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OCR for page 126
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 126 Basic Studies of Water on Deck M.Greco1, O.M.Faltinsen 1, M.Landrini2 of Marine Hydrodynamics—NTNU, Trondheim—Norway. 2 INSEAN, The Italian Ship Model Basin, Roma 1Department —Italy. ABSTRACT Extreme wave-body interactions may cause shipping and flowing of water on the main deck of ships (water on deck). In this paper, the role taken by some of the main geometric and kinematic parameters involved in the water on deck is carried out by using an approximate hydrodynamic model. In particular, the unsteady interaction between free surface and ship is analyzed by solving the inviscid two-dimensional fully nonlinear problem numerically. Both water on deck resembling dam breaking as well as due to plunging waves are investigated. INTRODUCTION In rough sea conditions, both moored vessels (such as a Floating Production Storage and Offloading Unit, FPSO) and ships in transit can suffer shipping of water on the deck. When a sufficient amount of water comes onto the deck, a flow with increasing velocity develops, possibly hitting obstacles on its way. Water impacting against the deck and superstructures may cause both high pressures in confined regions and contribute to global ship loads. Localized structural damages as well as excitation of global response of the ship are expected. The importance of hydroelasticity must then be assessed. Moreover, the fluid motion onto the deck may affect roll stability of smaller ships and cause capsizing. Different incidents occurred in the latest years to FPSO units that motivated experimental investigations and suggested some modifications of design rules. An overview of the most important ones and of the subsequent requirements of the Norwegian Petroleum Directorate are given by Ersdal & Kvitrud (2000). However, the numerous physical aspects determining the phenomenon make it difficult to clearly identify the design parameters relevant for the occurrence and severity of water on deck, and for its consequences to the ship. The effect of geometric parameters characterizing a ship bow is far from being clarified. Sometimes, it is not even clear whether they enhance or reduce the deck wetness. As an example, O'Dea & Walden (1984, experiments in regular waves) reported that a larger bow flare angle reduces the deck wetness, while Lloyd et al (1985, experiments in irregular waves) observed more frequent freeboard exceedances and deck wetness for more heavily flared bows. On this ground, fundamental investigations are necessary to improve this lack of knowledge and to develop numerical tools of practical use. The conventional way of estimating water on deck is to combine a probabilistic analysis (Ochi 1964) with a linear hydrodynamic analysis. It implies that the above water hull form is not included in the hydrodynamic analysis. The important hydrodynamic variable is the linear relative vertical motion between the ship and the water. Often only the incident wave and not the local wave accounting for the presence of the ship is used in this context. An effective freeboard is sometimes introduced for a ship with forward speed. This accounts empirically for the steady wave profile and the sinkage of the ship. Details of the flow near the ship require a local quantitative analysis of the specific conditions of interest. Maruo & Song (1994) studied the shipping of the water for high speed vessels by using a Slender Body Theory. This may also have relevance for slender ship bows at moderate forward speed. Buchner & Cozijn (1997) analyzed the bow deck wetness for moored ships, assuming a two-dimensional problem in the longitudinal ship direction. They presented both numerical simulations and experiments for a simple prototype problem but no comparison between simulations and measurements was presented. In head sea conditions, the most severe water on the authoritative version for attribution.
OCR for page 127
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 127 deck events are concentrated in the bow region. In these cases, after the water exceeds the freeboard, Buchner (1995) observed a marked similarity between the flow of the shipped water and the one generated after the breaking of a dam. Consistently, some authors studied the motion of the shipped water along the deck by shallow water models. The reliability of this type of approach is dependent on how the initial conditions as well as the inflow boundary conditions are determined. A sensitivity analysis in terms of the inflow velocities has been carried out by Mizogushi (1989) by comparing numerical results and experimental data for the S-175 container ship. The shallow water equations have been solved for the three-dimensional problem, using experimental results for the water height at the inflow boundary. In the simulation, the ship motion is not taken into account. The author concluded that the flow interactions and the efflux occurring between the deck area and the outer region represent important items in the water on deck phenomena. In this paper an attempt is made to study the fundamental aspects of deck wetness at the bow region of a FPSO unit in head sea conditions. It implies that forward speed effects are not investigated. The problem is idealized by considering the two-dimensional flow in the longitudinal plane of the ship and solving numerically the resulting fully nonlinear unsteady problem. After a preliminary validation, the effects of several physical parameters affecting type and severity of water on deck are discussed. The flow along the deck is also studied, with emphasis on pressures and impact loads with superstructures. ASSUMPTIONS AND MODELING Many physical aspects determine the considered problem. Wave-ship interactions (cf. figure 1.A) modify significantly the wave pattern with respect to that of the incident waves. This is related to local effects and to wave reflection, which in rough sea are highly nonlinear phenomena. Ship motion, figure 1.B, can either enhance or prevent the deck wetness occurrence. In this paper the heavy water on deck is analyzed by fully retaining the nonlinearities associated with body and free surface motions. The focus is on deck wetness at the bow region for a FPSO unit in head sea conditions. A blunt ship bow is assumed. Therefore, the problem is simplified by considering the two-dimensional flow in the longitudinal plane of the ship. Clearly, figure 1.C, three-dimensional effects are relevant, though less than for cases with forward speed. In the latter, the steady wave pattern is characterized by an increase of the water level due to a local disturbance and bow waves generation which by themselves decrease the effective freeboard, non uniformly along the longitudinal ship direction (Tulin & Wu 1996). Ship sinkage due to forward speed may also matter. However, a two-dimensional analysis can give important insights of the phenomenon and useful information about the effects of the parameters involved. Figure 1: Some physical aspects involved: Wave-body interaction (A), body motions (B), three-dimensional effects (C). The water motion is believed unaffected by the viscosity and a potential flow model is adopted. Actually, the edge of the deck could be a source of separation and vortex shedding, which are not presently modeled. Surface tension effects are neglected because of the relatively large spatial scales. Finally, the structural deformations are not considered and the body is assumed perfectly rigid. In the following, we consider the prototype two-dimensional problem sketched in figure 2. A frame of reference fixed with respect to the fluid at great depth is considered. The fluid domain Ω is bounded by the free surface, ∂ ΩFS, the instantaneous wetted portion of the ship, ∂ ΩBO, and a control surface ∂ Ω∞ in the the authoritative version for attribution.
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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 128 far field. Deep water conditions are considered, with incident Stokes waves approaching the body from left to right. Figure 2: Sketch of the numerical 2-D problem. The problem is governed by the Laplace equation for the velocity potential (1) is a point in the fluid domain and t is the time. Along the free surface and the impermeable boundaries, the where following kinematic constraint applies (2) where is the displacement velocity of the surface, and is the unit normal vector, pointing out of the fluid domain. The body motion is prescribed a priori. On the free surface, the dynamic condition requires constant pressure. Upon choosing a Lagrangian description of the free surface ∂ ΩFS, the kinematic and dynamic boundary conditions, respectively, read (3) Here pa is the atmospheric pressure, g is acceleration of gravity and ρ is the mass density of the fluid. The first equation satisfies condition (2), while the second follows from the Bernoulli equation. As usual in the mixed Eulerian-Lagrangian method for free surface flows (Longuet-Higgins and Cokelet 1976, Faltinsen 1977), the resulting problem is split into two sequential steps. In the first one, the kinetic problem for the velocity potential, with mixed Dirichlet-Neumann boundary conditions along • ΩFS and • ΩBO, is solved. In the second step, the free surface conditions are stepped forward in time to update geometry and boundary data. The kinetic problem is solved through the Green's second identity (4) where (5) θ is the inner angle at the point on the boundary, and (6) is the fundamental solution of the Laplace equation in two dimensions. The right-hand-side of (4) must be interpreted as a principal value integral when The right vertical and the horizontal portions of the surface ∂ Ω∞ are chosen far enough from the body to give negligible contributions. Along the vertical upstream barrier (left side in the sketch 2), both φ and ∂φ/∂n are specified by a truncated Fourier representation of the Stokes wave in deep water for arbitrary steepness (Bryant 1983). The vertical extent of the barrier is truncated at a suitable large depth, while the horizontal location is chosen far enough (order often the authoritative version for attribution. wavelengths) so that within the time scale of the simulation (at most the order of ten wave periods) disturbances due to wave reflection are small in proximity of the inflow boundary. Residuary (high frequency) effects are removed by using a damping layer technique (Israeli and Orszag 1989) close to the barrier. In particular the dynamic condition in (3) is modified by introducing a damping term proportional to φ−φsto , where φsto is the velocity potential of the Stokes wave. Similarly is done for the vertical component of the velocity in the kinematic condition. The Lagrangian drift of surface points is eliminated through periodic regridding of the upstream region. Invariance of the results have been checked by increasing the upstream length of the domain. Stokes wave conditions have been checked inside the domain Ω without body. For points on the boundary, where only φ (on the free surface) or ∂ φ/∂ n (on the body) are specified, the integral representation (4) provides the relevant integral equations to evaluate the remaining boundary data.
OCR for page 129
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 129 Once this is accomplished, φ and ∇ φ can be evaluated everywhere in Ω. The solution of the integral equations is obtained by using a panel method with piecewise linear shape functions both for the geometry and for the boundary data. The collocation points are taken at the edges of each element, resulting in a continuous distribution of the velocity potential along the free surface. The tangential velocity ∂ φ/∂ τ is simply determined by finite difference operators, while the normal velocity component is obtained from the integral equations. Higher order schemes (i.e. Landrini et al 1999) may lead to numerical difficulties at the body-free surface intersection point and therefore are not adopted here. Clearly, the use of a lower order method requires a finer discretization in region with high curvature of the boundary, or where the thickness of the fluid layer along the body is small. This has been achieved dynamically during the simulation by inserting new points where appropriate. The continuity of the potential is assumed at those points where the free surface meets a solid boundary. Though no rigorous justification is available, this procedure gives convergence of the numerical results under grid refinement (Dommermuth & Yue 1987). Occasionally, when the contact angle becomes too small, numerical problems still may occur and the jet-like flow is partially cut (Zhao & Faltinsen 1993). A standard Runge-Kutta fourth order algorithm is adopted to step forward in time the free surface evolution equations. This requires the solution of four integral equations each physical time step. Though less demanding schemes are conceivable, we preferred this for the simplicity in changing dynamically the time step. We found this crucial to keep under control the accuracy of the solution during the development of jet flows, impacts, and breaking waves. PHYSICAL INVESTIGATIONS Water on deck is a complex phenomenon and it is worth to identify separate stages of evolution, even if they are strongly connected with each other. In particular we consider: i) the run-up of the water at the bow, ii) the water shipping onto the deck, iii) the subsequent flow developing along the deck, and finally iv) the impact with ship equipments or deck house. Recently Greco et al (2000) studied separately some of these aspects by considering some suitable prototype problems both to achieve a validation of the numerical method, and to gain a first understanding of the phenomenon. For example, the study of the interaction of a solitary wave with a plane wall showed that weak nonlinear theories underpredicts the maximum run-up with respect to experiments and nonlinear solution, for wave amplitudes large enough. This suggests a limited applicability of simplified models to predict freeboard exceedance, at least without recurring to some empirical corrections. When the wave elevation exceeds the freeboard, the water can flow over the deck and, very often, the resulting flow field resembles the one after a dam breaking. Hence, we have further validated our model by studying this problem. Numerical results agreed well with small time expansion analytical solutions and experimental data. In the following, we analyze more directly the phenomenon of water on deck occurrence and the flow in case of impact with superstructures. Preliminary studies and validation As preliminary studies and validation of water on deck occurrence we consider the case of a fixed rectangular body under the action of regular waves. This case is used to define the proper treatment of the flow field when the freeboard is exceeded. In particular, we adopted a ‘Kutta' like condition, enforcing the flow to leave tangentially the bow when water reaches the instantaneous freeboard, f, (see figure 3). Once the freeboard is exceeded, the fluid velocity relative to the the ship determines whether the deck will be wetted or the water will be deviated in the opposite direction. Figure 3: ‘Kutta' condition enforced at the edge of the deck. Our numerical solution has been compared with the quasi-two-dimensional experiments by Cozijn (1995). A wavemaker was used to generate regular waves, interacting with a rectangular bottom mounted structure, placed at the opposite end of the tank. The freeboard f was 0.1 m, and the water on deck events were recorded the authoritative version for attribution.
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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 130 by a video camera. Figure 4 shows the comparison between numerical (solid lines) and experimental (circles) free surfaces profiles (for wave height H=0.128 m, frequency ω=5rad/s, experimental sequence coded 5:36:00–19 from the test No. A03). We have chosen the free surface configuration just before the shipping of water (see figure 4.1) and the following seven wave profiles, with a time interval of dt=0.04 s. The global behavior of the free surface is well reproduced by the numerical solution, confirming the efficiency of the adopted model and the limited effect of the sharp corner in the model (as we said we have not modeled the vortex shedding from the edge of the deck). Later in the evolution, the numerical solution predicts a fluid front moving faster than in the experimental case. Nevertheless, the water level along the deck is rather similar for the two results. A possible reason of the differences could be related to surface tension effects, which are not presently modeled. In facts, the thickness of the fluid layer is of order 0.01 m, and the high curvatures call for a more complex description of the dynamics of the contact point (cf. Dussan 1979). This is supported by the observation that the measured shape appears “rounded” and highly curved in proximity of the contact point. It would be relevant comparing experiments with larger scale. Anyway, since the deviation between the two results is strongly localized, it is believed unimportant in terms of effects on a superstructure hit by the water along the deck. Differences in the pressure over the structure are expected in a very small time initially that is unimportant from the structural reaction point of view. Figure 4: Water on deck of a rectangular structure due to incoming regular waves (H=0.128 m, ω=5 rad/s, initial freeboard f=0.1 m). Snapshots of the free surface. Experimental data (circles) are from Cozijn (1995). We now consider a more realistic set of parameters. In particular, we have chosen a FPSO unit in long and steep head sea regular waves (Buchner 1995) to fix the main parameters for our two-dimensional simulation. The draft of the ship is D=17.52 m, while the relative length and freeboard are respectively L/D=14.86 and f/D=0.507. In the experiments, a superstructure is located at a distance ds/D=2.05 from the bow. For simplicity, we have assumed a straight vertical bow and restrained the body motions. The following analysis is for the first water on deck occurrence caused by incident waves with wavelength λ/L=0.75 and height H/λ=0.09. Top plot of figure 5 shows the free surface just after the start of the water shipping and during the later evolution, after the impact with the superstructure. The four bottom plots report the time evolution of water level at locations A–D (shown in the top plot) along the deck and the numerical results are compared with the (three-dimensional) experiments by Buchner (1995). In the experimental set up the water level sensors were located along the centerplane of the ship. In this case, in spite of the three-dimensionality, the comparison indicates satisfactory numerical prediction of the propagation of the water front. It is observed that the scale of this experiment is quite larger than the one used by Cozijn, which supports our conjecture about the role of surface tension in the previous case. In more detail, in location A we observe a strong local effect associated with the first event of water on deck and a the authoritative version for attribution. rather large overprediction of hw with respect to the experiments. We recall that geometrical details of the bow and the actual wave induced body motions hamper the possibility of a finer comparison. The relative difference is smaller for locations B–C. For the two most advanced (C, D), the numerical results underpredict the experimental values. This is somehow consistent with an increasing three-dimensional behavior of the flow developing along the deck which gives additional contributions to the water
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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 131 level through the lateral ship sides. When the number of shipping events increases, the local wave steepness in the bow region decreases and in this case the resulting amount of water wetting the deck reduces, as well as the propagating flow velocity. Figure 5: Water level hw (meters) as a function of time (seconds) at locations A–D along the deck. Numerical results (solid lines) and 3D experiments (dashed lines) by Buchner (1995). twod is the time instant when the shipping starts. Effect of main geometric parameters A simplified parametric analysis of the deck wetness can be made taking the amount of shipped water, Q, as measure of the water on deck severity. Systematic variations both of body geometry and of incoming wave characteristics have been considered. In particular, we have considered the geometric parameters sketched in figure 6, with the draft of the ship as reference length. Finally, the amount of shipped water is made dimensionless by the amount of water Q0 associated with the incoming waves above the mean free surface level over a distance of a wavelength. Figure 6: Sketch of the main geometrical parameters considered. Figure 7: Influence of ship length and stem overhang on the bow deck wetness. At first we consider cases where the body motion is suppressed. We have considered four freeboard-to-wave height ratios, f/H=0.05, 0.24, 0.36 and 0.55, and we have analyzed the influence of the stem overhang angle α and of the length- to-draft ratio L/D of the ship, for H/λ=0.06 and λ/D=6.6. The results are presented in figure 7, where the relative amount of shipped water is plotted versus f/H. As expected Q/Q0 is strongly influenced by the freeboard of the ship. The length L does not affect directly the deck wetness severity, while it will through the length-to-wavelength ratio and then through the the authoritative version for attribution. induced body motions, here not considered. As expected, a positive bow stem overhang reduces the relative amount of ship-ped water due to a larger wave reflection by the ship. However, in the present case, the deck wetness severity does not change dramatically in the two considered cases (α=0°, 45°). This is more evident for larger values of f/H, which are the most interesting from a practical point of view. Figures 8–9 show the influence of steepness, H/λ,
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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 132 and wavelength-to-draft ratio, λ/D, on Q/Q0 by keeping the body parameters fixed. As expected, as H/λ increases, a larger amount of ship-ped water is computed, with an almost linear trend for small f/H. For larger f/H, a marked nonlinear behavior of Q/Q0 is observed as the wave steepness increases. This is natural because Q becomes stronger dependent on the wave crest flow, which will have an increased nonlinear behavior with increased wave steepness. In the second figure, the effect of the wavelength-to-draft ratio is shown for a constant wave steepness and zero stem overhang. The deck wetness severity changes a lot from case to case, though the nonlinearities associated with the incoming waves are the same. In particular, the worst conditions occur for large wavelength-to-draft ratios, for which a weaker wave reflection is observed, and which are also the more interesting from a practical point of view. Figure 8: Influence of nonlinearity of incoming waves on Figure 9: Influence of wavelength-to-draft ratio on the bow the bow deck wetness. deck wetness. Figure 10: History of the relative amount of shipped water Q/Q0. Cases A–E are described in table 1. The previous analysis considered the first water on deck occurrence. We now consider longer evolutions to analyze the history of shipping events. Figure 10 gives Q/Q0 as a function of the time, assuming as time origin the instant twod1 of the first event. A fixed f/H=0.24 is considered for the cases summarized in table 1. For all of them, we observe large changes of Q/Q0 with respect to the first water on deck occurrence. Interestingly, on a longer time-scale, it tends to reach a more defined value with almost the same periodicity as the incoming waves. Clearly, this result is not general because more realistic sea-state conditions are characterized by the interaction with irregular waves. However the results show that if two succeeding waves with nearly the same height and wavelength cause deck wetness, the last one gives the most severe condition. Table 1: Synopsis of cases considered for studying the history of water shipping. α λ/L H/λ case 00 A 0.33 0.064 450 B 0.33 0.064 00 C 0.33 0.095 the authoritative version for attribution. 00 D 0.05 0.095 00 E 0.67 0.095 In more detail, figure 10 shows that the worse deck wetnesses happen for the steeper conditions (cases C and E). The corresponding Q/Q0 tends almost to the same value, confirming the steepness as the most important wave parameter for long waves (in both cases the wavelength is large with respect to the draft). In case D, the steepness is the same but with shorter wavelength,
OCR for page 133
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 133 equal to the draft, and the shipped water is comparable to that computed for a longer less steeper wave (case A). Case B (same parameters as case A but with a=45°), shows a certain effect of the stem overhang in reducing the severity of the deck wetness. Figure 12: Influence of body motion. a) restrained body Figure 11: First four water on deck events for case A (cf. conditions, b) forced heave initially in phase with the tab. 1). For each event, free surface configurations are water at the bow, c) and d) forced heave initially out-of- plotted for the maximum freeboard exceedance (circles) phase with the water at the bow. and for zero flux entering the deck (solid lines). For the case A, figure 11 shows free surface profiles for the first four water on deck events. In particular, in each plot, two configurations are given: the one with maximum freeboard exceedance (circles) and the one with zero flux of water onto the deck (solid lines). In the four cases, the wave pattern in front of the body is not exactly the same because of the complex features of the reflected wave field. In spite of this, the wave forms in the very near field and on the deck attain a more defined pattern, consistently with the almost constant Q/Q0 previously reported. We define, conventionally, the beginning of the water shipping twod as the time just after the freeboard exceedance for which we have a positive inflow onto the deck. Further tlast means the time when the shipping of water on deck ends. The flux of water onto the deck is then zero. With these definitions, we observed that the time scale involved in a “water on deck cycle” does not change markedly the authoritative version for attribution. and remains (40%) of the wave period T. Effect of body motion Body motions play a major role in determining occurrence and severity of water on deck. Here, we did not solve for the wave induced motions of the ship but simply prescribed a priori the motion. In particular, since we are dealing with bow deck
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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 134 wetness in head sea, we have studied the effect of forced heave motion. The effect of the pitch angle both on the outer wave field and on the flow along the deck is neglected. Since the stem overhang angle had a small influence, we could argue that the pitch angle should not be important for shipping of water on the deck. But the pitch angle may have a larger effect for the flow on the deck. Top plot of figure 12 gives the first water on deck occurrence for a studied situation (case E from table 1) where we have considered a freeboard ratio f/H=0.55. In this case the body is constrained and of Q0. For the same parameters, plots b) through d) show the flow when a forced heave motion is excited just at the beginning of the water shipping, according to (7) H being the Heaviside function. The arrows in the plots indicate the direction of the heave motion. In particular, for the last case the shipping starts with a heave initially downwards. Heave is upwards in the end. In plot 12.b the motion is initially in phase with local wave motion. The amplitude-to-wave height ratio A/H is 0.25. The phenomenon appears qualitatively less severe. The amount of shipped water Q is only the 6% of Q0. However this nice situation is unlikely to occur in the case of a FPSO unit for the wave-body parameters we have chosen. Conditions of out- of-phase body motions are more reasonable and can make the water on deck much more severe than in the restrained body case. A heave amplitude A/H=0.25, third plot, increases the amount of shipped water with a factor 1.9 relative to case a), while for an amplitude equal to the one half of incoming waves, fourth plot, the factor becomes 3.2, to reach 6.2 in the case not shown with A/H=1. Occurrence of waves plunging on the deck The flow along the deck resembles the one after a dam breaking in the most common type of green water event. More recent experiments in irregular seas (MARINTEK 2000) showed that water on deck can also occur in the form of waves plunging directly on the deck. In this case, impacts with superstructures are likely to occur. This phenomenon appears like a ‘single' event associated with a very steep, almost breaking, incoming wave, usually with smaller background waves. Actually, we cannot classify this as ‘freak wave' but it is known that instability and modulation of wave groups in open sea can lead to the formation of steep highly energetic waves. Their interaction with structures is a known cause of highly nonlinear force components (Chaplin et al 1997, Welch et al 1999). Similar circumstances in complex combination with ship motions can cause these extreme events. So far, we have not tried to model more complex incoming waves than regular steep waves. In spite of this, we have analyzed some extreme cases to gain some insight, with some emphasis on the effect of body motion. The geometric parameters have been deduced from the MARINTEK experiments (L/D=13.75, f/D=0.8, ds/D=1.0625). Considering the limited role of the stem overhang, we have approximated the bow with a straight vertical wall. We have considered a wave train of long steep (eventually) regular waves with λ/L=1.022 and H/λ=0.095 and focused our attention on the interaction of the body with the leading wave which is characterized by higher steepness and strong tendency to break. This makes our analysis more consistent with the features observed in the experiments. Forced heave motion is excited at a time instant t0 with an amplitude A and a phase β, in the form (8) where T is the wave period. Wave generation starts at t=0, with the upstream section located 5 wavelengths ahead of the bow and the phase angle β is selected to give a sudden vertical displacement of the ship at t=t0. Some of cases studied, and discussed in the following, are summarized in table 2. Table 2: Plunging wave analysis: summary of cases presented. β case f/H A/H t0 /T twod/T a 0.6 0. – – 10.795 −900 b 0.6 0.5 10.626 10.841 −5 0 c 0.6 0.5 10.783 10.844 −100 c1 0.6 0.25 10.783 10.795 −20.5 0 c2 0.6 0.125 10.783 10.790 −110 d 0.6 0.5 10.783 10.783 e 0.5 0. – – 10.777 In figure 13, restrained body conditions are considered, and some free surface configurations are presented. The wave, reaching the bow, is steep and unsymmetric but its tendency to break is reduced during the run-up along the bow. The water shipping starts with already quite large horizontal velocities of the fluid making the phenomenon less similar to the dam break the authoritative version for attribution.
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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 135 ing problem. Though the shallower water conditions on the deck would amplify the original tendency to wave breaking, the fast motion of the wave front has opposite effect and is the main reason why the wave is not breaking before the water impact on the deck house. Figure 14: Plunging wave analysis: case b. Figure 13: Plunging wave analysis: case a. The heave motion largely affects the phenomenon. In the next figures 14–16, a heave with amplitude A/H=0.5 is considered (see table 2 for the other parameters). In the first case, the motion is excited with a phase such that the instantaneous freeboard at t=t0 is higher than the wave elevation at the bow. By ‘instantaneous freeboard' we consider the mean freeboard plus the change in vertical position due to heave. The upwards motion of the bow causes lower trough ahead of the breaking wave and a bow impact occurs. Air entrapment and (probably) a complex two phase flow are expected to occur. Upon neglecting these phenomena and “stretching” our simulation further, we observe that the shipping of water is not particularly severe. The upwards motion of the bow, in facts, limits the increase of the vertical velocity of the fluid after the impact. Figure 15: Plunging wave analysis: case c. Figure 16: Plunging wave analysis: case d. In figures 15 and 16 the heave motion is excited later than in the previous case b. This is done in both situations at the same instant but with different initial phase. This means a different instantaneous freeboard, in particular for the case d the wave elevation at the bow is equal to the instantaneous freeboard. The larger t0 eliminated the bow impact. However other interesting phenomena occurred. In case c, we observe an initial local breaking tendency of water along the deck. But this is prevented by an increase of the horizontal velocity of the contact point between water and deck. The subsequent flow is like the one after a dam breaking. In case d, the amount of shipped water is larger and the upwards motion of the ship results in a wave plunging onto the deck. The three considered situations could have quite different consequences on the ship. If we modify case c by taking a heave amplitude A/H=0.25 and A/H=0.125 but maintaining the authoritative version for attribution.
OCR for page 136
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 136 the same instantaneous freeboard at t=t0 we get the results shown in figures 17 and 18, respectively. As we can observe the water on deck is still quite serious but the consequences are more dangerous for the superstructure than for the bow or the deck. With A/H=0.25, in particular, the faster rate of the water region to become shallower steepens the wave propagating along the deck. A rather thin jet develops. The jet evolves faster than the water-deck contact point and eventually hits the superstructure. After the impact the simulation was continued by a local matching with the similarity solution by Zhang et al (1996) for an infinite asymmetric fluid wedge hitting a wall (see sketch at the top of figure 24). If the heave amplitude is further decreased (case c2, figure 18), the velocity of the wave front becomes larger relative to the plunging jet velocity. This implies that the impact with the superstructure occurs from the deck. The plunging wave hits the water mass rising along the vertical wall after the impact. This causes an air pocket to be formed. The relative velocity between developing plunging jet and the wave front depends on the rising rate of the deck. This has an important influence on the possibility of a plunging breaker hitting the superstructure. Figure 17: Plunging wave analysis: case c1. Figure 18: Plunging wave analysis: case c2. The analysis here considered is not complete and therefore no conclusive statement about the occurrence of the plunging wave event can be given. However, in the cases so far studied, the run-up along the bow eventually caused the more common dam breaking type of event. This fact supports the conjecture that a plunging wave on a deck completely dry is probably due to a “ready-to-break” wave rather than to bow interaction with incoming waves. the authoritative version for attribution. Figure 19: Set up (top) and simulation of the impact with a vertical structure following a dam breaking Impact with obstacles The fine details of the “external” flow field have a limited influence on the impact
OCR for page 137
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 137 because of the small time- and space-scales involved. Therefore, here we study the impact occurring with superstructures simplifying the initial conditions by using the flow generated after a dam breaking. The considered case is sketched in the top plot of figure 19. A reservoir of water with height h and length 2h, closed by a dam, is placed at a distance 3.366 h from a vertical obstacle. For t=0, the dam is suddenly removed and the flow develops along the horizontal deck, fig. 19.a, finally impacting against the vertical wall. The fluid is violenty deviated vertically upwards, fig. 19.b, rising along the wall in the form of a thin jet. At this stage, formation of spray and fragmentation of the free surface may occur. These finer details cannot be handled by the present method, though we believe they are not relevant to compute the structural loads. As time increases, under the reaction of the gravity, the fluid acceleration decreases and the upward velocity in the jet decreases until becomes negative. The motion of the water is reversed in a waterfall, fig. 19.c, overturning in the form of a large wave plunging onto the deck, fig. 19.d. The numerical simulation is eventually stopped due to numerical break-down. During the first stage of the impact against the wall, the flow resembles the one due to a (half) wedge of fluid hitting the structure, with the rest of the flow field roughly unchanged with respect to the case of infinite deck. The gravity plays a minor role since the vertical acceleration of the fluid around the contact point is (5g). In particular, in figure 20, the free surface close to the wall after the impact is shown in comparison with the zero gravity similarity solution by Zhang et al (1996), for an infinite wedge of fluid hitting a flat structure at 90°. The two solutions remain in qualitative agreement even for a non-dimensional time ∆τimp= 0.1338 after the impact. The two plots in fig. 21 show the evolution in time of the height of the water hw at the locations (x/h)A= 3.721 and (x/ h)B=4.542 along the deck shown in fig. 19. In agreement with the experiments (dashed lines), the numerical simulation (solid lines) shows a first stage characterized by a sudden rise of the water level hw, followed by a slower growth. The simulation is then stopped because of the surface breaking. Nevertheless, in section B, a third stage with a steeper increase of hw is captured. This is due to the water overturning which gives an additional contribution to the latest part of the evolution. Location A will be influenced later by this phenomenon. The measured data by Zhou et al (1999) are for an initial height of water h=0.6m. Due to lack of sufficient details about the experiments, the time when the experimental hw gets a non-zero value was set equal to the numerical one. The two types of curves fit quite well until breaking occurs. Figure 20: Free surface and pressure distribution during the initial stage of the impact. Solid lines: present numerical simulations; •: similarity solution from Zhang et al (1996); ∆τimp=τ−τimp . τimp=initial non-dimensional impact time. the authoritative version for attribution. We now discuss in more detail the impact pressures. Right plots of figure 20 present the pressure distributions corresponding to the free surfaces configurations on the left. According to the numerical results (solid lines), at each time instant the maximum value
OCR for page 138
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 138 of the pressure is located at the initial impact position and gets the highest values just after the impact. In the region of the thin jet along the wall, the pressure is almost equal to the atmospheric one, conventionally set to zero. The bullet symbols represent the pressure distribution for the zero gravity impact of a fluid wedge with a flat wall. In particular, this has been evaluated by solving numerically the boundary value problem for the velocity potential along the wall and taking the values of both φ and its normal derivative on the free surface from the similarity solution by Zhang et al (1996). At the beginning, the agreement between the two different results is good as for the corresponding free surfaces. For longer time, the pressure distributions seem to diverge faster than the free surface elevations. As can be expected, in the “exact” computations, the maximum pressure decreases while in the zero gravity case remains constant. In facts, as time and water level along the obstacle increase, the −ρ∂φ /∂ t contribution to the pressure reduces. In our simulation we have assumed a rigid wall but the pressure distribution could be influenced also by possible hydroelastic effects. In this case, it is important to introduce the generalized force ∫ pψnds, where ψn is an eigenmode for the local structural vibration. While the generalized forces related to the high initial values are modest ( z=0 is a structural node), smaller (but large enough) values of the pressure, distributed on a larger portion of the wall, may excite a hydroelastic response of the structure. On this ground, both the pressure distribution and its time evolution are important for structural analysis and hydroelastic effects should be considered if the time duration of the loading over the analyzed structural part is the same order or smaller than the highest natural period for the considered structural part (Faltinsen 1999). In figure 22, the dashed lines give the time evolution of the pressure measured by Zhou et al (1999) by a gauge, sketched at the top of the figure, with circular area of diameter 0.09 m centered at the location C along the wall (cf. top of fig. 19). We have shifted the time origin consistently with that discussed before for comparing the water level evolution. The dash-dotted lines are the numerical results at that location. The two curves get non-zero values almost at the same instant, confirming the global agreement between the numerical simulation and the experiment. A certain gap between theory and experiments is apparent, though. Mesh refinement and local regriding have been used to achieve invariance of the solution and to rule out the dependence Figure 21: Experimental (Zhou et al 1999, h =0.6 m) and Figure 22: Top: position of the pressure gauge in numerical water level hw (meters) at (x/h)A=3.721 and (x/ experiments by Zhou et al (1999, h=0.6 m). Bottom: h)B=4.542 as a function of time. Positions A and B experimental and numerical evolution of the pressure along definited in top plot of figure 19. the vertical wall (see top plot of figure 19). the authoritative version for attribution.
OCR for page 139
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 139 on the discretization parameters. On the other hand, the complexity of the experiment makes it difficult to identify the error sources because, according to authors comments, it was difficult to achieve repeatability of the results. It can be observed that, for the actual scales of the experiment, even a deck not perfectly dry (for example because of a previous experiment) can introduce significant differences in the measured data. This seems to be plausible also in the present case (cf. fig. 21). As said, though the experimental and numerical evolutions are globally in good agreement, close to the instant when the water level gets a non-zero value the numerics underpredicts the measured data. In particular, the measured hw gets a local maximum not present in the numerical results. These experimental features can be converted from a temporal to a spatial point of view. In particular they suggest a hump in the free surface close to the contact point. This is not visible in the dam breaking free surface profiles in Dressler (1954) and could be due to the presence of a layer of water before the dam breaks. Dam breaking experiments by Stansby et al (1998) show that, if the deck is not perfectly dry due to leakage (in those experiments a film of water with a thickness about 1–2 mm was downstream to the dam) a horizontal bulge of fluid develops just after the dam release giving rise to a very peculiar local flow. During the evolution, a hump becomes apparent in the most advanced flow region. Experimental water levels in Zhou et al (1999) are given only for one test case and our guess can not be confirmed by other comparisons. Anyway, according to the numerical simulation the experimental pressure curve is rather close to the pressure evolution at the lower location of the transducer (solid lines), indicated with the letter D in the sketch above figure 22. As previously discussed, water on deck can result in impact of fluid with superstructures. As an example of a complex interaction between the shipped water and the structures, we consider the first water on deck event for case e in table 2. The ship motion is restrained and the freeboard relative to the wave height is f/H=0.5. The flow evolution along the deck is presented in the top plot of figure 23. The motion of the fluid along the deck, resulting in a first impact against the deck house, is accompanied by a plunging wave hitting the upper part of the vertical wall. The related pressure distributions along the structure are estimated numerically by using the similarity solution as previously described Figure 24: Case e in table 2. Impact due to the plunging wave (solid lines) and to the wave front along the deck ( •) . Left: similarity solutions for free surface. Right: pressure on the wall. Initial impact positions in (0, 0). Figure 23: Case e in table 2. Top: flow onto the deck and impacts with the deck house from the deck and due to a plunging wave. Bottom: details of the plunging wave impact. the authoritative version for attribution.
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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 140 (i.e. by studying the zero gravity impact due to infinite fluid wedges) and are given in the right plot of figure 24. The corresponding free surface profiles are shown in the left plot, where the initial position of the impact for both cases coincides with (0,0). It is observed that the maximum non-dimensional pressure due to the plunging impact is significantly larger. The impact velocity itself is slightly higher than in the other situation This circumstance, and the fact that likely the plunging wave impact will occur far from a structural node, make this event more critical than the impact occurring at the deck level. CONCLUSIONS The phenomenon of bow deck wetness of a moored ship in regular head waves has been idealized and reduced to a simple two-dimensional wave-body interaction problem. The related unsteady fully nonlinear free surface flow has been solved numerically. Reasonably good agreement with experimental and analytical results enable us to use this simple model to gain some fundamental insights concerning the water on deck occurrence, the flow field over the deck and the impact with superstructures. In particular, an analysis on the parameters dependence of deck wetness has been carried out, showing that - For long wavelengths λ relative to the draft D, the wave steepness H/λ mainly determines water on deck occurrence and severity. The relative amount of shipped water depends nonlinearly on H/λ. - For small λ/L, where L is ship length, the bow wave reflection reduces or prevents the shipping of water, even for large wave steepnesses. - The stem overhang reduces the relative a-mount of shipped water, but its positive effect is less pronounced with respect to that of the freeboard. The occurrence of the less common “plunging wave water on deck” has also been investigated. Wave-body interaction by itself seems unable to cause a wave to plunge directly onto a completely dry deck, and the occurrence of this extreme and dangerous event appears more related to the interaction with a steep wave already prone to break. However, the influence of ship motion to enhance or reduce the severity cannot be excluded. It is fully realized that three-dimensional flow coupled with the ship dynamics have to be introduced in the future to predict quantitatively water on deck. ACKNOWLEDGEMENTS This research activity has taken place at the Strong Point Centre on Hydroelasticity in Trondheim, supported by NTNU and MARINTEK. The research has also been supported by the Italian Ministero dei Trasporti e della Navigazione through INSEAN Research Program 2000–02. The first author is a Ph.D. student at NTNU and is also associated with INSEAN. REFERENCES Bryant P.J. “Waves and wave groups in deep water”, in Nonlinear Waves, Cambridge University Press, 1983, pp. 100–115. Buchne r, B., “On the impact of green water loading on ship and offshore unit design”, Proc. Int. Symp. Practical Design of Ships and Mobile Units, PRADS'95, Seoul, The Society of Naval Architects of Korea, Vol. 1, 1995, pp. 430–443. Buchne r, B. & Cozijn, J.L., “An investigation into the numerical simulation of green water”, Proc. Int. Conference on the Behaviour of Offshore Structures, BOSS'97, Delft, Elsevier Science, Oxford, Vol. 2, 1997, pp. 113–125. Chaplin, J.R. Rainey, R.C.T. & Ye mm R.W., “Ring ing of a vertical Cylinder in Waves”, J. Fluid Mech, Vol. 350, 1997, pp. 119–147. Cozijn, J.L., “Development of a calculation tool for green water simulation”, MARIN Wageningen/Delft Univ. of Technology, the Netherlands, 1995. Domme rmuth D.G. & Yue D.K.P., “Numerical simulations of nonlinear axisymmetric flows with a free surface”, J. Fluid Mech., Vol. 178, 1987, pp. 195–219. Dressler, R.F., “Comparison of theories and experiments for the hydraulic dam-break wave”, Assemblée Générale de Rome, Int. assoc. of Hydrology, Vol. 3, 1954, pp. 319–328. Dussan, E.B.V, “On the spreading of liquids on solid the authoritative version for attribution.
OCR for page 141
About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 141 surfaces: static and dynamic contact lines”, Ann. Rev. Fluid Mech, Eds Lumley J.L., Van Dyke, M. & Reed H.L., 1979, Vol. 11, pp. 371–400. Ers dal, G. & Kvitrud, A. , “Green water on Norwegian production ships”, Proc. 10th Int. Conf. Offshore and Polar Engg, ISOPE'2000, Seattle, 2000. Faltinse n, O.M., “Numerical solutions of transient nonlinear free-surface motion outside or inside moving bodies”, Proc. 2nd Int. Conf. Num. Ship Hydrod., Berkeley, University Extension Publications, University of California, 1977, pp. 347–357. Faltinse n, O.M., “Water entry of a wedge by hydroelastic orthotropic plate theory”, J. Ship Research, Vol. 43, No. 3, 1999, pp. 180–193. Greco, M., Faltinsen, O. & Landrini, M., “An investigation of water on deck phenomena”, Proc. 15th Int. Workshop on Water Waves and Floating Bodies, Caesarea, Eds Miloh, T. & Zilman, G., 2000, pp. 55–58. Israeli, M. & Ors zag, S., “Approximation of radiation boundary conditions”, J. Comp. Phys., 1989, pp. 41:115–135. Landrini M., Grytoyr G., Faltinse n O.M. “A B-Spline based BEM for unsteady free surface flows”. J. Ship Res., Vol. 43, No. 1, 1999, pp. 1–12. Lloyd, A.R.J.M., Salsich, J.O. & Zseleczky, J.J., “The effect of bow shape on deck wetness in head seas”, The Royal Institution of Naval Architects, Trans. RINA, 1985, pp. 9–25. Longuet-Higgins, M.S. & Cokelet, E.D., “The deformation of steep surface waves on water. I A numerical method of computation”, Proc. Royal Society London A, Vol. 350, 1976. MARINTEK, Review, No. 1, April, 2000. Maruo, H. & Song, W., “Nonlinear analysis of bow wave breaking and deck wetness of a high speed ship by the parabolic approximation”, Proc. 20th Symp. on Naval Hydrod., Santa Barbara, National Academy Press, Washington D.C., 1994. Mizogushi, S., “Design of freeboard height with the numerical simulation on the shipping water”, Proc. Int. Symp. on Practical Design of Ships and Mobile Units, PRADS'89, Varna, Bulgarian Ship Hydrodynamics Centre, 1989, pp. 103–1, 8. Ochi, M.K., “Extreme Behavior of a Ship in Rough Seas—Slamming and Shipping of Green Water”, Annual Meeting of the Society of Naval Architects and Marine Engineerings, SNAME, New York, 1964, pp. 143–202. O'Dea, J.F. & Walden, D.A., “The effect of bow shape and nonlinearities on the prediction of large amplitude motion and deck wetness”, Proc. 15th Symp. on Naval Hydrod., Hamburg, National Academy Press, Washington D.C., 1984, pp. 163–176. Stansby, P.K., Chegini, A. & Barnes, T.C.D., J. Fluid Mech., Vol. 374, 1998, pp. 407–424. Tulin, M.P. & Wu, M., “Divergent bow waves”, Proc. 21st Symp. on Naval Hydrod., Trondheim, National Academy Press, Washington D.C., 1996, pp. 99–117. Welch, S., Levi, C., Fontaine , E. & Tulin, M.P., “Experimental study of the ringing response of a vertical cylinder in breaking wave groups”, Int. J. Off. Pol. Eng., Vol. 9, No. 4, 1999, pp. 276–282. Zhang, S., Yue, D.K.P. & Tanizawa, K., “Simulation of plunging wave impact on a vertical wall”, J. Fluid Mech., Vol. 327, 1996, pp. 221–254. Zhao, R. & Faltinsen, O., “Water entry of two-dimensional bodies”, J. Fluid Mech., Vol. 246, 1993, pp. 593–612. Zhou, Z.Q., De Kat, J.O. & Buchne r, B., “A nonlinear 3-D approach to simulate green water dynamics on deck”, Proc. 7th Int. Conf. Num. Ship Hydrod., Nantes, Ed. Piquet, J., 1999, pp. 5.1–1, 15. the authoritative version for attribution.
OCR for page 142
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About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as BASIC STUDIES OF WATER ON DECK 142 DISCUSSION G.Chahine Dynaflow, Inc., USA To enforce the Kutta Condition at the corners (intersection horizontal plane and vertical plane) did you have to use double nodes (nodes where there are 2 normals)? AUTHOR'S REPLY In the practical implementation we looked for a robust treatment of the flow at the corner at the initiation of and during the shipping of water. Before the shipping of water, the corner is not wetted and no special treatment is required. More relevant at this stage is the implementation of a decision criterium of shipping of water. In particular, we allow the fluid to leave tangentially the stem for a fraction of time step and evaluate the velocity component in the direction of the deck. In case of inward motion, the free surface is cut at the corner and the shipping of water starts. At this stage, the corner is fully wetted and a different treatment is adopted. We have not modeled the vortex shedding that, in principle, should take place there. Further, in the discretization of the integral equation, at the corner we define the unit normal vector as the vector along the bisector. This avoid the use of a double-node and, in a way, is equivalent to smooth the discontinuity of the geometry.  C.C.Mei, The Applied Dynamics of Ocean Surface Waves. Singapore: World Scientific (1983), pp.740. DISCUSSION D.K.P.Yue Massachusetts Institute of Technology, USA For both long and short waves, the authors find that stronger wave reflection by the ship causes less shipping of water on deck. In general, one would expect that stronger wave reflection leads to larger local wave height which produces more water on deck according to the dam-breaking theory. What causes this conflict? AUTHOR'S REPLY Consider two wave systems with the same steepness and different wavelengths, say L_a
Representative terms from entire chapter: