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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 713 Breaking Waves in the Ocean and around Ships M.P.Tulin1 and M.Landrini2 1Ocean Engineering Laboratory, UCSB, mpt@engineering.ucsb.edu 2INSEAN, The Italian Ship Model Basin, Italy, maulan@waves.insean.it ABSTRACT A wide ranging summary of research on ocean and ship breaking waves carried out within the Ocean Engineering Laboratory at the University of California, Santa Barbara, beginning in 1985, is presented in a coherent fashion. Emphasis is given to the role of naturally arising wave modulations in their crucial effect on wave breaking, as well as on the essential role which wave breaking plays in wave evolution. The OEL criterion for the onset of wave deformation and breaking is discussed and its mathematical derivation is given; its experimental validation through surface orbital velocity measurements is reviewed. The recent development of a gridless numerical simulation method for the study of post breaking wave flows is discussed, and examples of splashing flows are given, including bores, and bow breaking waves produced by a Wigley hull. Comparison with laboratory and field observations are made whenever possible. 1 FORWARD In 1985 we began our studies of breaking waves at the newly founded Ocean Engineering Laboratory within the University of California at Santa Barbara. We began with Raymond Cointe's analysis of the steady breaker above a submerged hydrofoil, Cointe and Tulin (1994), where good experimental data existed, Duncan (1983), and for which we obtained some predictions of breaking and breaker characteristics, based on a breaker model consisting of an essentially stagnant eddy, Figure 1, riding on the forward face of the leading wave in the wave train behind the hydrofoil and held in place by shear stresses along the dividing streamline. This model was quantified utilizing independent measurements of the turbulent shear stresses, and it was shown that the hydrostatic pressure acting on the dividing streamline underneath the eddy creates a trailing wave which largely cancels that trailing wave which would exist in the absence of breaking. The ‘wave' resistance of the hydrofoil thus manifests itself in the momentum flux of the residual trailing wave plus the breaker resistance, i.e. the momentum flux in the breaker wake. The total resistance for a fixed hydrofoil speed was shown to have a minimum corresponding to a particular value of the trailing wave steepness resulting in an onset condition for breaking. Figure 1: Physical model of a quasi-steady spilling breaker. In contrast to the quasi-steady hydrofoil breaker, energetic waves in the ocean are transient by their nature, and subsequently we have sought to gain a fundamental understanding of the latter, and of their effect on the nonlinear dynamics of ocean wave evolution. We early concluded, Li and Tulin (1992), that energetic breaking was closely the authoritative version for attribution. associated with these nonlinear dynamics, especially wave modulation (groups) caused by instability of the Benjamin-Feir type. As time went on we utilized various methods, as necessary, including analysis, exact numerical simulations, and a variety of experiments in our small and large wind wave tanks, Figure 2, including the use of radar. In response to needs of the Navy, we also

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 714 turned our attention to those breaking ship waves produced by the bow and which contributed to the pervasive surface wakes seen behind ships. For these studies we have relied heavily on nonlinear 2D+t numerical simulations, and these have been recently extended to the post-breaking regime dominated by surface turnover and splashing. Figure 2: The large OEL Wind-Wave Tank. It now seems time to present a wide-ranging summary of the various results which have been obtained. This is done first through an extended Overview, followed by two separate sections with more detail; Section 3 is devoted to the mechanism of breaking, and Section 4 to the numerical treatment of wave splashing and to splashing bow waves on ships and their near wake. We are concerned here with an exposition of our own work, and this should therefore not be construed as another attempt to review the widespread literature. For that there exist several recent reviews, Longuet-Higgins (1996) and Melville (1996). The first of these gives particular emphasis to theoretical work on the stability of steep waves, and the second to experimental and field observations. A large number of talented individuals connected with the OEL have taken part in the investigations which led to the present results, and this paper is fondly and gratefully dedicated to all of them, who have taught me so much. Their names will be found as authors of the OEL related papers in the References. During the last few years we have much benefited from a close working collaboration with my co-author, Maurizio Landrini, of INSEAN, Roma, who is now in charge of our numerical simulations of breaking and post breaking ship waves. He has written Section 4. The investigations reported on here were all supported by the Office of Naval Research, the investigations of ocean- like breakers by the Ocean Technology Division under a sequence of Program Officers (Gene Silva, Steve Ramberg, and Tom Swean jr.), and the investigations of ship breakers first by Jim Fein, and then by the Ship Hydrodynamics Program under Ed Rood. Their faithful support of the OEL from its first beginnings until the present time has been absolutely crucial to its development and its productivity, and we cannot thank them enough. the authoritative version for attribution.

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 715 2 OVERVIEW 2.1 1985 Our own view of both ocean and ship breakers has been radically transformed since we began our studies of breaking in 1985. The description of ocean waves was then, as it largely remains now, stochastic in nature, and the prototype wave was considered synonymous with the Stokes wave. Prevailing notions then related the breaking of energetic waves to the Stokes Limiting Wave, either through the Longuet-Higgins (1969, 1977, 1985) crest acceleration criterion, or through a wave steepness criterion determined experimentally; the Phillips energy spectrum, for ocean waves, which had enjoyed wide popularity, was entirely based on the notion that the saturated (breaking) wave at every scale shared the same geometry, depending only on g, as in the case of Stokes waves. Theorists had by then suggested that wave modulations of nonlinear origin might be crucial for ocean modeling, Yuen and Lake (1982); field observations of waves breaking at the peak of wave groups had been made, and systematic and insightful laboratory experiments on a large scale were being carried out. However, the significance of modulations for the breaking of waves, whether due to natural instability (group formation) or to wave-wave interactions, had not been well appreciated or understood. In fact, a small minority of the publishing wave community even accepted the existence of wave groups in the ocean, notable among them Su (1986), and Goda (1976). By 1985, a large body of literature existed on nonlinear wave interactions, originating in the pioneering work of Phillips (1960), on the four wave interaction mechanism, and in the form of the kinetic theory of waves by Hasselman (1962) and Zakharov (1968), and their new ideas were widely accepted within the theoretical ocean community and even incorporated into wave forecasting methods, Swamp Group (1985). In these kinetic equations the effect of breaking and wind pumping on the wave dynamics has been ignored. The reason has been given that these real processes are slower than wave interactions and therefore non-competitive. In wave forecasting this neglect would be fatal and therefore wave energy (or wave action) evolution is driven by wind pumping suitably reduced by energy dissipation due to breaking, all suitably parameterized in terms of the wave energy. Wave momentum considerations are neglected throughout, and this creates a vacuum regarding the evolution of wave frequency, a vacuum which is filled with certain largely empirical results from wave interaction calculations. This unsatisfactory situation continues, in fact, until the present time, although the effects of breaking on wave dynamics, particularly frequency downshifting, has been elucidated through experiments, Tulin and Waseda (1999), and theory, Tulin (1996). In 1985, the situation regarding the divergent bow waves from ships, and their breaking was under-developed and unsatisfactory. The only coherent view of these waves was as a part of the traditional Kelvin wave pattern, represented by the divergent portion of the ship's wave amplitude spectrum. The use of the Neumann-Kelvin method for the prediction of wave resistance continued this spectral representation of the divergent waves in that it incorporated the Havelock singularity based on linear considerations, as had the Michell theory and its slender body representation. Disquiet was injected into this situation by the experimental studies in Tokyo, Inui et al., (1979), and Miyata, (1980), which showed the appearance of a single almost straight wave crest at the bow, whose angle of inclination depended on the ship speed. Neither of these observed characteristics were understandable in terms of conventional theory, and it remained for nonlinear simulations, based on the 2D+t approximations, much later to reproduce and explain these results, Tulin and Wu (1996). The splashing, mixing, air entrainment, and noise generation which occurred in the aftermath of breaking of both energetic ocean waves and divergent bow waves, and are of such great interest today, had been seldom studied, with the exception of the hydraulic jump, or bore, about which a little was known. Our state of ignorance was nowhere more critically revealed and our curiosity prodded than by the stimulating reviews of Peregrine (1981), Melville (1996), the experiments of Bonmarin (1989), and our own experimental observations. Our own understanding of both ocean and ship breaking waves has been achieved in good part because of our high resolution simulation methods, LONGTANK and 2D +t, both based on boundary elements, and it was natural for us to extend simulations into the regime of splashing. This has required the use of a gridless Euler solver, based on a Smoothed Par 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 the authoritative version for attribution. BREAKING WAVES IN THE OCEAN AND AROUND SHIPS instability, wave group formation, and partial recurrence can be seen. LONGTANK simulation. large OEL Wind Wave Tank. The evolution of the instability until breaking and subsequent downshifting is demonstrated. Figure 4: Modulating waveforms in space, created by a wavemaker at the far left, for a sequence of times. The growth of the Figure 3: Wave energy density spectra at various fetches (9–36 m) from the wavemaker, measured with wave wires in the 716

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 717 ticle Hydrodynamics Lagrangian technique and which we call SPlasH. 2.2 Wave Modulation The instability growth and modulation of a nearly monochromatic wave has been demonstrated both in the laboratory, Lake et al. (1977), Melville (1982), Su and Green (1985) and Tulin and Waseda (1994), and in numerical simulations, Wang et al. (1994), and the connection with breaking has been conclusively demonstrated. The evolution of a wave system through modulation, breaking, and downshifting is seen through the measured spectra taken in the OEL wave tank (175′L×14′W×8′D), Figure 3. A time-distance simulation of a similar wave system, using the OEL numerical wave tank, LONGTANK, is shown in Figure 4. In the Ocean Engineering Laboratory (OEL) at UCSB we regularly produce waves breaking in self-excited wave groups for the purpose of studying the radar back scatter from a low grazing angle coherent radar. We have also used wave groups in the OEL tank to study the loadings on floating bridge structures, Welch et al. (1996), and to study the response of flexible vertical cylindrical structures to excitation in deformed and breaking waves, Welch et al. (1999). Do such wave group modulations occur in the ocean, and do they cause wave breaking? The evidence is that they do. The earliest observations and connections between ocean wave groups and breaking include: Donelan et al. (1972), Rye (1974), Goda (1976), Su et al. (1982) and Holthuijsen and Herbers (1986). Despite these, there has been much controversy amongst ocean engineers, where the concept of a stochastic sea has proven so useful and has therefore become deeply imbedded. Most of all, the visualization and measurement of wave groups has not been clear-cut, and a clear overall picture of how they exist in the ocean and under which conditions has not yet been obtained. Recently, however, low grazing angle (6°) coherent radars when viewing wind waves in the field, have produced remarkable range-time R−t maps which unmistakably show the existence of wave groups, Werle (1995), Smith and Poulter (1993), Ivanov (1993). Figure 5 is an actual radar image together with an explanatory sketch. These maps show bursts of scattering along straight lines in the R−t diagram and the analysis of these shows that the slope of the lines conforms to the law cg=dR/dt and the temporal spacing of the bursts is 2T. A large number of these maps have been obtained in sheltered waters, but they have been seen in the open ocean (Ivanov) and more experiments are underway. Figure 5: Top: range-time radar return from a low grazing angle, coherent radar mounted on a cliff overlooking Lock Linnhe; the bursts are breaking events. Bottom: a range-time schematic allowing a quantitative analysis of the upper figure. There is of course a theoretical basis for these nonlinear wave groups. In the first place, it had been shown long ago by Benjamin and Feir (1967) that monochromatic waves are unstable to small sidebands, although the theory does not allow for spatial variations in the wave system, only temporal. The space-time evolution of the resulting wave train was later studied, both theoretically and experimentally by Lake et al. (1977); they employed the weakly nonlinear model (the nonlinear cubic Schrodinger equation, NLS) first introduced by Benny and Newell (1967). Later, exact nonlinear simulations showing wave group modulations were first carried out by Dold and Peregrine (1986), and subsequently a technique was developed at the OEL for simulations of very long wave trains, specifically for the purpose of understanding breaking in wave groups. These allowed long time exact inviscid studies of wave instability growth, wave the authoritative version for attribution.

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 718 group formation, and wave breaking, see Wang et al. (1994). Figure 6: Temporal wave records in the large OEL Wind Wave Tank. Fetches from the wavemaker: 27 m (upper); 18 m (middle); 5 m (lower). Strong grouping and breaking (upper), and energy concentration under the group peak. What are the possible effects of wave group modulations? Most important, wave energy becomes concentrated under the peak of the envelope, and this effect is strongest near breaking as shown in Figure 6, a wave elevation time series taken in the OEL tank of modulating wave groups near breaking. A summary of available information on this concentration effect of grouping is shown in Figure 7, where data on the relation between and the wave steepness before modulation, a0k0, are shown, based on laboratory measurements, Su and Green (1985), and simulations, Wang et al. (1994). Under realistic conditions the wave height at the peak of the envelope is seen to be almost double its initial value. This alone, points up the potential importance of wave groups for us. Connected with this fact, there is ample evidence connecting breaking with wave groups, and it is during breaking that heightened crests and steepened wave faces occur. We discuss these subsequently. Figure 7: The wave height at breaking Hmk0/2 vs. the initial wave steepness a0k0, from tank experiments, Su and Green (1985), and simulations by LONGTANK. 2.3 Wave Deformation Surface wave kinematics approaching breaking and during the breaking process change dramatically from those we expect based on quasi-sinusoidal wave forms. First of all, the almost circular motion of individual surface particles becomes disrupted, resulting in a much larger down-wave drift than Stokes, which varies as (ak)2. This effect, in fact, occurs before breaking according to LONGTANK simulations, see some examples in Figure 8. Then, as the face progressively steepens in time, see Figure 9, the motion of a large part of it becomes directed toward the normal to the surface, and the particles are accelerated down tank, see Figure 10. Figure 8: Top: Particle trajectories, non-breaking modulating wave, LONGTANK. The dots · are separated by constant intervals in time. Note occasional large excursions, when group peaks have passed by. Bottom: Horizontal, u, vs. vertical, v, velocity for a particle, which eventually made a large excursion. As a result of both increase in H and the rise of the crest, the amplitude of the crest is much increased, see Figure 9. Of course, this crest rise is very important in the estimation of platform requirements. A crest rise factor, Amax/H, in the range 0·74 to 0·80 has been found in LONGTANK simulations. This is larger than the value of 0·6 suggested for engineering applications, Stansberg (1991), but close to the value 0·77 quoted by Kjeldsen (1990) for breaking waves in the authoritative version for attribution. his tank experiments. These deforming waves found in LONGTANK simulations have also been observed in tank measure

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 719 ments at the OEL. A set of wave profiles measured with a 14 wire linear array in the large UCSB wave tank is shown in Figure 11 compared with a simulation; the wave length is 2·3 m and the modulating waves were measured at about 100′ from the wavemaker. Figure 9: Wave Deformation from Inception (left) until Jet Formation (right), LONGTANK. Figure 10: The velocity field within a breaking wave with jet, LONGTANK. Figure 11: A comparison of breaking wave profiles measured in the large OEL wind-wave tank (dots) with LONGTANK simulations (lines). The connection between asymmetric deformation (front face steepening) and the breaking of extreme waves at sea was first made by Kjeldsen and Myrhaug (1980), based on observations. They claimed that extreme waves are accompanied by a shallow trough and a steepened wave face; they defined a deformation parameter, see Figure 12, and a critical minimum value for breaking. The LONGTANK simulation of wave breaking in wave groups and the OEL measurements clearly confirm the K-M ocean observations, and partially confirm the value of their deformation parameter see Wang et al. (1994); in the light of the breaking criterion found in LONGTANK, the deformation has to be viewed as a consequence of the breaking process and not the cause of it; and the K-M deformation parameter serves to quantify the deformation just before jet formation, and not as a criterion for breaking. Figure 12: Extreme wave, deformation sketch after Kjeldsen and Myrhaug (1980). For breaking: [η′/H] [λ/λ′]>3·5(K−M), and 3·48−5·29 (LONGTANK) 2.4 Inception and Mechanism of Breaking We now have a clear understanding of the exceptional role played by wave modulation in the initiation of wave deformation and breaking, and furthermore we possess a simple criterion for the initiation of this process as the wave passes through the modulation peak. In Section 3 of this paper we present theory which explains the mechanism associated with this criterion, and also its experimental verification through surface particle tracking experiments. As we have noted, wave modulation in the ocean is a result of wave instability of the Benjamin-Feir type, and this newly described mechanism leads to breaking at group-averaged steepnesses as low as about 0·1. This explains those field observation, Holthuijsen and Herbers (1986), Weissman et al. (1984), which came to our attention at the beginning of our work, and 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 the authoritative version for attribution. plunging jet. Figure 13: Wave breaking in LONGTANK, BREAKING WAVES IN THE OCEAN AND AROUND SHIPS a0k0=0·28. (a): waveforms in the initial stage of breaking. (b): the formation of a 720

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 722 pation. After all, the waves only exist because of the wind, and the most systematic measurements of wave growth (fetch limited growth) reveal a scaling of the fetch laws with the wind speed. We know, too, that for moderate to equilibrium wave systems the breaking dissipation is of the same order as the wind pumping. We have ourselves tried to understand and quantify the role of real effects in the dynamic evolution of waves. Our first attempts, described in (A) below involved the use of the modified complex evolution equation (NLS), where we have introduced the breaking dissipation as a right-hand forcing term, as others before us, Trulsen and Dysthe (1990); Hara and Mei (1991). Unlike most previous work, however, we have introduced a dissipation dependent upon (ka)4; we had discovered that only for this particular power law dependency on steepness would the breaking dissipation balance the wind input for waves in equilibrium, provided that Toba's law of individual waves is invoked. Later we have confirmed that this law is consistent with the observed fetch laws. In the mid 90's we have realized the great importance of momentum considerations in addition to energy. In particular, we discovered that the dual conservation laws for energy and momentum can be combined to produce a law for wave speed evolution, and can quantify downshifting, Tulin (1996). This is briefly described in (B) below. (A) The NLS allows the calculation of possible shapes of the equilibrium modulational envelope; however, without consideration of wind pumping and dissipation, the envelope of groups which actually arise in nature cannot be predicted. Therefore, Li and Tulin (1993), used this model equation including real effects to predict envelope shapes in general. A major result of this study was a prediction of the number of waves in a spatial group as a function of the ratio of maximum and minimum wave steepness, in the group. For values of this ratio between about 0·3 and 0·08, the number of waves varies between about 2 and 7. These results are consistent with the observations of Holthuijsen and Herbers (1986). Note that the number of waves in the temporal group would be twice the quoted numbers. Later, Li and Tulin (1994) thoroughly considered the long time evolution of sideband systems with wind and dissipation, using their modified NLS and dynamic systems analysis; they discussed the strange attractors and possibilities of chaotic behavior. (B) As breaking proceeds, water flows into the plunging jet, carrying with it, both kinetic and potential energy E and momentum q. Upon splashing, these are lost to the organized wave motion and these losses must be accounted for in the further evolution of the wave. This requires conservation laws for both wave energy and momentum, which may be constructed on the basis of weak non-linearity. We have first done this on a heuristic basis, Tulin (1996), and furthermore shown that a set of evolution equations for E-c correspond to the original, E-q pair. More recently we have derived these same conservation laws from variational considerations including a rigorous inclusion of breaking in the variational formulation; this was done by introducing, in addition to the usual wave Lagrangian, a work function representing the effect of breaking and wind input on the flow dynamics, and by a rigorous definition of this function, Tulin and Li (1999). In addition, we have shown that our set of conservation laws for energy and wave speed can be combined into the complex Landau-Ginzberg or NLS equation, giving the latter new meaning, and extending it to include the effect of breaking in a rigorous way. Here is the newly modified NLS, valid for all (X, T): (1) where (X, T) are the long time and space scales, A=aeiθ, where θ is the wave phase, and Db are the rate of the wind Db~g|A|4 where the constants of proportionality follow pumping and breaking dissipation, respectively; from the field observations; A consequence of the law for the wave celerity is that breaking leads to a continual increase in the wave speed (frequency downshifting) at a rate controlled by the breaking process. The way in which this works was studied in our large wind-wave tank through laboratory observations of wave group evolution, including breaking effects, Tulin and Waseda (1999). A crucial aspect is the cooperation of breaking dissipation and momentum loss acting together with near- neighbor energy transfer in the discretized spectrum, this transfer being due to detuned resonance acting over a limited time (less than 50 wave periods). This is a very different point of view than is commonly incorporated in large-scale wave prediction modeling, where the conservation of wave momentum is ignored. the authoritative version for attribution.

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 723 2.6 On the Modeling of Splashing, and its Consequences We are indebted to the early, careful, experimental observations of Bonmarin (1989) for some knowledge of the splashing process, which follows wave breaking. These depended heavily on photographic visualization. Recently we have undertaken detailed studies of the entire splashing process and its mechanics, making use of a high-resolution numerical simulation, SPlasH. This is described in Section 4. The simulations reveal the presence of the partial forward ricochet of the plunging jet, the collapse of the cavity under the jet, and a backward counter-jet created in reaction, and of the eventual formation of a dipole structure with downward momentum behind the splashing breaker. These all seem consistent with Bonmarin's observations. We have systematically studied the special case of the propagating bore in shallow water, as this has been much studied in the past, and therefore offers an opportunity for validation of the simulation. We have found the simulated propagation of the bore in excellent agreement with predictions based on mass and momentum conservation. The simulation method allows for arbitrary variations in the fluid density, and we intend to extend the present simulations to include the presence of entrained air in the large scale flow structures and to the eventual fate of this air. 2.7 Surface Tension Effects; Microbreakers The local curvatures in the wave geometry are largest at the crest of a deformed wave in its later pre-jet stage and in the jet itself, whose thickness may only be a few percent of the wave amplitude. We have carried out LONGTANK simulations of modulating and breaking waves, including the effect of surface tension, for waves of varying length down to 25 cm. Scaling shows that the effect of surface tension increases with reduction in the length of the wave. For those waves, simulations showed that surface tension does not effect the breaking criterion and has insignificant effect on the wave deformation except in reducing somewhat the largest curvatures at the crest of the wave prior to breaking. Surface tension (or the length scale) does, however, have a pronounced effect on the jetting process for waves shorter than about 2 m in length. For 1 m waves the jet is considerably rounded and weakened, which explains both visual and radar observations of breaking waves in the OEL large wind-wave tank, where much less energetic splashing is observed for waves 1 m and shorter. For waves between about 25 and 75 cm the jet does not appear at all, to be replaced by a forward facing bulge growing out of the wave crest, Figure 15. Figure 15: The effect of surface tension on the jet development. Wave profiles were computed with LONGTANK. Microbreakers have wavelength below 1 m and surface tension reduces the jet to a bulge. Non-jetting waves of this kind have been called micro-breakers. Unlike energetic breakers where the shape continually evolves in transient fashion, continually changing over a time interval of about one wave period, the microbreaker can propagate for a considerable distance without significant change of shape. This can be seen in radar tracks in range-time diagrams, since the bulges on the microbreaker crests provide a relatively strong specular radar return, Fuchs and Tulin (2000), Figure 16. A striking feature of the microbreaker bulge is the appearance at the foot of the bulge of a small region of very high curvature, see Figure 15, It is well known that vorticity is generated at a free surface with curvature, and there is ample evidence, Duncan (1994, 1996), Dommermuth and Mui (1996), that a strong effect of this kind does originate at the foot of the bulge, causing dissipation and probably resulting eventually, for very short waves (sub-microbreakers), in the separation of the bulge from the main wave flow through a strong shear layer, forming a cap on top of the wave, as noted by Ebuchi et al. (1987). This small scale flow then bears a family relation to the steady hydrofoil breaker, Figure 1, differing in its more or less symmetric placement over the wave crest. Ebuchi and Toba's waves were wind generated, and it may very well be that the wind stress plays an important role in allowing the propagation of this unusual wave. The microbreaker itself is the consequence of the authoritative version for attribution.

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 735 It can be seen, that a portion of the particles originally in the first plunger are now riding on the moving front of the propagating wave, while the others are captured by the cavities and effectively mixed with other portions of fluid. In following cycles, all the tracked particles will be captured by the rotating structures and fresher particles will feed the bore front. A more quantitative analysis is obtained by the two bottom plots, where the components of the velocity of the center of mass of the jet-particles (18) are plotted in time. The free fall stage in the first part of the jet-evolution can be clearly seen in the right plot, and accompanied by an almost steady translation in the horizontal direction (cf. left plot). The instant of the impact is evidenced by the sharp changes of Vc,x, Vc,y. The first drops down to less than 25% of its initial value: this corresponds to the splitting of the jet into two portions with backward and forward motion, balancing each other and thus reducing Vc,x. The vertical component bounces up, changing in sign, because both portions are moving upward, and then oscillates consistently with the orbital motion of the particles entrapped in the cavities. Eventually, Vc,y attains an almost zero value, with small amplitude oscillations related to the orbital motion inside the vortical structures rotating in an uncorrelated way. After some oscillations with decreasing amplitude, due both to the orbital motion and to the multiple splash-up, the horizontal component attains the uniform velocity of the moving piston. We have analyzed several cases, in the range For larger U, a more vigorous formation of the backward plunging jet and MODE B interaction has been observed, with initial formation of stronger dipole-like structures, followed by a predominance of MODE A interaction with vortical regions more confined to the free surface. A slower motion of the piston leads to less pronounced splash-up cycles, producing weaker vortical structures of opposite sign, mainly due to MODE A interaction. In the present simulations, for dipole-like structures are never observed. In all cases, MODE B appears connected with the initial breaking events, when the most energetic plungers are generated. Plots in Figure 27 give an indication of this transformation by comparing some global characteristics of the plunging jet and those of the corresponding splash-up, for two values of the piston velocity Many parameters are involved and we simply report some of them, just to show the complexity of the underlying dynamics involved. All the quantities are referred to a unit slice of fluid in the direction orthogonal to the plane of motion. The mass in the jet at the touch down is 2·55 kg and 0·618 kg for the two cases, respectively, and the corresponding cavity entrapped is 45·4 cm2 and 13·9 cm2. Despite the rather large differences in jet masses, the masses involved in the splash-ups are comparable: 4·51 kg and 3·45 kg. This is probably due to the difference in the impacting angle (21.1° vs. 18·6°), implying different depths of penetration of the jet. It is remarkable that the mass of the authoritative version for attribution. Figure 27: The first plunging event and the corresponding splash-up for (top) and (bottom). The configuration of the splash-up is reported when the upward-raised water reaches the maximum height.

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 736 the splash-up is greater than the original jet which gave rise to it. The entire history of the bore from initiation of the motion involves the pile up of the water at the piston, the propagation of the wave, overturning and initial splashing followed by a series of splashes. Finally the splashes ceases and a surfing eddy remains on the face of the equilibrium bore, analogous to the breaker above a hydrofoil, Figure 1; this was observed in our simulations and in experiments reported by Peregrine (1983). In this condition, the velocity of the breaking front attains a constant value: i.e., in the frame of reference of the front, the velocities far upstream and sufficiently downstream are constant. For hydraulic jumps, by using mass and momentum conservation principles, we can relate the dimensionless velocity of the bore (19) to the upstream, h0, and downstream, h1, water levels. In Table 1, data from SPlasH simulations are collected and compared with the theoretical prediction. The agreement is generally within 1%, and 2% at maximum, quite reasonable for such complex, non-steady dissipating flows. Table 1: Comparison between the computed velocity, uSPlasH, of the bore front and the theoretical value, uth, predicted by global mass and momentum conservation principles, cf. Equation (19). Velocities are expressed in m/s. h1/h0 h1 F0 uth uSPlasH 0·95 0·212 2·12 1·82 1·80 1·81 0·9 0·21 2·1 1·8 1·78 1·78 0·8 0·19 1·9 1·66 1·64 1·63 0·7 0·18 1·8 1·59 1·57 1·57 0·6 0·17 1·7 1·52 1·50 1·47 0·5 0·15 1·5 1·37 1·36 1·38 The result strongly suggests that inertial (Euler) effects may dominate the observable phenomena, including strong mixing of surface water when passing through the bore. 4.3 Breaking Bow Waves Breaking ship waves have always captured the interest of hydrodynamicists and naval architects because of their role in contributing to the resistance of the hull. More recent, however, is the interest of the Navy in the very long narrow wakes behind ships which can be observed remotely, and which originate abeam the ship through extensive breaking of diverging bow and stern waves. The following analysis of this problem is limited to fine ships with a sharp stem, for which fundamental understanding can be gained by an approximate quasi-three dimensional model based on the idea that longitudinal gradients of relevant flow quantities are small compared with vertical and transverse gradients. The introduction of this seminal idea is not new, and a historical recollection of slender-body theory for ship hydrodynamics is given by Maruo (1989) and by Fontaine and Tulin (1998). More specifically relevant to our purposes, nonlinear versions of this parabolized model for ship flows have been presented by Faltinsen and Zhao (1991), Maruo and Song (1994) and Tulin and Wu (1996). In the last two, the calculations are carried out in two dimensions, vertical and transverse, and successively in time and the method has been consistently named 2D+t. Tulin and Wu (1994) presented a thorough analysis of the genesis of diverging bow waves, and a detailed parameter investigation is reported in Wu (1997). In particular, the method exhibited the advantage of high resolution, sufficient to capture breaking, and even to trace the jet overturning. An example of this ability is presented in Figure 28, where free surface profiles for successive timesteps are superimposed, showing one of the key features of the slender-hull bow wave system: the splash, Tulin and Wu (1994). For this ship (a Wigley hull), the free-surface flow is not much decelerated before the stem, but upon reaching it, is deviated sharply upwards, rises on and eventually levels off and falls down. An entire thin sheet is formed in this process and appears as a splash on either side of the hull. The relaxation of these splashes is the prime source of divergent waves. In the present case, the large beam-to-length ratio, B/L=0·2, makes the radiated waves large enough to break, following an evolution similar in many ways to that described in Sections 2–3, with crest-rising, front-steepening and jet formation. Some genuine three-dimensional effects, such as upstream influence and breaking before the bow, become increasingly important as the ship becomes full 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 the authoritative version for attribution. BREAKING WAVES IN THE OCEAN AND AROUND SHIPS Figure 28: 2D+t simulation of wave patterns generated by a Wigley hull (B/L=0·2, D/L=0·1) for increasing Froude number 737

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 738 and slow and are lost to this parabolized theory. Figure 29: Perspective view (from upstream) of the wave pattern generated by a Wigley hull (B/L=0·1, D/L=0·1, Fr=0·46). BEM computation (blue lines) starts from the bow, up to x/L=0. Black solid lines are made by the uppermost layer of fluid particles from SPlasH computations, initialized by BEM. In the following we apply a version of SPlasH to study the splashing of the forward wave generated by a Wigley hull. The computation is initiated by a 2D+t code based on BEM, and continues up to the detection of jet overturning. Once this is accomplished, a sub-domain, surrounding the breaking area, is defined and filled in with particles. The initial velocity and pressure of the particles is determined by BEM but, so far, we have not fully coupled the two algorithms and, on the outer SPH boundary, stationary conditions have been imposed during all the following evolution. Therefore, preliminary sensitivity tests have been conducted to determine the vertical and horizontal dimensions of the SPH domain to assure the invariance of the results during the entire simulation time. A perspective view of this compound simulation is presented in Figure 29. The view point is located upstream the bow, and y−z configurations for increasing time are plotted in a 3D fashion by using x/L=Ut/L as longitudinal coordinate, with the origin located midship. Ship cross sections (red lines) are plotted only for those slices obtained by BEM, and the corresponding free surfaces are represented by blue lines. For this Froude number, Fr=0·46, we observe a substantial splash at the bow, with height comparable with the draft and maximum located around 25% L from the stem line. The gravitational collapse of the splash is accompanied by the radiation of a wave, which steepens and forms a plunging jet, almost touching the free surface for x/L=0. From this configuration on, the SPH computation has been started and the resulting free-surface pattern is reported in the same figure by black lines. Only the layer of particles at the free surface is plotted and its horizontal extent corresponds to the actual computational domain used by SPlasH. A more detailed analysis is given in Figure 30, where an enlarged view of the wave pattern during the splash (top) is presented together with some selected particle distributions (bottom). Upon adopting a stern viewpoint, we observe that the breaking crest is propagating with an almost unchanged phase speed, though the simulation time is rather short to be conclusive and 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 the authoritative version for attribution. BREAKING WAVES IN THE OCEAN AND AROUND SHIPS Figure 31: Particle distributions from Figure 30 showing splashing and formation of vortical regions. Only the uppermost layer of particles from SPlasH computations is plotted. Red colored sections are reported in Figure 31. Figure 30: Breaking of the bow wave generated by a Wigley hull (B/L=0·1, D/L=0·1, Fr=0·46). Details from Figure 29. 739

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 740 the analysis hampered by the spreading of particles around the crest emerging from the breaker. More clear is the presence of a slower trace, originating from the splash-up, and left on the free surface behind the breaking crest. The analysis of particle distributions, colored according to the vertical distance from the highest point, reveals its origin. The impacting jet, plot A, causes splash-up and cavity formation, plot B. A backward-facing jet is created, with a MODE B interaction, plot C, and eventually resulting in a dipole structure left behind the propagating crest, plot D through E. This dipole is weaker with respect to those observed for breaking bores with large U. Furthermore, the overall downward-backward motion of the fluid past the crest corresponds to the possible self-convection of the dipole, which remains close to the free surface. In any case, because of its vortical nature, the dipole structure does not follow the breaking crest, leaving its own signature on the free surface. An analysis of the motion of the particles initially forming the plunging jet shows strong similarities with that presented in Figure 26, for the breaking bore. Also in this case, the jet flows in a backward stream, captured in the nascent cavity, and another one feeding the splash-up. It is worth stressing, in this case, that the fluid particles emerging forward from the splash-up in plots B-C, are then “surfing” the wave crest after the splashing event, plots D-F, thus resembling the weak eddy steadily moving with the hydrofoil breaking wave, as discussed at the beginning of this paper. We found a striking similarity between the present results and the experimental observations by Lamarre, reported in Melville (1996, Figure 2). Clearly, on a longer time scale, the degassing stage will be profoundly effected by the two- phase nature of the bubbly flow created during the breaking. For bow-flared ships and practical Froude numbers, wave breaking takes place much closer to, or at the bow. This has been shown in Tulin and Wu (1996, Figure 13), where the first stage of the gravitational collapse of the splash at the bow, with formation of a strong plunger, has been computed for a frigate ship. Although the application of the SPH technique to study these flow conditions requires some improvements (e.g. modeling of curved boundaries, and matching with an outer solution for long-time simulations), we can already get some basic insights by considering the flow forced by an inclined piston. In particular, we selected an angle of 66·8° by taking the slope at the waterline of the DTMB Model 5415 (cf. top plot in Figure 32) in the bow region. In our simulations we used a constant horizontal velocity, For the actual ship and Fr=0.41, the expansion velocity of bow cross sections is even higher but drops down sharply. In the bottom plots of Figure 32, we observe a quick piling up of the water against the piston, with a maximum run-up of about 2h0, and the formation of a rather thick jet, eventually collapsing down in the form of a plunger. In contrast to previous results, the impacting fluid is not creating a crater, even though a portion of the impacting jet is still deflected inside the nascent cavity. The evolution on a longer time scale is significantly effected by finite depth, and is less relevant to the ship problem. Figure 32: Top: sketch of the model problem adopted to study a breaking splash. Bottom plots: evolution of the splash the authoritative version for attribution. breaking near the piston. Jet particles are marked in black to put in evidence the ricochet.

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 741 Much remains to be done on ship breakers but the present results are very encouraging. ACKNOWLEDGEMENTS The ONR sponsors have been gratefully acknowledged in the Forward. I would mention especially our constructive relationship with Dr. Ed Rood, who provided the incentive for the present splashing studies. Among the many OEL researchers who have contributed to our understanding of breaking waves I would specifically mention four of those who not only received their PhD's at UCSB, but stayed on as Post Graduate Researchers: Pei Wang (Mrs. Y.T.Yao), who conceived of the special domain decomposition technique and implemented it for the high resolution simulation of long wave trains and the breaking process- she is the Mother of LONGTANK; Yi Tao Yao, who utilized LONGTANK so brilliantly in studies of wave breaking, who discovered the breaking criterion, simulated surface tension effects, and who collaborated in a host of studies of ocean wave behavior; J.J.Li, who made many deep and scholarly mathematical studies of wave dynamical behavior in connection with ocean wave modeling; Takuji Waseda, who developed much of our experimental systems for wind wave studies, pioneered in the development of our experimental wave group capability, and carried out a series of profound experimental studies of wave breaking; Ming Wu, who continued the 2D+t simulations of ship bow waves initiated by Maruo and Song, and used them intensively and fruitfully in systematic studies of bow waves and of deck wetness. I am very grateful to my old colleague, Professor Hajimi Maruo, of Yokahama, for spending two productive years at the OEL, during which time the 2D+t simulations were begun. I must also thank Dr. Emmanuel Fontaine, now of IFP in Paris, for his vital role while a visiting researcher at the OEL in initiating our work on gridless Euler simulations, and his early explorations and use of the SPH method in splashing studies. I thank my colleague and co-author Maurizio Landrini for agreeing to the present collaboration which brings him from Rome to the OEL for six months a year, together with Andrea Colagrossi of Rome, who has participated in many of the SPlasH computations. The successful collaboration was made possible only through the agreement of Admiral U.Grazioli, the President of INSEAN, and of Dr. U.Bulgarelli, the Scientific Director, to whom we are grateful. Finally, many thanks to the tireless staff of the OEL, who have helped in the preparation of this paper, and especially Ms. Sandra Jeppesen, and Mr. Daniel Matsiev. REFERENCES Belytschko T., Lu Y.Y. and Gu L. (1994), Element free galerkin, Int. J. Num. Meth. Eng., Vol. 37, pp. 229–256. Benjamin B.T and Feir J.E. (1967), The disintegration of wave trains in deep water, J. Fluid Mech., Vol. 27, pp. 417–430. Benney D.J, and Newell A.C. (1967), The propagation of nonlinear wave envelopes, J. Math. Phys., Vol. 46, pp. 133–139. Bonmarin P. (1989), Geometric properties of deepwater breaking waves, J. Fluid Mech., Vol. 209, pp. 405–433. Cointe R. and Tulin M.P. (1994), A theory of steady breakers, J. Fluid Mech., Vol. 276. Di Lisio R., Grenier E. and Pulvirenti M. (1998), The convergence of the SPH Method, Computers Math. Applic., Vol. 35, No. 1/2, pp. 95–102. Dold J.W. and Peregrine D.H. (1986), Water-wave modulation, Proc. 20th Int. Conf. on Costal Eng., Taipei,Vol. 1, pp. 163–175. Dommermuth D., and Mui R. (1996), The vortical structure of a near breaking gravity-capillary wave, Proc. 20th ONR Symp. on Naval Hydrodynamics, pp. 551–567, Natl. Academy Press. Donelan M., Longuet-Higgins M.S and Turner J.S. (1972), Periodicity in Whitecaps, Nature, Vol. 20, pp. 449–451. Duncan J. (1983), The breaking and non-breaking wave resistance of a two dimensional hydrofoil, J. Fluid Mech., Vol. 126, pp. 507–520. Duncan J., Philomin V., Qiao H. and Kimmel J. (1994), The formation of a spilling breaker Phys. of Fluids, Vol. 6. Duncan J., Philomin V., Qiao H. (1996), The transition to turbulence in a spilling breaker, Proc. 20th ONR Symp. on Naval Hydrodynamics, pp. 530– 567, Natl. the authoritative version for attribution.

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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 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 BREAKING WAVES IN THE OCEAN AND AROUND SHIPS 745 DISCUSSION B.Beck University of Michigan, USA Looking at your figures, it appears that in the 2D+t computations, the diverging wave crest is a straight line. The figure using SPH in the cross flow plane seems to indicate the breaking wave front slows down so that the diverging wave crest is parabolic. Could you please comment on this? AUTHOR'S REPLY In Figures 30 and 31, the growing separation of the dipole vortical structure and the cresting breaker can be seen. The vortical structure is slower and would appear to follow a roughly parabolic track, although its asymptotic motion cannot yet be known from the limited calculations which we have done. The crest of the divergent wave is faster and its structure spreads with time. This corresponds physically to the growth in extent of the broken water just behind the crest and which would appear as a foam scar in the ocean. In front of this scar is the small surfing breaker. This marks the front of the broken wave. A line drawn through the center of the surfing eddies back to the touchdown of the original jet would appear to be reasonably straight. Again, further calculations in time must be made to understand the fate of the surfing eddy. It would seem reasonable to expect that it must eventually disappear. the authoritative version for attribution.