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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
SESSION 11
WAVY/FREE-SURFACE FLOW: SHIP MOTIONS

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
The Simulation of Ship Motions
H.B.Bingham, F.T.Korsmeyer, J.N.Newman, and G.E.Osborne
(Massachusetts Institute of Technology, USA)
Abstract
A three-dimensional panel method is used to solve the linearized ship motions problem for a ship traveling with steady forward speed through quasi-random incident waves. The exact initialboundary-value problem is linearized about a uniform flow, and recast as an integral equation using the transient free-surface Green function. This integral equation is discretized in space by using planar panels, on which the potential is assumed to be constant, and in time by using the trapezoid rule. Collocation is performed at the centroids of each panel.
A technique for approximating the asymptotics of the solution is presented and used to reduce the required length of the computations. Results are shown for a Wigley hull, with and without a steady forward speed. The calculated hydrodynamic coefficients are compared to experiments, as well as to the calculations of two frequency-domain solutions, and a simulation is performed of the ship traveling through a Pierson-Moskowitz sea.
1
Introduction
Since the time when Haskind [6] and Cummins [3] put the superposition principle for transient ship motions on a solid foundation, much work has been focused on calculating the requisite impulseresponse functions, or the analogous frequency-response functions. The linearized theory (elegantly reviewed by Ogilvie [22]), has been implemented for practical ship motions calculations, originally using strip theory methods, and more recently by way of three-dimensional panel methods. For solving problems involving a steady forward speed, the most promising panel methods fall into two categories: solutions using the transient free-surface Green function, where only the ship surface is discretized; and Rankine methods, where singularities are distributed on both the hull and a portion of the free-surface. The present work falls into the first category. Frequency-domain results using Rankine methods have been presented in the last few years, (e.g.: [2]; [20]) and Nakos et al. [19] have recently extended their Rankine, frequency-domain method to the time domain. Solutions using the transient free-surface Green function have been reported by several investigators (e.g.: [9]; [10]; and [14]). More recently the body-exact formulation (where the body boundary condition is applied on the exact instantaneous position of the hull, while the free-surface condition remains linearized) has been used to extend the technique to large amplitude body motions (e.g.: [1]; [16]; [17]).
We have developed a computer code for the transient hydrodynamic analysis of ships and other bodies. The material in this paper is based on our experience in developing and using this code which is called TiMIT (Time domain MIT). We discuss theoretical and practical issues which are germane to the robust and efficient calculation of impulse-response functions, and their subsequent use in performing a simulation of a ship traveling in a seaway. In Sections 2 and 3, we review the linearized formulation of the transient ship motions problem. Section 4 describes the numerical solution, and presents some techniques which can be used to improve the computational efficiency. In particular, we show how a knowledge of the asymptotic behavior of the solution, combined with an appropriate choice of an impulse, may be used to reduce the required length of the computational record. Section 5 presents the results of a simulation of a Wigley hull travel-

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ing through a Pierson-Moskowitz spectrum of incident waves. Finally, some concluding remarks appear in Section 6.
2
Equations of Motion
During normal operating conditions the ship's weight, along with the steady, hydrodynamic sinkage force and trim moment acting on it, are all balanced by the hydrostatic pressure acting on the hull, while its steady resistance is overcome by the propulsion. These forces are in balance, and we will focus upon the unsteady perturbations about this equilibrium condition. Through Newton's law, the dynamics of a ship's unsteady oscillations are governed by a balance between the inertia of the ship and the external forces acting upon it. This balance is complicated by the existence of radiated waves, as a consequence both of the ship's own motions and its scattering of the incident waves. This means that waves generated by the ship at any given time will persist indefinitely and, in principle, affect the ship at all subsequent times, a situation which is described mathematically by a convolution integral. Having assumed that the system is linear, the equations of motion may be written in a form which is essentially identical to the model proposed by Cummins [3]:
j=1, 2, …, 6, (1)
where the exciting forces on the right-hand side may be determined from:
(2)
as proposed by King [8].
In equation (1), the ship's displacement from its mean position in each of its six rigid-body modes of motion is given by xk, and the overdots indicate differentiation with respect to time. The excitation of the ship is provided by ζ(t), a time history of the incident wave elevation at some prescribed reference point on the free-surface. The ship's inertia matrix is Mjk, and the linearized hydrostatic restoring force coefficients are given
Figure 1: The reference frames and surfaces of the problem.
by Cjk. The hydrodynamic coefficients and the kernel of the convolution on the left-hand side of (1), and the kernel of the convolution on the right-hand side of (2), make up a set of radiation and diffraction impulse-response functions: the combination of ajk, bjk, cjk, and (t) is the force on the ship in the jth direction due to an impulsive motion in mode k, while the function KjD(t,β) is the force on the ship in the jth direction due to a uni-directional impulsive wave elevation incident from a heading angle of β. Equation (1) differs from the usual form only because the free-surface memory is represented as a convolution with the ship's displacement rather than with its velocity. This choice is convenient in practice because it produces a kernel which vanishes for large time (see Section 4).
With the hydrodynamic and hydrostatic coefficients in hand, a simulation of the ship translating in an ambient wave field may be carried out by integrating in time the above system of six coupled differential equations.
3
Hydrodynamics
3.1
The Exact Problem
Consider a three-dimensional body in a semiinfinite fluid with a free-surface, as shown in Figure 1. The ship moves through an incident wave field with velocity , and is allowed to perform small unsteady oscillations about its mean position in any of its six degrees of freedom. The fluid is assumed to be ideal and the flow irrotational, free of separation or lifting effects. Two

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Proceedings of the Sixth International Conference on Numerical Ship Hydrodynamics
coordinate systems will be employed in the ensuing derivations: the system is fixed in space, and the system is fixed to the mean position of the ship. At t=0, these two coordinate systems coincide.
Subject to the above assumptions, the fluid velocity may be described by the gradient of a scalar velocity potential, . Conservation of mass requires that this potential satisfy the Laplace equation everywhere in the fluid:
▽2Φ=0. (3)
The pressure in the fluid, , is given by Bernoulli's equation,
(4)
where g is the acceleration due to gravity, ρ is the fluid density, and pa is the atmospheric pressure, which is assumed to be constant. (Partial differentiation is indicated when the independent variables x,y,z,t appear as subscripts.) If surface tension is neglected and the pressure on the free-surface is set equal to zero, a combined free-surface boundary condition may be written:
on z0=ζ, (5)
where ζ(x0,y0,t) is the unknown free-surface elevation. Since the free-surface condition is second order in time, two initial conditions are required, and it will suffice to let
on z0=0, for t<T0 (6)
On the submerged portion of the hull the normal components of the fluid velocity and the ship velocity must be equal:
on Sb(t), (7)
where Sb(t) is the exact position of the ship surface, , is the velocity of a point on the ship, and is the unit vector normal to the ship surface. Because of the initial conditions, fluid motions caused by the ship will go to zero at spatial infinity for all finite time,
3.2
Linearization
In order to make further progress both the free-surface and the body boundary conditions, as well as the Bernoulli equation, will be linearized. Let us now use the coordinate system fixed to the mean position of the ship, which is traveling along the x0-axis with a constant speed U. We will assume that the ship was accelerated to this speed at some time in the past and that all transients due to this acceleration have decayed to zero. The total velocity potential, in the ship fixed reference frame, will be decomposed as follows:
(8)
The combination of and is the potential due to the steady-state limit of the ship's uniform translation at forward speed U. This will be referred to as the steady problem. The radiation problem ensues when this translating ship is forced with some prescribed motion in a single rigid body mode k. The potential due to this motion is . If the steadily translating (but otherwise motionless) ship encounters an incident wave system with potential , the scattering of those waves by the ship will be described by the potential . This is the diffraction problem. Note that in the moving coordinate system, the fluid velocities in the far field will tend to those of the free stream and the undisturbed incident wave:
where i is the unit vector in the x-direction. Velocities described by the function in the above decomposition are assumed to be , while the remaining potentials describe velocities which are small perturbations to this basis flow. Far from the ship the basis flow must tend to the free stream, however the choice of is not unique.
If the decomposition of equation (8) is used in equations (7) and (5), the free-surface and body boundary conditions may be linearized about the mean positions of the ship and free-surface boundaries. The simplest choice of a basis flow, and the one that will be used here, is the freestream alone:
This choice leads to the familiar Neumann-Kelvin linearization of the pressure, the free-surface condition and the body boundary conditions:
(9)

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(10)
(11)
In equations (9) and (10), is used to represent any of the above mentioned perturbation potentials and the linearized body boundary conditions in equation (11) are to be applied on , the mean position of the ship surface. The generalized unit normal nk is defined by
(12)
The steady and the unsteady potentials are coupled through the presence of the so-called m-terms in the body boundary condition. For this linearization the m-terms simply reduce to
mk=(0,0,0,0,Un3, −Un2)
Other linearizations can be derived by making a different choice of basis flow,
3.3
The Integral Equation
The foregoing initial-boundary-value problem can be recast as an integral equation by making use of the transient free-surface Green function. The three perturbation problems described above all satisfy the same boundary-value problem, with the exception of the body boundary condition. Consequently, the same integral equation may be used to solve for any of these potentials. The integral equation is derived by applying Green's theorem to the transient free-surface Green function and the time derivative of the potential. This Green function is derived in Wehausen [24]:
(13)
where
(14)
(15)
and J0 is the Bessel function of order zero. It is straightforward to verify that this Green function satisfies the complete linearized initial-boundary-value problem, with the exception of the body boundary condition, equation (11). The result of these manipulations is the following integral equation:
(16)
where the ship waterline is the intersection of and the z=0 plane, and the arguments of the Green function in the convolution integrals are retarded . This equation is identical to that used by King et al. [9], or Lin & Yue [16], for example.
The following sections will discuss in more detail the solution of the perturbation potentials.
3.4
Forced Motion Problems
The steady perturbation potential, when solved as the limit of a transient problem, and the radiation potentials are all solutions to similar (and in some cases identical) forced motion problems. The only difference being that in the steady problem it is the steady-state limit which is of interest, while in the radiation problem we seek a transient response.
3.4.1
The Radiation Problem
For each radiation problem, the steadily translating ship is moved impulsively in mode k, and the force on the ship in mode j (i.e. the corresponding radiation impulse-response function) is calculated. In principle, there are any number of possible forcings for this problem. An impulsive motion of the ship in any derivative (or integral) of its motion will do; or the motion need not even be impulsive, as long as a convenient Fourier transform exists with which to reconstruct the impulse-response from the non-impulse-response [9]. It is clear from the boundary condition that the solution to one impulsive radiation problem is related to that of any other through some number of time derivatives. For example, the body

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boundary condition corresponding to an impulsive acceleration of the ship is
(17)
where H(t) is the Heaviside step function and r(t) is the ramp function r(t)=tH(t). The body boundary condition produced by an impulse in the ship's velocity is the time derivative of equation (17)
(18)
where δ(t) is the Dirac delta function. Because this is a linear system, the two solutions are related in the same way, and it can be shown that any canonical radiation potential satisfies
where is the radiation potential due to an impulse in the nth derivative of the ship's motion. This relationship is not surprising given that the same information can be constructed from any canonical radiation potential. Specifically, the potential due to an arbitrary motion of the ship in mode k, Φk, can be written as a convolution of with a time history of the nth derivative of the ship's motion in that mode, dnxk/dtn.
(19)
The force on the ship in mode j due to this arbitrary motion in mode k is found by integrating the consequent linearized pressure over the body surface
. (20)
This form is computationally inconvenient, since it involves a spatial derivative of the potential, and the calculation may be simplified by using a variant of Stokes' theorem attributed to Tuck [23]:
(21)
where is the unit vector tangent to the mean waterline. For a wall sided ship, the line integral is identically zero and we may write
. (22)
Given the form of the body boundary condition for this problem, it is natural to consider each radiation potential to be the sum of three terms:
(23)
where x(t) and must be thought of as the generalized functions appropriate to an impulse in the nth derivative of the ship's motion. The two time constants, Nk and , are solutions to pressure release type problems. They satisfy the following pair of boundary value problems:
(24)
and may be calculated from the following pair of integral equations
The transient or memory potential, , will depend upon the type of impulse the ship is forced to undergo.
Generally, this problem is solved by giving the ship an impulse in its velocity, which means setting and letting n=1 in equation (19). Because the position of the ship is changed by this choice of an impulse, we must expect it to produce a steady-state limit to the memory potential, . Physically, when the ship is displaced from its original position by the impulse, a change occurs in the steady wave pattern which must be reflected in the large time limit of the radiation potential. This is undesirable from a computational standpoint because it tends to increases the required length of the computations (see Section 4). A simple way to avoid a non-zero steady-state limit is to prescribe a forced motion which will bring the ship back to its original position. An impulse in displacement, i.e. letting x(t)=δ(t), is the obvious choice to satisfy this requirement. The body boundary condition in this case becomes
and the solution will be
(25)
In contrast to using an impulsive velocity, this will produce a radiation problem with a memory term that tends to zero at large time.

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An integral equation for the impulsive displacement memory potential, , (where the super- and sub-scripts have been omitted for clarity) may be derived by inserting the decomposition of the potential into equation (16). The result is similar to the equation derived by Liapis [15] for the impulsive velocity memory potential:
(26)
where is always non-zero.
Using the decomposition of equation (25) through equation (19) and into equation (22), the complete radiation impulse-response function, which consists of the constant coefficients ajk, bjk and cjk combined with the memory function , may be expressed in terms of the the general canonical radiation potentials as follows:
(27)
where the coefficient, , has appeared because , and it must be included with cjk when n=0.
It has been pointed out in the past that the coefficient ajk is a genuine added-mass coefficient which is independent of both time (or frequency) and forward speed. The coefficients bjk and cjk are, on the other hand, functions of the forward
Figure 2: Wigley hull at Fn = 0.3, impulsive pitch displacement memory function.
Figure 3: Wigley hull at Fn = 0.3, impulsive pitch velocity memory function.
Figure 4: Comparison of the pitch-pitch added-mass coefficient calculated by impulsive displacement and by impulsive velocity of the ship.

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Figure 5: Wigley hull at Fn=0.0, heave-heave added-mass coefficient.
speed. It can be shown by applying Green's theorem to Nk and , and using the boundary conditions which they satisfy, that the constants bjk satisfy the following relations,
bjk=0 for j=k (28)
bjk+bkj=0 for j≠k
Some sample calculations of the pitch memory functions due to both an impulsive displacement and an impulsive velocity appear in Figures 2 and 3. Both calculations have been made for a Wigley hull at a Froude number 0.3.
It can be shown that the impulsive velocity and the impulsive displacement memory functions are related in the same way as the corresponding radiation potentials. That is,
This means that it is possible to calculate from , instead of solving the impulsive displacement radiation problem directly, and this may in practice be more convenient.
If the motion of the ship is considered to be time harmonic at frequency ω, then the force on the ship may be written in complex form as
Fjk=(ω2 Ajk(ω)−iω Bjk(ω)−cjk) xj,
and the impulse-response functions calculated using the canonical radiation potentials at any n, are related to the more familiar frequency-response functions (i.e. the added-mass and
Figure 6: Wigley hull at Fn =0.0, heave-heave damping coefficient
Figure 7: Wigley hull at Fn =0.0, pitch-pitch added-mass coefficient.
Figure 8: Wigley hull at Fn=0.0, pitch-pitch damping coefficient.

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Figure 9: Wigley hull at Fn=0.3, heave-heave added-mass coefficient.
damping coefficients) through a Fourier transform:
(29)
Figure 4 shows a comparison between the added-mass coefficients calculated using the two memory functions shown in Figures 2 and 3. As expected, the results are practically identical.
Calculations made using this method are compared to frequency domain calculations using WAMIT at zero forward speed in Figures 5 through 8. The two solutions for a Wigley hull are in excellent agreement. Comparisons between these two computer codes for more complicated bodies, as well as a description of WAMIT, may be found in Korsmeyer, et al. [10]. Figures 9 through 16 show the results of calculations made for the Wigley hull at a Froude number of 0.3. These results are compared both to the experiments of Journée [7], and to calculations made using the Rankine panel method SWAN [ 20]. The difference between these two sets of calculated results may be attributed to the fact that SWAN uses a linearization about the double-body flow rather than the Neumann-Kelvin linearization which is used in TiMIT.
3.4.2
The Steady Problem
The steady perturbation potential, , can be calculated as the steady-state limit of a particular radiation problem: that of an impulsive accelera
Figure 10: Wigley hull at Fn = 0.3, heave-heave damping coefficient.
Figure 11: Wigley hull at Fn = 0.3, heave-pitch added-mass coefficient.
Figure 12: Wigley hull at Fn=0.3, heave-pitch damping coefficient.

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Figure 13: Wigley hull at Fn=0.3, pitch-heave added mass coefficient.
Figure 14: Wigley hull at Fn=0.3, pitch-heave damping coefficient.
Figure 15: Wigley hull at Fn=0.3, pitch-pitch added-mass coefficient.
tion of the ship to a forward speed U. The boundary conditions for this problem are equations (10) and (17), which in the limit of large time will become the steady-state Neumann-Kelvin conditions
Figure 16: Wigley hull at Fn=0.3, pitch-pitch damping coefficient.
This approach is somewhat indirect, since the Green function for this problem is known (i.e. the Kelvin wave-source potential). The most direct way of calculating the steady potential would be to apply Green's theorem, with the Kelvin wave source potential, and to solve the resulting integral equation for . However, efforts to calculate the Kelvin wave source potential in a robust and efficient way have not yet been entirely successful. The transient approach, although computationally expensive, is an alternative.
3.5
The Diffraction Problem
The diffraction problem, that of finding the velocity potential for the case of the ship fixed to its mean position in the presence of an incident wave, may be solved to find the transient exciting forces. When the diffraction problem is forced by an impulsive wave elevation, the computed transient forces may be related to impulse-response functions. These impulse-response functions are the kernels of convolutions which can be used to compute the exciting forces and moments which appear on the right-hand side of the equations of motion (1) given an arbitrary, known, incident wave elevation.

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Figure 5. The Series-60, Cb=0.7, hull and computational domain used for solution of the forward speed steady flow.
convergence with the grid density may be observed, especially with respect to the longer length scales of the solution. Shorter diverging wave components are de-emphasized by the coarse discretization mainly due to the application of space-filtering. Figure 6 also shows the pressure distribution over the hull surface which may be seen to be satisfactorily converged.
The time histories of the wave induced forces and moments on the vessel are illustrated in figure 7, as predicted by both the fine and coarse discretizations. All curves show sharp initial transients which settle rapidly to a decaying oscillation at period , corresponding to the —singularity which is excited due to the impulsive start of the forward motion. The amplitude of these transient oscillations is, however, smaller than related predictions of Lin and Yue (1990).
The aforementioned “critical” oscillations are much more pronounced in the next simulation which considers the same series-60 hull at Froude number F=0.2. Figure 8 shows the time history of the resultant forces and moments, as predicted by four different computational domains. All grids are of the same density with 50 panels along the waterline length. Two different values of the free surface extent in the transverse direction (YOUT) and of the transverse width of the
Figure 6. Wave patterns for two discretizations of the Series-60, Cb=0.7, hull in steady motion at a Froude number of 0.3.

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Figure 7. Time history for the Series-60, Cb=0.7, hull in steady motion at a Froude number of 0.3.
artificial beach (CW) are considered. All records oscillate at at approximately the same critical period, Tc, and about the same mean value, but with desperately different amplitudes and phases. Further numerical experimentation has shown that the precise form of the critical transient oscillation depends, almost exclusively, on YOUT and Cw with the general tendency to decrease as the size of the free surface domain and artificial beach increase.
The —singularity of the forward speed problem which is the underlying cause of the persistent transient oscillations has been been extensively studied over the years. For elementary flows due to translating wave singularities it may be proven that, within linear theory, there exist a wave component with vanishing group velocity whose energy is trapped in the near-field. However, such studies concentrate on the Neumann linearization of the free surface conditions which is not the framework of the present study. Moreover, the presence of the truncated free surface and wave absorbing layer is expected to have a significant effect on the “trapped” wave component whose wavelength is typically much larger than the overall size of the computational domain.
From the practical standpoint, the transient effect of the critical oscillations causes a significant delay to the onset of the corresponding steady-state and, therefore, increases the computational effort needed for the accurate prediction of the steady-state forces. The idea of filtering any such persisting transients appears very attractive and it is proposed for further study.

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Figure 8. Time history for the Series-60, Cb=0.7, hull in steady motion at a Froude number of 0.2.
Figure 9. Wave pattern for the Series-60, Cb=0.7, hull in steady motion at a Froude number of 0.2

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Figure 10. Wave pattern for the modified Wigley hull at a Froude number of 0.3.
Figure 11. Time history for the modified Wigley hull in steady motion at a Froude number of 0.3.

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Figure 12. Wave pattern for the modified Wigley hull at a Froude number of 0.2.
Figure 13. Time history for the modified Wigley hull in steady motion at a Froude number of 0.2.

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The insensitivity of all the above conclusions to the hull shape is illustrated by simulations of the modified Wigley model in steady forward motion at Froude numbers 0.3 and 0.2. Like for the Series-60 hull, the simulations are carried out to time using a grid density corresponding to 50 panels within the waterline length and a time step of Δt=0.03. Figures 9–13 show the predictions for the wave pattern at the final time step and the time history of the forces and moments acting on the hull. Once again, the critical transient oscillations in the force record are much more pronounced at the low speed.
7. FORCED HARMONIC MOTION AT CONSTANT FORWARD SPEED
The simulation of forced harmonic oscillations of
Figure 14. Added mass and damping for the modified Wigley hull at a Froude number of 0.3.

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ships advancing at a constant forward speed is the subject of this section. The vessel is considered to be at rest for all time t≤0, when it suddenly starts moving forward at full speed while oscillating with amplitude given by (5.1).
The wave flow due to the forced harmonic heave and pitch motion of the modified Wigley model and the Series-60, cB=0.7, is solved by using both the coarse and fine discretizations of section 6. The time history of the resultant forces and moments is evaluated by integrating the pressure,
(7.1)
over the hull surface. Notice that the last two terms of (6.1) are not included in (7.1) since they do not contribute to the steady-state oscillatory forces.
Figure 15. Added mass and damping for the Series-60, Cb=0.7, hull at a Froude number of 0.2.

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Simulations are carried out until steady-state is reached and the force records are fitted by appropriate sinusoidal functions in order to identify the “frequency domain” added mass and damping coefficients. Figures 14 and 15 compare the predictions of the present time domain solution algorithm SWAN-2 to computations by the frequency domain solver SWAN-1 (see Nakos and Sclavounos (1990)), using identical linearization of the free surface and body boundary conditions. Very satisfactory agreement is found both for the modified Wigley model at Froude number 0.3 and the Series-60 hull at Froude number 0.2.
8. FREE MOTION AT CONSTANT FORWARD SPEED
This section presents free motion simulations for ships advancing at a constant forward speed. The vessels start impulsively at t=0 and retain freedom of motion in heave and pitch. Incident head seas can be imposed for the simulation of the resulting ship motions. If the ambient sea state is calm, the simulation becomes a free-decay test where the ship transits from the rest, or static equilibrium, position towards her dynamic equilibrium position, defined by the appropriate steady-state sinkage and trim.
The equation of motion of the hull as a rigid body read as follows,
(8.1)
where M is the inertia matrix, C are the hydrostatic restoring coefficients, are the rigidbody displacements, and is the resultant hydrodynamic force or moment. The initial conditions for the ship can vary. It may be started from a calm rest position or given a specified initial heave and/or pitch displacement and/or velocity. Currently, the time-stepping algrorithm for the integration of the equations of motion (8.1) is based on a semi-implicit backwards differentiation scheme which is only first-order accurate. As discussed at the end of this section, a higher-order scheme with more favorable stability properties is necessary and it is currently being designed.
Within this study three cases of free motion simulation are examined. The first case, illustrated in Figure 16, shows the heave and pitch response of a modified Wigley hull which starts impulsively at t=0 from its calm-water position with a forward speed corresponding to Froude number F=0.3. For all practical purposes, at time the initial transients have decayed and the solution has approached the corresponding steady-state sinkage and trim.
Figure 16. Modified Wigley hull started impulsively from rest and tending towards a forward speed equilibrium position at a Froude number of 0.3.

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Figure 17. Modified Wigley hull started impulsively and dropped from an initial height, ξ3/LWL=0.01 at three speeds.
Figure 18. Heave and pitch RAO's for the modified wigley hull at a Froude number of 0.3 as predicted by two methods.

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The behavior of the transients is worthy of note. Two distinct frequencies are evident is the transient response of figure 16. The shorter period is the natural period of oscillation for the ship at this Froude number and corresponds to the peak in the Response Amplitude Operator curve predicted by the frequency domain solver SWAN-1. The longer period corresponds to the critical transient oscillation due to the —singularity, discussed in preceding sections and it is excited by the impulsive start of the ship.
Subsequently simulations of the same modified Wigley model were carried out at Froude numbers 0.2, 0.3, and 0.4, but this time the vessel was given an initial heave displacement equal to 1% of its waterline length. The predicted motion records are illustrated in figure 17. These tests are instructive in examining the decay of the natural modes of oscillation, and indicate how quickly a hull may reach the steady-state limit, given an initial heave and pitch displacement. The natural period of oscillation of the vessel is evident in the transients excited during the latter simulations. However, the scale of the oscillations at the natural frequency is such that nearly overshadows the critical transient oscillations at period T0=8πF. The critical oscillations are most noticeable at the lowest Froude number but also affect the modulation of the motions at the higher speeds. Although there was no initial pitch for these freedecay tests, pitch motion did result due to hydrodynamic cross-coupling, which is amplified at higher speeds. After a sufficiently long interval of time the heave and pitch tend towards the sinkage and trim, respectively, appropriate for these speeds.
The predictions of the steady-state sinkage and trim by the present transient solver are considered superior to related computations made by steady-state solvers like SWAN-1. That is because the present transient approach incorporates, to leading-order, the effect of the sinkage and trim on the resultant force and moment due to the inclusion of the m-terms in the body boundary condition (2.5). Steady-state solvers may also account for the same effect, but only if they are employed iteratively with respect to the dynamic equilibrium position of the hull.
The last simulation test shows the heave and pitch response of the modified Wigley model advancing at Froude number 0.3 through multichromatic head seas. The time records of the resulting heave and pitch motions are illustrated in figure 18. The simulation is carried out for a sufficiently long interval of time so that transients have decayed and the motion records may be safely transformed into the frequency domain. The resultant Response Amplitude Operator is compared to related predictions of the frequency domain solver SWAN-1.
The predictions of both methods compare very favorably except for the peak magnitude of the heave response amplitude. Numerical convergence difficulties are experienced with respect to the time-step size at the peak of heave response, which may be attributed to numerical dissipation errors in the time-stepping algorithm for the integration of the equations of motion. Efforts are currently focused on analyzing this phenomenon and prescribing a higher-order algorithm for the solution of the equations of motion.
9. SUMMARY AND CONCLUSIONS
The design and implementation are presented of a Rankine Panel Method for the solution of transient wave-body interactions in three-dimensions. In the presence of forward speed the free surface and hull boundary condition are linearized about the double-body flow. The space discretization is based on the approximation of all unknowns in terms of the biquadratic B-spline functions and the time evolution is performed by a neutrally stable time-stepping algorithm. An artificial wave-absorbing beach is designed and employed for the damping of reflections due to the finite free surface computational domain.
Wave flows due to forced and free motions of realistic ship hulls are computed with and without mean forward speed. The simulations are continued until steady-state is reached and the resulting steady-state forces are compared to predictions by well-established frequency-domain solvers. The comparison is excellent for all cases, owing to the optimal design of the time-steeping algorithm and the effectiveness of the employed wave absorption device.
In the presence of forward speed, the onset of steady-state conditions is significantly delayed due to the —singularity of the corresponding frequency domain problem. The time-records of forces and motions are found to be “contaminated” by a slowly-decaying oscillation

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at the critical frequency ωU/g=0.25, which is excited by the impulsive start-up of the vessel's motion. The wavelengths associated with this critical transient oscillation are often too long to be handled by the adopted extend of free surface computational domain and absorbing artificial beach.
Future extensions of the present formulation and solution scheme include relaxation of the body boundary condition linearization, in order to model large-amplitude ship motions, and solution of the wave flow past ships advancing along arbitrary curvilinear paths which addresses the problem of ship maneuvering in the presence of ambient waves.
ACKNOWLEDGEMENTS
This study has been supported by Det Norske Veritas Research AS.
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