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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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 1 Modern Seakeeping Computations for Ships R.Beck (University of Michigan) A.Reed (David Taylor Model Basin, Carderock Division, Naval Surface Warfare Center) ABSTRACT Current computational methods for solving seakeeping problems of ships with forward speed are reviewed. A brief historical perspective is given to show the interdependency and development of the different ship motion theories that are presently being used. These are placed in context by a discussion of the taxonomy of seakeeping computations relative to the fully-nonlinear incompressible free-surface viscous flow problem. The state-of-the-art in computational seakeeping of ships is discussed. In general, the accuracy of the solution must be balanced against the computational effort. The advanced codes give more detailed and better solutions, but they require super computers or the equivalent. Fully and partially nonlinear inviscid computations for wave diffraction, and added mass and damping are described and a few examples are provided to illustrate the impact of the various levels of complexity of the calculations on the accuracy of results compared to experimental results. Finally, a series of state-of-the-art issues are raised: computationally efficient numerical methods, large amplitude motions and capsizing, horizontal plane motions (coupling between seakeeping and maneuvering), finite depth in the littorals, and validation and verification of codes for extreme motions. 1 INTRODUCTION Modern seakeeping computations are used in all aspects of engineering for the marine environment. They have become a standard design tool; they are used in simulators; and they are used operationally to predict the motions of a vessel in real time. Modern seakeeping computations are performed using a wide variety of techniques—from simple strip theory to extremely complex fully nonlinear unsteady RANS computations. To cover all aspects would require a book, not a short paper. Consequently, we are going to limit the discussion to ships at forward speed. This largely eliminates any discussion of the computational techniques developed by the offshore oil industry in order to compute wave loads and motions of offshore structures. We do not want to minimize the contributions of the offshore industry which have been substantial (some might even argue that modern computational techniques have been driven by the needs of the offshore industry), but the focus of this symposium is naval hydrodynamics with its emphasis on ships at forward speed. Modern seakeeping computations are far from a mature engineering science. There are several aspects to ship seakeeping that make it one of the most challenging problems in the marine hydrodynamics field. It has all the complexities of wave resistance or maneuvering problems with the addition of unsteadiness due to incident waves. The ultimate goal, of course, is a unified theory of resistance, maneuvering, and seakeeping. Historically and for a variety of reasons, each of the fields have developed independently. At present, they are still separated and it will probably be twenty years before computations are truly unified. Unfortunately, design problems will not wait and designers are constantly pushing for better computations. In this paper, we want to summarize the present state-of-the-art in seakeeping computations and then point out major research issues that need to be addressed. The major difficulties in seakeeping computations are the nonlinearities. There are nonlinearities associated with the fluid in the form of viscosity and the velocity squared terms in the pressure equation. The free surface causes nonlinear behavior due to the nature of the free-surface boundary conditions and the nonlinear behavior of the incident waves. Finally, the body geometry often causes nonlinear hydrostatic restoring forces and nonlinear behavior at the body/free- surface intersection line. The only good news is that because of forward speed ships tend to be long and slender with smooth variations along their length. This geometric feature of typical ships is the basis of many approximations that have allowed a significant amount of progress to date. Recently, seakeeping computations for ships operating in the littoral region have become of interest. Offshore computations are often done in finite depth, but it is unusual for ships. Most theories and computations have been for infinitely deep water. Many theories could be extended to finite depth in a relatively straightforward manner. For example, replacing the deep water Green function with a finite depth Green function can extend 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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 2 linear strip theory ship motion programs. Unfortunately, because of the nonlinearities associate with shallow water waves, current effects, and non-uniform bottom topography, the linear predictions may not be accurate. Specialized approaches will eventually have to be developed. Throughout this paper we shall assume that water is incompressible and the density is constant. The compressibility of water may be an important factor in underwater explosions and impact problems, but for general seakeeping studies the incompressible assumption is sufficient. On the other hand, water in the ocean does not have constant density. Under limited circumstances, the internal waves that are set up because of the density gradient in the water can have an influence on ship performance. However, in typical situations the density variation in the vicinity of the ship is negligible and the constant density assumption is justified. This paper starts with an historical review of approaches to seakeeping predictions and a taxonomical discussion of the various approximations that are made to obtain tractable seakeeping problems for solution. This is followed by a discussion of contemporary calculation methods, which begins with a discussion of the seakeeping viscous flow and potential flow boundary value problems, various approximations to the solution of the potential flow problem, examples of some of these solutions, and a discussion of the derived quantities: structural loads, green water on deck, and added resistance in waves. Finally the paper concludes with a discussion of major research issues: efficient numerical methods, large amplitude motions and capsizing, horizontal plane motions (coupling between seakeeping and maneuvering), finite depth in the littorals, and verification and validation. 2 BACKGROUND This review begins with an historical review of the computational approaches to the seakeeping of ships. These are placed in context by a discussion of seakeeping fluid dynamics problems as a taxonomy, starting with the most general incompressible fluid dynamics problems and progressing through a sequence of approximations and assumptions resulting in more and more tractable problems, which may or may not successfully model the physical reality. Historical Approaches to Seakeeping The computation of ship motions has a long history starting with Froude's (Froude 1861) original work on rolling. Detailed histories of the development can be found in many sources including Newman (1978), Maruo (1989) and Ogilvie (1977). Modern computations began with two developments in the 1950's. The first was the use of random process theory to determine the statistics of the ship responses in a seaway. The second was the development of linear ship motion theories to predict the responses of the ship to regular waves. The seminal paper of St. Denis and Pierson (1953) proposed a method to predict the statistics of ship responses to a realistic seaway. Using spectral methods developed in other fields, they related the spectral density of ship responses to the input ocean wave spectrum. Two assumptions are critical: 1) the sea surface is an ergodic, Gaussian random processes with zero mean and 2) the ship can be represented by a linear system. The first assumption enables the probability density function of the ship responses to be completely characterized by the variance, which is simply the area under the spectral density of the response. Once the probability density function for a given response is known, all the desired statistics of the response can easily be determined. The linear system assumption allows the spectral density of any given response to be found by multiplying the incident wave spectrum by the square of the response amplitude operator (or RAO) of the desired response. In other fields, the RAO is often called the transfer function or the linear system function. At any single frequency, the RAO is the amplitude and phase of the desired response to regular incident waves acting on the vessel at the given frequency. In order to use the St. Denis and Pierson approach, the input wave spectrum and the RAO's for the vessel must be known. Having good wave spectral information is critical in order to obtain good ship response estimates. Naval architects usually rely on oceanographers to provide this information and much research has been done in the area. New satellite tracking techniques are being developed that will allow real time wave spectral estimates for any point in the ocean. Because of limited space, we will not discuss wave spectra in this review; it will be assumed that the necessary wave spectra and/or wave time histories are available. The RAO's can be determined either experimentally or analytically. Almost all of the analytic work has neglected viscosity and used potential flow. Except for some empirical viscous corrections, seakeeping computations have all been potential flow until approximately the last five years. The 1950's saw the start of the development of analytic prediction techniques. The first theories built on the thin-ship approximation of Michell (1898). The thin-ship approximation assumes that the beam of the ship is small relative to the length and draft. The thin-ship approximation was examined critically by Peters and Stoker (1957). They used a systematic perturbation procedure with the ship's beam and unsteady motions were assumed to be of the same small order of mag 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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 3 nitude. The first-order theory was rather trivial in that it balanced hydrodynamic forces due to the undisturbed incident wave pressure field (the Froude-Krylov exciting force) and the hydrostatic restoring forces with the ship's mass times acceleration term. Thus, to first order there is an unbounded resonance in heave, pitch and roll because of the lack of hydrodynamic damping. Newman (1961) avoided the shortcomings of Peters and Stoker by introducing refinements using a systematic expansion in multiple small parameters and a more accurate statement of the body boundary condition. Computed results from his theory did not compare well with experiments. The problem is that typical ship hulls with both the beam and draft small relative to the length are closer to slender bodies than thin ships. Although slender-body theory has been used in aerodynamics since Munk (1924) studied the flow around airships, it was not until the 1950's that slender-body theory was applied to ships, first to steady forward motion and then to unsteady motions. Rigorous slender-body theories were originally developed by several researchers (Joosen 1964, Newman 1964, Newman and Tuck 1964, Mauro 1966) using a long wave assumption that the incident wavelength is on the order of the ship length. Unfortunately, as with thin-ship theory, most nontrivial hydrodynamic effects are higher order compared to the Froude-Krylov exciting force and the hydrostatic restoring force. Moreover, to leading order the predicted motions are non-resonant because the inertial force due to the body mass is of higher order. At the same time that the long wave slender-body theories were being investigated, an alternative strip slender-body was being also being studied. Korvin-Kroukovsky (1955) (or a sequel by Korvin-Kroukovsky and Jacobs (1957)) did the initial work. Using a combination of slender-body theory and good physical insight, they developed a theory for heave and pitch that was suitable for numerical computations on the newly emerging digital computers. Strip theory was the first ship motion theory that gave results with enough engineering accuracy that the predicted motions were useful for design. A modified strip theory of Gerritsma and Beukelman (1967) was shown to give good agreement with experiments for head seas. In the late 1960's more comprehensive strip theories were developed by several researchers; most widely cited is Salvesen, et al. (1970). Using a combination of mathematics and judicious assumptions, these researchers ingeniously arrived at a form of strip theory that today is still the most widely used method for seakeeping computations of ships. A mathematically consistent approach to strip theory was developed by Ogilvie and Tuck (1969) [or see Ogilvie (1977)]. They made a short wavelength approximation and carried out a systematic analysis for the slender-body problem to determine the added mass and damping in heave and pitch. At zero speed the results reduce to pure strip theory. Many of the forward-speed correction terms are similar to Salvesen, et al. (1970) but there are also some integral terms over the free surface that make evaluation of the Ogilvie-Tuck coefficients very difficult to compute. The rational approach to strip theory also involves changes in the formulation for the diffraction exciting forces. Because of the high frequency (short- wavelength) of the incident waves, the diffraction potential is no longer slowly varying along the ship length. A solution must be sought as a product of a highly oscillatory longitudinal function times a slowly varying solution of the Helmholtz equation. Troesch (1976) examined the case in non-head seas. For head seas the problem is singular and special analysis is required (cf. Faltinsen 1972, Maruo and Sasaki 1974, or Ogilvie 1977). Strip theory is a short wavelength theory and slender-body theory is a long wavelength theory. Attempts have been made to bridge the gap and find a theory that was valid over a wider frequency range. The interpolation theory of Maruo (1970) and the unified theory of Newman (1978) are typical examples. For short wavelengths the results reduce to strip theory and for long wavelengths the results of slender-body theory are recovered. The velocity potential in the inner region includes a particular solution that is equivalent to the strip theory result and a homogeneous component that after matching with the outer solution accounts for interactions along the hull length in a manner similar to long wave slender-body theory. Comparisons with experimental results by Sclavounos (1990) have indicated improved predictions relative to strip theory predictions. Recent work to be presented by Kashiwagi, et al. (2000) at this Symposium shows that for a VLCC 1 the unified and strip theories give essentially equivalent predictions for heave and pitch motions at a variety of heading angles. The vertical bending moments at station five of a container ship are slightly better predicted by unified theory. The next developments had to await the arrival of faster and larger computers. Using added mass and damping tables, ship motion predictions from Korvin-Kroukovsky's original strip theory could be calculated by hand. Since then, the sophistication of seakeeping theories has paralleled the growth of computational power. At times the available computational power was greater than our ability to use it productively and at other times researchers have been waiting for larger and faster computers. Today, the most advanced techniques are beyond the capacity of readily available computers and wide spread verification will have to await further increases the authoritative version for attribution. 1Very Large Crude Carrier

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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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 4 in computational power. By the late 1970's the Neumann-Kelvin approach was starting to be used. In the Neumann-Kelvin approach the body boundary condition is applied on the mean position of the exact body surface and the linearized free-surface boundary condition is used. The traditional approach to solving the Neumann-Kelvin problem is to use boundary integral methods in which the solution is formulated in terms of integrals of fundamental singularities (source and dipoles) over the surface surrounding the fluid domain. Normally, the integral equation would have to be applied over all surfaces surrounding the fluid domain. However, by combining the fundamental singularities with other analytic functions, it is possible to develop Green functions that satisfy all the boundary conditions of the problem except on the body surface. In this case, the governing integral equation need only be solved on the body surface. For wave problems, free-surface Green functions have been established for many different cases (for example cf. Wehausen and Laitone 1960, Newman 1985a, Telste and Noblesse 1986). In general, the greater the complexity of the problem, the more difficult it is to evaluate the Green function. For example, finite depth Green functions are harder to compute than infinite depth Green functions; evaluation of forward-speed Green functions requires more effort than zero forward speed. Hess and Smith (1964) pioneered boundary element methods for flows without a free surface (equivalent to a double- body flow with a rigid free surface). Using just a source distribution, they subdivided the body surface into N flat quadrilaterals over which the source strength was assumed constant. Satisfying the body boundary condition at the center of each quadrilateral (also called a node, control, or collocation point) resulted in a system of N linear equations for the unknown source strengths. By knowing the source strength, the velocities and pressure at each control point can easily be determined. The flat quadrilaterals were often called panels and now the term “panel methods” has come to mean any solution technique in which the body surface (and possibly other surfaces of the problem) has been subdivided. Higher order panel methods involve the use of panels that are not flat and/or singularity distribution strengths that are not constant over a panel. A Galerkin procedure can be used to satisfy the integral equation in an integrated sense over each panel. In panel methods, two tasks require almost all of the computational effort. The first is setting up the influence matrix that requires multiple evaluations of the Green function for the problem. The second is solving the resulting linear system of equations. For small problems, direct solvers such as L-U decomposition work fine. As the problem becomes larger, an iterative technique such as GMRES (Saad and Schultz 1986) is more appropriate. Direct solvers require on the order of N3 operations while iterative solvers are on the order of N2. However, there is a set up time and thus direct solvers work better for small problems; the exact trade off point depends on the computer system and the specific program. For very large problems order N methods such as fast multipole acceleration (cf. Scorpio and Beck 1998) or pre-corrected Fast Fourier Transform (cf. Kring, et al. 1999) may be necessary. An optimized numerical approach will balance the number of panels, the time spent setting up the influence matrix, and the cost of solving the system of linear equations in order to obtain a desired level of accuracy. It should be pointed out that boundary element methods, while the most popular, are not the only methods avail able to solve the Neumann-Kelvin problem. Examples of finite element or finite difference approaches are given by Bai, et al. (1992) or Wu, et al. (1996). These methods have significantly more unknowns, but the matrix that must be inverted is very sparse. The total computational effort and accuracy of the solution relative to panel methods depends on the details of the code. The Neumann-Kelvin approach was first used by the offshore industry since strip theory could not possibly work for the vessel geometry typically used in the exploration and production of offshore oil and gas. The original codes used lower order panel methods and the zero-speed, free-surface Green function in the frequency domain. Several commercial codes are available, the first probably being Garrison (1978) and the most widely used is WAMIT (Korsmeyer, et al. 1988). The codes have been extended to include second order mean drift and slowly vary forces. The difficulty in extending the offshore work to ships is the forward speed. The forward-speed free-surface Green function in the frequency domain is extremely difficult to compute. The first attempt was by Chang (1977), with later work by Inglis and Price (1981), Guevel and Bougis(1982), Wu and Eatock-Taylor (1987), and Iwashita and Ohkusu (1992). Chen, et al. (2000) are presenting more recent work at this Symposium. An alternative to working with the frequency-domain Green functions is to work in the time domain. The original work on the time-domain Green function is credited to Finkelstein (1957). For fully linear problems at constant or zero forward speed, the time-domain and frequency-domain solutions are related by Fourier transforms and are, therefore, complementary (for examples of time domain computations see Beck and Magee 1990, Bingham, et al. 1994, or Korsmeyer and Bingham 1998). Working in one domain or the other might have advantages for a particular problem. The time domain requires the evaluation of convolution integrals over all 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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 5 previous time steps; this takes both computer time and memory. The time-domain Green function is similar to the zero- speed frequency-do main Green function and its evaluation requires approximately the same amount of effort. At zero speed the conventional frequency domain computations are faster because of the convolution integrals. These integrals require many time steps for adequate resolution where as the frequency domain requires only a few frequencies. However, at forward speed the situation is reversed. In the time domain the Green function does not change and run time is approximately the same as zero forward speed. In the frequency domain, the forward-speed Green function is much more complex with greatly increased computer time. An inconsistent, but further refinement to the Neumann-Kelvin problem is to satisfy the hull boundary condition on the exact wetted surface of the body while retaining the linearized free-surface boundary condition. This body-exact problem is a time variant linear system and the frequency domain and the time domains are no longer simply related. Except for some very simple cases, the body-exact problem must be solved in the time domain. The hydrodynamic forces acting on a vessel undergoing sinusoidal motions are no longer simply sinusoidal; the results typically have a mean shift with the presence of second order and higher harmonics. Beck and Magee (1990), Magee (1994), Lin and Yue (1990), or Shin, et al. (1997) give examples of this approach. The Neumann-Kelvin and body-exact approaches are linearizations about the free-stream velocity. This is not the only possibility. In the so-called double-body or “Dawson's Approach” (Dawson 1977, Sclavounos 1996), the linearization is about the double-body flow. The boundary conditions on the body remain the same as in the Neumann-Kelvin approach but the free-surface boundary conditions are significantly altered. Because the free-surface boundary conditions are a function of the geometrically dependent double body flow, a single free-surface Green function is no longer applicable. The resultant body value problem is typically solved using a distribution of simple Rankine sources over both the body and calm water surfaces. Nakos and Sclavounos (1990a, 1990b) are examples of the method applied to seakeeping problems. Bertram (1998) gives a variant of the method in which he uses the calm water free surface and potential as the basis flow. As the body-exact approach is a refinement of the Neumann-Kelvin method, the weak scatter hypothesis of Pawlowski (1992) is a further refinement of the Dawson approach. Assuming the ship disturbance is small relative to the incident waves, the linearization of the ship generated wave disturbance can be done around the ambient wave profile with a body-exact condition on the ship hull. Sclavounos, et al. (1997), and Huang and Sclavounos (1998) have used this method in the SWAN 2 code. Both the body-exact and weak scatter approaches to seakeeping computations treat the body boundary condition properly, but the free-surface boundary condition has been “linearized” in some sense. A third alternative is to keep the fully nonlinear free-surface boundary conditions. Fully nonlinear computations can be performed in a variety of ways. For steady forward motion, an iterative procedure can be used in which the boundary conditions are initially applied on the calm water plane and the solution iterated until the fully nonlinear conditions are satisfied on the exact free surface. Convergence of the iteration procedure can be a problem but successful solutions have been obtained by among others Jensen, et al. (1989), Raven (1993, 1998), Scullen and Tuck (1995), and Scullen (1998). For unsteady problems, time-stepping solutions must be used. Spectral methods have been applied to water wave problems and to wave diffraction by two-dimensional and/or simple geometries (see for example Chapman 1979, Dommermuth, et al. 1988, Liu, et al. 1992). Longuet-Higgins and Cokelet (1976) introduced the mixed Euler-Lagrange method for solving two-dimensional fully nonlinear water wave problems. In this time-stepping procedure two major tasks must be completed at each time step. The first is to solve a mixed boundary value in an Eulerian frame. The potential is known on the free-surface and the normal velocities are known on the body surface from the body boundary condition. In the Lagrangian phase, the fully nonlinear freesurface boundary conditions are used to track the freesurface amplitude and the value of the potential on the free surface. The rigid body equations of motion are used to update the body position in space and the normal velocity on the body surface is given by the body boundary condition. The method has been applied to a wide variety of two- and three-dimensional water wave problems, both with and without a body present. Among the researchers who applied the method to two-dimensional problems are Faltinsen (1977), Vinje and Brevig (1981), Baker, et al. (1982), and more recently Grosenbaugh and Yeung (1988), and Cointe, et al. (1990). Three-dimensional problems have been investigated by Lin, et al. (1984), Dommermuth and Yue (1987), Zhou and Gu (1990), Cao, et al. (1991), Scorpio, et al. (1996), Beck (1999), and Subramani (2000). Three-dimensional, fully nonlinear calculations are computationally very intensive. A compromise approach is to solve the nonlinear problem in the cross flow plane and pseudo-time step the solution in the downstream direction. Fontaine and Tulin (1998) give a history of the method that they call 2D+t. The idea has been used in planing boat problems for many years. Using an approach apparently first proposed by Cummins the authoritative version for attribution.

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The inner region solution is then matched to the outer region to include both diverging and transverse wave systems. Maruo and Song (1994) used fully nonlinear free-surface boundary conditions in the cross flow plane to investigate bow wave breaking. More recent work of Wu, et al. (2000) applied the method to study deck wetness. Two problems with the Euler-Lagrange method have limited its application. As discovered by Longuet-Higgins and Cokelet in the first application of the method, the stability of time stepping of the free surface can be a problem. The numerical techniques, panel size, and time-step size must all balance or the free surface can become unstable and the calculations break down. Smoothing, regridding, and artificial damping have all been applied to try and alleviate the problem. The other major difficulty is wave breaking. Wave breaking is a natural phenomenon that occurs very often but unfortunately causes the Euler-Lagrange method to break down. The most troublesome waves are the bow and stern waves of high-speed ships. Any region with a great deal of flare will tend to cause overturning and hence breaking of the local wave crest. This local breaking may have no effect on the global hydrodynamic forces acting on the ship but can cause the computations to stop. Because of the problems associated with fully nonlinear computations, several researchers have been examining what we shall call “blending methods.” These methods are a blend of linear and nonlinear theories. In these theories the equations of motion are integrated in the time domain, with the hydrostatic and Froude-Krylov forces integrated over the exact wetted surface. The added mass and damping are found using a linear theory, typically a strip approach. A detailed discussion of the different theories and comparisons with experiments can be found in the ISSC report on Extreme Hull Girder Loading (ISSC 2000). The blending theories are used because they are fast and allow long time records to be generated with engineering accuracy. Finally, the most recent approach to seakeeping is to solve the Reynolds Averaged Navier-Stokes equations in the time domain (so called unsteady RANS). This is a new area of research and results are just starting to be presented (cf. Wilson, et al. 1998, Gentaz, et al. 1999). Normally, RANS codes are iterated until a steady state solution is obtained. In unsteady RANS, iteration is still used at each time step but the global solution is made time accurate by using a time-stepping method. Not enough results are yet available to arrive at any conclusions and much more work remains to be done. Taxonomy of Seakeeping Computations At the present time, active research in the area of predicting ship motions is continuing on panel methods, both fully nonlinear and double body methods, blending methods and the application of unsteady RANS. For design purposes a naval architect has a wide choice of methods with which to do seakeeping computations; the choices are no longer limited to strip theory and its derivatives. However, it should be pointed out that even with the availability of a wide selection of computational methods, probably 80 percent of all design related calculations for ships at forward speed are still made using strip theory. Strip theory has the advantage of being fast, reliable, and able to accommodate a wide range of hull forms. It is a method that is hard to beat for conventional ships at moderate speeds. However, for higher speed vessels, highly non-wallsided hull forms, wave loads or extreme motions, the comparisons with experiments are much poorer; this has been the primary motivation for the development of more advanced theories. In order to try and put some relative order into all the different modern seakeeping computational methods and marine hydrodynamics in general, we present Figure 1. The governing equations in the fluid for the general, three-dimensional, incompressible, constant density fluid flow problem are the continuity equation and the three components of the Navier-Stokes equations. These equations result in a system of four, coupled nonlinear partial differential equations for the four unknowns of pressure and the three components of velocity. To obtain a unique solution requires boundary conditions on all surfaces surrounding the fluid—the wetted surface of the body, the free surface, the bottom, and the surfaces at infinity. On solid surfaces such as the body surface there are two boundary conditions. The first is the kinematic condition of no flow through the surface. And the second is a no slip condition on the tangential velocity. These are applied on the continuously changing wetted surface of the vessel. On the free surface there is a kinematic condition and a dynamic condition of constant pressure with no shear stress. The free-surface boundary conditions are applied on the unknown free-surface amplitude, which must also be determined as part of the solution. On the bottom boundary for finite depth there is a kinematic condition, or in infinitely deep water the disturbance velocities must go to zero. At infinity, incident waves are prescribed and there is a radiation condition of outgoing waves on the ship-generated waves. This general problem is highly nonlinear in both 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. MODERN SEAKEEPING COMPUTATIONS FOR SHIPS Fig. 1 Taxonomy of hydrodynamics problems for seakeeping. 7

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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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 8 the governing equations and boundary conditions; at the present time, it is beyond the computational state-of-the-art. Consequently, approximations must be made in order to have a tractable mathematical problem. For discussion purposes, we have tried to put all the different available computational techniques into the broad framework shown in Figure 1. Figure 1 categorizes the different approaches that can be taken to solve the general three-dimensional, incompressible, constant-density marine hydrodynamics problem. While not all of the boxes are applicable for seakeeping computations, they have all been kept for completeness of the figure and to indicate that there are additional possibilities that might have application to seakeeping problems. The techniques to solve the general three-dimensional problem can be divided into two major categories—viscous and inviscid flow approximations. Viscous flow approximations attempt to model viscous effects by keeping some form of the viscous terms in the Navier-Stokes equations. The biggest difficulty is the turbulence in the high Reynolds number flows associated with typical marine problems. Direct Numerical Simulation (DNS) solves the Navier-Stokes equations directly including turbulence. DNS is so computationally intensive that it has only been applied to very simple problems such as flow in a rectangular channel. At the other extreme is Stoke's flow which keeps only the pressure and viscous terms in the Navier-Stokes equations. Stokes' flow is essentially a very low Reynolds number approximation, so it is useful in lubrication problems and to model the swimming of microorganisms. It is not particularly useful in high Reynolds number seakeeping problems. High Reynolds number flows are characterized by the viscous effects being confined to a region near the body and a viscous wake. Boundary layer approximations give reasonable results up to the separation point but cannot be carrier further. At present there are two methods to compute “average” viscous flow—Large Eddy Simulation (LES) and Reynolds Averaged Navier-Stokes equations (RANS). Each approach has its strengths and weaknesses. Turbulence modeling for RANS requires averaging over all velocity fluctuation states. The state-of-the-art is that RANS models fail in regions of significant anisotropy, such as portions of the flow influenced by rigid no-slip boundaries and free surfaces. LES methods model only small-scale fluctuations while directly computing the large scale ones. To the extent that small-scale fluctuations are locally isotropic, LES computations are potentially more accurate than RANS computations, but are achieved at significantly more cost (cf. Dommermuth, et al. 1998 for an example of a LES calculation of the steady flow about a ship bow). RANS has typically been used to investigate interior flows in ducts and exterior flows around bodies. LES has been used to study the interactions of different scales of turbulence in open flows such as occur in ship wakes. Reynolds Averaged Navier-Stokes equations are derived by assuming that all the velocity components can be approximated by a mean component plus a highly oscillatory, small amplitude, zero mean component that represents the turbulence. These are substituted into the Navier-Stokes equations that are then time averaged over a suitable time scale. The resulting equations for the mean flow are identical to the original Navier-Stokes equations except for the addition of second order inertial terms in the oscillatory velocities that do not time average to zero. These so-called Reynolds stress terms represent the influence of the turbulence on the mean flow field. While there are numerous numerical models for the Reynolds stress terms, none of them are entirely satisfactory. None of the present turbulence models can properly account for the anisotropy of the turbulence near the free surface that can have important effects in the wake region. RANS codes are state-of-the-art; they are used for steady resistance calculations and work is proceeding on unsteady RANS that includes incident waves and ship motions (cf. Wilson, et al. 1998, and Gentaz, et al. 1999). Yeung and his colleagues (Yeung, et al. 1998, Roddier, et al. 2000, Yeung, et al. 2000) have started using unsteady RANS in roll damping computations. In the long term, unsteady RANS will probably be matched with fully nonlinear potential flow computations in the far field to give a complete solution. The box labeled empirical approximations under the viscous flow branch in Figure 1 is included because designers must have answers and viscous flow calculations often are not applicable or are too computationally expensive. Many empirical methods have been developed in which theory is used to develop a framework with unknown coefficients that must be determined by experiments and full-scale measurements. Classic examples are: 1) The maneuvering simulation equations that use stability derivatives to estimate hydrodynamic forces. 2) The use of Morison's equation to approximate the wave exciting forces on circular cylinders in regular waves using an inertial coefficient for the added mass effects and a drag coefficient for the viscous component of the load. The coefficients are strongly dependent on the frequency of the waves and the diameter of the cylinder as expressed in the Kooligan-Carpenter number (cf. Sarpkaya and Isaacson, 1981). 3) The empirical roll damping models that are used to estimate the increase in roll damping due to viscous effects in potential flow ship motion calculations. Finally, on the viscous flow side of Figure 1 is a special box labeled Smooth Particle Hydrodynamics. Smooth Particle Hydrodynamics is a relatively new 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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 9 technique to compute fluid flows, and its application to seakeeping problems has not yet been determined. Monaghan, et al. (1994) have used it to simulate two-dimensional free-surface flows. Fontaine, et al. (2000) show some interesting results for the sloshing problem and the post-breaking behavior of water waves. The inviscid flow models neglect viscosity completely and are indicated by the second major branch of Figure 1. In this case, the Navier-Stokes equations reduce to the Euler equations and some of the boundary conditions have to be modified. Namely, the no-slip boundary condition can no longer be met on rigid surfaces such as the wetted surface of the ship. In addition, the boundary condition of zero shear stress on the free surface is not applicable. Even this reduced problem is very hard to solve and further simplifications are necessary. For rotational flows, in which vorticity is present, the vorticity equations and vortex methods can be used. These techniques have found limited application in roll damping computations and separated flows around circular cylinders. As indicated by the second large branch under Inviscid Flow in Figure 1, the most widely used technique is potential flow. The vortex theorems show that for an inviscid, constant density fluid started from rest no vorticity can be present. In this case, the fluid velocities can be written in terms of the gradient of a scalar velocity potential. The governing equation in the fluid flow is found by substitution of the gradient of the velocity potential into the continuity equation. The resulting Laplace equation is a linear partial differential equation that depends only on space variables and is independent of time. Unique solutions of the Laplace equation require boundary conditions on all surfaces surrounding the fluid domain. Integrating the Euler equation results in the Bernoulli equation that relates the pressure to the time derivative and gradients of the velocity potential. Thus, the potential flow assumption has allowed the problem to be reduced from solving four coupled, nonlinear partial differential equations to solving a single linear partial differential equation for the velocity potential. The only nonlinearities left in the problem are in the boundary conditions. The kinematic body boundary condition may be stated such that at each point on the hull-wetted surface the normal velocity of the water must equal the normal velocity of the hull. This condition is linear except that is must be applied on the exact wetted surface. This leads to a time variant system for which traditional linear system theory is not valid. The major nonlinearities in the general potential flow problem are in the free-surface boundary conditions that involve the square of the fluid velocities and products of the fluid velocities with the unknown free-surface amplitude. Consequently, the general potential flow problem with a free surface is very difficult to solve and still further simplifications have in the past been found necessary. The most obvious simplification is to eliminate all the nonlinearities by eliminating the free surface. The boxes on the far-left side of Figure 1 are use to indicate these infinite fluid problems. Infinite fluid problems are useful in many areas of marine hydrodynamics including submarine work, propeller work, and the study of flow around appendages. However, in seakeeping research they are of little use except as crude approximations or limiting values. In general, the effects of the free surface are too important to neglect. Only recently has the computer power been available that makes it feasible to attempt to solve the fully nonlinear problem using the exact body boundary conditions and the fully nonlinear free-surface boundary conditions. As previously discussed, results have been obtained for a limited number of hull forms in moderate seas. The principal difficulties here are numerical stability of the time stepping method and the local breaking waves. In Figure 1, the boxes under the exact potential flow problem represent the different approximations that are available today. The greatest degree of approximation is in the box to the left and the least is on the right hand side. In general, computational times increase as one moves to the right, but there are no hard and fast rules. For example, flat-ship theory is similar to the Neumann-Kelvin problem in computational difficulty. Two sets of approximations have to be made. The first deals with the free-surface boundary conditions and the second with the body boundary conditions. The four sets of vertical lines represent different levels of approximation to the freesurface boundary conditions. The individual boxes are different techniques to meet the body boundary condition. By far, the most widely used technique is to linearize the free-surface boundary condition about the free-stream velocity, U0, and satisfy it on the calm water plane. This allows the use of the free-surface Green functions and as discussed in the previous history, many different theories have resulted. The different theories can be broken down into at least four basic approaches. If the beam or draft is much smaller than the length, the body boundary condition can be met on a flat plane. For small beam the body boundary condition can be satisfied on the centerplane and a thin-ship theory results. Thin-ship theory tends to produce added mass and damping coefficients that are too small and is rarely used. For small draft, flat-ship theory satisfies the body boundary condition on the calm water plane. The resulting equations are similar to lifting surface theory in aerodynamics. A flat-ship theory (cf. Lai 1994, Lai and Troesch 1995) has been used to solve planning boat problems where con 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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 10 ventional strip theory fails. The next level of approximation assumes that the beam and draft are smoothly varying functions and small relative to the length. This results in transverse derivatives being an order of magnitude larger than derivatives in the longitudinal direction. In this case, the three-dimensional problem can be reduced to a series of two-dimensional problems in the transverse or “cross-flow” plane. Depending on the assumed orders of magnitude of forward speed and wave frequency, different theories result. Strip theory is a high-frequency theory and for slender-body theory the wavelength is on the order of the ship length. Unified theory links the two theories into a single theory valid for a wider range of frequencies. In the Neumann-Kelvin theory the body boundary condition is satisfied on the mean position of the body (i.e. the wetted surface up to the calm waterline). Because the body boundary condition is satisfied on the mean position of the hull rather than the exact wetted surface, certain “mj” terms arise in the body boundary condition. The mj terms are difficult to compute because they involve higher derivatives of the constant forward-speed perturbation potential. For this reason, a further simplification is often used in which the mj terms are approximated using just the angle of attack corrections that are independent of the forward-speed potential. Neumann-Kelvin theory is truly three-dimensional and is typically solved using panel methods in either the frequency domain or the time domain. Neumann-Kelvin theory is widely used in the offshore industry for offshore structures such as semi's and TLP's that are highly three-dimensional. It can be extended to second order mean and slow drift forces. In the body exact method, the body boundary conditions are applied on the exact wetted surface of the body while retaining the linearized free-surface boundary condition. This results in a time varying linear system rather than a time invariant system. Consequently, the usual application of random process theory will not work and the body exact problem is usually solved in the time domain using the time dependent Green function. As previously discussed, the basis flow for the linearization of the free-surface boundary condition does not have to be the free stream. In Dawson's method or the double-body formulation, the linearization is about the double-body flow. The resulting free-surface boundary conditions are applied on a known position, but they involve complex functions of the usually numerically determined double-body flow. Even though the unknown free-surface displacement has been eliminated and the free-surface boundary conditions are applied on a known surface, the remaining boundary value problem is still difficult to solve. Rankine source methods are used with sources distributed over both the free surface and the body surface. The radiation conditions at the edge of the computational domain must be carefully considered to avoid wave reflection. Sclavounos, et al. (1997) use an absorbing boundary in SWAN 2. The weak scatter formulation goes one step further and applies the boundary conditions on the incident wave disturbed free surface and the instantaneous body wetted surface (cf. Huang and Sclavounos 1998) The final two boxes represent solution techniques that are still nonlinear but have been reduced in scope in order to make them more tractable. In the 2D+t methods, the fully nonlinear problem is solved in the cross flow plane with a hyperbolic marching used in the longitudinal direction starting at the bow. The blending methods have little rational basis. They are an engineering solution that combines the nonlinearities that are easily computed (typically nonlinear hydrostatics and the Froude-Krylov exciting force) with linear hydrodynamics. For head seas it appears that the primary nonlinearities are the hydrostatics and the Froude-Krylov exciting force. If a program has these two components correct, predictions for large amplitude motions are improved. 3 CONTEMPORARY CALCULATION METHODS At the present time, the majority of design seakeeping computations for ships at forward speed probably still involve the use of strip theory. That does not mean to imply that the more advanced theories that we discussed in section 2 are not important. As ship geometry becomes more complex and the design speed increases, the advanced methods will find more and more applications. This transition will be accelerated only by the availability of cheaper and faster computers. The technical literature on strip theory, long wave slender-body theory and unified theory is immense and we shall not discuss them any further in this paper. Unsteady Viscous Flow As just discussed in Section 2, the viscous flow about a ship is governed by the Navier-Stokes and continuity equations: (1) (2) where the ui, i=1, 2, 3 are the x-, y-, and z-components of the velocity, P is the pressure, gi is the is the component of the gravitational acceleration g in the xi-direction, and where in accordance with the Einstein summation convention double the authoritative version for attribution. subscripts within a term imply summation over that index.

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Thus, the velocity ui must account for the total velocity including the time dependent components at wave encounter frequency and the turbulent velocity components with much faster variation both temporally and spatially. This results in a computationally intractable problem for fluid volumes the size of a ship on the ocean's surface. The commonly accepted way in which this problem is made tractable is to decompose the velocities and pressures into slowly varying and rapidly varying components. Doing this, one obtains and where the “overbar” represents a Reynolds average taken over a time/spatial scale large relative to the scale of the turbulence and the primed quantities account for the velocities and pressure at turbulent scales. Substituting this decomposition of the velocity and pressure into (1) and (2), one obtains the Reynolds-averaged Navier-Stokes (RANS) and continuity equations: (3) (4) where is the Reynolds stress tensor. The RANS equations must be solved subject to boundary conditions on the ship's hull, the free surface, the far (from the ship) fluid boundary, and on the bottom of the fluid domain. As discussed earlier, there are both a kinematic and a “no slip” condition on the hull surface. On the free surface, there is a kinematic condition of no fluid flow through the surface, a dynamic condition that requires that the pressure equal the atmospheric pressure, and, assuming no wind, a no shear condition. On the far fluid surface boundary, there is either a no disturbance condition or a no wave reflection condition, depending on how far the far boundary is from the ship and how long a time the simulation is being run. On the bottom, there is either a kinematic condition or the disturbance must go to zero as the depth goes to infinity. The no shear condition on the free surface does not mean that there are no viscous effects at the free surface. Due to the nature of wave flow there is an inherent natural shear, which, even for small amplitude linear waves, results in a thin viscous layer near the free surface (cf. Mei 1983). Fortunately, the gradients due to the wave motion are small compared to the gradients near the hull where the no slip condition is applied. Thus, the gradients in the waves may be neglected with no significant consequences if one is only interested in the body forces and the flow local to the ship. The viscous nature of the wave flow is only important over length scales greater than several characteristic wavelengths and time scales greater than several characteristic wave periods. This common wisdom may not hold if the waves are steep and there are significant nonlinearities. Equations (3) and (4) constitute four equations for 13 unknowns, the three velocities, the pressure, and the nine components of the Reynolds stress tensor. Thus, the equations are not closed. To obtain closure, the Reynolds stress tensor is usually related to the mean velocities by an eddy viscosity. This generally involves the introduction of another variable such as the turbulent kinetic energy and an equation relating the turbulent kinetic energy to the mean velocities and the eddy viscosity. Speziale (1992) provides a survey of Reynolds stress models. The use of unsteady RANS to solve the viscous formulation of the seakeeping problem is in its nascent stage at present. Wilson, et al. (1998) present the results of RANS simulations for both a Wigley hull form and DTMB Model 5415 fixed in head seas at a single Froude number, wave frequency, and wave elevation for each model. The results are largely inconclusive. Gentaz et al. (1999) present the results of RANS calculations of forced oscillation motions for a hemisphere at zero speed, and forced heave and pitch of a Series 60 model at a single Froude number over a range of wave frequencies. The added mass and damping predictions for the Series 60 model are shown in Figures 2 and 3. These results include comparisons of the RANS predictions (called “Present meth.” on the figures) with two grid densities, against potential flow calculations and experimental results. For these vertical plane motions, the RANS and potential flow predictions compare quite reasonably with the experimental results. A not surprising result as potential flow methods have for many years been providing adequate predictions of vertical plane motions. For monohull ships, roll is the mode of motion where viscous effects have the greatest significance—through viscous roll damping. Although the efforts reflected in Wilson, et al. (1998) do not explicitly mention it, this is the direction in which the RANS efforts at the University of Iowa are headed.2 Multihull vessels, such as SWATH ships can have significant viscous damping in vertical plane motions. Yeung, et al. (1998, 2000) and Roddier, et al. (2000) present the responses, and added mass and damping from both two-dimensional experiments and two-dimensional RANS and potential flow calculations for rectangular cylinders fitted with bilge keels. Yeung and his colleagues apply two methods to the solution the authoritative version for attribution. 2Private communication.

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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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 35 dom seas to predict the true statistical responses of the ship in random seas. It may well be that monochromatic wave testing will easily elicit parametric resonant responses that are not realistic in random seas. This would be the case if the group must contain a particularly large number of waves for the specific extreme response to occur. A final issue relating to validation is that of scale effects between model- and full-scale measurements. While validation must almost by definition be against model-scale experiments, the true issue at hand must be to predict the performance of full-scale ships in a seaway. There are contributors to the equations of motion such as viscous roll damping that may have significant variation from model-scale tests to full-scale trials. The means of accounting for these factors in validation is not clear. The best solution would be to have a physics-based—as opposed to an empirically-based—model for the phenomena, so that the model automatically accounts for the scale. At present this is naive, but it may indicate a direction that computational models must go. 5 CONCLUSIONS There is a wide variety of computer codes available to perform seakeeping computations in low to moderate sea states. Strip theory is still the most widely used. However, in many situations it does not give adequate results and more advanced techniques must be used. Strip theory's principal weaknesses are the lack of three-dimensional effects, the inability to account for the above-water hull form, the forward speed corrections, and the lack of viscous effects. In theory, unsteady RANS with fully nonlinear free-surface boundary conditions can account for all of these, but it is very intensive computationally and few results are presently available. Potential flow methods need only ignore the viscous effects. Advanced potential flow methods have shown marked improvement over strip theory predictions. For predictions in the littorals, new wave climatologies are required, and computer codes adapted for finite depth and non-uniform bottoms are needed. Finite depth computations can be accomplished by simply changing the Green function in existing seakeeping codes. No codes presently available can adequately predict the behavior of ships in extreme seas including green water on deck, broaching, slamming and capsizing. This is beyond state-of-the-art codes. Much more research into breaking waves and the development of new analytic and numerical techniques needs to done. There is a critical need for more validation data. Experimental data, both model and full scale, with enough detail and accuracy to be useful for validation studies is time consuming and costly to obtain. While there is sufficient data for the linear range (small amplitudes) in head seas, fewer results are accessible for non-head seas. Very limited data of validation quality for the variation of seakeeping responses with wave height is available. For extreme sea states in which the responses are inherently random, the proper form of validation data is not even known. ACKNOWLEDGMENTS We offer apologies to researchers in the seakeeping field whose work has not been cited, we could not begin to reference all of the works in the field. To the reader, we urge you to see the many references in our cited works to get a sense of the true breadth of the work in the field. RFB's work was support by grants from the Office of Naval Research. Computing support for the computations reflected in many of the figures came from the U.S. Department of Defense High Performance Computing Modernization Program and the National Partnership for Advanced Computational Infrastructure. We owe thanks to John F.O'Dea, Katherine McCreight, Francis Noblesse, and Ed Rood who provided discussions and suggestions on the content of the paper; and Phil Alman, Lew Thomas, Woei-Min Lin, Dennis Woolaver, and Martin Dipper who provided up to date data and references for the paper. Finally, we would like to thank Vickie Kline, Kay Adams, Traci Meadows and Luella Miller who provided support for the preparation of the paper, and Suzanne Reed who again wielded the editor's pencil. 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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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 36 Baitis, A.E. & L.V.Schmidt (1989) Ship roll stabilization in the U.S. Navy. Naval Eng. J., 101(3):43–53. Baitis, E., D.A.Woolaver & T.A.Beck (1983) Rudder roll stabilization for Coast Guard cutters and frigates. Naval Eng. J., 95(3):267–82. Baker, G.R., D.I.Meiron & S.A.Orszag (1982) Gen eralized vortex methods for free-surface flow problems. J. Fluid Mech., 123:477–501. Bandyopadhyay, B. & C.C.Hsiung (1994) Mechanism of broaching-to of ships from the perspective of nonlinear dynamics. 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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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 43 seas. To be published in J. Ship Res. Wyatt, D.C. (2000) Development and assessment of a nonlinear wave prediction methodology for surface vessels. J. Ship Res., 44(2):96–107. Xü, H. & D.Yue (1992) Computations of fully-nonlinear three-dimensional water waves. Proc. 19th Symp. Naval Hydro., pp. 177–201. Yang, C. & R .C.Ertekin (1992) Numerical simulations of nonlinear wave diffraction by a vertical cylinder. J. Offshore Mech. Arctic Eng., 114:36–44. Yeung, R.W. (1982) The transient heaving motion of floating cylinders. J. Eng. Maths., 16:97–119. Yeung, R.W. & S.H.Kim (1984) A new development in the theory of oscillating and translating slender ships. Proc. 15th Symp. Naval Hydro., pp. 195– 218. Yeung, R.W., S.-W.Liao & D.Roddier (1998) Hydrodynamic coefficients of rolling rectangular cylinders. Int'l. J. Offshore & Polar Eng., 8(4):241–50. Yeung, R.W., D.Roddier, B.Aless andrini, L.Ge ntaz & S .-W.Liao (2000) On roll hydrodynamics of cylinders fitted with bilge keels. Proc. 23rd Symp. Naval Hydro., 18 p. Zhou, Z. & M.Gu (1990) A numerical research of nonlinear body-wave interactions. Proc. 18th Symp. Naval Hydro., pp. 103–18. APPENDIX—CORRECTION TO THE ADDED RESISTANCE OF LIN AND REED (1976) There is an error in the ranges of integration in the definition of the component of added resistance that results from the interaction of the radiation-diffraction waves with themselves, given in Lin and Reed (1974). It is corrected here: and for τ ≤1/4 and τ>1/4, respectively. The constants ± Γ and π±∆ appearing in the limits of integration correspond to the zeros of dθ/du for λ1, and π± corresponds to the zeros of dθ/du for λ2, where θ(u) is derived from: The Kochin function H(u, λ), u0, and λp are defined in Lin and Reed (1976). 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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 44 DISCUSSION U.P.Bulgarelli Istituto Nazionale per Studi ed Esperienze di Architettura Navale, Italy I presume that a unified theory of resistance, maneuvering, and seakeeping is based on unsteady RANSE. To do that one point should be stressed much more than in the past, the treatment of the free surface that is not yet enough accurate. AUTHOR'S REPLY The authors appreciate the interest and stimulating questions the discussers have provided on this complicated subject. As with the paper, our responses will leave as many questions as they answer. U.P.Bulgarelli. points out that free surface resolution is a problem in RANS computations. For seakeeping calculations in extreme seas it is critical that an accurate free-surface elevation be predicted because the free surface will impact the above water portion of the hull. It remains to be seen to what extent this issue can be resolved in unsteady RANS—often, this is still an issue for steady RANS DISCUSSION L.J.Doctors University of South Wales, Australia I would like to first say that I enjoyed the presentation very much. As I have come to expect from these two authors over the years. In reference to slender hulls, you refer to the 2D +t (two-dimensional plus time) method. Do you believe this approach will work for a catermaran in which the vessel as a whole is not slender—even though the individual demihulls maybe slender? AUTHOR'S REPLY L.J.Doctors asks whether or not 2D+t methods can be applied to multi-hull vehicles which overall are not slender even though the individual demi-hulls may be slender. We are not sure, but we think there would be deficiencies for two reasons. First, the 2D+t theory does not contain the transverse waves and these surely will affect the unsteady loads on the second hull. Secondly, the second hull is in the far field of the first hull. In 2D+t theory we would expect inaccuracies in the far-field wave predictions that in turn would lead to deficiencies in the wave load predictions on the second hull. DISCUSSION L.Eca Instituto Superior Tecnico, Spain The authors definition of verification: “The demonstration that the code is ‘reasonably' bug free and that the output is numerically correct” suggest the following comments: Although we are aware that there is still a lot of debate on the proper definition of verification, the present definition does not mention the need to quantify the error and/or the uncertainty of the verification procedure, which we believe to be essential in such a process. It could also be mentioned in the paper that the verification of a complex flow is not a straight forward exercise and that, in general, it is very costly and time consuming. AUTHOR'S REPLY L.Eca raises the issue of numerical error and uncertainty in the verification process. We agree that the quantification of numerical errors and uncertainty are imperative for verification and validation of a code. Whether error analysis belongs in verification or validation is not that important—the key is that is must be done. In reality, it probably should be part of both processes. DISCUSSION M.Tulin University of California, USA I just want to congratulate Bob Beck on this very valuable paper and thank him for its preparation. I say that particularly because of the clear emphasis and focus he gave to large amplitude motions and the complexities of that regime. He mentioned SPH (particle tracking methods) which are in their beginning. I think these hold enormous potential for the future because of their innate ability to deal with large surface deformations, breaking, splashing, and vortical structures. He also gave emphasis to validation 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 MODERN SEAKEEPING COMPUTATIONS FOR SHIPS 45 and the importance of data. It is striking how much data we have in the smallest amplitude regimes and so little (virtually no systematic data) for very large ship motions. AUTHOR'S REPLY M.Tulin lauds SPH and endorses our observation that there is a paucity of large amplitude motion data for validation and verification purposes. We are pleased that he supports our position that much more validation of large amplitude ship motions must be done. SPH has demonstrated interesting possibilities in two-dimensional computations. Questions remain as to the validation of the SPH predictions for physical quantities such as pressure and particle velocity. In addition, the extension of the method to external three-dimensional flows could prove to be problematic. the authoritative version for attribution.