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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 NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 746 Numerical and Experimental Study of the Wave Breaking Generated by a Submerged Hydrofoil A.Iafrati, A.Olivieri, F.Pistani, E.Campana (CETENA S.P.A., Italy) ABSTRACT In the present paper the two-dimensional wavy flow generated by an hydrofoil moving beneath the free surface is experimentally observed and numerically studied. The numerical investigation is performed by means of a finite difference Navier-Stokes solver. The free surface is embedded in the computational domain and the flow either in air and in water are computed. The Navier-Stokes solver is coupled with a Level Set technique to captures the interface location. The presence of the hydrofoil is taken into account either by introducing suitable body forces on the grid points inside the body contour or by a new domain decomposition approach, developed to concentrate computational efforts in the free surface region. Experimental study concerns the wave-breaking dominated by the ripples formation. The stages of the evolution of a breaking generated after the onset of a capillary wave train are visualized. For a fixed Froude number and angle of attack, the depth of the hydrofoil has been gradually varied, until the condition for incipient breaking has been reached. Depending on the condition of the experiment, the wave breaking can start from the forward face of the second or third wave crests, hence propagating to the first wave, leading to the full developed event. INTRODUCTION The knowledge of the mechanisms responsible of the breaking waves is of great importance for the comprehension of many natural phenomena and the development of several engineering processes. There are so many problems related with breaking waves that a complete list is hard to compile. To be confined to those related with ships, breaking waves are produced by almost any marine vehicle, and are relevant in the definition of their operative conditions. Beside of being responsible of the increase of the ship's resistance, breaking waves play a relevant role in active and passive ship detection problems. The hydrodynamic noise produced by the breakers can lower to a great extent the efficiency the ship's detection equipment, usually located inside the bulb. Although the problem may be solved by increasing the depth of the sonar dome, this not always represent a winning hydrodynamic solution. On the other side, breaking waves are responsible of possible detection of the ship from synthetic aperture radar (SAR) images of the sea surface. Furthermore, breaking waves are always in close connection with vorticity and turbulence production at the free surface, as well as the generation of a bubbly near wake of the ship, again a relevant signature problem, and a great effort is currently devoted toward the understanding and modelization of these phenomenon (see for example Reference 1). A long, and far from being complete, list of references could be write down. So many researches have contributed to our basic understanding of the breaking phenomena that we have to confine ourself just to some of previous studies. The flow structure near the ship when breaking events occur has not been deeply investigated. In Miyata & Inui (1984) the problem has been reviewed and, more recently, Dong et al. (1997) performed detailed particle image velocimetry (PIV) measurements of the flow about a ship model, carefully analysing the wave structure near the bow and the mechanism of free surface vorticity production. The flow structure of 2D spilling breakers has been more extensively studied. Hydrofoil generated spilling breakers have been experimentally investigated with great accuracy by Battjes & Sakay (1981), Duncan (1981, 1983), Mori (1986), Duncan & Dimas (1996), Lin & Rockwell (1996). In particular the work of Duncan's research group, represents a reference point, especially for the 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 NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 747 case of the towed hydrofoil. Theoretical studies suggested that the flow just below the breakers is turbulent, and suggested the existance of a shear layer beneath the breaking wave, see Peregrine & Svensen (1978), Longuet-Higgins (1994a), Longuet-Higgins (1994b), Cointe & Tulin (1994). The numerical description of the breaking phenomena via moving grid approaches is not straightforward. Recently, new techniques describing the two-phase flow of both air and water have been developed, allowing for a complete description of the breaking and post-breaking event. In Sussman et al. (1994) an additional variable, i.e. the signed normal distance from the interface, is introduced and the free surface location is identified as the zero level set (LS) of this quantity. The distance is a continuous function across the interface, reinitialized at each time step. The capability of this approach to deal with complex flows in which topological changes of the interface occur has been proved by some recent papers by Azcueta et al. (1999), Vogt & Larsson (1999), Iafrati et al. (2000), Iafrati & Campana (2000). The purpose of this paper is to report recent developments at INSEAN in the numerical and experimental investigation on breaking waves, in the framework of a cooperative project involving ONR, IIHR and DTMB. Present experimental study concerns the wave-breaking rising from the formation of capillary waves on the forward face of the gravity wave. The problem has been treated theoretically by Longuet-Higgins (1992). Preliminary observation of the occurrence of these ripples and of the breakdown of this type of flow are reported in the following. The numerical approach is here used to study the inception of the breaking produced by a submerged hydrofoil, with particular reference to the velocity and pressure fields. Several numerical schemes have been adopted, ranging from a simple inviscid rotational formulation (mainly used for verification purposes), to the solution of the Navier-Stokes equations in the full domain. A Navier-Stokes solver in generalized coordinates, together with a Level Set technique, used to follow the free surface dynamics, has been developed. This approach can lead to free surface instabilities in regions where the grid is highly skewed, unless an high grid refinement is used (Iafrati et al. 2000). This in turn implies that some difficulties may be encountered when studying the free surface flow induced by bodies moving close to the interface, due to the distortion of the body fitted grid. Nevertheless, when attention is mainly devoted to the free surface dynamics rather than to a detailed description of the flow about the body, the above problem has been overcome, either by using an orthogonal grid and introducing suitable “body forces” that mimic the presence of the solid boundary, or developing a new approach based on a domain decomposition technique. EXPERIMENTAL INVESTIGATION Experimental system and techniques The experiments have been carried out at INSEAN basin n.2 (220 m long, 9 m wide and 3.5 m deep). The towed hydrofoil is a NACA 0012 profile made of composite material, whose chord and span are respectively 0.4 m and 2 m. The hydrofoil is connected to the carriage by two vertical, surface piercing, side struts. Variation of the angle of attack and rotation along the z axis are allowed. Images of the generated wave pattern have been taken using a video camera and pictures have been subsequently extracted. Moreover, a submerged video camera has been applied to visualize the flow around the hydrofoil. A fluorescent substance, introduced upstream the hydrofoil by a thin duct, has been used to enhance the vortical structures leaving the rear part of the hydrofoil. The light source needed for the underwater images has been provided by an 800 watt photo-floodlight with Fresnel lens placed close to the surface on the rear part of the hydrofoil. A flat mirror mounted below the hydrofoil has been used to increase the amount of light (Fig. 1). The tests have been carried out at a constant Froude number of 0.177, while the hydrofoil has a 10 angle of attack (nose up). To detect the onset of the wave breaking, the hydrofoil depth has been slowly decreased during the carriage run, following the evolution of the breaking from the initial stage up to its complete development. Figure 1: Sketch of the experimental apparatus (side view). 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 NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 749 for waves of wavelength of about 20 cm, as in the present study. Pictures acquired by the underwater camera show the presence of vortex shedding from the rear part of the hydrofoil. This is due to the combined effect of relatively low Reynolds number and high angle of attack (Fig. 6). The effects of this separation will be discussed later. Figure 6: Vortex shedding from the upper side of the hydrofoil. NUMERICAL MODELING Navier-Stokes solver for the two-phase flow The two-phase flow is modeled as the flow of a single fluid whose density and viscosity smoothly changes across the interface. By assuming both phases to be incompressible, in an Eulerian frame of reference the local fluid properties changes with time only due to the interface motion. If surface tension and turbulence effects are neglected, the unsteady non-dimensional Navier-Stokes equations for an incompressible fluid in generalized coordinates are: (1) (2) where ui is the i-th cartesian velocity component and δij is the Kronecker delta. The quantity (3) is the volume flux normal to the ξm iso-surface and J−1 is the inverse of the Jacobian. In Eq. (2) the gravity term is written in non-dimensional form, being the Froude number and Ur and Lr reference values for velocity and length, respectively. In the diffusive term is the reference Reynolds number being µ w the values of density and dynamic viscosity in water that are also used as reference values. The quantity (4) is the mesh skewness tensor. The numerical solution of the Navier-Stokes equations is achieved through a finite difference solver on a non staggered grid similar to that suggested by Zang et al. (1994). Cartesian velocities and pressure are defined at the cell centers whereas volume fluxes are defined at the mid point of the cell faces. For the computation of the convective terms and to enforce the continuity, fluxes at cell faces are evaluated by using a quadratic upwind scheme (QUICK) to interpolate cartesian velocities. The momentum equation is integrated in time with a semi-implicit scheme: explicit terms are computed with a variable time step Adam-Bashfort scheme while a Crank-Nicolson discretization is employed for the implicit terms. Since the grid is time independent, the discretized form of Eq. (2) is (5) where ∆t=tk+1−tk and represents the convective terms at the step k, Ri is the gradient operator in the authoritative version for attribution. curvilinear coordinates, DI and DE are the diagonal and off-diagonal diffusive operator. The use of an explicit scheme for the convective terms limits the time step: this is chosen so that the maximum Courant number all along the computational domain is

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This means that when refining the grid the solution changes not only due to the accuracy in the description of the fluid flow but also due to the changes in the body shape. This question will be discussed more deeply when analysing the numerical results. Domain decomposition For the purposes of the present work, we focus our attention toward the processes near the free surface. According to this, an accurate description of the flow field about the body is not really needed. Furthermore, when considering high Reynolds number flows, the full Navier-Stokes approach becomes very expensive and, moreover, a turbulence model should be included to accurately predict the flow about the body. The above consideration suggested to develop a zonal approach, decomposing the fluid domain in an upper region, near the free surface, and a lower region, where the body is located. In the body (lower) region, an inviscid flow model is assumed. The flow is described via a velocity potential and a suitable Kutta condition can be used to describe the rotational flow about the hydrofoil. In the free surface (upper) region, the flow is described by the Navier-Stokes equations. This decomposition allows to concentrate the computational effort, to a great extent devoted to the solution of the Navier-Stokes equations, in a small domain enclosing the free surface (see Fig 7). Figure 7: The decomposition of the computational domain in a lower (body) region, computed via a BEM solver, and an upper (free surface) region, where the Navier-Stokes solver is coupled with a Level Set technique for solving the air-water flow. Viscous and the inviscid rotational solutions are fully coupled at each time step, with a simple and effective procedure. The potential flow region is resolved first. Neumann boundary conditions are applied on the two sides of the computational domain (inflow and outflow of the body region), on the bottom and on the body contour, while a Dirichlet boundary condition is applied onto the matching surface and a Kutta condition is imposed at the trailing edge of the hydrofoil. The solution of the flow in the body region provides the normal and tangential velocity components at the matching surface. This velocity is used as a boundary condition for the Navier-Stokes solver in the free surface region. At the end of the advancement in time the Navier-Stokes solver provides the pressure field on the matching surface that is used to update the velocity potential via the unsteady Bernoulli's equation. In the following, additional details about the potential solution and the coupling procedure are discussed. Although the coupling procedure here suggested could work even in the three-dimensional case, details below refers to the two- dimensional case. The potential domain is limited on the top by the matching surface, on the two sides by the inlet and outlet vertical sections, and by the solid boundaries, that is by the body and/or the bottom. When the flow about an hydrofoil is investigated, a Kutta condition is enforced at the trailing edge. To this aim, a vortex line, with a uniform distribution of the vorticity density γ, is introduced within the hydrofoil, ranging from the leading edge to the trailing edge. The vorticity density is fixed, so that the average of the velocities at the midpoint of the two panels at the trailing edge is parallel to the vortex line. A further simplification is introduced, in that the vortex shedding, characterizing the initial transient, is not accounted for in this model. This simplified model is acceptable when attention is mainly focused in the final quasi-steady solution. As stated above, the velocity potential is assigned on the matching surface by integrating the unsteady Bernoulli's equation that, in a frame of reference attached to the body, takes the following form: (16) where p is the pressure value coming from the Navier-Stokes solution, is the velocity potential in the absolute frame of reference, uv is the velocity field induced by the vortex and UB is the velocity of the frame of reference, that is the authoritative version for attribution. attached to the moving body or to the moving bottom. All along the other boundaries, the normal derivative of is assigned. On the moving bottom and/or on the

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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 NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 756 wavelength, (ii) poor convergence in terms of wave amplitude, (iii) a phase shift similar for all the grids and (iv) a spike in the wave profile at x=6 for the medium grid result. Investigating about these points, attention has been focused on close up views of the velocity field in the leading edge region, obtained with the three grids, depicted in Fig. 13. The velocity field in this region shows relevant changes when the grid is refined. In fact, during this process, due to the zero order model used for assign body forces, substantial differences in the computational shape of the solid boundary occurs. Nevertheless, although the above limit of the body force approach can justify the poor convergence, the phase shift is almost the same for all the grids. As a carefull look at the vorticity shedded from the trailing edge of the hydrofoil reveals (Fig. 14), the interactions between the vorticity field beneath the free surface and the wave profile (with some dipole rising up toward the free surface) are responsible for the spike in the wave elevation at x=6. The flow separation occuring from the suction side, can significantly alter the wave amplitude and phase. Figure 14: Vorticity contours behind the hydrofoil: from the top to the bottom t=18, t=20, t=22 On the basis of the above consideration, a fair comparison with experiments performed at high Reynolds number cannot be established unless a turbulence modeling, beside to an improved body force formulation, is introduced. DD results The problems encountered in the correct prediction of phase and wave amplitude, lead to the development of the domain decomposition approach that uses the inviscid rotational flow model and the boundary element technique to describe the flow about the lifting body whereas uses the Navier-Stokes solver coupled with the Level-Set technique to describe the flow in the free surface region, where complex interface topologies may develop in breaking condition. Numerical simulation have been carried out by assuming the matching surface at y=−0.2. In the Navier-Stokes region a 256x96 grid is employed with grid points suitably clustered about x=1 in the horizontal direction and about y=0 in the vertical direction. In this case a uniform vertical grid spacing ∆y=0.005 is used in y ∈ (−0.2, 0.2), whereas δ=0.03 is used as the half width of the jump. In Fig. 15, the free surface profile obtained by the DD approach is compared versus the fully non linear BEM result and with the experimental data by Duncan (1983) for the same conditions as before. Figure 15: Comparison among the free surface profiles obtained by the DD approach (solid), the full BEM (dashed) and the experimental data by Duncan (1983) (dot) With respect the FNS results, the DD allows a much better description of the first trough, even though an excessive damping of the following waves appears, and wave phase and lenght are in good agreement with experiments too. As already stated, reasons for this excessive damping are not yet really understood, although it is believed to be related to the explicit integration of the Bernoulli's equation and to the approximation of the pressure field given by Eq. (9). However, it has been verified that, as long as the matching surface is deep enough with respect to the wave troughs, its position does not significantly affects the solution. 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 NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 757 Submerged hydrofoil: breaking regime In spite of the limits of the numerical solver, simulations of the breaking wave produced by an hydrofoil have been attempted, aimed at the verification of the capability of the solver to predict and model the breaking wave phenomena. For a non-dimensional submergence of 0.9113, Duncan (1983) observed that a weak spilling breaker is present, reducing the following wave system. The comparison between the numerical solution and the experimental data is reported in Fig. 16. It shows that the amplitude of the first crest is overpredicted and also the following wave system is not damped, but for numerical effects. However, the close up view shows that a good agreement is achieved in terms of the slopes of the front and back of the first wave, meaning that something is occurring. Differences are due to the low resolution adopted in the numerical simulations, that is the wave tries to break but the resolution does not allow to correctly capture the establishment of the breaker. As a consequence, the dissipative effects played by the breaker on the following wave are not modeled. Figure 16: Top: Wave profile behind a hydrofoil at a non-dimensional submergence 0.9113. Bottom: Close up view of the first wave. In this condition, a breaking condition was observed by Duncan (1983) (dots). The adopted grid resolution allow numerical solutions to (solid) capture the asymmetry but is not able to fully resolve the breaker region. If the hydrofoil submergence is further reduced, the intensity of the wave breaking grows. For a non-dimensional submergence 0.783, Duncan (1983) observed an intense wave breaking with a high dissipative effect on the following wave. The comparison between the numerical solution and the experimental data for this submergence is reported in Fig. 17: in this computation, due to the more pronunced wave trough, the matching surface has been located at y=−0.36. As for the previous case the first crest is largely overpredicted and no dissipative effects on the following wave are predicted. Also in this case, however, the strong asymmetry of the first wave can be noted. Figure 17: Wave profile behind a hydrofoil at a non-dimensional submergence 0.783. An intense breaking was observed by Duncan (1983) (dot) whereas the poor resolution does not allow the numerical approach (solid) to capture neither the breaker nor the dissipative effects on the following wave. the authoritative version for attribution. Figure 18: Dynamic pressure distribution in non-breaking (top) and breaking condition: 0.911 (center), 0.783 (bottom). Even though the free surface elevation is the most obvious quantity to check, additional important information can be provided by looking at the dynamic pressure field. In Fig. 18 the contours of the dynamic pressure are shown for the non breaking case and for the two breaking case.

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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 NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 758 It is worth to notice that, despite the poor resolution, the distribution of the dynamic pressure changes significantly passing from the non-breaking to the breaking stage. Unfortunately, the poor resolution does not allow the water to splash down and to form the breaker region. CONCLUDING REMARKS Numerical and experimental studies have been carried out at INSEAN on the 2D wave breaking flow produced by a towed hydrofoil. Numerical simulations of the two phase flow in air and water have been performed, and validation studies have been conducted in the case of the flow on a bottom bump, producing a plunging breaker. The formation of the jet, the splash-up and the post breaking event are in a qualitative agreement with experimental observation. Hence, the flow past the submerged hydrofoil has been simulated and the condition for the onset of the breaking have been investigated. This has resulted in a detailed analysis of the computed velocity and pressure fields. A Navier-Stokes solver in generalized coordinates, together with a Level Set technique, used to follow the free surface dynamics, has been used and two different numerical codes have been developed, based on the body force and on a domain decomposition approaches respectively. The body forces approach has been developed in view of dealing with the flow about multibody configurations as it is the case of a ship with appendages. This approach has proved to be useful, although a higher order model for the assignement of the body forces, and hence of the way in which the shape of the body is represented, is needed. In order to gain insight into the dynamics of the free surface, the domain decomposition approach has proved to be promising, focusing attention and computational efforts in the free surface region. With reference to the quasi-steady breaking produced by the hydrofoil, numerical results discussed here suggest that, although the numerical techniques are able to detect the inception of the breaking, the adopted grid were too coarse to resolve the flow. Possible extension of the work is the inclusion of surface tension effects, allowing a comparison with the set of experimental data. The emphasis of the experimental work is on understanding the conditions under which capillary waves may force the breaking on the folowing wave train, subsequently forcing the extension of the breaking area to the forward waves. The work is largerly under development and a new system has already been designed for reproducing the experiments, making also quantitative measurements. ACKNOWLEDGEMENTS This work was supported by the Ministero Trasporti e Navigazione in the frame of the INSEAN research plan 2000– 02. REFERENCES ——, Proceedings of the ONR 2000 free surface turbulence and bubbly flows workshop, California Institute of Technology, Pasadena (USA), 2000. Adalsteinsson D. and Sethian J.A., The fast construed on of extension velocities in level set methods, J. Comput. Phys., vol. 148, 2–22, 1999. Azcueta, R., Muzaferija, S., Peric, M. and Yoo, S.D., Computation of flows around hydrofoils under the free surface, Proceedings of the 7th Int. Conf. on Num. Ship Hydro., ed. Office of Naval Research, Nantes, FRANCE, 1999. Battjes J.A., Sakai T., Velocity field in a steady breaker, J. Fluid Mech., vol. 111, 421–437, 1981. Bonmarin P., Geometric properties of deep-water breaking waves J. Fluid Mech., vol. 209, 405–433, 1989. Cointe R., Tulin M., A theory of steady breakers, J. Fluid Mech., vol. 276, 1–20, 1994. Dommermuth D., Innis G., Luth T., Novikov E., Schalageter E. and Talcott J., “Numerical simulation of bow waves,” Proceedings of 22nd Symposium on Naval Hydrodynamics, Office of Naval Research, 1998. Dommermuth D., Mui R., The vortical structure of a wave breaking gravity-capillarity wave, Proceedings of 20th Symposium on Naval Hydrodynamics, ed. Office of Naval Research, Washington D.C.., 1994. Dong R.R., Katz J. and Huang T.T., On the structure of bow waves on a ship model, J. Fluid Mech., vol. 346, 77–115, 1997. Duncan, J.H., An experimental investigation of breaking waves produced by a towed hydrofoil, Proc. R. Soc. Lond., vol. A 377, 331–348, 1981. Duncan, J.H., The breaking and non-breaking wave resistance of a two-dimensional hydrofoil, J. Fluid Mech., vol. 126, 507–520, 1983. Duncan J.H., Dimas A.A., Surface ripples due to steady breaking waves, J. Fluid Mech., vol. 329, 309–339, 1996. Duncan J.H., Philomin V. and Qiao H., The transition to turbulence in a spilling breaker, Proceedings of the 20th Symposium on Naval Hydrodynamics, ed. Office of Naval Research, Washington, D.C., 1994. Iafrati A., Campana E.F., A level-set technique applied to complex free surface flows, Proceedings of ASME FEDSM'00, Boston (MA), USA, 2000. Iafrati A., Di Mascio A. and Campana E.F., A level-set technique applied to unsteady free surface flows, Int. J. 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 NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 759 for Num. Meth. in Fluids, in press, 2000. Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S. and Zanetti, G., Modelling merging and fragmentation in multiphase flows with SURFER, J. Comput. Phys., vol. 113, 134–147, 1994. Longuett-Higgins M.S., Capillary rollers and bores, J. Fluid Mech., vol. 240, 659–679, 1992. Longuett-Higgins M.S., The initiation of spilling breakers, Int. Sym. Waves—Physical and Numerical Modeling, U. of British Columbia, Vancouver, Canada, 1994. Longuett-Higgins M.S., Shear instabilities in spilling breakers, Proc. R. Soc. Lond., vol. A 446, 399–409, 1994. Lin J.-C., Rockwell D., Evolution of quasy-steady breaking wave, J. Fluid Mech., vol. 302, 29–44, 1996. Miyata H., Inui T. Nonlinear ship waves, Adv. Appl. Mech., vol. 24, 215–288, 1984. Mori, K.-h., Sub-breaking waves and critical condition for their appearance, J. Soc. Nav. Arch. Japan, vol. 159, 1–8, 1986. Muzaferija, S., Peric, M., Sames, P. and Schellin, T., A two-fluid Navier-Stokes solver to simulate water-entry, Proceedings of the 22nd Symposium on Naval Hydrodynamics, ed. Office of Naval Research, Washington, D.C., 1998. Peregrine D.H., Svendson, I.A., Spilling breakers, bores and hydraulic jumps, Proceedings of 16th Coastal Engng Conf, Hamburg, Germany, 1978. Rosenfeld M., Kwak D. and Vinokur M., A fractional step solution method for the unsteady incompressible Navier-Stokes equations in generalized coordinate systems, J. Comput. Phys., vol. 94, 102–137, 1991. Sussman M., Smereka P. and Osher S., A level-set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., vol. 114, 146– 159, 1994. Unverdi S.O. and Tryggvason G., A front-tracking method for viscous, incompressible multi-fluid flows, J. Comput. Phys., vol. 100, 25–37, 1992. van der Vorst, H.A., Bi-CGSTAB: A fast and smoothly converging variant to Bi-CG for the solution of nonlinear system, SIAM J Sci. Statist. Comput., vol. 13, 631–644, 1992. Vogt, M. and Larsson, L., Level set methods for predicting viscous free-surface flows, Proceedings of the 7th Int. Conf. on Num. Ship Hydro., ed. Office of Naval Research, Nantes, FRANCE, 1999. Zang, Y., Street, R.L. and Koseff, J.R., A non-staggered grid, fractional step method for time-dependent incompressible Navier-Stokes equations in curvilinear coordinates, J. Comput. Phys., vol. 114, 18–33, 1994. 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 NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 760 DISCUSSION J.H.Duncan Naval Surface Warfare Center, Carderock USA I find the photographs in Figure 5 particularly interesting. In my experiments with surface-tension-dominated unsteady breakers with wavelengths of about 1 meter, a bulge forms at the crest with capillary waves upstream of the leading edge (toe) of the bulge. The transition to turbulent flow seems to be initiated by flow separation at the toe. In your photographs, it appears that the transition to turbulent flow is initiated in the region of capillary waves upstream of the toe. In particular, it appears that the capillary waves are breaking first. Would the authors please comment further on this phenomenon. It would also be very interesting if they can provide a qualitative description of the temporal evolution of the free surface after the first appearance of these breaking capillary waves. In the numerical calculations, the authors have attacked an exceedingly difficult problem. As they pointed out, increased resolution will be needed to accurately compute the flow. However, given the present resolution, the authors have compared the behavior of the following wavetrain under breaking conditions to experimental measurements. Have they also examined the vertical distribution of horizontal velocity in the following wavetrain to look for evidence of the wake found near the free surface in the experiments? AUTHOR'S REPLY We thank Prof. Duncan for the questions and for calling our attention to some flow details that deserve some more comments. About the different breaking mechanisms of the capillary waves, a possible explanation lay in the different water quality. Indeed, in the experiment carried out by Prof. Duncan, the quality of the water is frequently cleaned through filtering and the value of the surface tension is assessed with great accuracy. On the other side, being the present experiment carried out in a large towing tank (220 m long), the presence of dust on the free surface cannot be avoided, as it may be seen in Fig. 5. This can cause the growing of instabilities of the capillary wave front, eventually leading to the difference in the breaking event. A qualitative description of the observed temporal evolution of the free surface, after the appearance of capillary waves, is sketched in Fig. 19 below. Fig. 19—Sketch of the 3-dimensional instabilities (top view). The arrow shows the velocity of the hydrofoil (represented with a thick black ribbon), the solid (dashed) lines represent the gravity (capillary) waves. If the depth of the hydrofoil is large enough, some capillary waves appear on the forward face of the second and third crests (Fig. 19a). When the depth of the hydrofoil is reduced, three dimensional instabilities appear (Fig. 19b), eventually leading to wave breaking (Fig. 19c). Finally, depending on the depth, the breaking may also propagate to the first crest. Concerning the last question raised by Prof. Duncan, due to the poor resolution used in the calculation here presented, the computed wake past the breaking cannot be seen. However, a calculation for the bump case with a more refined grid (640x256) at low Reynolds number (Re=1000) has been carried out. Results show that a slackness is operated the authoritative version for attribution. by the breaker (Fig. 20) at least beneath the first crest.

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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 NUMERICAL AND EXPERIMENTAL STUDY OF THE WAVE BREAKING GENERATED BY A SUBMERGED HYDROFOIL 761 Fig. 20—Vorticity contours for a spilling breaking condition. The black ribbon represents the free surface location and the distribution of the velocity is shown along some vertical lines. Fig. 20 also shows the intense counter-clockwise vorticity originating close to the toe of the bulge. Due to the low Reynolds number, vorticity is rapidly diffused into the fluid domain. DISCUSSION D.Dommermuth Science Applications International Corp., USA The authors have developed a unique procedure for modeling breaking waves. Could they please compare the domain decomposition method that is described in their paper to the Schwarz alternating method (Schwarz, 1890)? AUTHOR'S REPLY We thank Prof. Dommermuth for the interesting question. The Domain Decomposition (DD) approach we have used in the paper does not need an overlapping region and is assigned on the matching surface. To apply the Schwarz alternating method an overlapping is needed and the normal component of the velocity must be exchanged between the subdomains. In contrast with the former approach, the latter algorithm does not require an explicit time integration for the exchanged variable. Nevertheless, some subiterations are necessary, whose number depend on the extension of the overlapping region. As a consequence, an a priori comparison of the two different approaches in terms of computational efficiency do not permit to establish which is the best choice. The development of the Schwarz method, in order to compare the two approaches in terms of CPU time and accuracy, is a part of ongoing activity. the authoritative version for attribution.