Twenty-Fourth Symposium on Naval Hydrodynamics (2003)

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24eh Symposium on Naval Hydrodynamics Fukuoka, JAPAN, 8-13 July 2002 Nonlinear Green Water Effects On Ship Motions and Structural Loads Daniel A. Liut, Kenneth W. Weems, and Woei-Min Lin (Science Applications international Corporation, USA) ABSTRACT Green water on deck can have very important, nonlinear effects on ship motions and structural loads. Such occurrences can not only affect the global motions and loads on ships but also cause damage on deck structures. In this paper, a novel finite-volume strategy is presented to simulate three- dimensional green-water events, mainly on ship platforms. The equations of conservation of mass and momentum are solved in the time domain. Shallow-water assumptions are made, and viscous effects are ignored. The current green-water method was developed in the framework of the nonlinear 3-D time-domain ship-motion simulation program LAMP (Large Amplitude Motion Program) to account for water-on-deck effects. The method can handle a variety of boundary and initial conditions, and it is capable of supporting arbitrary motions and general geometries of a ship deck. This approach has been validated with available experimental data and has been successfully integrated with the LAMP System. INTRODUCTION The objective of the work presented in this paper was to develop a sophisticated green-water-on-deck model that could be integrated directly into a time- domain ship motion calculation, thus allowing green- water effects to be included in the calculation of nonlinear ship motions and loads. In order to attain this, an approach was selected in which the ship motions and green-water-on-deck calculations run concurrently in the time domain. The ship motions code in the present effort is the Large Amplitude Motions Program (LAMP) System. At each time step of the LAMP calculation, the ship motion and wave definition are used to compute the relative motion at the deck edge (often called the deck edge exceedance), the deck tilt, and the deck acceleration, which are then passed to the green-water-on-deck calculation. The green-water calculation is subsequently updated after which the green-water forces are passed back to the ship motions calculation to be used in solving the equations of motions and computing sectional loads. f Ship Moron Coincident WaYe ~BodyBound~y~ f DeckMotion ~ `~ Condition J (Edge Exceedanc:) u ~:~.V"~2 I'm ~.~.~2'~ ;5'"~.~'~'F'~ ~~ ~7 :'~'~"~'~ '.~ ~~ i' .~ .~. i. ~~ ~~ ~ ~~' ~~.~ I've ~~.~"~'<~ "'~r'>~'~'~"~"~'~>'~'~"~i ~~:~'~""'~ i''" ~~ ~~ ~~ :' ~S~ ~ 'o'er —~ T~ ~~ + :: ' '?~'~S'S~T'~ ~:~ I ~ ~wi~oU~o ~~ ~~ ill ~~ :~_r~—ore ~~.~uv~ator~ : .i .~ I', - ~ .~i. . . ~,~i~:~.~:~ .: is. A,. _,. we. -,:,.:: -I .~^ .- t ~~ ~~ - i, ~ -I ~ -a: ~ ~~ ill ,~ ~~ At , : ~~ ~ I,,. Eli ~~ . ~~ hi. ~~,~ ~~ ~.~.; ~~ ~ ~~ ~~, ~ .~: . ail TO: :ii*: : :~ ~ ~~;~ A it ~T`.~: ~ ~~ to ~ i,~'~2~ lu~rn:~n slim sac .~: >~i ~ ; ~ in: all i, p:f~: -:: ¢° :.. ~ ::' A At i! ,~ , i.~,.:, ~~;~i~~ it_ _~ j~j~,j~:,~ ~~ ~ ., ~~ if_ ., I, '2 2~ '.2.~ ,. . I'd'','': j~.~'~"~-.,~it' SS:'Ni~i,~'~" a.' ii"'': Ti~:i"i~ ~j'SS~'~"~:~ Ajar ~ ~'~''i'~'LL.'S~''~ ~.~'S~S'~''"'.~:~i''?. : i: ~,'S,~:~: :~: A: i": :'~''~'S'S,2~' ~',~':4 ~ ~ j j ~ ~~ ~ ~ ~ j ~ j' j i, ~ ~ ~ 'S ~ '. ~ ~ j ,. :''-: :_ ' ~ :: ' ' ~ :: S~ an>: :: Ian ~~ i~ ~~ .~ ~. ci:~i~i-i~-~i~ ~~ >~; ~ cu .~hon~ ~~-~ . ~ .. -all i ~ j j ~~ ..~jj~,~,,.~.~.~ .,,, . ~i ..~ ~.,.~jjjjj ~ j j.jj ,.,~., j.j.~.~ jiW., jj.~ji j.~ ~.~j~,> ~~.~ ~ r ~ r _ Deck Pressure Boundary Forces Wave Forces Extends Forces Figure 1: Structure of the LAMP System with Green Water On Deck For this approach to be successful, the green-water-on-deck calculation is required to be reasonably fast, robust, and capable of calculating the flow of a deck that is moving with large-amplitude six-degrees-of-freedom motions. A multi-level implementation was selected incorporating both a semi-empirical "baseline" model and a general solution of the 3-D shallow-water flow equations. The former model is intended to provide a fast estimate of green-water effects while the later provides a higher fidelity solution of the water-on- deck problem. This latter higher-fidelity model constitutes the principle focus of the present paper.

The numerical strategy adopted for solving the 3-D shallow-water flow equations is based on a novel finite-volume approach. The flow is assumed to be incompressible and viscous effects are ignored. Given the complexity inherent in the shallow-water problem compound by the fact that the calculations would in many cases involve large-amplitude motions, one of the major goals of the new development was to achieve the robustness necessary to handle the general situation of a platform that is constantly moving and exchanging water with the environment. With this in mind, a numerical method was developed capable of assuring a stable solution. This paper begins with a discussion of the LAMP System and the incorporation of the green- water calculations. In subsequent sections, the solution to the shallow-water equations of mass and motion are summarized, including some comments related to the stability of the solution. Next the alternative semi-empirical approach is described. This is followed by a discussion of computational results, including validation data. The paper is closed with some final remarks. THE LAMP SYSTEM The LAMP System is a time-domain simulation model specifically developed for computing the motions and loads of a ship operating in extreme sea conditions. With its general nonlinear time-domain approach and solution of the 3-D flow field, it is well suited for incorporating a nonlinear green-water-on- deck calculation model. Wave-Body Hydrodynamics One of the most important features of the LAMP System is a 3-D body-nonlinear approach for solving the wave-body interaction problem in the time- domain. The computational model is based on a potential-flow "body-nonlinear" approach (tin and Yue, 1990, 1993; and Lin et al., 1994~. In contrast to the linear approach in which the body boundary condition is satisfied on the portion of the hull under the mean water surface, the body-nonlinear approach satisfies the body boundary condition exactly on the portion of the instantaneous body surface below the incident wave surface. It is assumed that both the radiation and diffraction waves are small compared to the incident wave so that the free-surface boundary conditions can be linearized with respect to the incident-wave surface. A complete body boundary condition is applied incorporating forward speed, the ship motion (radiation), and the incident wave (diffraction) effects. The solution of this nonlinear wave-body using a "panel" method gives the velocity potential over the hull surface. Bernoulli's equation can then be used to compute the hull pressure distribution including the second-order velocity terms. The hydrostatic restoring force is also computed on the wetted hull up to the incident wave. In this formulation, both the body motions and the incident waves can be large relative to the draft of the ship. Several variations of Lin and Yue's original body-nonlinear approach have been developed and are currently available in the LAMP System. In addition to the general body-nonlinear approach described above, a 3-D body-linear approach has been implemented, which solves for the velocity potential only on the mean wetted hull surface and computes the hydrostatic restoring forces, along with Froude-Krylov wave forces, from hull water-plane quantities. An approximate body-linear formulation has also been implemented, which combines the body-linear solution of the disturbance potential with body-nonlinear hydrostatic-restoring and Froude- Krylov wave forces. This latter approach captures the significant nonlinear effects of most ship-wave problems at a fraction of the computation effort of the general body-nonlinear formulation. Mixed Source Formulation To solve the ship-wave interaction problem described above, a hybrid numerical approach has been developed that uses both transient Green functions and Rankine sources (tin et al., 1999~. This approach has been implemented in the LAMP System as the "mixed-source formulation." In the mixed- source formulation, the fluid domain is split into two domains as shown in Figure 2. The outer domain is solved with transient Green functions distributed over an arbitrarily shaped matching surface, while the inner domain is solved using Rankine sources. The advantage of this formulation is that Rankine sources behave much better than transient Green functions near the body and free surface juncture, and that the matching surface can be selected to guarantee good numerical behavior of the transient Green functions. The transient Green functions satisfy both the linearized free-surface boundary condition and the radiation condition, allowing the matching surface to be placed fairly close to the body. This numerical scheme has resulted in robust motion and load predictions for hull forms with non-wall-sided geometries. 2

~ ~: /Sf Figure 2: Mixed Source Formulation Non-Pressure Forces In order to calculate the time-domain six-degree-of- freedom coupled motions for any ship heading and speed, LAMP also includes models for non-pressure forces including viscous roll damping, propeller thrust, bilge keels, rudder and anti-rolling fins, mooring cables, and other systems. For oblique-sea cases, a PID (Proportional, Integral, and Derivative) course-keeping rudder control algorithm and a rudder servo model are implemented. Because of the time- domain approach, these non-pressure force models can include arbitrary nonlinear dependency on the motions, etc. Adjustable viscous roll-damping models are available that allow the roll damping to be "tuned" to match experimental values by simulating roll decay tests. Equations of Motion Once the hydrodynamic and non-pressure forces have been computed, the general 6-DOF equations of motion are solved in the time domain by either a fourth-order Runge-Kutta algorithm or a predictor- corrector scheme. Since the forces on the right-hand side of the equations of motion include the instantaneous added mass, an estimated added-mass term is added to both sides of the equations of motion to achieve numerical stability. Sectional Loads In addition to motions, LAMP calculates the time- domain wave-induced main girder loads, including the vertical and lateral shear forces and bending moments, torsional moments, and compression forces, at any cross-section along the length of the ship. Structural loads can be computed using rigid- body or finite-element beam models and can include the whipping responses to bottom or flare slams as well as wave induced loads (Weems, et al. 1998~. Interface to Green-water Calculation At each time step, LAMP uses the ship rigid body motion, the incident wave definition, and the hull pressure distribution to compute the relative motion of the edge of the deck to the wave surface. The hull pressure is used to predict the disturbance wave or 'pile-up" of the free surface due to the presence of the ship. This relative wave height (or deck exceedance) and its relative flow velocity are passed to the green-water-on-deck calculation module in order to define suitable inflow and outflow boundary conditions. The ship rigid body motion, velocity, and acceleration vectors are also passed in order to define the tilt of the deck and inertial terms in the green- water-on-deck equations. Based on these data, the green-water-on- deck calculation is then advanced to the current LAMP time calculation. The deck pressure and edge forces due to green water are passed back to LAMP where they are integrated and added to the right hand side of the equations of motion as well as being used in the sectional-load calculations. GREEN-WATER FORMULATION Conservation Of Mass Given a control volume CV and the corresponding control surface CS enveloping it, Reynold's Transport Theorem can be used to express the principle of conservation of mass as do at ipdVol+ J.p(vren)ds=0 (~1~) where S is the surface of CS, Sol represents the volume of CV, p is the fluid density, t is time, m is the mass of fluid inside CV, vr is the flow velocity on CS, and n is the normal-to-S unit vector given point wise on S [n = Ox, y, z)]. The first step to solving the shallow-water problem with the present strategy is to divide the computational domain into a set of vertical hexahedrals elements (close-volume elements with quadrilateral faces), which are contiguously connected, as shown in Figure 3.a. 3

~ ——~ —it i f~% x -A—h~ ~ ~ `~ (a) d, / ce ~d3 , , d2 (b) ~ 1 Figure 3: Finite volume element e with adjacent elements qs. The subscript s characterizes each of the four sides of element e. The corresponding numbering convention for s is given in part b, where the characteristic area Ce for a generic element e is shown, along with each side do. As seen in this figure, each element e has a fluid elevation he' measured from the geometric center of the base of the corresponding hexahedral element to its top. Also, for each element, a characteristic area Ce is defined, which is the projection of the base area of element e onto a surface normal to the z axis, which contains the geometric center of the hexahedral's base. The vertical axis z is set to be parallel to the acceleration of gravity. Four lateral surfaces As define each side s of each element e (s = 1, 2, 3, 4), which remain always vertical. As shown in Figure 3.a, the subscript qs denotes the adjacent element q to side s of a given element e. If an element had a triangular base, one of its four side surfaces As would be collapsed to a line, a situation that is perfectly acceptable with the present method. The normal vectors to the lateral areas, n, are always normal to the gravity vector. The flow velocity measured on each lateral surface As is represented by the vectors vr, which are always parallel to the undisturbed water level. If the principle stated in equation (1) is applied to a generic element e, conservation of mass can be expressed as: Ha [pCh]e+[p~,vr5 ens As ~ O (2, where vr is the average normal velocity to each lateral surface As of element e. Each element has a characteristic flow velocity Ye, with two horizontal components vx and vy. As stated above, the platform for the shallow-water occurrence is allowed to move with six degrees of freedom following the deck of a ship. Thus, for each element, the base horizontal surface Ce will typically be a function of time. Regarding the differentials as finite differences and taking into account the incompressibility condition, equation (2) can then be written as: [h/\C+C/\h]e+ [~(Vrx;vry~s·(nx;ny)s hs ds ~ —O (3) s=1 e where the bars account for average values between time step k and k -1, and where do are the four sides s corresponding to the horizontal surface Ce (see Figure 3.b). The subscripts x and y indicate vector components parallel to Ce in the corresponding x and y directions (see Figure 3.a). Expanding the finite differences, equation (3) can be written as: [ - k+1 k+1 —k+1 ( k+1 k)1 t[~( - — t+1 (— — Jk+1 h—k+1 dk+l] ~ o (4) where the superscript k is the time step counter, and /\Ck+l=Ck+l-Ck. Rearranging terms and after some numerical treatment, equation (4) can be written in the compact form Meje hekj+1 +~Mejq hq =W) (5) s=1 where 4

M J (~`C kj_~ +1 + C kj_~ +1 1 (6) [ at [ v c h]+ ~ Vs (VrS · nS ) As ~ = (13) Mj ~tQkj,+1 (7) _ |yz (n dS)- | b dVol eqS 8 eqS W' hi' Ckj l+1 6kj_l-1 d At fit ~ (Qk,-l +1 hki-l +1 + ~ Qk~~l +1 hkj-l +1 ) 4 ~ (Qeqs hqs + Qees he ) s= where Beet =(Vx;vy)e.(nx;ny)e des Qeq5 = (VX; vy )q ~ (nx; ny )e des ~eqS 3 42 Ye ~eg,~ 61leqs (9) (10) (1 1) The variable ~e iS the vertical acceleration of element e (which is described in the next section), whereas /~qS is the difference in fluid level between element e and element qS. The superscript j is an iterative counter; an iterative process is needed to solve the algebraic equations defined by (5) since the coefficients defined by (7) and (8) are a function of the elevations h, and the other two sets of unknowns given by the vectors v. Conservation Of Momentum CSe C ve where ~ is the vertical acceleration given by ~ = g + a~ (14) Thus the vertical acceleration ~ includes the acceleration of gravity g and the vertical acceleration az induced by the vertical motion of the platform considered. The body accelerations represented by b are the horizontal accelerations (normal to the gravity vector) induced by the motion of the platform combined with the effect of the local slope of the platform ~nz (nX;ny) . Solving the integrals in equation (13), they yield [Ch3V+vh3C+vCah +~Vs (vr, ·ns)As~ = (15) _ [ r ~ h2 d n + b C h] Regarding the differentials as finite differences, equation (15) can be written as: r—- LC h ~v+v h /`C+ v C Ah +^t~Vs (Vr·nS) h5d55 _ Given a control volume CV, Reynold's Transport s=~ s (16) Theorem applied to the conservation of momentum principle yields the following formulation: dt ~t |vpdVol+ |pv~vr~n~dS (12) CS CV t- 2 ~ h5 ds nS-bC h &| s=] where, as before, bars account for average values = Jp(-nds)- JpbdKol between time step h end k-1. Expanding the finite differences, this expression can take the following compact form: where b is a horizontal acceleration vector (described below), and p is pressure. If the same control volume defined for the conservation of mass formulation is considered for each element, and taking into account the incompressibility assumption, equation (12) can be written as: Reie Vej l +~Rejq vgj =sej (17) s= where s

~ (—kj '+1 h—~j+1 h—kj+1 ~Ckj l+1 (18) +C—kj_,+1 Ah kj+l ) Ri =_ Dsj~ eqs 2 SJ = Ce j he i (Ve —bei-l At) 2 ~ VqS Ds Pe (20) s=l (19) Ds = 2 (he deS Vre ·neS +hq5 deS VrqS ·neS ) (21) The term pk; is the force term due to the hydrostatic pressure, which is given by —kj 1_kj + —2 j+ —kj + _kj i+ Pe = A Ye At ~ (heq5)k ds us (22) s=1 where Max is the maximum value of the vertical acceleration (which includes the acceleration of gravity), and where 427max he is an over- conservative estimate of the maximum velocity that could flow between two adjacent elements. Approximating Ce as Ce = de do, such as, do = Min[Max(d~, d3 ), Max(d2, d4 )] (25) where do, d2, d3, d4 are the sides of Ce as defined in Figure 3.b, then equation (24) can be written as: At < (h k+} _|^he |) ~ h 3/ 2 (26) max max where /\he =hei+i -trek. The fluid elevation hma,` is the maximum value of fluid elevation for the shallow-water assumptions to hold. A maximum value of lkhe can be estimated using the conservation of mass equation (see equation (3)) as Ce |Ahe | < At Max t(42 Ymax he ~ ds he be (27) As in the case of the equations of mass, the system of equations defined by (17) is solved iteratively for v since the coefficients given by equations (18)-(22) are a function of v and h. In the overall scheme, at each iteration j, first the fluid elevations he are solved, next the x components of the vectors ve, followed by the y components of the same vectors. This process is repeated, within the 12 h3/2 same time step, until convergence is achieved. |/`he| < At v Ymax max (28) Minimum Time Step To attain stability in solving the conservation of momentum equations, the main diagonal of the corresponding set of algebraic equations must be predominant at all times. For the formulation proposed in this paper, the following conservative criterion was adopted: |Ree | > Max~Reqs ~ (23) If the term Face in equation (18) is neglected, the previous criterion can be expressed as: Ce he + Ce (he —he ) At Maxt(~/2 Ymax h5 ~d5 he he where similar assumptions as those taken into consideration for equation (24) were adopted. If Ce is defined as Ce = de do, and using the concept of hma<, inequality (27) can be written as de Replacing inequality (28) in (26), the latter yields At < hk+~ dG (29) Ymax max > (24) Evidently, if he can take any value between zero and ham the only stable solution would be the trivial solution in which At = 0. Therefore, to render the calculation possible while at the same time stable, a minimum value of fluid level hmin must be defined such that when the fluid elevation of an element e drops below this value, that element should be considered dry. This minimum value will determine both the speed and the precision of the calculation Equation (29) can be rewritten in terms of hmin as: 6

At ~ hmin 2 ,~h3l2 (30) Ymax max It remains to be determined what would constitute a good estimate for hum. To this end, a suitable estimate for de must first be found. If the computational grid has elements such that their individual Ce areas are comparable, the ratio de_LIN will be chosen as an estimate for the smallest value of de (which will determine the maximum time step to be used). The ratio L I N is computed as: ~ ~ (Ni Nj ) (31) where Li denotes lengths taken along the grid i directions, Lj designates lengths measured along the corresponding j directions (see Figure 3), and where Ni is the i-wise grid dimension, whereas Nj is the j- . . . . wise grlc . c .lmenslon. If hm`= were arbitrarily small, At could take arbitrarily large values. But it is desirable that hmaX be allowed to take the largest possible values to extend as much as possible the stable range for the computations of he. To this end, it is taken as a criterion that hma,` be an order of magnitude larger than de. This criterion was chosen to be de/hmaX-_LINi/2, which ensures that de be an order of magnitude smaller than hm`nc, whereas hm`= is an order of magnitude smaller than L. This can be expressed as where 6= ~ (34) Replacing equations (32) and (33) into inequality (30), the latter yields 2 a/ 2Lrymax which sets the maximum value of At that can be used to ensure a stable solution. The conditions required for this criterion can be summarized as: 1) The individual area of each element of the computational grid must be of the same order of magnitude. 2) The maximum fluid elevation hmaX = Ll~ should never be exceeded. It can also be proven that the stability condition of inequality (35) deduced for the conservation-of-momentum is sufficient to satisfy the stability of the conservation-of-mass equations as well. Semi-Empirical Model hma~: = ~ L = '~ L (32) 61 621, i L (33) Fe=p:CH a, +VH al <37' al ]e As mentioned in the introduction, the green-water calculation has been implemented as a multi-level approach. In addition to the f~nite-volume method described so far, an alternative statistics-based calculation was developed. With this approach, an expedite though coarse prediction of green-water effects can be produced. In relation to the finite volume approach, this alternative method provides a means to predict green-water forces for those elements in which the fluid level may exceed hen,, for which shallow-water assumptions cease to be valid. Statistical data can be used to estimate the water-on-deck as a function of freeboard exceedance. With this information, and following a similar approach proposed by Buchner (1995), the forces on deck can roughly be estimated as F - Jp b dVol = a |V P dVol (36) cv cv which for each element can be expressed as where V is the flow velocity on deck, and H is the green-water elevation. Two of the three components of V are the horizontal flow components, whereas the third is given by dH/dt. The two horizontal components of V and the elevation H are computed by interpolation from the flow velocity and water elevation on the boundary of the computational grid. The relationship between H and the water level on the grid boundary is obtained from statistical data. Figure 4 illustrates such a relationship as reported by Zhouetal. (1999~. 7

15 ,.;. ,.;,0 so .= ~ 10 cad ~ s 3 ~ o . ,~ ~ 0 o ~ Jeff .—~ 1 1 1 ~ s 10 15 Exceedance of freeboard "meters] Figure 4: Relation of water height on deck boundary and freeboard exceedance (Zhou et al., 1999~. VALIDATION Part of the present effort has been directed toward establishing the accuracy of the green-water model in relation to both theoretical and experimental data. To this end, as a first step, a linear analytical solution to the shallow-water problem was considered. One example of this solution is given by Stoker (1957), who presents the evolution of the level of water behind a dam after the dam is suddenly removed. In one of the examples given by Stoker, a dam whose initial water height is 10 meters is removed, and the corresponding water profile is computed one second after the event. The same case was modeled with the present numerical model, using a finite-volume grid of 41 elements in a row. Both the analytical and numerical solutions are plotted in Figure 5, where a good agreement between the solutions can be observed. The main difference between both solutions appears in the smoother backpropagation of the surface perturbation computed by the non-linear approach. 12 10 —8 ~ 6 to 4 con 2 o ! Liut et al.: Non-Linear Sol. ~ x Stocker: Linear Sol. -90 -60 -30 0 Distance from dam (m) , __~ : == t = 1 30 60 90 Figure 5: Comparison of present numerical model with the linear analytical solution of a breaking dam. As mentioned above, comparisons have also been made with experimental results. In this regard, some experiments produced by Zhou et al. (1999) were considered. One of those experiments is depicted in Figure 6. Part a of that figure is a schematic of the experimental set up. A flap is located at a certain distance from the back of the tank, separating a region with water from an initially empty region. Suddenly the flap is removed, and the water is allowed to flow freely until hitting an impact plate where the water bounces back. Different instances of the water distribution can be seen in Figure 7. The water level is constantly recorded at a probing point 1.525 meters ahead of the removable flap. Some of the experiments done with this tank setup were modeled with the numerical approach presented in this paper. In Figure 6, one of these experiments (experiment Nr. 4487001) is described. A 21- element finite-volume grid was used for this calculation. Figure 6.b compares the experimental outcome, the current numerical calculation, and a numerical calculation done by Zhou et al. (1999) using a shallow-water model based on Glimm's method (Grimm, 1965). As it can be observed in the corresponding plots, a good agreement was obtained with the experiment. r Impact Flow | plate Area , ' ~ 1.525 m ~ . ~ Probe Flap Reservoir Area 0.6 _ 0.4 - _ _ _ l 0) 0.3- ___ I l 1 <~' 0.2- 0.1 O - --1--- l.OOOm ~ 1.200m 1 to L L i ~ _ · ' 1 1 —Zhou et al -- Simulation l l ~ Zhou et al -- Experiment l l 0 5 1 --- ut et al. - Simulation ~ i L — | L ;N ~ I /:v] I I ,. 2.020 m (a) r L 1 1 1 1 1 1 I I ~ I 1 1 1 1 1 1 1 0 1 2 3 4 5 Time (sec.) (b) 6 7 Figure 6: Comparison of present numerical model with the linear analytical solution of a breaking dam. 8

(I z Ins - ·1 _os~ to (I Figure 7: Different stages of water distribution after removing the flap of the experiment described in Figure 6, as computed by the present finite-volume scheme. LAMP BASED CALCULATIONS As mentioned in the introduction, the present approach was developed to provide green-water-on- deck calculations within the LAMP ship-motion and load-calculation environment. The approach is fully integrated in the LAMP System, and several studies have been carried out in which green-water calculations for different ships in a varied range of sea conditions have yielded satisfactory results. At every time step, LAMP calculates the relative motion at the deck edge and passes it, along with the rigid-body ship motions, to the green-water calculation supervisor. This supervisor updates the green-water calculation and returns the pressure distribution over the deck plus any boundary forces. The integrated pressure and boundary forces are added to the hydrodynamic and other forces on the right hand side of the equations of motions, which are then integrated to get the ship new position. The deck pressure and boundary forces are also included in LAMP's loads calculations. When setting up an integrated LAMP-green- water problem, a variety of boundary conditions can be defined over the green-water computational domain. For example in Figure 8, a superstructure comprising a central deckhouse and a full width bulkhead are modeled using "infinitely high" walls (i.e. always taller than the green-water maximum level). Also, bulwarks have been modeled on both sides of the ship. These bulwarks were set to have a rather small height above the deck. The ship was set to move in head-storm conditions. In the second and third slides (slides b and c) it can be appreciated how 9

the water pours over the rim of the bow bulwarks. In the last three slides (slides c-e), it can also be seen that the boundary conditions defining the superstructure effectively isolate the interior of the superstructure from the incoming water. ~2 0-04 (a) 0.04 y MY ~0 (C) 0.04 y 2 MY ~2 0 ~ n no , . 0.04,` ~ 0.02 z MY non ~-0.02 0 —.02 0-04 (e) 0.04 y Figure 8: Shallow-water on deck with different boundary conditions. Head-seas conditions. 2 The effect of shallow water on the motion of ships has been an important part of the research done with this numerical tool. In Figure 9 parts a and b, the effect of green-water on deck is shown for a CG47 cruiser sailing on regular and head-storm seas, respectively. The wave height of the regular-sea conditions is 14.32 meters, whereas the significant wave height of the head-storm conditions is 9.75 meters. In both cases the ship speed is 10 knots. In this figure, the results from the current finite-volume model are shown together with the results from the green-water semi-empirical model, and the effect that a mere hydrostatic force would produce if the water completely covered the deck. It can be observed that all three models have the tendency to reduce the vertical bending moment. This is the expected behavior since the net downward forces due to the 10

green water partially offset the large hydrostatic restoring moment generated by the bow submergence. 1 .0E-04 8.0E-05 a) 6.0E-05 4.0E-05 Q) 2.0E-05 ._ I) O.OE+OO m -ME 05 ~ ~.OE-05 Cat -6.0E-05 a) > -8.0E-05 -1 .OE-O' O _ No Green Water Effects Semi-empirical —Finite-volume —Hydrostatic (a) 8.0E-05 - 6.0E-05- ~ 4.0E-05- o 2.0E-05- . - O.OE+OO- a' -2.0E-05 - m <15 ~.OE-05- : -6.0E-05 - .OE-05- 1 no no - No Green Water Effects Semi-empirical Finite-volume Hydrostatic (b) Time ~ Figure 9: Shallow-water effect on the bending moment of a CG47 cruiser ship (a) in regular sea and (b) in a head storm sea. The bending moment is non- dimensionalized by p g L4 p . For these cases, the semi-empirical model was run ignoring the flow vertical accelerations aHlOt. This conservative restriction was used for comparisons with the hydrostatic model. Thus the similar magnitude of the maxima these two models exhibit. In Figure 10 the effect of the green-water on deck for the pitch motion is visualized. Though the impact of the green water seems to be very small on the ship pitch motion, the plots seem to indicate that the green water does induce a lag and an increase in the pitch amplitude, though both very small. This is consistent with the increase of the pitch moment of inertia of the ship induced by the extra water on deck. 2.0E-O, - 1.5E-01 1.0E-01 _% ~ 5.0E-02 - ° O.OE+OO Q -5.0E-02 -1.0E-01 -1.5E-01 0.0 No Green Water Effects Semi-empirical —Finit~volume —Hydrostatic. 5.0 1~.0 15.0 Time (see) ] f ~~ V Figure 10: Shallow-water effect on the pitch motion 15 of a CG47 cruiser ship in a head storm sea. Finally, Figures 11 and 12 show results from green-water-on-deck calculations for a large (LBP=30O meters) modern containership. Figure 11 part a depicts the ship motion, incident wave surface, and computed green-water elevations for the ship in large, regular head seas. The wave height is 11.0 meters, the wavelength is 300 meters, and the ship speed is 20 knots. The green-water surface is above the incident wave surface because of the significant predicted "pile-up" created by the ship's extreme bow flare. Part b shows results for the same wave but at a bow-quartering condition. The shape of the green-water surface looks consistent with the corresponding sea condition for each case. Note the piling up of water at the front of the containers, which is treated as a wall in the green-water calculation. 11

(a) (b) Part c of the same figure depicts a head-sea situation in which the combination of a large seaway, the extensive bow and stern flare, and the ship's dynamic properties result in a very large roll response through a phenomena known as parametric rolling. This takes place in large waves when the principle wave encounter period is approximately one half of the roll natural period. For this case, the wave encounter period is 10.4 seconds, which corresponds to a wave period of 13.9 seconds, whereas the roll natural period is 22 seconds. The nonlinear coupling between the heave, pitch, and roll degrees of freedom channels a kinetic energy transfer from the pitch to the roll mode. An advantage of the current approach is that it includes a direct calculation of the dynamic pressure distribution over the deck. Figure 12 shows the instantaneous deck pressure at one time step of the regular head sea case shown in part a above. The calculation of the deck pressure allows the approach to be used for evaluating deck and hatch cover loads directly or by using the pressure to create load data sets for detailed structural analysis. .. _ ~ . ~ ~ 9.0909 8.4848 7.8788 7.2727 6.6667 6.0606 t 5.4545 4.8485 4.2424 3.6364 3.0303 2.4242 1.8182 1.2121 0.6061 0.0000 (c) Figure 11: Water on deck for modern container ship: (a) Large regular head seas (b) Large regular oblique seas (c) Parametric rolling condition Figure 12: Pressure distribution on deck due to green water. The pressure units are kN/m2. 12

FINAL REMARKS The numerical model presented in this paper has proven an effective tool to compute green-water occurrences with shallow-water assumptions, for static or moving platforms. The method has been validated with theoretical solutions and experimental results. Several applications have been tested using the LAMP System as a motion and hydrodynamic platform. The outcome of these applications indicates that the technique is very suitable for computing shallow-water-on-deck situations for a variety of sea conditions, ship types and motions, and different deck layouts. The calculation is robust and accurate, with a reasonable computational effort. The approach demands a fairly simple grid structure, and offers a broad flexibility in defining different types of boundary conditions. ACKNOWLEDGEMENTS The development of the LAMP System has been supported by the U.S. Navy, the Defense Advanced Research Projects Agency (DARPA), the U.S. Coast Guard, the American Bureau of Shipping (ABS), and SAIC. The green-water development has been supported by the Office of Naval Research (ONR) under program manager Dr. Patrick Purtell, by the US Coast Guard under the Program manager Mr. Peter Minnick, and by ABS under program manager Dr. Yung Shin. REFERNCES Buchner, B., "On the Impact of Green Water Loading on Ship and Offshore Unit Design,'' in Proceedings of the Sixth Symposium on Practical Design of Ships and Mobile Units, September 17-22. 1995. on.1.430- 1.443. , ~~ Glimm, J., "Solutions in the Large for Nonlinear Hyperbolic Systems of Equations," Communications on Pure and Applied Mathematics, Vol. 18, 1965, pp.697-715. Lin, W.M., and Yue, D.K.P., "Numerical Solutions for Large-Amplitude Ship Motions in the Time- Domain," in Proceedings of the Eighteenth Symposium of Naval Hydrodynamics, The University of Michigan, U.S.A, 1990. Lin, W.M., and Yue, D.K.P., "Time-Domain Analysis for Floating Bodies in Mild-Slope Waves of Large Amplitude," in Proceedings of the Eighth International Workshop on Water Waves and Floating Bodies, Newfoundland, Canada, 1993. Lin, W.M., Meinhold, M., Salvesen, N., and Yue, D.K.P., "Large-Amplitude Ship Motions and Wave Loads for Ship Design," in Proceedings of the Twentieth Symposium of Naval Hydrodynamics The University of California, Santa Barbara, U.S.A., 1994. Lin, W.M., Zhang, S., Weems, K., and Yue, D.K.P., "A Mixed Source Formulation for Nonlinear Ship- Motion and Wave-Load Simulations," in Proceedings of the Seventh International Conference on Numerical Ship Hydrodynamics, Nantes, France, 1999. Stoker, J.J., "Water Waves," Pure and Applied Mathematics, Vol. 9, The Mathematical Theory and Applications, Institute of Mathematical Sciences, New York University, U. S. A., 1 95 7, pp. 29 1-3 14. Weems, K., Zhang, S., Lin, W.M., Shin, Y.S, and Bennett, J., "Structural Dynamic Loadings Due to Impact and Whipping," Proceedings of the Seventh International Symposium on Practical Design of Ships and Mobile Units, The Hague, The Netherlands, 1998. Zhou, Z.Q., De Kat, J.O., and Buchner, B. "A Nonlinear 3-D Approach to Simulate Green Water Dynamics on Deck", in Proceedings of 7th Numerical Simulation Hydrodynamics, 1999. 13

DISCUSSION Dr. Ole A. Hermundstad MARINTEK, Norway It is interesting to see a numerical method for green water calculations being applied to ships with forward speed. The shallow water formulation is similar to that applied in (1) and (2), but the present method allows for triangular elements, which give some more flexibility in the geometric modelling. The authors use a 3D body-nonlinear method for the global ship motion problem, while the quasi- 3D shallow water formulation is used for the local green water domain. However, the paper does not precisely describe the interaction between the local and global domains along their common border, namely the deck edge. The water flow on the deck is strongly dependent on the conditions at the boundaries. It is not clear from the paper if the method assumes a nonzero horizontal flow across the bulwark or not. From the transverse waves on the green water surface in Figure 8 it seems that the horizontal velocity is zero and that the boundary conditions are similar to those of a dam-breaking problem with time- dependent reservoir height. If this is the case the method will generally underestimate the velocity of the flow on the deck and be unconservative if it were to form input for load-calculations on superstructure and deck-mounted equipment. If the authors really use a nonzero horizontal velocity it would be interesting to know how it is obtained from the LAMP results. Another issue that requires some more attention is the flow at obstacles, such as the walls of the superstructure. The shallow water method will give a pile-up of water when the flow meets a wall, but one of the fundamental assumptions of the formulation is obviously violated in that case. I.e. the vertical velocity component is no longer small compared to the horizontal components. Another model is therefore needed to predict slamming pressures on the wall. The method, as presented in the paper, may be used to predict pressures on decks and hatch covers as well as hull girder loads caused by green water. However, it seems to need further development before it can be used to calculate slamming loads on superstructure and containers. (1) Hellan, 0., Hermundstad, O.A. and Stansberg, C.T. "Designing for wave impact on bow and deck structures" Proceedings of the 1 1th ISOPE Conference, Stavanger, Norway, 2001. (2) Stansberg, C.T., Hellan, 0., Hoff, J.R. and Moe, V. "Green sea and water impact on FPSO: Numerical predictions validated against model tests". Proceedings of the 21St OMAE Conference, Oslo, Norway, 2002. AUTHORS' REPLY We would like to thank Dr. Hermundstad for his comments. The papers quoted in the discussion have a valuable contribution to the green water study. In our current model, a fully nonlinear set of differential equations describing the shallow- water phenomena is solved, with a special care for different types of boundary conditions and obstacles. The methodology uses a finite- volume approach that solves the equations in the time domain. Special cares were taken to satisfy numerical stability requirements. From the content of the aforementioned papers, it seems that the methodologies used in their work involve elaborate semi-empirical strategies that do not share much in common with our physics- based approach. The green water model does take information of the environment as computed by LAMP. To that effect, LAMP provides the velocity and wave elevation with respect to the deck as inputs to the green-water element at the deck edge. This information is passed to the green-water solver. The type of information used depends on the types of cases, such as the water flowing in or out of the deck and its being higher or lower than the water deck elevation computed fin LAMP on the deck boundary. Regarding the accuracy of the method when green water moves over obstacles, it is true that if an obstacle were relatively high with respect to the local water elevation (on a given point of the green-water grid) the substantial vertical component of the fluid velocity at this location would be ignored. This limitation is based on the model utilized, which solves for the shallow- water assumptions. To capture the vertical flow components even a jet, a fully 3D calculation would be needed. A 3D capability is currently under development at SAIC.

In principle, we do not see any particular restriction on computing impact loads, provided the shallow-water assumptions are still valid. For cases involving large vertical flow velocity or formation of jet, the current approach cannot be used to calculate the impact load accurately. DISCUSSION Allen Engle, Naval Surface Warfare Center Carderock, USA A green water prediction capability is a welcome addition to the LAMP suite of programs. The need to include such affects within a motions and loads prediction methodology are brought out in the paper. However, I might add that water on deck is an issue not only for the conditions mentioned, but must also be considered as part of a rigorous treatment of the capsize problem, where the combination of a ship operating in stern quartering seas with water shipping on deck can exacerbate an already precarious situation. With this in mind I have the following questions for the authors, 1. Does the current formulation allow for the user to easily simulate the shipping of water on deck for all headings? 2. At the 22nd Symposium on Naval Hydrodynamics, a paper presented by Huang et. al. incorporated a correction factor to account for viscous and other effects not explicitly included as part of the water shipping on deck problem. For the treatment at hand it is stated that viscous effects are ignored. Could the authors comment on the degree to which the physics of the water shipping problem dictates the need to account for viscous effects? 3. Have the authors performed any convergence studies to identify an appropriate level of grid size for simulating the green water problem? and are there any potential problems in using different grid sizes in defining the hull and the deck regions? AUTHORS' REPLY We would like to thank Mr. Engle for his valuable discussions also. 1. Yes, it does! The current green water approach allows for an easy definition of all types of boundary conditions in addition to obstacles inside the grid. LAMP takes care of automatically providing the needed data for different ship headings and sea conditions. Nevertheless, the green-water model implementation can work in a standalone mode, in which case, through an appropriate input file, the user can specify any arbitrary motion of the grid and boundary characteristics, both in terms of obstacles and water elevations, and regarding incoming or exiting water flow. 2. The model currently has a methodology to estimate viscous effects using a Von Karman type of analysis, which will be the subject of fixture research and validation. 3. Convergence and stability is one of the major accomplishments of this work. From the beginning of this development, effort was directed to study the stability conditions of the numerical method. As a result, the program automatically computes the (maximum) time step below which numerical stability is ensured. It can also automatically generate its own grid satisfying convergence requirements. Research was also conducted to ensure the stability of the solution beyond shallow-water assumptions. The reason for this is that whereas for a given ship and sea conditions it would be expected that the resulting ship motion and interaction with the environment (computed by LAMP) would generate shallow-water conditions during most part of the calculation, there could be particular instances (during the calculation) when the motion of the ship or some unexpectedly high incidence waves could induce water elevations on deck that could go beyond shallow-water assumptions. In such cases the solution will still remain stable, though at the expected expense of accuracy. Note that a single run may encounter several green-water occurrences. Even if in some particular case, shallow-water assumptions may be compromised, the solution would always remain stable and it will still be accurate for all the other cases. This model has been involved in the study of the influence of water on deck on ship motions. This is already available in the work done by Vadim et al (1) quoted below/ (1) Vadim, B.L., Liut, D.A., Weems, K.M., Young, S.Y., "Non-Linear Ship-Roll Simulation with Water-On-Deck," Proceedings of the 2002 Stability Workshop, Web Institute, New York, Glen Cover, October 13-16, 2002.