is the reason to choose the focused wave f o r the green water input.
4.2 Green water-on-deck validation
I n this subsection, the nonlinear wave-structure interac-tions are performed and compared w i t h experimental measurements. As mentioned before, the results o f t w o -dimensional simulations w i t h 2 - D O F body motions have been analyzed previously [ 2 5 ] . Here the results o f t w o -dimensional simulations w i t h 3 - D O F body motions are
J Mar Sci Teclinol ( 2 0 1 4 ) 1 9 : 3 0 2 - 3 1 3 309 0.1 ~§ 0.0 -0.1 A 0.00 -0.02 10 15 tIT 20 25
Fig. 7 Input velocities and wavemaker position for wave generation
0.5 i 0-0 -0,5 0.6
t
0.0 -0.6 1t
0 0.6 0.0 -0.6 0.5 ^ f 0.0 -0.5 o Exp THINC/SW - - THINC .4=0.07m .v//i=27.4 .V.-7J=17.5 xlh=2.(, 10 20 25Fig. 8 Comparison of the focused wave profile between computation and experiment with the focusing amplitude A = 0.07 m
focused and compared w i t h the experimental data. The discrepancies between experimental and numerical results are discussed.
Computations are carried out using the numerical wave tank w i t h the floating body placed at x = 7.0 m away f r o m the wavemaker boundary as shown i n F i g . 1. However, several simplifications are made to save the computation time. First, waves are generated by prescribing an i n f l o w velocity similar to a piston-like wavemaker instead o f simulating the wavemaker motion. Second, a damping zone is adopted to shorten the tank length other than the
absorbing wavemaker i n the experiment. I n the computa-tion, a stationary Cartesian grid is employed w i t h a grid number o f 648(a-) x 248(3') ^nd a m i n i m u m grid spacing o f
— A y = 0.003 m around the body and the f r e e surface. The time step is dynamically determined to satisfy the stability criterion o f a Courant-Friedrichs-Lewy condition w i t h the total simulation time up to 30.0 s. A l l the cases presented i n the paper are run on a normal PC.
Figure 9 presents the comparison o f the simulated and measured body motions, free surface profiles and the impact pressure on the deckhouse. B o t h the original T H I N C scheme and the T H I N C / S W are used f o r the f r e e surface/interface treatment. The 2 - D O F body motions w i t h the sway m o t i o n fixed is displayed i n c o l u m n (a), w h i l e the 3 - D O F body motions w i t h aU modes free are displayed i n column (b). The present numerical results are f o u n d to be i n good agreement w i t h the experimental data. Comparing column (a) w i t h c o l u m n (b), fitfle difference can be f o u n d f o r the heave m o t i o n , r o l l m o t i o n and free surface elevation before the body. B y checking carefully, the m a x i m u m heave m o t i o n i n positive direction f o r the case o f 3 - D O F is a l i t t i e larger than that o f 2-DOF. Meanwhile, f o r the case o f 3-DOF, the numerical model predicts the peak c l o c k w i s e r o l l m o t i o n less accurately. However, the trend o f the r o l l m o t i o n has been captured by the present model. For the sway m o t i o n , discrepancy between the numerical results and the experimental data can be f o u n d f r o m its m a x i m u m sway m o t i o n . The sway motion reaches its m a x i m u m value almost at the end o f the water impact according to the heave m o t i o n and r o l l m o t i o n . A t that moment, the flow structure around the body is extremely complex i n c l u d i n g the water-air interaction, bubbles, turbulence and structure vibration and so on. I t is very hard to obtain the exact hydrodynamics f o r c e acting on the body i n such complex flow field. Comparing the two free surface capturing schemes, the predicted results repeat each other quite w e l l except f o r m i n o r difference at the peak values. For the 2 - D O F r o l l m o t i o n , the T H I N C scheme underestimates the m a x i m u m clockwise r o l l motions. W h i l e f o r the 3 - D O F r o l l m o t i o n , the T H I N C scheme underestimates the m a x i -m u -m anticlockwise r o l l -motions and the -m a x i -m u -m sway motion.
The i m p u l s i v e impact pressures on the deckhouse are shown at the b o t t o m o f F i g . 9. Notice that t w o peaks phenomenon o f the pressure variation can be predicted w e l l b y the present model. However, the exact peak pressure values are underestimated by both the original T H I N C scheme and the T H I N C / S W scheme. The first peak is caused by the first high-speed water impact along the deck, w h i l e the other is caused by waterfall along the vertical deckhouse; the second peak is larger than the first peak. I n another w o r d , the w a t e r f a l l is more dangerous f o r structure safety. M e a n w h i l e , more D O F body motions appear to
310 J Mar Sci Teclinol (2014) 19:302-313 (a)
J5
0,4 è ïs 0,0 -0,4 0 Exp T H I N C / S W T H I N C [_|^=0.07m 1 1 P - D O P ll 1 Hf.Txv Ï 0 Exp T H I N C / S W T H I N C 1 . , . . • • . . . 1 1 . 1 . , . . 0I
10 -10 5 1 L" 0 ^ 0 g I F PL, 0 20 25 16.0 16,5 20 r ' • • ' • 1 . • - • 1 . - • • 1 • • • • 20 -| Elevation atx/h= 12,7 tIT 20 25 (b) -0.4 17.0(/r
17.5 IS.O O Exp T H I N C / S W - - T H I N C 0 -1 h 1 ^ 20 25 20 25 Ehvalion Ö/.V'7J=12.7 I IS.OFig. 9 Comparison of body motions and impact pressure due to focused waves with the focusing amplitude A = 0.07 m; a 2-DOF; b 3-DOF
decrease the impact pressure, where the pealc pressure f o r 2 - D O F is larger than that o f 3-DOF. The effect o f D O F on the body motions w i l l be analyzed i n the next section. A l t h o u g h the exact impact force is predicted less accurately here, there m i g h t be several possible reasons f o r this problem. First could be the g r i d resolution selected. Second could be the accuracy o f the coupling o f w a v e - b o d y interactions.
T o investigate the convergence o f the model, a fine mesh (798 x 324) w i t h m i n i m u m mesh A-T = A y = 0.002 m is used to simulate the nonlinear w a v e - b o d y interactions. Computed body motions show that they are convergent and grid-independent results are obtained f o r both the fine g r i d and the coarse grid. The coarse grid means the g r i d used above w i t h the m i n i m u m mesh Ax- = A y — 0.003 m . The comparison o f t h e sway motions between the fine grid and the coarse grid is presented i n
F i g . 10a. I t can be noticed that i m p r o v e d sway m o d o n is obtained using the fine grid as the simulation dme is over 207^. I t can be explained by the complex flow structure. Fine g r i d is necessary to capturing the hydrodynamics force acting on the body i n such complex fiow structure.
Figure 10b, c shows the numerical results o f the impact pressure w i t h both the fine g r i d and the coarse grid. I t can be seen that the numerical prediction o f the first peak pressure w i t h fine resolution is higher than that w i t h coarse resolution, especially f o r the 3 - D O F case. W h i l e f o r the second peak pressure, l i t d e improvement can be noticed. This indicates that local quantities such as the first peak pressure are obviously i m p r o v e d as the g r i d resolution increases, but the second peak pressure is not so sensidve to the grid resolution. Further study is warranted f o r improvement i n the accuracy o f the c o u p l i n g o f wave-body interactions.
J Mar Sci Teciinol (2014) 19:302-313 311
4.3 E f f e c t o f D O F on impact pressure
I n this subsection, aspects o f body motions w i t h d i f f e r e n t DOFs are simulated numerically. The effect o f D O F on the subsequent motions o f the floating body is considered, and the impact pressures are also predicted as f o l l o w s . The input wave conditions are the same as the w a v e - b o d y interactions i n Sect. 4.2. Several aspects are considered, including the heave, r o l l , and sway motions, the impact pressure and the mass o f water on deck.
The numerical results f o r different D O F body motions are presented i n F i g . 11. The results f o r the cases o f m a x i m u m and m i n i m u m D O F are drawn with bold lines. I t shows that all the predictions f o r all DOFs are similar except f o r the peak values i n the v i c i n i t y o f focusing position. For the heave motion, the m a x i m u m heave m o t i o n occurs when the floating body moves f r e e l y w i t h all
the modes free. Outside the focusing location, there is Httie divergence compared w i t h different DOFs. For the case o f 1-DOF, the body should suffer f r o m the m a x i m u m heave response as there is only one D O F to absorb the wave energy. However, the heave m o t i o n o f 1-DOF is com-pressed mostly comparing w i t h that o f other DOFs. This may be caused by the gravity effect o f water on deck when the heave motion m o v i n g vertically i n the same direction as the gravity. It indicates that water on deck significantiy influences the heave motion o f the floating body. W h i l e f o r the r o l l and sway motions, the m a x i m u m motion occurs when the floating body moves w i t h 1-DOF. I t indicates that f o r more D O F case, the wave energy is transfeiTcd into more directional motions.
The effect o f D O F on the impact pressure is also con-sidered, as shown i n F i g . l i d . The results show that the m a x i m u m peak pressure occurs when the body is fixed. As the D O F increases, the peak pressure decreases rapidly.
(a) o Exp Fine grid Coarse grid (b) 1.5 2-DOF o Exp Fine grid • Coarse grid o Exp Fine grid • Coarse grid (c) 1.5
ê
1.0 '& 0.5 p.. 0.0 : 3-DOF O Exp Fine grid 'Coarse grid O Exp Fine grid 'Coarse grid f ' 1 , , , 1 , , . . 16.8 16,9 17.0 17,1 tIT 17.2 0 I D O F -3 D 0 F - 2D0F heavu f3.-;ed 2D0F roll Exed I D O F -3 D 0 F - 2D0F heavu f3.-;ed 2D0F roll Exed L(c) ; • , • I 18 20 22 24 ODOF 3 DOFI D O F heave only— IDOF roll only IDOF sway only - 2 D 0 F roll fixed 2 D 0 F sway fixed — — 2DOF heave fixed
ilT Fig. 10 Effecl of the grid resolution on wave-body interactions:
a sway motion; b 2-DOF impact pressure; c 3-DOF impact pressure
Fig. 11 Prediction of body motions, impact pressure and mass of water on deck for different DOF
312 J Mar Sci Technol (2014) 19:302-313
Table 3 Maximum of pressure and green water mass for different
DOF
DOF number Max (P) (kpa) Max (M„/Mo)
0-DOF 1.917 0.21104
1-DOF, heave only 1.424 0,21554 1-DOF, sway only 1.22 0.1696 1-DOF, roll only 1.17 0.152268 2-DOF, sway fixed 1.02 0.136916 2-DOF, roll fixed 1.019 0.160566 2-DOF, heave fixed 0.806 0.1304
3-DOF 0.799 0,0778 * Max(pr6ssure) o Max(M^/M„)*10 1 - * * Ó : — I — I — I — I — I — 1 I I I I < I I I 1 I I 1 0 1 2 3 DOF
Fig. 12 Relation of the peak impact pressure and the peak mass of water on deck multiplied by 10
The peak pressure of 0-DOF is more than twice that of 3-DOF. Therefore, for a 3-DOF case, there is about 20 % reduction of impact pressure as compared to a 2-DOF case, while 40 and 60 % reduction compared to a 1-DOF and 0-DOF cases, respectively. The mass of water on deck is shown in Fig. l i e . The parameter MJMQ is considered, where Mg is the mass of water on deck and MQ is the mass of the floating body, respectively. The relationship between the mass of water on deck and DOF shares the same trend as that of impact pressure. The mass of water on deck decreases with increasing DOF. To explain this phenome-non clearly, the peak value of the impact pressure and the peak mass of water on deck are shown in Table 3. The results show that the two column data obey the same ten-dency that the value decreases with increasing DOF. The maximum value is over twice bigger than the minimum value of each column. Meanwhile, the impact pressure value is about ten times as that for the mass of water on deck. Here, since they have different physical units, so only the values are considered. To vaHdate this assumption, the peak impact pressure and the peak mass of water on deck multiplied by 10 is drawn in Fig. 12. It is shown that a part of data coincides with each other and they share the same tendency that the impact pressure and the mass of water on deck decreases with increasing DOF. Therefore, it can be
concluded that more water on deck leads to higher impact pressure.
5 Conclusions
A CIP-based model has been proposed for investigating green water loading on a floating structure with degree of freedom effects. The model governed by the N-S equations is solved by a CIP-based high-order finite difference method on a fixed Cartesian grid system, which leads to a robust flow solver for the governing equations. The SOR algorithm is used for the velocity-pressure coupling, the immersed boundary method is employed to deal with complex geometries. Fluid-structure interaction is treated as a multi-phase flow problem with water, air and solid phase solving one set of governing equations. A VOF-type scheme, the THINC/SW scheme is implemented in the model to accurately define the free surface configuration.
In order to validate the mnnerical results, physical experiments have been performed in a 2D wave tank. Waves are generated using a wave focusing theory and its interacting with a box-shaped floating body is studied. Wave profile along the tank and motions of the body are measured, impact pressure on a vertical deckhouse along the body deck is predicted. Qualitative and quantitative comparisons between numerical results and laboratory data have been presented. Special attentions are given to the free surface profile, nonlinear body responses and impact pressure due to water on deck numerically and experi-mentaUy. Results show'that the CIP-based model can be used to reproduce the extreme wave nonlinear interacting with the floating body, where the distorted free surface such as wave breaking and water-air mixing, and large amplitude body motions occur. Fairly good agreement is obtained for the prediction of the free surface profile and the body motions numerically and experimentally. The predicted impact pressure due to water on deck shows that the same tendency with laboratory results for both 2-DOF and 3-DOF. However, the peak impact pressure due to water on deck is under predicted for both 2-DOF and 3-DOF. The effect of grid resolutions on the results is checked by changing the grid sizes. Results indicate that no obvious difference could be seen in the global body motions except for the sway motion, but the local first peak pressure due to water on deck, especially for the 3-DOF case is improved with finer giids. Comparison ofthe effects of the free surface capturing for wave generation and wave-body interactions are also presented. No remarkable difference is observed.
As an application of the numerical model, aspects of body motions with different DOF are studied numerically. The predicted results showed that the maximum peak
J Mar Sci Teciinol (2014) 19:302-313 313
pressure occurs when the body is fixed ( 0 - D O F case). Meanwhile, the peak pressure decreases rapidly w i t h the increase o f D O F . The peak pressure o f 0 D O F is over twice times as much as that o f 3-DOF. Comparing w i t h other body motions, the heave motion of the floating body is influenced and suppressed significantly by water on deck. The effect o f D O F on the impact pressure is also consid-ered. As a result, the peak pressure decreases rapidly w i t h the increasing D O F . For a 3-DOF body, there is about 20 % reduction o f impact pressure as compared to a 2 - D O F floating body, w h i l e 40 and 60 % reduction compared to a 1-DOF and 0 - D O F body, respectively. Therefore, the body w i t h more D O F could be safer f o r wave impact. I n addi-tion, the mass o f water on deck is introduced to relate w i t h the peak value o f the impact pressure. I t is f o u n d that they coincide w i t h another and share the same trench that both the impact pressure and the mass of water on deck decreases w i t h the increasing D O F . I n other w o r d , more water on deck leads to higher impact pressure.
Acknowledgments The laboratory experiments of this study were performed during the author's visit to RIAM at Kyushu University. Financial support from Kyushu University for the author is gratefully acknowledged. Special Acknowledgments should be given to Dr Changhong Hu. This work is jointly supported by the Fundamental Research Funds for the Centi-al Universities (2012QNA4020), the National Natural Science Foundation of China (No. 51209184), Key Laboratory of Water-Sediment Sciences and Water Disaster Preven-tion of Hunan Province (Grant no. 2013SS03), and the Zhejiang Open Foundation of the Most Important Subjects.
References
1. Buchner B (2002) Green water on ship-type offshore structures. PhD Thesis, Delft University of Technology
2. Cox DT, Ortega JA (2002) Laboratory observations of green water overtopping a fixed deck. Ocean Eng 29:1827-1840 3. Ariyarathne K, Chang KA, Mercier R (2012) Green water impact
pressure on a tbi-ee-dimensional model structure. Exp Fluids 53:1879-1894
4. Chang KA, Ariyarathne K, Mercier R (2011) Three-dimensional green water velocity on a model structure. Exp Fluids 51:327-345
5. Greco M , Landrini M , Faltinsen OM (2004) Impact flows and loads on ship-deck structures. J Fluid Struct 19:251-275 6. Faltinsen OM, Landrini M , Greco M (2004) Slamming in marine
applications. J Eng Math 48:187-217
7. Greco M , Faltinsen OM, Landrini M (2005) Shipping of water on two-dimensional structure, J Fluid Mech 525:309-333
8. Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110:399^06
9. Koshizuka S, Tamako H, Oka Y (1995) A particle method for incompressible viscous flow with fluid fragmentation. Comput Fluid Dyn J 4:29-46
10. Koshizuka S, Oka Y (1996) Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123:421-434
11. Sueyoshi M, Kashiwagi M , Naito S (2008) Numerical simulation of wave-induced nonlinear motions of a two-dimensional floating body by the moving particle semi-implicit method. J Mar Sci Technol 13:85-94
12. Shibata K, Koshizuka S (2007) Numerical analysis of shipping water impact on a deck using a particle method. Ocean Eng 34:585-593
13. Shibata K, Koshizuka S, Tanzawa K (2009) Thi-ee-dimensional numerical analysis of shipping water onto a moving ship using a particle method. J Mar Sci Technol 14(2):214-227
14. Rudman M , Cleary PW (2013) Rogue wave impact on a tension leg platform: the effect of wave incidence angle and mooring line tension. Ocean Eng 61:123-138
15. Khayyer A, Gotoh H, Shao SD (2009) Enhanced predictions of wave impact pressure by improved incompressible SPH methods. Appl Ocean Res 31:111-131
16. Khayyer A, Gotoh H (2012) A 3D higher order Laplacian model for enhancement and stabilization of pressure calculation in 3D MPS-based simulations. Appl Ocean Res 37:120-126
17. Hirt CW, Nichols BD (1981) Volume of fluid (VOF) method for the dynamics of free surface boundaries. J Comput Phys 39:201-225
18. Youngs DL (1982) Time-dependent multi-material flow with large fluid distortion. In: Morton KW, Baines MJ (eds) Numerical methods for fluid dynamics, vol 24. Academic Press, New York, pp 273-285
19. Xiao F, Honma Y, Kono T (2005) A simple algebraic interface capturing scheme using hyperbolic tangent function. Int J Numer Meth Fl 48:1023-1040
20. Yokoi K (2007) Efficient implementation of THINC scheme: a simple and practical smoothed VOF algorithm. J Comput Phys 226:1985-2002
21. Xiao F, l i S, Chen C (2011) Revisit to the THING scheme: a simple algebraic VOF algorithm. J Comput Phys 230:7086-7092 22. Kleefsman KMT, Fekken G, Veldman AEP, Iwanowski B, Buchner B (2005) A volume-of-fluid based simulation method for wave impact problems. J Comput Phys 206:363-393
23. Yamasaki J, Miyata H, Kanai A (2005) Finite-difference simu-lation of green water impact on fixed and moving bodies. J Mar Sci Technol 10:1-10
24. Hu CH, Kashiwagi M (2009) Two-dimensional numerical sim-ulation and experiment on strongly nonlinear wave-body inter-actions. J Mar Sci Technol 14(2):200-213
25. Zhao XZ, Hu CH (2012) Numerical and experimental study on a 2-D floating body under extreme wave conditions. Appl Ocean Res 35:1-13
26. Liao KP, Hu CH (2013) A coupled FDM-FEM method for free surface flow interaction with thin elastic plate. J Mar Sci Technol 18(1):1-I1
27. Hu GH, Kashiwagi M (2004) A CIP-based method for numerical simulations of violent free surface flows. J Mar Sci Technol 9:143-157
28. Yabe T, Xiao F, Utsumi T (2001) The constrained interpolation profile method for multiphase analysis. J Comput Phys
169:556-593
29. Hirt CW (1993) Volume-fraction techniques: powerful tool for wind engineering. J Wind Eng Ind Aerod 46-47:327-338 30. Zalesak ST (1979) Fully multi-dimensional flux corrected
trans-port algorithm for fluid flow. J Comput Phys 31:35-62
