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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 techniquesâfrom 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.

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.

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

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 âpanel methodsâ 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.

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 âDawson's Approachâ (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 âlinearizedâ 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.

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 6 (1956), Ogilvie (1972) studied the waves produced by a fine ship's bow using a linear free-surface boundary condition. Chapman (1976) used the full nonlinear free-surface boundary condition to investigate a yawed flat plate. Only the divergent waves are simulated by the method and thus it is most appropriate for high-speed ships. Yeung and Kim (1984) developed a special inner region Green function that meets a linear free-surface boundary condition with the forward-speed terms. 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 âblending methods.â 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 fluidâthe 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.

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

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 categoriesâviscous 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 âaverageâ viscous flowâLarge 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.

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.

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 âcross-flowâ 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 âmjâ 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.

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 11 Equations (1) and (2) must satisfy a kinematic and a no-slip condition on the body. These equations apply to any viscous flow, laminar or turbulent. As we are interested in the flow about ships with forward speed in a seaway, the flow will be turbulent over substantially all of the ship's surface. 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 âoverbarâ 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 âno slipâ 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 âPresent meth.â 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 significanceâthrough 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.

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 12 of the RANS equations, the free-surface random vortex method (FSRVM) and the boundary-fitted finite-difference method (BFFDM). They perform purely inviscid calculations by turning off the viscous terms in FSRVM. Yeung, et al. (2000) shows much better agreement between the viscous FSRVM predictions and experimental measurements than between the inviscid FSRVM predictions and experimental measurements for roll added mass and damping (A44 and B44) and roll-sway coupled added mass and damping (A 24 and B24). The viscous FSRVM predictions of A44 and B44 agree quite well with where b is the half beam. For the viscous FSRVM calculations of A44 diverge experiments for from the experimental results and converge to the inviscid FSRVM calculations; the B44 predictions diverge from the experiments but do not converge to the inviscid results. The viscous FSRVM predictions of A24 and B24 agree with the experiments for the mid frequencies, but not for either the lower or higher frequencies; they are closer to the experimental measurements than the inviscid FSRVM predictions. While these results are encouraging, it should be recognized that they are for the linear regime with small roll angles. It remains to be seen how well the method works for large amplitude motions. Fig. 2 Added mass and damping coefficients (A33 & B33) Fig. 3 Added mass and damping coefficients (A55 & B55) for a Series 60 CB =0.6 model in forced heave, Fn=0.20, for a Series 60 CB=0.6 model in forced pitch, Fn =0.20, Re=8.3Ã105 , |Î¾3|/L=0.01 (from Gentaz, et al. 1999). Re=8.3Ã105 , |Î¾5|=1.5Â° (from Gentaz, et al. 1999). The unsteady RANS computations are so new and so computationally intensive that only Wilson et al. (1998) and Genatz et al. (1999) show three-dimensional computational results. Consequently, in the remainder of this section we shall focus on potential flow computations. Potential Flow Formulation For potential flow computations the fluid is assumed to be inviscid, homogeneous and incompressible. Surface tension on the free surface is neglected. A right-hand coordinate system, Oxyz is translating in the negative x-direction relative to a space-fixed frame. The time-dependent velocity of translation is given by U0(t); for steady forward speed, U0(t)=U0. We will write the governing equation and boundary conditions in the time domain. For frequency domain computations, the time dependence is replaced by exp(iÏ t) and it is understood that only the real part is to be used. The Oxyz axis system is oriented such that the z=0 plane corresponds to the calm water level with z positive upwards. The total velocity potential of the flow can then be expressed as (5) the authoritative version for attribution.

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 13 where is the perturbation potential. Both and must satisfy the Laplace equation in the fluid domain: (6) Boundary conditions must be applied on all surfaces surrounding the fluid domain: the free surface (SF), the body surface (SH), the bottom (SB) and the surrounding surface at infinity (SInf ). The kinematic body boundary condition is applied on the instantaneous position of the body wetted surface: (7) where n=(n1, n2, n3) is the unit normal vector into the surface (out of the fluid domain) and VH is the velocity relative to the moving coordinate system of a point on the body surface including rotational effects. The subscripts 1, 2, and 3 refer to the x-, y-, and z-axis directions, respectively. There is also a kinematic condition applied on the bottom: (8) where VB is the velocity of the bottom relative to the Oxyz system. For an infinitely deep ocean, equation (8) becomes (9) In Rankine source methods, finite depth will increase the computational time because of the additional unknowns necessary to meet the bottom boundary condition, but there is no increase in computational difficulty. Flat bottoms are very easy to compute since an image system can be added to the influence matrix for relatively small computational cost. On the other hand, with the typical Green function approach, the finite depth Green function is significantly harder to compute than an infinite depth Green function. Finite depth Green functions are almost always derived for flat bottoms so that arbitrary bottom contours can not be accommodated. On the instantaneous free surface both the kinematic and dynamic conditions must be satisfied. The kinematic condition is (10) where z=Î·(x, y; t) is the free-surface elevation. The dynamic condition requires that the pressure everywhere on the free surface equals the ambient pressure, P a. The ambient pressure is assumed known and may be a function of space and time; normally it would be set equal to zero. Using Bernoulli's equation, the dynamic condition can be written as: (11) where Ï is the fluid density and g the gravitational acceleration. In the time domain, the initial values of the potential and free-surface elevation must be specified. Normally, it is assumed that the computations start from rest such that (12) In the frequency domain no initial conditions are necessary. Boundary conditions at infinity will also be necessary. In the frequency domain, the waves due to the body disturbance, including diffracted waves, must be outgoing at infinity. In the time domain, for an initial value problem with no incident waves, the fluid disturbance must vanish at infinity: (13) The hydrodynamic forces acting on the body are found by integrating the pressure over the instantaneous wetted surface. The generalized force acting on the body in the jth direction of the body axis system is thus given by: (14) where nj is the generalized unit normal into the hull defined as (15) where n is the unit normal to body surface (out of fluid), and with the body fixed coordinate system. The subscripts, j=1, 2, 3 correspond to the directions of the axis, and j=4, 5, 6 are the moments about the axis respectively. The pressure in the moving coordinate system is given by Bernoulli's equation: the authoritative version for attribution. (16a) or (16b) where is the time derivative of the potential following a moving point on the body and v is the velocity of that point relative to the Oxyz coordinate system.

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 14 Motions by Fully Nonlinear Potential Flow The fully nonlinear potential flow problem has been solved by a number of researchers in both two and three dimensions using a variety of methods. For unsteady sea-keeping problems the most common technique is to use the mixed Euler-Lagrange, or MEL, method originally developed by Longuet-Higgins and Cokelet (1976). As previously explained, this method uses the free-surface boundary conditions in the form of (10) and (11) to time step the potential on the free surface and the free-surface amplitude. At each time step, a mixed boundary value problem must be solved in which the potential is known on the free surface and the normal derivative of the potential is known on the remaining surfaces. On the body, is determined from the kinematic body boundary condition (7). For prescribed motions VH is given; for free-body motions, the equations of motion must be solved concurrently with the MEL time stepping in order to determine the wetted surface and VH for the next time step. Solving the equations of motion adds further complexity to the problem since before the equations of motion can be integrated the hydrodynamic forces acting on the vessel (14) must be determined. As can be seen from (14) and (16), to find the hydrodynamic forces the time derivative of the potential, must be known at the present time step. The conventional technique is to use backward differencing, but at times this may not be accurate enough. An alternative is to set up an auxiliary problem to solve for directly at each time step. The auxiliary problem is similar to the original mixed-boundary value problem except that the boundary conditions have been altered. Thus, if the inverse of the influence matrix used to solve the mixed boundary value problem is known, the auxiliary problem can be solved with little additional computational burden. However, iterative solvers such as GMRES do not compute the inverse matrix and the solution to the auxiliary problem requires the same effort as the original problem, effectively doubling the computer time. Vinje and Brevig (1981), Yeung (1982), Kang and Gong (1990), Cointe, et al. (1990), and Beck, et al. (1994) provide examples of solving an auxiliary problem. Standard techniques can be used to solve the mixed-boundary value problem that results at each time step in the MEL approach. Most codes use some form of panel method with Rankine sources distributed over both the free surface and the body. For constant finite depth a simple image system can be used to satisfy the bottom boundary condition. For varying bottom geometry singularities must also be distributed over the bottom surface. The boundary conditions at infinity are often a problem. For the diffracted and radiated waves, there must be a radiation condition. The incident waves, if they are present, must be introduced on the upstream boundaries. Without special care waves can reflect from the edge of the computational domain. In linear problems that use free-surface Green functions, the computational overhead is zero because the Green function automatically satisfies the radiation condition at infinity. However, regardless of whether the problem is linear or nonlinear, in Rankine panel methods the computational burden to meet the far-field boundary conditions can be severe. Periodic boundary conditions assume the solution is periodic in space so that the values of the unknowns on one vertical boundary can be set equal to those on the other vertical boundary of the computational domain. The technique is easy to implement, but the assumption of periodicity can be problematic. Yang and Ertekin (1992) and others have directly applied a Sommerfeld type radiation condition. This works well in two-dimensional linear problems, but in three dimensions varying wave directions and obtaining accurate estimates of the local phase speed can cause difficulties. Several authors have used matching boundary conditions (for example see Dommermuth and Yue 1987 or Lin, et al. 1999) to match a Rankine inner solution to a linear solution in the outer domain that is modeled with free-surface Green functions, thus satisfying the radiation condition because of the far-field free-surface Green functions. This matching requires that the potential, its normal derivatives, and the wave height agree at the boundary between the inner and outer domains. However, achieving this matching between the nonlinear inner solution and linear outer solution can be a problem. Probably the most common technique to prevent wave reflection is to use absorbing boundary conditions, or beaches. Extra terms are added to the free-surface boundary conditions, either one or both of (10) and (11), to cause wave dissipation. The design and use of absorbing boundary conditions is reviewed in Israeli and Orszag (1981), Cointe, et al. (1990), Cao, et al. (1993a), or Clement (1996). Typically, the boundary conditions have coefficients that must be tuned to the wave frequency. While numerical beaches might be the best solution we have at present, they are far from satisfactory. They add a significant number of unknowns on the free surface that must be determined at each time step. Even the best numerical beaches still have a few percent wave reflection. The ideal far-field boundary condition would be one that had zero wave reflection for waves propagating in all directions and did not add significantly to the computational overhead. For forward-speed problems where Ï=ÏeU0/g> 1/4, it is often possible to âget away withâ open boundary conditions on the side and downstream boundaries. While not mathematically correct, it seems that depending on the details of the numerical algorithm it is sometimes possible to neglect the downstream boundaries the authoritative version for attribution.

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 15 with little effect on the solution at the body. The upstream boundary is another problem. For Ï>1/4 it must be taken far enough ahead of the ship to not be affected by the ship generated waves. If there are no incident waves, the perturbation potential and unsteady wave height are taken equal to zero. For Ï<1/4 there does not seem to be a good technique to prevent wave reflection from the upstream boundary except to move the boundary far upstream. Absorbing layers can be used in radiation problems if they can be formulated not to make waves. In fully nonlinear computations, the incident wave field should also be fully nonlinear. However, many codes use higher order Stokes' waves. Ferrant (1999) uses a Fourier series to represent the waves and then solves for the diffraction potential. Scorpio, et al. (1996) use a fully nonlinear approach in which the incident wave field is precomputed in a two-dimensional wave tank. The appropriate normal velocity and wave height are then applied on the upstream boundary of the computational domain. The computations in the two-dimensional wave tank use a wave maker at the head of the tank and an absorbing layer at the downstream end. The tank can be as long as necessary to get a good wave record without an excessive use of computer time. The same calculations carried out in three dimensions would be prohibitive. The panel method computations themselves have been made with both lower and higher order methods. The lower order methods with constant source strengths over quadrilateral panels are not as accurate but seem to be more robust for complex hull shapes. For simple body shapes such as spheres and Wigley hulls, the higher order panel methods with bi- linear or quadratic source strength distributions over either flat or curved panels appear to be superior. Examples of higher order panel methods used in nonlinear computations are Ferrant (1996, 1999) or XÃ¼ and Yue (1992). In a series of papers, Beck and his colleagues (Cao, et al. 1991, Cao, et al. 1993b, Beck, et al. 1994, Scorpio, et al. 1996, Beck 1999, Subramani 2000) used a desingularized method in which the source panels are located outside the fluid domain and thus the kernel in the governing integral equation is desingularized. In addition, because of the desingularization the panel distribution can be replaced by simple point singularities. The use of point singularities greatly reduces the complexity of the computer code because the integrals over panels that normally have to be evaluated to set up the influence matrix are replaced by simple summations. Regardless of the method used to set up the influence matrix, the resulting system of equations must be solved at each time step. This typically takes approximately one half of the CPU time. As previously discussed, depending on the size of the problem, direct solvers, iterative solvers, or order-N methods might be most appropriate. Fully nonlinear computations have advantages in that they automatically including forward-speed effects and nonlinear interactions. No special treatments such as the mj terms in the body boundary condition or the modified free- surface boundary condition in the double body approaches are needed to include forward-speed effects. Nonlinear effects such as added resistance, mean drift forces, or the difference between the hogging and sagging midship bending moment that normally require extensive additional computations naturally fall out of fully nonlinear computations. The problem with fully nonlinear computations is wave breaking and the stability of the time stepping method. In their original work, Longuet-Higgins and Cokelet (1976) encountered a sawtooth instability of the free surface and employed a smoothing technique to suppress its growth. For axisymmetric problems, Dommermuth and Yue (1987) found high wave number instabilities. They postulated that they were the result of the concentration of Lagrangian markers in the region of higher gradients that cause the local Courant stability condition to be violated as the wave steepened. They eliminated the instability by regridding the free surface after each time step. Park and Troesch (1992) have investigated in detail the stability of time stepping for a variety of two- and three- dimensional problems and reached a number of conclusions. They found that the stability depends upon the geometry of the specific problem, the closure at infinity, and the time integration method. They also note that three-dimensional problems tend to be more stable than two-dimensional problems. Explicit Euler time-stepping schemes are unconditionally unstable, while implicit-like and implicit Euler schemes and fourth-order Runge-Kutta schemes are conditionally stable. They defined a local stability index Ïgât2/âx which if exceeded leads to an unstable solution. The limit of the local stability index depends upon the geometry and the time-marching scheme used. In fully nonlinear computations, wave breaking is also a problem because it forces the calculations to stop. Even small, local areas of breaking such as the spray region in the bow or corner of the transom stern will cause computational instability. Although these local areas of breaking may be of no interest since they have minimal influence on the global forces acting on the vessel, they must be properly accounted for. MEL has been used to study plunging breakers up until the time of impact on the front face of the wave. Post breaking behavior is not well understood; level set methods (Sussman and Dommermuth 2000) and smooth particle tracking (cf. Fontaine, et al. 2000) have been used to try and simulate it. (In their steady flow simulations, Sussman and Dommermuth (2000) report difficulty in getting the level set function to extend inside the body for tracking the authoritative version for attribution.

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 16 the hull/free-surface intersection.) Most of the work has been devoted to trying to suppress the wave breaking so that the calculations can continue. Haussling and Coleman (1979) and Subramani, et al. (1998a) use a pressure distribution on the free surface in the vicinity of wave breaking to suppress breaking. Subramani, et al. (1998b) developed a local curvature criterion to detect areas in which the wave is about to break. A pressure distribution is then used to remove energy from the wave system and prevent breaking. An alternative possibility is to fair through the free surface and remove the nodes that are about to break. Wang, et al. (1994) use a peeling technique in fully nonlinear simulations of breaking in a numerical wave tank. Subramani and Beck (2000) continue to use the curvature criterion to identify areas of breaking and then fair through the offending nodes using a cubic spline. Subramani (2000) has successfully suppressed bow and transom stern wave breaking for three hull forms. To our knowledge, fully nonlinear wave diffraction and seakeeping computations have only been performed on simple mathematical shapes such as axisymmetric bodies, circular cylinders, and the Wigley hull. Many researchers have computed the fully nonlinear steady flow (see for example Raven 1998, Wyatt 2000, Subramani 2000) about arbitrary hull forms. Figure 4 is taken from Subramani (2000) and shows the steady wave contours about a naval combatant hull form, DTMB Model 54153. The computations were started from rest using the desingularized approach with point sources. The intent is to extend the computations to include nonlinear incident waves but only a few preliminary calculations have been made with a Series 60 hull. The problem is that the present code uses fixed stations and that presents difficulties as the water surface moves along the raked bow or the transom comes in and out of the water. The Wigley hull has a plum bow and stern so that computations are much more straightforward. Scorpio (1997) has computed both the fully nonlinear hydrodynamic radiation-diffraction forces on the Wigley hull. To illustrate the importance of the various linear and nonlinear components of the force, Figure 5 from Scorpio, et al. (1996) shows the hydrodynamic force acting on the modified Wigley hull III used by JournÃ©e (1992) due to forced pitch motion at a frequency of The model was accelerated from rest, brought to a constant forward speed and then at a nondimensional time of 14 the pitch motion was slowly built up to an amplitude of 1.5Â°. In the figure, the total force and its five separate components as listed in (16b) are plotted. The first component and also the largest during the initial start-up is due to the time derivative of the potential following each individual node The second term is the linear pressure due to the forward speed This is the usual linear pressure component in steady forward motion problems. It is the dominant component for surge and heave forces. The third is the gravity term due to the change in the integration of â Ïgz over the wetted surface. For steady forward speed this term is proportional to the wave elevation squared and is small and negative. In unsteady motions, this term includes the hydrostatic restoring force and can be the largest component. The fourth component is the velocity squared term which is also small and negative. The final component is the correction for the moving nodes. The nodes are distributed between the keel and the free surface. As a result of the changing water surface elevation along the hull the nodes are redistributed along each station. For typical computations this component is relatively small; for steady wave resistance this component will go to zero as the wave elevation becomes constant. Fig. 4 Comparison of measured and computed wave contours for DTMB Model 5415 at Fn=0.28, (from Subramani 2000). The starting sequence of first accelerating the model up to speed and then starting the unsteady motions was found by the authoritative version for attribution. Scorpio, et al. (1996) to minimize the transients compared to starting both the forward speed and the unsteady motions at t*=0, where In addition, this technique reduces the computational time since the acceleration to steady forward speed is run first. All the unsteady cases then use this as a starting point by restarting the code, in this case at t*=14. All the computations have to be carried out to t*=28. However, the unsteady motion calculations need only be done from 3Complete details of this transom stern hull form can be found on the world wide web at http://www50.navy.mil/5415.

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 17 t*=14 to t*=28. If the calculations had been started from zero, each frequency would have to be run from t*=0 to t*=28. Fig. 5 Force components for Modified Wigley hull III due to pitch excitation, Fn=0.3, x5=1.5Â°, (from Scorpio, et al. 1996). Examining the start-up phase, t*<14, in Figure 5, several characteristics typical of the wave resistance problem can be seen. First, the surge force (positive out the stern) is equivalent to wave resistance. Initially the term is the largest and then decays to zero as the model comes up to speed. As the model reaches steady speed, the linear term is, as expected, the largest component. The components due to the âÏgz term and to are the same order of magnitude and both are negative. There is a fairly large sinkage (or negative heave) force acting on the model at steady forward speed. The trim (or pitch) moment starts off as bow up, reverses to bow down and then goes to approximately zero. This is consistent with experimental measurements on the standard Wigley hull that found almost no trim angle for a Froude number of 0.3. In the computations, the model could have been given the freedom to respond to the sinkage force and trim moment, but this is inconsistent with the experiments of JournÃ©e that were conducted fixed about the calm waterline. As with the surge force, the biggest component of the heave force is the term. At t*=14 the forced pitch motion begins. As can be seen, the hydrodynamic forces acting on the model quickly build to a steady state. For surge the two largest components of the force are still the linear terms ( and ). While not obvious because there is not enough time for the steady results to settle down completely, there is a mean shift between the steady portion (t*<14) and the unsteady portion of the curve. This mean shift is the added resistance due to pitch motion. As previously mentioned, one of the advantages of fully nonlinear calculations is that higher order quantities such as added resistance are automatically accounted for. The character of the heave force is similar to the surge in that the linear components are the most prominent and there is a significant mean shift. Again, the mean shift changes due to the unsteady motions. There also appears to be a low- frequency component to the heave force that is a carry over from the original startup. As expected, the pitch moment is dominated by the âÏgz term. This component is a combination of the usual linear hydrostatic restoring moment and the nonlinear effect caused by the varying wetted surface. The other components of the force are all relatively small. The time histories of the unsteady forces and moments, such as those plotted in Figure 5, can be used to determine the equivalent linear added mass and damping coefficients. As shown in JournÃ©e (1992), the equivalent linear coefficients are found by equating the actual unsteady force time history to the linear representation of the force using added mass and damping coefficients. Since the time histories are not perfectly linear, some care must be taken to ensure an equivalent the authoritative version for attribution. linearization. Scorpio, et al. (1996) found that a Fourier series or a least square fit of the data over several cycles yielded the same results. Figures 6 and 7 show the added mass and damping coefficients as a function of frequency for the modified Wigley hull in at a Froude number of 0.3. Figure 6 is for forced heave and Figure 7 is for forced pitch. The experimental results and the other numerical calculations were taken from JournÃ©e (1992). The strip theory results are computed using the modified strip theory coefficients of Salvesen et al. (1970) with a close fit method to compute the two-dimensional added mass and damping. The three-dimensional hybrid results were computed using the three-dimensional zero-speed coefficients calculated by WAMIT and modified for forward speed by the Salvesen et al. (1970) forward-speed corrections. In general, the two linearized computations are fairly close indicating that three-dimensional effects are not too impor

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 Fn=0.3, x3/L=0.00833 (from Scorpio, et al. 1996). Fig. 6 Experimental and theoretical added mass and damping coefficients for forced heave of Modified Wigley hull III, 18

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. x5=1.5Â° (from Scorpio, et al. 1996). MODERN SEAKEEPING COMPUTATIONS FOR SHIPS Fig. 7 Experimental and theoretical added mass and damping coefficients for forced pitch of Modified Wigley hull III, Fn =0.3, 19

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 20 tant for this relatively slender model. As can be seen, the fully nonlinear computations show much better agreement with experiments than either of the linearized computations. The added mass and damping due to heave (A33 and B33) are both a little low but the trends or the curves agree with the trends shown by the experiments. The heave damping is over predicted by the linearized computations. The pitch moments due to heave (A53 and B53) show excellent agreement in contrast to the linearized predictions that are poor. As with heave, the pitch added mass and damping (A55, B55) are under predicted by the fully nonlinear computations relative to the experiments. However, for pitch damping the nonlinear computations agree with experiments much better than the linear theory predictions. For the pitch added mass, the nonlinear computations are approximately the same as the linear results. The reasons for the under predictions are not known. It is expected that the damping predictions would be low because viscous effects are neglected. The fully nonlinear pitch cross-coupling coefficients (A35, B35 ) show very good agreement with experiments. It appears that the cross coupling coefficients are better predicted than the heave and pitch added mass and damping. Improvements similar to those shown in Figures 6 and 7 for nonlinear theories over linear results have been shown by the double body approaches. Nakos and Sclavounos (1990b) used the SWAN code to produce similar figures for the modified Wigley hull I. SWAN is a frequency domain code that is linearized about a double body basis flow. Consistent with the free-surface linearization, the body boundary condition is satisfied on the mean position with the mj terms based on the free stream plus double body flow. Rankine singularities are used to satisfy the free-surface and body boundary conditions. An equivalent time domain code, SWAN 2, has been extended to nonlinear ship motions using the weak scatterer formulation. Figure 8 from Huang and Sclavounos (1998) presents the heave and pitch response of the S-175 container ship in head seas for a Froude number of 0.275. Four sets of results are shown. The first is from the linear SWAN 2. The second is a quasi-nonlinear result in which the nonlinear hydrostatic and Froude-Krylov exciting forces are combined with the linear hydrodynamic forces of SWAN 2 to calculate the motions. This is similar to the blending methods that combine linear and nonlinear terms. Predictions using the weak scatterer formulation are labeled nonlinear SWAN 2. All of the predictions may be compared with the experimental results of Dalzell, et al. (1992). While not perfect, the nonlinear computations greatly improve the agreement between theory and experiment. The attractive idea of only using those nonlinear terms that can easily be computed helps the comparison but is clearly not enough. This demonstrates the importance of including all nonlinearities if accurate results are to be obtained for hulls with bow flare and stern overhang. Fig. 8 Heave and pitch response amplitude operator (RAO) for S-175 container ship in head seas at Fn= 0.275 (from Huang and Sclavounos 1998). the authoritative version for attribution.

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 21 Fig. 9 Selected time steps from deck wetness cycle simulation for the O'Dea & Walden (1984) Frigate, Fn=0.3, Î»/L=1.2 (from Wu, et al. 2000). Derived Quantities Derived quantities are those responses of a ship that are a result of ship motions or the radiated-diffracted waves. These include factors such as accelerations at the bow, green water on deck, slamming, structural loads, and added resistance in waves. We shall touch briefly on green water on deck, structural loads, and added resistance in waves. Green Water on Deck An example of 2D+t theory is the deck wetness study of Wu, et al. (2000). Building on the work of Maruo and Song (1994), Wu, et al. use order of magnitude arguments to reduce the three-dimensional, fully nonlinear problem to a two- dimensional nonlinear problem in the cross-flow plane. The body boundary condition is satisfied on the exact wetted cross section. A two-dimensionally nonlinear free-surface boundary condition is met on the instantaneous wave elevation augmented by the incident wave elevation. The boundary value problem in the cross-flow plane is solved using an mixed Euler-Lagrange approach to march the solution forward in time. In a sense the 2D+t method is a blend of the fully nonlinear MEL method and the weak scatterer formulation. As with the weak scatterer, the presence of a ship does not effect the incident waves that are used as a basis on which to solve the radiation-diffraction problems. The MEL approach is then used in the cross flow plane. This eliminates any transverse waves, but for high the authoritative version for attribution.

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 22 speed ships and in the bow region this is a good approximation. Fig. 10 Dynamic process leading to deck wetness (from Wu, et al. 2000). Figure 9, which is from Wu, et al. (2000), shows the time and space evolution of the bow wave for a heaving and pitching frigate tested by O'Dea and Walden (1984). The ship motion amplitudes and phase angles relative to the incident wave were taken from the experimental results. The Froude number is 0.3 and for this particular case of maximum motions in head seas, Î»/L=1.2. The sequence shown in Figure 9 is the bow downward phase just after the bow has reached its maximum up position. The incident wave is initially near the bow and then moves aft as time marches forward. As the bow moves down, the water is pushed upward and outward causing an overturning breaker that eventually tops the level of the ship's deck. Figure 10 summarizes the temporal evolution of the previous time sequence. As can be seen, there is a large dynamic rise to the water surface that is completely neglected if the traditional relative motion between the deck edge and the incident wave amplitude is used to predict water on deck. Structural Loads Purely linear theories predict that the hogging and sagging bending moments acting on a ship's structure in waves are identical. Experiments and full-scale measurements have shown that in fact the sagging moment tends to be larger than the hogging moment. Figure 11 is taken from the soon-to-be-released report of the ISSC committee on Extreme Hull Girder Loading (ISSC 2000). It shows the sagging (positive) and hogging (negative) bending moments along the S-175 containership model. The O-model figure is the original S-175 and the M-model is a modified version with increased bow flare. The increased flare was all above the waterline and thus would not affect linear computations. The Froude number is 0.25 and the incident regular head waves have a Î»/L=1.2 and a wave amplitude to ship length ratio of 1/60. The experiments were conducted at both The Ship Research Institute of Japan and the China Ship Scientific Research Center. As can be seen from the key, many different organizations participated in the study and ran their nonlinear seakeeping codes. Most of the codes used time domain analysis and a blending approach that included nonlinear hydrostatics and some nonlinear hydrodynamics. Details of the different computational techniques can be found in ISSC (2000). The calm water bending moment as measured by Ueno and Watanabe (1987) is include to give a scale to the unsteady bending moments. As can be seen the sagging moment is significantly larger than the hogging moment. Considering the many approximations made by the different codes, the analytic predictions are surprisingly consistent. Except for the hogging predicted by one code the differences between the codes are consistent with the scatter found in the experiments. The large increase in sagging moment caused by the increased flare is correctly predicted by the theories. Adde d Resistance in Waves Added resistance in waves is the mean increase in resistance force resulting from radiation of energy away from a ship operating in a seaway. This mean shift was clearly shown in Figure 5 as the increase in the mean of the unsteady surge force due to pitch motion. In linear theory, added resistance is the steady second-order force resulting from radiation of energy contained in the first order radiation-diffraction waves. The equivalent force on a body at zero speed is known as the drift force. Historically, drift forces have been of much more interest due to their importance in the design of offshore structures and their moorings. Added resistance has not been of concern in U.S. Navy ship designs due to the ships' excess power, resulting from the fact that they normally operate at speeds significantly less than their full-power speed. In fact Navy ship operators report resorting to voluntary speed reductions well before speed loss due to added resistance become significant. Higher fuel prices may force the Navy to take added resistance into consideration in future ship designs. The one class of vessels for which added resistance is considered is that of sailing yachts. The IMS Rule (Claughton 1999) includes a generic assessment of added resistance in its rating rule for determining a sailing yacht's handicap. IACC the authoritative version for attribution. America's Cup yachts are designed taking into account added resistance for the specific environment in which the craft will be sailing. Sclavounos and Nakos (1993) describe their efforts toward predicting the added resistance of one of these vessels.

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 23 Fig. 11 Non-linear sagging (positive) and hogging (negative) bending moments of the original and modified S-175 container ship moving in regular waves, Î»/L= 1.2, Î·a=L/60, Fn =0.25 (from ISSC 2000). Watanabe (1938) and Havelock (1942) first performed analysis of the the interaction between an incident wave system and ship motions, and Havelock (1940) analyzed the mean force on a fixed body in waves; these three efforts together confirmed the existence of a drift force. Maruo (1960) provided the first complete analysis of the drift force acting on a ship at zero speed. Newman (1967) further investigated the drift force problem and included the drifting-yaw moment, all by use of momentum theory. Ogilvie (1983) presents an excellent survey paper on drift forces. Huijsmans and Sierevogel (1994), Prins and Hermans (1994, 1996), and Lee and Sun (1997) provide examples of contemporary drift force calculations, including the effect of currents, which are the equivalent of added resistance as ship speed approaches zero. Strom-Tejsen, et al. (1973), provides a survey paper that summarizes the early work in added resistance. Gerritsma and Bueklman (1972) is most representative of these methods, most of which relate the added resistance to a phase lag between incident waves and vessel motions. Lin and Reed (1976) extend the drift force calculations of Newman (1967) to added resistance by means of momentum theory and repeated use of the principal of stationary phase. Kashiwagi (1992) extends the momentum theory approach of Lin and Reed to include the second-order steady yaw moment, avoiding the repeated use of the stationary phase method by the use of Parseval's theorem. Chapter 9 of the report of the Loads and Responses Committee of the 22nd ITTC (ITTC 1999a) provides a summary of more recent added resistance research. The added resistance formulations of Lin and Reed and Kaskiwagi involve far-field representations of the wave field and are in many ways analogous to the measurement and prediction of steady wave resistance. The added resistance results from two sets of wave-wave interactions. The first results from the interaction of the ship generated radiation-diffraction waves interacting with incident waves. The second contribution results from the interaction of the radiation-diffraction waves with themselves.4 Such approaches to added resistance prediction were necessary in the days when strip-theory methods were the only means of predicting ship motions. With the advent of three-dimensional methods for the prediction of seakeeping, it has become feasible to predict added resistance by means of direct pressure integration over a ship's hull surface. Such an approach must encompass the hull surface up to the relative wave height above the ship's steady waterline. This approach has been taken by Sclavounos and Nakos (1993), Sierevogel, et al. (1996), and Bunnik and Hermans (1999). As in the case of wave resistance, accurate pressure integration over the hull surface can be problematic in cases where the wetted surface is ambiguous or difficult to determine accurately, such as in the case of transom stern ships. In these cases, far-field spectral methods such as Lin and Reed and Kaskiwagi may still be the preferred means of determining added resistance in waves. 4 MAJOR RESEARCH ISSUES With the advent of the 21st century, the U.S. Navy is responding to a changing world where it must perform new missions in new regions of the ocean (O'Keefe, et al. 1992, CNO 1997). The new DD-21 destroyer, which is under design, is being pushed toward lower signatures, which results in tumble-home hull forms whose sides retreat inward above the waterline rather than flare the authoritative version for attribution. 4There is an error in this term in Lin and Reed (1976). It is corrected in the Appendix.

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 24 outward, Figure 12. The tumble-home hull forms and their topside shaping result in wave-piercing bows and wide, shallow transoms, Figure 13. One of DD-21's major missions is to provide naval gunfire support to forces ashore, which means that the ship will be operating in the littorals. What are the implications of these changes? The move to tumble-home hull forms results in ships with extremely non-linear hydrostatics and that in turn poses the risk of seakeeping characteristics significantly different from those of traditional naval combatants. The move into the littorals also raises the prospect of environments that are quite different from the ones for which naval vessels have traditionally been designed and characterized. Fig. 13 Profile of a notional tumble-home ship with shallow transom immersion. Fig. 12 Midship section of a notional tumble-home ship. Figures 14 and 15 provide examples of the nonlinearities observed during seakeeping model tests of a notional tumble-home combatant. The two figures show the vertical-bow accelerations in head seas for a tumblehome model in regular waves with a Î»/L of 1.0 for two wave steepnesses, 1/45 and 1/30, and at a Froude number of 0.21. As can be seen, the accelerations appear to be inverted trochoids, with the steeper waves providing âsharperâ troughs. The acceleration time histories are not symmetrical about the zero acceleration line. The negative accelerations are 25 to 50 percent lower than the crests. These accelerations are clearly not âÏ2 times some sinusoidal displacement. Examination of the heave and pitch displacement time histories shows that both are asymmetric about the zero displacement line, though much more sinusoidal in appearance. At lower wave steepnesses, the heave displacement is almost symmetric and becomes less symmetric as the wave slope increases. Pitch displacement is skewed at the lowest wave steepnesses and becomes more so as the wave steepness increases. Examination of bow and stern displacement time histories shows that stern displacements are almost sinusoidal and symmetric about zero. Bow displacements are asymmetric with the peak displacements as much as 50 percent higher than the troughs. Fig. 15 Vertical bow acceleration for a tumble-home Fig. 14 Vertical bow acceleration for a tumble-home monohull model in head seas, Fn=0.21, Î»/L=1.0, 1/30 wave monohull model in head seas, Fn=0.21, Î»/L=1.0, 1/45 wave slope (courtesy of Martin Dipper). slope (courtesy of Martin Dipper). the authoritative version for attribution.

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 25 An examination of Figure 13 provides some insight as to a possible cause of this nonlinear pitch response. As the model pitches down by the bow the stern comes out of the water and the waterline becomes shorter, resulting in a significant reduction in the pitch restoring moment. Conversely, when the model pitches up by the bow, the stern is immersed and the waterline length remains essentially constant, leaving the bow-up pitch restoring moment constant at the calm water value. Thus the pitch restoring moment is quite asymmetricâthere is little resistance to bow-down pitch and significant resistance to bow-up pitch. That the resulting pitch responses are asymmetric is no surprise. In extreme following seas, the tumble-home model has shown a greater tendency to âsurfâ than traditional combatant hull forms. As a consequence, the model will on occasion have its bow plunge into the back of the preceding wave. This is when there is the greatest possibility of broaching, leaving the model beam-to in extreme seas. This poses a risk of extreme rolling if not actually capsizing. These issues point out what has long been recognized in the R&D communityâthat design and analysis tools for evaluating non-traditional hull shapes do not exist. The Naval Studies Board of the National Research Council has identified seakeeping as one of the three critical hydrodynamic technologies for future naval platforms (NSB 1997, 2000). Based on these observations and the discussions of Sections 2 and 3, we identify six specific technology areas on which seakeeping research needs to be focused: 1) efficient numerical methods, 2) large amplitude motions and capsizing, 3) nonlinear dynamics, 4) horizontal plane motions, 5) finite depth in the littorals, and 6) verification and validation. While this list is certainly not all encompassing, it is hoped that it will provide ideas and research direction for those who wish to focus their efforts on problems of Navy relevance. Efficient Nume rical Methods To impact ship design, computational methods must be integrated into the design process. This in turn means that the computer codes must be fast enough that they can provide results in no more than an hour or two of computing time. While there has been work done on improving the computational efficiency of computer codes for ship hydrodynamics problems, a tremendous amount of work still needs to be done. The work to date on computational efficiency has resulted in a large enough literature base that this subject is worthy of a paper of its ownâtime and space obviously preclude this. Although much of the preceding discussion has concentrated on time domain simulations, there are still many situations in which frequency domain computations are used. However, the computational complexity of the linear three- dimensional time periodic forward-speed Green function is one of the reasons that the linear problem of a ship transiting in waves has not been widely embraced as have the steady three-dimensional Kelvin wave and zero-speed seakeeping problems. A great many investigators have formulated the Green function for this problem: Hanaoka (1953), Haskind (1953), Havelock (1958), Newman (1959), and Wehausen and Laitone (1960), to name a few. Chang (1977), Inglis and Price (1981), Wu and Eatock-Taylor (1987), Iwashita and Ohkusu (1992), and Ba and Guilbaud (1995), Guilbaud, et al. (2000) have all developed codes for and presented the results of three-dimensional seakeeping calculations. However, these codes have not gained either wide or prolonged application. Faster and more efficient computational methods are still required. Newman has published a number of papers on the approximation of the steady Kelvin wake Green function (Newman 1987a, 1987b) and the zero-speed frequency-domain radiation-diffraction Green function (Newman 1985a, 1985b) by means of rational polynomials and Chebyshev polynomials. These representations adjust the number of terms in the polynomial approximations with the distance from the singularity so as to maintain consistent accuracy while minimizing computational effort. Due to having two parameters (Froude number and nondimensional oscillation frequency), an extension of these polynomial approximations to the forward speed radiation-diffraction Green function will be more complicated than that required for the steady motion and zero-speed radiation-diffraction Green functions. However, if such an approach could be successfully developed, these approximations could significantly reduce the computational time for evaluating the forward-speed radiation-diffraction Green function. Noblesse and two coauthors have produced a number of papers (Noblesse, et al. 1997, Chen and Noblesse 1998, Noblesse, et al. 1999, Chen 1999) presenting a unified Fourier/spectral theory of the Green functions for the three principal problems of ship hydrodynamics: the steady Kelvin wake problem, the frequency-domain zero-speed radiation-diffraction problem, and the frequency-domain forward-speed radiation-diffraction problem. Noblesse, Yang and Chen's work employs the same form of Green function for each problem, only the dispersion function in the kernel of their wave number integrals changes. Noblesse, Yang and Chen call this Fourier/spectral representation of the Green function the Fourier-Kochin representation. Their representation is a generalization of a common representation of the Green function for the steady Kelvin wave problem that results from a kinematic analysis of the waves in the Kelvin wave pattern. Like the steady the authoritative version for attribution.

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 26 Kelvin wave Green function, their Green function is decomposed into a singular term (1/R +1/R â²) representing the inverse of the distances from the singularity point and its image in the free surface to the field point, a far-field term where the waves are represented by a single Fourier integral, and a non-oscillatory near-field term represented by a double Fourier integral. Computationally, these methods should be very efficientâquite comparable to the wave resistance Green function. Chen (1999) and Chen, et al. (2000) present a traditional Green function analysis, while Chen and Noblesse (1998), and Noblesse, et al. (1999) present what they call a âsuper Green functionâ, G. They call these super Green functions because they perform the spatial integration over the panel geometries before the wave number integration, obtaining a spectral representation of the Green function that is more amenable to numerical evaluation in wave number space. The idea behind the super Green function is not new (cf. Reed, et al. 1990), and is, in fact, old enough that we are not sure of its origins. The benefit of G is that it has much better behavior near the origin than the traditional Green function, G. This is because the amplitude function, A, that results from the spatial integration within the wave number integral has improved asymptotic characteristics. For Î¶+z<0, A decays exponentially as k â â and for Î¶+ z=0, A â 0 like 1/k as k â â, which means that G is finite at the origin in physical space and is slowly varying as the origin is approached. In G, A is finite as k â â, which means that G is singular at the origin in physical space and oscillates rapidly as the origin is approached. Another means of speeding the solution of forward-speed radiation-diffraction problems, an approach that works with both Rankine and linear Neumann-Kelvin singularities, is the use of high-order panel methods, (cf. Newman 1986, Hsin, et al. 1993, Lee, et al. 1996). These methods use conformal panels shaped to the contours of the vessel under analysis. The singularity strengths on these conformal panels can be constant or can also vary with location on the panel. These high- order panels allow significant reductions in the number of unknowns that must be solved for in the boundary value problems. A common approach employed by both Hsin, et al. and Lee, et al. is to use B-splines to fit the hull surface and to represent the potential on the surface of the body. Data shown in Lee, et al. (1996) for two zero speed examples shows that for comparable accuracy, the execution time for high-order panels is 0.3 to 7 percent of that required for flat panels with constant strength singularities. Experience shows the high order panel methods work well for the cylinders and spheres that make up many offshore platforms. However, anecdotal evidence indicates that it is much more difficult to obtain consistent, converged results with high order panel methods on ships. We hypothesize that this is a result of the compound curvature that is commonly found in ships. If these convergence difficulties can be overcome, then high order panel methods have the possibility of reducing computation time by almost two orders of magnitude. As introduced in Section 2, Nabors, et al. (1994), Scorpio, et al. (1996), Scorpio (1997), Scorpio, et al. (1998), and Kring, et al. (1999) have developed order N methods for solving radiation-diffraction problems. These methods, based on the work of Greengard (1987), use multipole expansions to reduce the number of calculations that must be performed by consolidating near-by panels for computing their influence at field points far-away. Nabors, et al. (1994) and Kring, et al. (1999) use these methods to obtain the solutions via traditional BEM solutions. Scorpio, et al. (1996), Scorpio (1997), and Scorpio, et al. (1998) solve the forward speed radiation-diffraction problem by means of desingularized methods where point singularities off the body and free surface are employed in lieu of continuous distributions of panels on the body and free surface. Without preconditioning, Scorpio (1997) reports execution times with the multipole accelerated method that are 25 percent of that of the unaccelerated method. With preconditioning, execution times are 20 percent of the preconditioned unaccelerated method's time, and 5 percent of the non-preconditioned unaccelerated method's time. In addition to multipole acceleration, Nabors, et al. (1994) and Kring, et al. (1999) employ a precorrected FFT (pFFT) technique to further sparsify the matrix equation that must be solved and speed the computation. This technique uses FFT convolution over the spatial grid to compute the potential over the field due to the singularities distributed over the grid. The potential at distant field points is still evaluated by multipole acceleration. Kring, et al. (1999) report execution times with the pFFT approach that are 3.6 to 5.3 percent of the times obtained using a traditional high density BEM solver. Unsteady RANS calculations are extremely time consuming. If any computational efficiency is going to be attained, unsteady RANS calculations need to employ multigrid/multiblock techniques. However, even with these methods, unsteady RANS will continue to be extremely time consuming. It may well be that the most appropriate approach to the unsteady RANS problem will be to couple the RANS solution near the ship to a potential flow solution away from the ship. This could be done by the mixed source technique currently employed in LAMP (Lin, et al. 1999) or by solution of the linear far-field problem recently formulated by Noblesse (2000). Ultimately, there may be frequency domain, and possibly time domain, approaches like the steady the authoritative version for attribution.

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 27 forward-speed Havelock-Dawson approach (Scragg and Talcott 1990), where Rankine singularities are placed on the body, and forward-speed radiation-diffraction singularities are placed on a limited region of the undisturbed free surface near the body. The method allows one to âget away withâ paneling only a limited region of the free surface because the singularities satisfy both the linearized free-surface and the radiation boundary conditions, resulting in a âself-limitingâ distribution of singularities on the free surface. The Havelock-Dawson approach is computationally very efficient. On the body and from the body to the free surface, the Rankine singularity influence functions are evaluated very efficiently. On the free surface, the panel-to-panel influence functions are known constants like 4Ï. This leaves the free surface to body influence-function computations as the only ones involving the free-surface Green function, a relatively small number of evaluations. The result is a boundary element method that can be quite efficient over all. Scragg (1999) demonstrated this for the Havelock- Dawson solution of the steady forward-speed problem. In the case of time domain solutions, the convolution integrals constitute a time consuming portion of the computation. Clement (1999) presents a method of speeding the evaluation of the zero-speed time-domain transient Green function by taking advantage of a fourth order ordinary differential equation (ODE) satisfied by that Green function. Similarly, Ba and Guilbaud (1995) present a method that employs a first order ODE to speed the computation of the forward-speed frequency-domain radiation-diffraction Green function. Thus, while it remains to be seen if the time-domain ODE approach can be extended to forward speed, the work of Ba and Guilbaud indicates that there may be some possibility of this. If the ODE acceleration approach can be extended to the forward-speed time-domain radiation-diffraction Green function, it has the potential for significantly speeding up the time-domain solution of this problem. In the frequency domain, Rankine singularity methods (cf. Kring, et al. 1996 and Landrini, et al. 1999) have limitations on Ï, with Ï not being too much smaller than 1/4. Kring (1998) shows that this limitation need not apply in the time-domain solution of either the zero-speed or forward-speed radiation-diffraction problems. However, the free-surface gridding of these solutions must be tailored to the values of Ï and the Froude number for which a solution is soughtâthe same free-surface gridding cannot be used across a broad range of ship speeds and wave frequencies. The classical forward-speed radiation-diffraction Green function (cf. Wehausen and Laitone 1960) has a singularity at Ï=1/4. This corresponds to the boundary between the regime where the radiated waves completely surround the ship because it is moving more slowly than the group velocity of the radiated waves, and the case where the waves are left behind the ship in a cusp-like pattern because the ship is moving faster than the group velocity of the radiated waves. For many years it was speculated that at Ï=1/4, the wave energy was trapped near the ship and that the motions would blow up due to the unbounded energy. Grue and Palm (1985) performed an analysis of the two-dimensional problem for a submerged cylinder and showed that the forces remain bounded as Ï â 1/4. Liu and Yue (1993) reformulated the two- dimensional solution of Grue and Palm and extend it to a body penetrating the free surface. Finally Liu and Yue analyzed the classical three-dimensional forward-speed radiation-diffraction Green function and showed that its behavior in the limit Ï â 1/4 is such that the potential exists as long as a submerged body has a nonzero cross sectional area, or a body intersecting the free surface has a finite waterplane area. They concluded that the singularity of the Green function is an artifact of a point singularity. Subsequently, Kring (1998) has used a time domain computation to successfully obtain solutions in regular waves at and around Ï =1/4. Depending on the Froude number, Kring's solutions have âspikesâ in the damping coefficients at Ï =1/4, but the damping remains finite. Using the frequency domain Green function of the forward-speed radiation-diffraction problem, Chen, et al. (2000) show similar finite spikes in the added mass and damping coefficients at Ï=1/4 for both a submerged sphere and a Wigley hull form. Large Amplitude Motions and Capsiz ing The Navy is concerned with large amplitude motions from two perspectives. First, in moderately high sea states the concern is with habitability and crew effectiveness. In extreme sea states, the concern becomes ship survivability. This latter concern has become a significant issue with the consideration of tumble-home hull forms for DD-21 (Holzer 2000). Accelerations due to ship motions affect humans by causing fatigue, loss of performance and extreme discomfort and/or illness. An assessment of these potential effects can be obtained by applying the criteria of ISO Standard 2631/1 and /3 (ANSI 1985a, 1985b), which provides an evaluation of the effects of human exposure to whole body vibrations. The 1/3-octave analysis for human performance includes criteria for seasickness at the lower frequencies. The analysis is accomplished by dividing the acceleration power spectral density (PSD) into 1/3-octave bands over the frequency range of interest and determining the rms energy within each band. The rms energy within each 1/3-octave band is compared with the sea sickness/proficiency limits for a given length of exposure as established by the ISO standard. the authoritative version for attribution.

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 28 The rms energy within a 1/3-octave band, a1/3, is obtained by integrating the acceleration spectrum, SAcc, between the corresponding 1/3 octave frequency limits, The upper and lower frequency intervals are defined by fi+1=Î±fi, where Î± is and the center frequency, is given by In terms of the initial starting center frequency, and A typical starting center frequency is 0.01 Hz. Ship survivability is judged by two criteria, one related to structural integrity and the other related to susceptibility to capsizing. Structural integrity is tied to the primary structural loads that result from ship motions and secondary loads that result from wave impacts. The hull structural girder must be designed to withstand the primary loads that result in a seaway, including the structural fatigue that results from experiencing hundreds of thousands of load cycles over the life of the ship. Secondary and tertiary loads affect the local structure and result in failures of the local plating and/or substructure due to extreme local pressures. Wave impacts can also result in whipping of the primary structure that in turn can affect its fatigue life. Computer codes such as LAMP (described below) are used to predict primary bending loads and to some extent secondary and tertiary loads. As this paper is focusing on seakeeping, we shall now discuss various aspects of large amplitude motions leading to capsizing. Capsizing One of the earliest papers to develop a classification of capsizing was Oakley et al. (1974). Based on their observations of model scale capsizing, they found the following modes of capsizing: 1. Low-cycle resonance. 2. Pure loss of stability. 3. Broaching. The report of the Specialist Committee on Stability of the 22nd ITTC (1999b) refines the causes of capsizing, separating out the following causes: 1. Static Loss of Stability. 2. Dynamic Loss of Stability. (a) Dynamic rolling. (b) Parametric excitation or low cycle resonance. (c) Resonant excitation. (d) Impact excitation. (e) Bifurcation. 3. Broaching. (a) Successive over-taking waves. (b) Low frequency, large amplitude yaw motions. (c) A single wave. 4. Other factors. In the ITTC classification of capsizing, static loss of stability occurs at high speed (Fnâ 0.4) and short waves (Î»/L â1), such that the ship speed matches the phase speed of the wave, and the static stability of the ship poised on a wave crest is reduced to the point that the ship capsizesâno dynamics are involved (cf. Tatano, et al. 1990, Umeda, et al 1990, Vermeer 1990). Dynamic loss of stability occurs when roll motions become so large that the righting arm becomes zero or negative and the ship capsizesâthis usually involves nonlinear seakeeping. Broaching involves large heading excursions from the desired course, and the ship capsizes because of dynamic loss of stability or at high enough speeds, the momentum of the ship can force extreme roll due to high resistance to lateral motion. Other factors are issues such as water on deck that results in a loss of static stability, shifting cargo, or extreme wind loads in beam seas. Dynamic rolling as a cause of capsizing is an asymmetric roll that results from all six modes of motion being coupled. The ship takes on extreme roll to leeward when on a wave crest in following or quartering seas, and rolls back to windward in the wave trough. This cycle repeats with increasing roll amplitude during subsequent wave encounters, until it capsizes. Dynamic roll results from the loss of stability on wave crests that is not compensated for in the troughs. Dynamic roll will typically occur at wave periods significantly greater than the roll natural period. Senjanovi (1994) shows how this type of superharmonic excitation could occur. Parametric excitation is resonant roll that occurs as a result of heave and pitch coupling with roll in following and quartering waves whose lengths are about equal to the ship length. The roll is symmetric and the encounter frequency is half the roll natural period. Quite large roll angles can occur even in fairly small amplitude waves and with small vertical the authoritative version for attribution. plane motions (Oakley et al. 1974, de Kat and Paulling 1989). The analyses of Nayfeh (1988) and Senjanovi (1994) show how this type of subharmonic excitation could occur. Most recently Nevis and Valerio (2000) have completed an analysis of parametric excitation for a ship with three-degrees-of-freedom (heave, pitch, and roll). They find that there are distinctly different behaviors in oblique seas that do not occur in following seas. Resonant excitation occurs in quartering and beam seas when the ship encounters waves whose periods match the roll natural period. In a narrow banded spectrum, there may be enough successive waves with the right period to cause capsizing. Shen and Huang (2000) present an analysis using a stochastic differential equation to show how long it will take to capsize under this type of excitation in irregular waves. Impact excitation occurs when the impact of steep breaking waves against the side of a ship in beam seas causes the the ship to take

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 29 on extreme roll. The resulting heel angles may be large enough to cause the ship to capsize. Bifurcation is nonlinear dynamic phenomena that occurs when as the result of a perturbation, the motions jump from one stable state to another, causing a sharp increase in roll amplitude. Bifurcation has been studied analytically and in laboratory experiments in regular waves. Roll and Cross-Flow Drag Many studies of ship rolling have been performed using two-dimensional models. Bishop, et al. (1990), Falzarano (1994), and Oh and Nayfeh (1996) make the point that either linear or nonlinear coupling of roll with other modes of motion can have a significant impact on the prediction of roll motions and the point at which capsizing occurs. Bishop, et al. (1990) conclude that the neglect of other degrees of-freedom can result in an optimistic prediction of vessel stabilityâa higher GM is required for dynamic stability than would be predicted by a simpler model, while Falzarano (1994) shows that a single-degree-of-freedom roll model predicts much higher maximum roll than a multiple-degree-of-freedom model. Chen, et al. (1999) studied the coupled heave-sway-roll equations for a two-dimensional ship in beam seas. By nondimensionalization and rescaling the problem in a wave fixed coordinate system, they are able to reduce the problem to a single-degree-of-freedom problem dominated by roll. The problem has parameters which are functions of yg and zg, the positions of the center of gravity. By singular perturbation analysis, the movement of the center of gravity is removed as a higher order effect and the analysis proceeds. Chen, et al. study a fishing vessel that has capsized twice, it is is shown to loose stability at Â±32.2Â° of roll. They believe that a similar analysis should be possible for six-degrees-of-freedom. Another of the factors having a significant effect on roll response is viscous roll damping. As discussed earlier, roll is the mode of motion where viscous effects are likely to be most significant. Brook and Standing (1990) present a brief survey of empirical roll damping formulations and discuss experimental techniques for determining roll damping, albeit for barges. Yeung, et al. (1998), Roddier, et al. (2000) and Yeung, et al. (2000) present a method for computing two- dimensional roll damping, albeit for small roll amplitudes. Spyrou and Thompson (2000) present a method for determining the damping for large amplitude roll; in particular, they take into account the nonlinearities in the restoring moment in their derivation so that accurate values of the damping are available even in situations where the restoring moment is vanishing. The cross flow drag on a ship is critical to simulating the yaw and sway motions of a vessel in extreme motions. These forces have generally been treated empirically. TÃ¸nnessen, et al. (1999) present the results of a finite-difference simulation of the two-dimensional Navier-Stokes equations for blunt bodies with sharp corners and flat plates. They find that an eddy viscosity must be introduced in order to eliminate numerical instabilities and that the nature of the solution can be quite sensitive to mesh density and time step, particularly near the corners. Thus, one must be careful to make sure that one has a converged solution. TÃ¸nnessen, et al.'s method predicts the Strouhal numbers satisfactorily, but in general the drag predictions exceed experimental values by significant amounts, although with less error than other methods. Similar methods need to be developed for three-dimensional problems for ships with forward speed. Nonlinear Dynamics and Bifurcation Serious work on nonlinear dynamics and bifurcation applied to ship rolling appears to have started in earnest around 1990. Nayfeh and Sanchez (1990) and Sanchez and Nayfeh (1990) are two papers that introduced this work. In these papers, analytical-numerical models were applied to study the two-dimensional roll and capsizing of ships under mild conditions. Nayfeh and Sanchez (1990) have shown bifurcation in beam seas, where the roll will jump from a moderate level to capsizing conditions under slight perturbations. Sanchez and Nayfeh used analog computations to study the roll behavior of a ship as wave slope and excitation frequency were varied. They present the bifurcation diagrams for the motions near resonance, for simulations with no roll bias and for roll with biases of Â±6Â°. The biases lead to quite different behavior from that with no bias and from each other. Francescutto and Nabergoj (1990) study an idealized nonlinear differential equation for the roll response of a ship. They have a cubic term corresponding to the hydrostatic restoring moment and they subject the system to narrow band excitation. As is to be expected, they obtain responses at both the linear natural frequency and a subharmonic frequency at approximately one-third the natural frequency. Their solutions demonstrate the classic bifurcations that equations with nonlinear damping exhibit. At the same time, Falzarano and Troesch (1990) and Falzarano, et al. (1990) were studying the nonlinear motions of a small fishing vessel that is known to have capsized twice. Falzarano and Troesch (1990) study the effects of water on deck and increased damping on the motions of the vessel as a function of wave amplitude and frequency. They employ âgeometricâ methods to examine the responses of both forced motions in the absence of damping and damped motion in the absence of excita the authoritative version for attribution.

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 30 tion. Based on this analysis they develop PoincareÃ© maps showing behavior of the roll motion as a function of the initial conditions (initial roll angle and initial angular velocity). These plots serve to define the regions of stability where the motions remain bounded and those regions of instability where the motion grows without bound until capsize is predicted. Falzarano, et al. (1990) perform a similar analysis for a vessel undergoing a slow steady turn using both a one-degree-of- freedom and a three-degrees of-freedom model with heading as a parameter. The motions are shown to have responses at both the linear natural frequency and at twice this frequency. It is found that the behavior differs between the one-degree- of-freedom and three-degrees-of-freedom models with the three-degrees-of-freedom model showing differences related to the forward speed that are absent in the one-degree-of-freedom model. The results of Chen, et al. (1999) for the coupled heave-sway-roll equations of a two-dimensional ship in beam seas are interpreted in the context of PoincareÃ© maps. They select a value for a parameter such as height of the center-of-gravity and then form maps for roll displacement and roll angular velocity. Such analyses need to be extended to the full six- degrees-of-freedom motions of a ship in an arbitrary seaway. State-of-the-Art in Large Amplitude Motion Predictions The Navy has two computer codes that it has been using for computation of large amplitude motions, FREDYN and LAMP. The code FREDYN (cf. de Kat and Paulling 1989, de Kat 1994, de Kat, et al. 1994) has been used extensively to assess the capsize susceptibility of ships. LAMP (cf. Lin, et al. 1997, 1998b, 1999) has been used primarily to predict structural loads (cf. Weems, et al 1998, Lin, et al. 1998a), although it is now beginning to be used for capsizing assessments. FREDYN is a blended time-domain strip-theory code that includes nonlinear hydrostatics and Froude-Krylov exciting forces, with linear radiation-diffraction forces computed using conformal mapping. In addition to normal seakeeping forces, the code includes viscous roll damping, empirical cross-flow drag, total resistance including that due to the orbital velocity of the waves, propeller thrust, side force due to the rudder, and wind forces on the hull and superstructure. The portion of the code associated with the linear and nonlinear maneuvering forces is customizable for the specific hull form under consideration. The code contains a proportional, integral and derivative (PID) auto pilot to keep the ship on a straight course. The simplicity of FREDYN allows the computations to be performed effectively on a PC. The use of conformal mapping techniques for radiation-diffraction computations in FREDYN restricts the range of section shapes for which accurate geometric fits can be obtained. There appears to be a paucity of published data on the validation of FREDYN for the full six-degrees-of-freedom motions in high sea states. LAMP5 computes six-degrees-of-freedom ship motions and wave loads in large-amplitude waves. The method employed is a mixed source formulation, where the fluid domain is divided into inner and outer domains, separated by a matching surface. In the inner domain, the body, free surface, and matching boundary are all discretized with Rankine singularities. The inner domain free-surface problem is solved under the weak scattering assumption, where it is assumed that the radiation-diffraction waves are small compared to the incident waves. The body boundary condition is satisfied on the instantaneous wetted hull under the incident wave profile. At each time step, local free-surface elevations are used to transform the body geometry into a computational domain with a deformed body and a flat free surface. By linearizing the free-surface boundary conditions about this incident wave surface, the solution is simplified. In the outer domain a linear problem is solved using linearized free-surface transient Green functions distributed only on the matching surface. In this approach the correct hydrostatic and Froude-Krylov wave forces are readily included. This extension is important especially for transom-stern ships and when the amplitude of the incident wave is large. As the matching boundary is fixed and never deformed, the convolution integrals over past time are greatly simplified and accelerated. Viscous and lifting effects are included in LAMP using semi-empirical formulas. At each time step, viscous and lifting forces and moments acting on the ship are calculated and included in the equations of motion as part of the forcing functions. A PID-control auto-pilot system is also included for course-keeping purposes. For these partially nonlinear calculations, a super computer or parallel processor machine is required. Published seakeeping results for LAMP are limited to head seas. The correlations between predictions and experiments are too limited to draw any firm conclusions about the validity of the code. Large amplitude motions and capsizing of smaller vessels such as fishing boats requires the consideration of additional factors in the computations. These include the shipping of water on deck, the motions and forces due to water on deck, the escape of water from the deck, and the hydrodynamic forces resulting from the deck being under water. Grochowalski, et al. (1998) and Huang, et al. (1998) present a large amplitude motions program the authoritative version for attribution. 5LAMP is in fact three different computer codes, LAMP 1, LAMP 2, and LAMP 4, for computing ship motions. Each code performs computations with different levels of linearization. LAMP 1 is a fully linear and LAMP 4 is nonlinear. In the present discussion, when we discuss LAMP, we are describing LAMP 4.

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 31 that takes these factors into account. Their model is a blended three-dimensional model that includes nonlinear hydrostatic restoring forces and Froude-Krylov exciting forces, viscous damping and cross flow drag terms, the forces due to the rudder, and an auto pilot. Water shipping on deck is solved as a dam breaking hydraulics problem whenever the deck edge is below the instantaneous free-surface level. The motion of water on deck is analyzed as a shallow water flow. The freeing of water on the deck is treated empirically as a bore exceeding the height of the bulwark. The forces on the submerged deck are also treated empirically as a resistance to the motion of the vessel if the relative motion of the deck is opposed to the flow velocity in the incident wave. Presumably, if the hydrodynamic forces were computed on the instantaneous wetted surface rather than on the mean wetted surface, these forces on the submerged deck would be accounted for automatically. The flow of green water onto, on, and off of the deck of naval vessels is an area where research is needed. To investigate the flow of water on deck Huang and Hsiung (1996) employ a finite difference scheme to solve the Euler equations under a shallow-water flow assumption. They present two- and three-dimensional flow calculations which compare quite favorably with model-scale experiments. Fekken, et al. (1999) use a volume-of-fluid method to track the location of the free surface in a simulation of the Navier-Stokes equations for the flow of water on deck. They present comparisons of wave profiles and pressures from experiments and computations for three-dimensional flow on a model foredeck. Horizontal Plane Motions We have chosen to refer to the subject of horizontal plane motions for this section because we wish to emphasize the coupling between ship maneuvering and seakeeping. Two issues come to mind immediately: the first is the use of the rudder to provide roll stabilizationâin lieu of roll fins, and the second is the relation of broaching to surging, with the loss of directional control, leading to possible capsizing. Rudder Roll Stabiliz ation Baitis (1980), Baitis, et al. (1983), and Baitis and Schmidt (1989) present their experience in using the rudder to control roll. The principal is simple in that if high enough rudder turn rates can be achieved, then the rudder can produce a useful roll moment that can be used in lieu of or to supplement roll fins. Such systems have been installed on several U.S. Coast Guard cutters and on several ships of the DDG-51 class. The analog system first installed on the USCG Hamilton produced a 30 to 40 percent reduction in roll motion using the original turning gear. A second installation using a digital controller and a modified steering system achieved roll reductions of up to 70 percent. This shows the benefit of designing such a system from the start as opposed to trying to execute it as a retro fit. A steering system designed for rudder roll stabilization will need to be capable of attaining rudder rates as high as 21Â°/sec. Baitis and Schmidt (1989) present a simple analysis of rudder induced motions and a controller algorithm, all based on linear system characteristics. An analysis based on contemporary nonlinear motions models would be appropriate. Parametric analysis of the design space in terms of a ship's gross geometric properties (length, beam, draft, and displacement) is required to determine those regions in which rudder roll stabilization can work effectively. For rudder roll stabilization to work, there must be sufficient separation between the frequency at which vessel sway-yaw responses due to rudder oscillations start and the roll natural frequency. Otherwise, the rudder's action will affect the steering characteristics of the ship or not provide roll stabilization. Broaching Grochowalski, et al. (1998) show poor agreement between predicted capsizes and experimental observations in following and quartering seas. This is because the predicted surge, sway and yaw do not agree well with the experimental results. This may be largely due to the inadequacy of the controller. However, the surge displacement of the vessel over time tends not to follow the trends of experiments. If the predictions are sinusoidal about a mean, then the experiments show a forward drift (which may or may not have a superimposed sinusoidal motion) or vice versa. De Kat and Thomas (1998) show similar discrepancies in the instantaneous forward speed predicted with FREDYN compared to model experiments. Unfortunately, they do not present predictions of other modes of motion corresponding to their experimental results. Using phase-plane analysis, Vassalos and Spyrou (1990) analyze the effects of trim by the bow which can negatively affect directional stability. This loss of directional stability can occur in waves, and the large rudder angles required to control the heading can result in significant heel angle or biases resulting in susceptibility to capsizing. Umeda (1990) examines the susceptibility of ships to surf using phase-plane analysis. It is found that the results of regular wave analysis are not valid in irregular seas. In particular he identifies particular Froude numbers and wavelength to ship-length ratios where surging can be expected. He finds that reducing propeller rpm can deter a tendency to surf. However, once a wave has âcapturedâ the ship, there is little that can be done to con the authoritative version for attribution.

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 32 trol the ship. Bandyopadhyay and Hsuing (1994) and Spyrou (1996, 1997, 2000) perform analysis of similar conditions using coupled surge-sway-roll-yaw equations. Using phase plane analysis, they show that there is a bifurcation that results in the ship being captured by a wave and losing directional stability leading to broaching, loss of dynamic stability and possible capsizing. Umeda, et al. (1997) reports on experiments that largely validate the conclusions of the preceding models. They conclude that the model must include roll, and that a model with just the surge-sway-yaw modes of motion is not adequate. Interestingly, they conclude that broaching occurs when there are no stable periodic motions. Vassalos and Maimum (1994) examine heave-pitch-roll-yaw coupled motions and show that broaching results from strong coupling between lateral and vertical plane motions in extreme seas. In following seas, harmonic and superharmonic roll, excited by heave and pitch, is a precursor to capsizing that can occur if there is a loss of directional control that results in the ship becoming beam on to the seas. The impact of an asymmetric underwater body due to roll is investigated by Renilson and Manwarring (2000). Using experiments and CFD they demonstrate a steady hydrodynamic coupling between roll and yaw moment. Using the results of this analysis they show that there is a significant reduction in the wave steepnesses at which broaching occurs over a range of wavelength to ship-length ratios from 1.0 to 1.5. Umeda and Matsuda (2000) propose an anti-broaching device for small craft that is in many ways akin to rudder roll stabilization. They propose that the steering system be designed with high enough turning rates and large enough deflection angles so that the rudder can be used as a drogue to slow surge and to provide directional stability. Model tests with a high gain controller show that this can be effective in preventing broaching. The controller must be coupled with a pitch gyro so that the anti-broaching mode is only entered when there is a threat of a broach; otherwise, the large rudder angle could cause dynamic behaviors as bad as those that are to be prevented. In the case of tumble-home hull forms, the wide shallow transoms of these hull forms appear to promote a tendency to surf in following and quartering seas, a particularly dangerous situation. Under these conditions, models with wide shallow transoms seem to surge at slightly more than the phase speed of the wave, resulting in little relative velocity over the rudders and a possible loss of directional control. Finite Depth Proble ms in the Littorals The littorals are characterized by waters shallow enough that the form of the ambient waves is affected by the finite depth and currents. In particular, the phase and group velocities of the waves are modified by the presence of the bottom. This means that the classic wave climatologies are no longer valid and that the radiated and diffracted waves are also affected. Thus, computational methods intended for use in littoral problems with flat bottoms must use the appropriate two- or three-dimensional finite-depth Green functions, not the infinite depth Green functions previously discussed. While the finite depth Green functions are significantly harder to compute, they do not require any different logic and can usually simply be inserted into an existing infinite depth program. The frequency-do main, forward speed, radiation-diffraction theory of Noblesse, Chen, and Yang (cf. Noblesse, et al. 1997, Chen & Noblesse 1998, Noblesse, et al. 1999), can be extended to finite depth by substituting the appropriate dispersion relation into their wave number integrals. For Rankine singularity methods, the flat bottom boundary condition can easily be met by using an image singularity placed at the appropriate distance beneath the bottom. Again, the required modifications to an existing program are minimal. The frequency-domain Green function and Rankine source methods all assume a flat bottom, which may not be realistic in the littorals. We are not aware of any computational results for non-uniform bottom topography. Computations could be performed using Rankine source methods in which the free surface, bottom surface and hull surface are all panelized. Korsmeyer, et al. (1993) have used this approach to compute the interaction forces between ships moving along straight paths over a non-uniform bottom. As a rule-of-thumb, finite depth effects become important when the water depth is less than half the incident wavelength (h0< Î»/2). Offshore engineering programs (for example see WAMIT 1999) all include finite depth effects because the bodies are often so large that the radiation forces must include bottom effects even if the incident wavelengths are short enough that they may be considered as deep water waves. For ships, strip theory programs are easily modified for finite depth effects by replacing the infinite depth, two-dimensional Green function subroutines with those for finite depth. The finite depth programs take longer to run, but the computer time is still short. Kim (1968), Takaki (1978), and Beck and Troesch (1989) give examples of these types of calculations. Takaki (1978) and Tasai, et al. (1978) give experimental results and compare them with calculations. In general, the analytical results agree reasonably well with the experiments, showing agreements similar to those found for strip theory in infinite depth. For the two cases they studied (water-depth to draft ratios of 1.5 and 2.1), finite depth effects significantly alter the motion responses from infinite depth and can not be the authoritative version for attribution.

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 33 neglected. Tasai, et al. (1978) found significant wall effects in their experiments due to the finite tank width. The limit of finite depth computations as the depth goes to zero is shallow water. In many ways linear shallow water computations are easier than finite depth since in shallow water the waves are no longer dispersive and many of the computations are simplified. The major difficulty is to properly account for the flow in the small gap between the bottom and hull bottom. Tuck (1970) and Beck and Tuck (1972) developed a slender-body theory for seakeeping in shallow water. In their work, the incident wavelength must be on the order of the ship length and much greater than the depth. As a general rule, this means that the wavelength is 10 times the water depth. Unfortunately, linear shallow water waves are not stable (for example see Chakrabarti 1994) and quickly break down. Nonlinear effects tend to dominate shallow water waves and they can not be ignored. Shallow water theories that rely on linear shallow water incident waves seem to be doomed to failure; fully nonlinear or at least higher order computations will eventually have to be done. There has been some progress in developing a wave climatic model for waves in coastal regions (cf. Lin and Perrie 1997a, 1997b, 1999). However, much work remains to be done in developing climatic models and characterizing the environment in the littorals. Particularly these models must include the source terms that originate as waves in deep water and propagate into the finite depth coastal regions, and the effects of currents and wind on the development of waves in these regions. There is a need for a finite depth seakeeping code that can accommodate the nonlinear wave inputs that are likely to result from realistic descriptions of the littoral wave environment. This will require a time domain model, and significant efforts in the area of computational efficiency will be warranted due to the additional complexity of the finite depth transient Green function. An example of coupling a coastal climatic model to seakeeping predictions is presented in Lin and Thomas (2000). They present examples wave climatology predictions and assessments of large amplitude motions and capsizing using FREDYN (de Kat 1994, de Kat, et al. 1994) for two cases: the Gulf of St. Lawrence, and a Pacific Ocean seamount. In principal the FREDYN code violates the requirement for a finite depth Green function seakeeping code. However, as the water depths in both example cases are such that h0>Î»/2 (h0 â100 m for the Gulf of St. Lawrence and h0=500 m for the Pacific Ocean seamount), there is no significance to the fact that FREDYN does not use the finite depth Green function for these calculations. It is not clear whether or not the wave climatology input into FREDYN is in reality nonlinear. However, due to the relatively deep waters considered, finite depth related wave nonlinearities are probably not an issue either. Verification and Validation The development of complex computer codes always raises the risk of errors. These may be a consequence of faulty modeling of the physics of the problem, errors in the computer code, or explicit or implicit approximations that limit the validity of a code's predictions to a certain class of vehicle geometries. There is a three-stage process that all codes should go through before they are ready for routine use in the support of design: verification, validation, and accreditation. This process shows that a code is capable of solving a particular physical problem for a particular class of vehicle. Beck, et al. (1996) discuss these issues in the context of CFD codes for ships. Rood (1996) presents the validation process that is being used in the case of the free-surface RANS codes that are being developed under ONR sponsorship. The definitions of verification, validation, and accreditation from Beck, et al. (1996) are as follows: Verificationâthe demonstration that the code is âreasonablyâ bug free and the output is numerically correct. For complex codes verification is very difficult because the correct answers are not known and there are literally thousands of possible cases to check. Usually codes are verified using internal consistency and convergence checks, computing known analytic solutions for simple test geometries, and testing against other codes. None of this can guarantee a bug-free code for all possibilities (in fact, the first law of computing is that all large codes contain bugs), but hopefully the bugs that do remain are inconsequential. Validationâthe comparison of the numerical predictions with physical results. Validation is usually done by comparison with detailed experiments. On occasion full-scale experiments can be used for validation, but full-scale results are usually not accurate enough. The required precision of the experiments can not be over emphasized. If there is a difference between theory and experiment, the confidence level of the experimental results must be such that one can be sure that the problem lies with the theoretical predictions. There are many horror stories of wasted effort trying to find fault with the numerical computations, only to discover that the experimental results were inaccurate. In the ideal world there should be a synergism between numerical computations and experiments so that both are improved in parallel. Accreditationâthe certification that a specific code is acceptable for a certain type of design problem. After accreditation the code can be made available to designers to solve the accredited type of problem. Accreditation must be very specific or there will be a tendency to apply the code to problems for which it was not intended. the authoritative version for attribution.

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 34 Implementation of these three stages requires money and time, neither of which are normally available. Often the validation requires model-scale tests. These tests can take a significant portion of the time for code development and cost at least as much as the code development. If full-scale data is also required, the costs can escalate even further. Too often the approach is to try the code on a problem before the code is ready, resulting in the possibility of misinformation, leading to bad design decisions and problems that may last forty years or moreâthe lifetime of a ship. A search of the literature found very little guidance on how validation, let alone validation in extreme seas should be approached. Chapter 10 of the report of the Seakeeping Committee of the 20th ITTC (1993) discusses âUncertainty Analysis and Validation Proceduresâ and Section 11.3 of the report of the Seakeeping Committee of the 21st ITTC (1996) discusses âValidation of Seakeeping Calculations.â The report of the 20th ITTC lists a number of steps for what they call a âQuality Plan.â The first steps of this plan are associated with what we call verification, but steps 6 through 12 relate to validation. They are as follows: 6. Tests against analytical or exact solutions. 7. Tests against asymptotic formulae. 8. Comparisons against test results for basic geometric shapes. 9. Comparisons against other programs 10. Preprocessing: grid generation, topography size and order of panels, and convergence tests for element distribution. 11. Post processing: computations from regular to irregular waves, and data reduction. 12. Systematic accuracy analysis, and determination of sensitivity of final results to individual error sources. These steps all relate to linear small amplitude processes. It is disappointing that the 21st ITTC provides even less detail on validation, although they do state that based on their examination of current practice that âmore work needs to be done.â Thus it would appear seakeeping validation is an area in which the ITTC could profitably focus in the future. The issue of how to validate a code for seakeeping in high sea states and/or in the littorals is not well defined. Most of the validation work that has been published involves comparisons of linear frequency domain quantities such as added mass and damping, and response amplitude operators (cf. JournÃ©e 1992). Such data certainly provides the first test that any seakeeping code must pass. However, it is far from sufficient for extreme conditions. Hjort (1993) reports on an effort to assess software for stability calculations. Although the effort described is only related to hydrostatics, there are a number of conclusions that are relevant to large amplitude motion predictions, particularly because hydrostatic restoring forces are so significant to many components of ship motions. The three conclusions that appear most significant are: 1) A graphics capability for verification of the geometry should be mandatory, 2) An approved system should be based on test calculations for at least two ships, and 3) Validation should include a sensitivity test by varying the number of sections used in in the geometry. Ohkusu (1998) argues that the use of integrated quantities such as added mass and damping or ship motion responses is inappropriate for validation. He feels that integrated quantities do not identify the deficiencies in a theoretical formulation and that they combine many factors that can hide errors in both the experiments and theory. Ohkusu argues in favor of measurements of quantities such as hull pressures and local wave fields as providing better validation and more insight into the hydrodynamic processes involved. While these claims may well be correct, the required experiments and experimental measurements will be much more complicated, will take much longer to perform, and will be more costly. In examining the seakeeping of ship models in extreme seas, one is dealing with a stochastic process where the seas are random or pseudo random, but possibly repeatable if the wave maker can be restarted in the same location within the sea state record. Blume (1993) discusses the issues of model testing for ship capsizing when one is dealing with random processes. One of the difficulties is that there are a number of variables in the test that are random and out of the control of the test engineer even if the seas are repeatable. These are primarily related to the controllability of the ship model, which will invariably take on different speeds and angles to the sea from run to run. Dealing with these sorts of variability requires many repeated runs to obtain statistical validity of the extreme responses. It is known that âcriticalâ extreme responsesâslamming, green water on deck, capsizingâare often the result of large steep waves that are grouped (Tika and Pauling 1990, Blume 1993), and that a certain minimum number of waves must be in the group to elicit the response. If indeed it is the case that a group of waves must be encountered to elicit the critical response, then from the perspective of code validation, it may be that large amplitude monochromatic wavesâthe large amplitude equivalent of regular wavesâare an efficient means of obtaining the necessary experimental data for code validation. Such laboratory waves are easily repeated and should eliminate one of the uncertainties of validation experiments. It must be cautioned that such monochromatic wave testing will not eliminate the need for experiments in ran the authoritative version for attribution.

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-basedâas opposed to an empirically-basedâmodel 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):43â53. Baitis, E., D.A.Woolaver & T.A.Beck (1983) Rudder roll stabilization for Coast Guard cutters and frigates. Naval Eng. J., 95(3):267â82. Baker, G.R., D.I.Meiron & S.A.Orszag (1982) Gen eralized vortex methods for free-surface flow problems. J. Fluid Mech., 123:477â501. Bandyopadhyay, B. & C.C.Hsiung (1994) Mechanism of broaching-to of ships from the perspective of nonlinear dynamics. 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Proc. 14th Chesapeake Sailing Yacht Symp., Annapolis, MD, pp. 1â20. Cle ment, A. (1996) Coupling of two absorbing boundary conditions for 2D time-domain simulations of free-surface gravity waves. J. Comp. Phys., 126:139â51. Cle ment, A.H. (1999) Using differential properties of the Green function in seakeeping computational codes. Proc. 7th Intl. Conf. Num. Ship Hydro., Paris, France 15 p. CNO (1997) Forwardâ¦from the seaâthe Navy operational concept. U.S. Department of the Navy, Washington, D.C., 8 p. (Available online at <http:// www.chinfo.navy.mil/navpalib/policy/fromsea/ffseanoc.html>.) the authoritative version for attribution.

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 37 Cointe, R., P.Geyer, B.Kring, B.Molin & M.Tramoni (1990) Nonlinear and linear motions of a rectangular barge in a perfect fluid. Proc. 18th Symp. Naval Hydro, pp. 85â99. Cummins , W.E. (1956) The wave resistance of a floating slender body. Ph. D. Thesis, American Univ., Washington, D.C., 66+v p. Dalzell, J.F., W.L.Thomas, III & W.T.Lee (1992) Correlation of model data with analytical load predictions for three high-speed ships. DTMB Report CARDEROCKDIV-SHD-1374â02, 222+ix p. Dawson, C.W. 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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):96â107. XÃ¼, H. & D.Yue (1992) Computations of fully-nonlinear three-dimensional water waves. Proc. 19th Symp. Naval Hydro., pp. 177â201. Yang, C. & R .C.Ertekin (1992) Numerical simulations of nonlinear wave diffraction by a vertical cylinder. J. Offshore Mech. Arctic Eng., 114:36â44. Yeung, R.W. (1982) The transient heaving motion of floating cylinders. J. Eng. Maths., 16:97â119. 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. 195â 218. Yeung, R.W., S.-W.Liao & D.Roddier (1998) Hydrodynamic coefficients of rolling rectangular cylinders. Int'l. J. Offshore & Polar Eng., 8(4):241â50. 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. 103â18. APPENDIXâCORRECTION 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 Ï â¤1/4 and Ï>1/4, respectively. The constants Â± Î and ÏÂ±â appearing in the limits of integration correspond to the zeros of dÎ¸/du for Î»1, and ÏÂ± corresponds to the zeros of dÎ¸/du for Î»2, where Î¸(u) is derived from: The Kochin function H(u, Î»), u0, and Î»p are defined in Lin and Reed (1976). the authoritative version for attribution.

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 RANSâoften, 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 slenderâeven 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: âThe demonstration that the code is âreasonably' bug free and that the output is numerically correctâ 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 importantâthe 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.

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.