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• Waves and Forces Caused by Oscillation of a Floating Body Determined through a Unified Nonlinear Shallow-Water Theory 993-1005

The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001

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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 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 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 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.

OCR for page 126

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 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,

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 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 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 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

OCR for page 126

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 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 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 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

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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 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 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 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.

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