M.Ohkusu (Kyushu University, Japan)

Some of theoretical methods to predict wave-induced motion and wave load of ships running at forward speed in waves are reviewed. We focus on validation of those theories by means of detailed experimental data on hydrodynamic pressure on the hull surface and/or wave elevation close to the hull surface. Traditional experiments comparing the predicted ship motions and global hydrodynamic force with the measured neither demonstrate the advantage of the advanced theoretical methods nor pinpoint the deficiency of them. It is because the ship motion and the global hydrodynamic force are the result of plenty factors integrated and do not provide high grade information on hydrodynamics involved.

Since the late 1980’s vigorous effort has been made by many research workers to develop rational theoretical methods for predicting wave-induced motion and wave loads of ships. Most of them are boundary integral methods based on rational analysis of the free surface flow including the nonlinear effect. Some of them will be reviewed in this article. They attempt rigorous analysis no matter how complicated their implementation and they are computationally involved. The final goal of those methods is to understand and simulate the seakeeping of ships moving at forward speed in high waves.

Despite such progress achieved in the theoretical methods, experiments on which their validity is to be tested remains classical; most of seakeeping experiments are the ship motion test and the forced motion test for the global hydrodynamic force. Those experiments provide only low grade information on hydrodynamics of ship-wave interaction. Advantage of the advanced theoretical methods over the primitive methods will not be fully displayed in comparison with the result of those experiment; both the former and the latter methods are often presented by many authors to predict the ship motion and the global load with rather adequate accuracy. Though the advanced methods may predict some nonlinear phenomena which the primitive theories can not do so, quantitative information of such phenomena will not be obtained on the ordinary ship motion test.

Hydrodynamically correct theory must be the theory that is able to account for the flow field, hydrodynamic pressure distribution and wave elevation around a body. We may say that they are more hydrodynamic phenomena than the resulting body motion and global force. The ship motion is an integrated effect with which a lot of factors not only hydrodynamical but also mechanical are involved; fine prediction of the ship motion is an outcome of the correct account of hydrodynamics of elementary process of ship-wave interaction. The global hydrodynamic force is also an integrated result of hydrodynamic pressure which does not directly reflect the hydrodynamics.

The present author’s proposal is that hydrodynamically advanced theory or method must be tested on ‘hydrodynamic’ experiment rather than the ship motion experiment or the global force experiment. Hydrodynamic experiment means the experiment to understand basic hydrodynamics of the ship-wave interaction such as the measurement of the distribution of hydrodynamic pressure on the ship surface or the measurement of the wave field around the ship. We readily understand implication of this type of experiment if we recall a big role of the flow visualization rather than the drag measurement played in the progress of hydrodynamics of the body-flow interaction. Of course flow visualization is impossible with a ship in waves. Measurement of the hydrodynamic pressure and the wave field will be substitutes for it.

Objective of this article is to investigate validity of the modelling of the flow in the theoretical methods for ship motions by hydrodynamic experiment. We may expect it will demonstrate the advantage of the rigorous analysis and pinpoint where they are to be improved if any.

Regrettably few experimental data meeting our requirement are available; only several examples are found and compared with the prediction of the theoretical methods. Actually no experiment has ever been attempted to exemplify quantitatively the non-

Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.

Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.

Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
Validation of Theoretical Methods for Ship Motions by Means of Experiment
M.Ohkusu (Kyushu University, Japan)
Abstract
Some of theoretical methods to predict wave-induced motion and wave load of ships running at forward speed in waves are reviewed. We focus on validation of those theories by means of detailed experimental data on hydrodynamic pressure on the hull surface and/or wave elevation close to the hull surface. Traditional experiments comparing the predicted ship motions and global hydrodynamic force with the measured neither demonstrate the advantage of the advanced theoretical methods nor pinpoint the deficiency of them. It is because the ship motion and the global hydrodynamic force are the result of plenty factors integrated and do not provide high grade information on hydrodynamics involved.
1. Introduction
Since the late 1980’s vigorous effort has been made by many research workers to develop rational theoretical methods for predicting wave-induced motion and wave loads of ships. Most of them are boundary integral methods based on rational analysis of the free surface flow including the nonlinear effect. Some of them will be reviewed in this article. They attempt rigorous analysis no matter how complicated their implementation and they are computationally involved. The final goal of those methods is to understand and simulate the seakeeping of ships moving at forward speed in high waves.
Despite such progress achieved in the theoretical methods, experiments on which their validity is to be tested remains classical; most of seakeeping experiments are the ship motion test and the forced motion test for the global hydrodynamic force. Those experiments provide only low grade information on hydrodynamics of ship-wave interaction. Advantage of the advanced theoretical methods over the primitive methods will not be fully displayed in comparison with the result of those experiment; both the former and the latter methods are often presented by many authors to predict the ship motion and the global load with rather adequate accuracy. Though the advanced methods may predict some nonlinear phenomena which the primitive theories can not do so, quantitative information of such phenomena will not be obtained on the ordinary ship motion test.
Hydrodynamically correct theory must be the theory that is able to account for the flow field, hydrodynamic pressure distribution and wave elevation around a body. We may say that they are more hydrodynamic phenomena than the resulting body motion and global force. The ship motion is an integrated effect with which a lot of factors not only hydrodynamical but also mechanical are involved; fine prediction of the ship motion is an outcome of the correct account of hydrodynamics of elementary process of ship-wave interaction. The global hydrodynamic force is also an integrated result of hydrodynamic pressure which does not directly reflect the hydrodynamics.
The present author’s proposal is that hydrodynamically advanced theory or method must be tested on ‘hydrodynamic’ experiment rather than the ship motion experiment or the global force experiment. Hydrodynamic experiment means the experiment to understand basic hydrodynamics of the ship-wave interaction such as the measurement of the distribution of hydrodynamic pressure on the ship surface or the measurement of the wave field around the ship. We readily understand implication of this type of experiment if we recall a big role of the flow visualization rather than the drag measurement played in the progress of hydrodynamics of the body-flow interaction. Of course flow visualization is impossible with a ship in waves. Measurement of the hydrodynamic pressure and the wave field will be substitutes for it.
Objective of this article is to investigate validity of the modelling of the flow in the theoretical methods for ship motions by hydrodynamic experiment. We may expect it will demonstrate the advantage of the rigorous analysis and pinpoint where they are to be improved if any.
Regrettably few experimental data meeting our requirement are available; only several examples are found and compared with the prediction of the theoretical methods. Actually no experiment has ever been attempted to exemplify quantitatively the non-

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
linear effect of seakeeping which some of the theoretical methods are able to predict. So the methods whose validation is investigated in this article against the experimental results are mostly linear ones. We have to wait for the future work to confirm validity of the most sophisticated results of the nonlinear theories.
2. Linear Theory I
Correct modelling of the interaction of the steady disturbance on the water surface produced by the constant forward speed of a ship with the unsteady flow produced mainly by the ship-waves interaction is not straightforward in formulating a theoretical approach to predict the wave-induced motions and wave loads of the ship. The total flow produced by the ship in waves is obviously decomposed into the steady flow and the other flow naturally unsteady. It is appropriate to attempt the following form for the velocity potential Φ of the total flow.
(1)
The flow is described in the right-hand reference frame fixed to the ship moving at the constant speed U into the positive x direction; the x axis coincides with the ship’s center line and the x−y plane is the mean water surface; the z axis is taken vertically upward. The second term of (1) corresponds to the uniform flow relative to the ship, the third term represents the steady flow and the fourth term the unsteady flow component.
Wave elevation ζ is also a superposition of the steady component ηS and the unsteady component η.
(2)
Magnitude of the unsteady flow is determined by magnitude of the incident waves and the resulting ship motions. Magnitude of the steady flow is supposed to be dependent on ship geometry such as its slenderness. It means that rational basis of the linearization of will be independent of that of . When we discuss their interaction without introducing any assumption on the ship geometry (practical hull forms are never slender), the most reasonable choice of will be a full nonlinear solution. Yet we like to somehow avoid the full nonlinear solution and use physically correct but simpler solution; too complicated mathematical expression of might lead to unnecessary difficulty of any linear theory of the unsteady flow .
The choice of the steady flow model, if we have no mathematical basis, will inevitably be decided on physical argument or arbitrary; consequently various approaches will be possible to incorporate the effect of into . Their validity therefore must be carefully tested on ‘hydrodynamic’ experiment.
A popular way to account for the steady flow effect allowing the relatively easy formulation of the unsteady flow is the introduction of the low order solution of the steady flow. The simplest choice is to assume the steady disturbance is zero; the steady flow around the ship is assumed to be only the uniform relative flow U in deriving the free surface conditions.
We linearize the free surface conditions and the hull surface condition, with respect to the velocity potential and the corresponding wave elevation η. The wave-induced ship motions are assumed to be of the order Ο (η). will be a solution of the boundary value problem:
(3)
(4)
(5)
(6)
and the radiation condition.
(4) and (5) are the free surface conditions satisfied on the mean water surface z=0. (6) is the body boundary condition imposed on SB representing the ship hull surface immersed under z=0 at the ship’s mean position, a is the motion vector of a point r(x, y, z) on the hull surface; r(x, y, z) moves due to the wave-induced ship motions. V is the steady flow velocity around the ship given by
(7)
n is the unit vector normal to the boundary surface and directing outward from the fluid. The third term of the body boundary condition on SB is to correct the difference of the steady flow velocity on SB from that on the exact instantaneous hull surface. The fourth compensates the effect of the variation Δn of

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
the unit normal vector n which is produced by the ship’s rotational motions (roll, pitch and yaw).
We have a computational problem when evaluating the right side of the body boundary condition (6). It includes the second derivative of which is singular at the corner on the hull surface and have to be carefully evaluated at the location of large curvature (Zhao and Faltinesen (1989)). In the time domain computation it might be better not to linearize the body boundary condition but to satisfy it on the exact instantaneous position of the body to avoid this problem though it is inconsistent.
The body boundary condition (6) in this formulation might look strange because it includes the effect of notwithstanding we ignore it in the free surface conditions. Perhaps a consistent way is to neglect in the third and the fourth terms in (6). However it puts us in an uncomfortable situation: we feel the steady flow by the ship forward speed is never the relative uniform flow U even to the practically lowest order of approximation. To follow the formalism of the rational strip theory (Ogilvie and Tuck (1969)) will be a remedy for this. The rational strip theory claims consistently that the effect of is of higher order in the free surface condition, while it is not to be ignored in the body boundary condition. In this case, however, the choice of will return as a point in question.
The Green’s second identity gives in the form
(8)
where r=(x, y, z) and r′=(x′, y′, z′). G is a function satisfying the Laplace equation and having a singularity of the form 1/|r−r′|. Integration of (8) is with respect to r′. SF represents the free surface (z=0 plane); S0 is a control surface surrounding the ship, located away from it and below z=0. If the field point r is located on one of the smooth boundary surfaces, 4π in the right side of equation (8) is replaced by 2π.
Hereafter in this section we concentrate on analysis in the frequency domain: the time dependency is in the form of eiωt as ηeiωt and aeiωt. Elimination of η from (4) and (5) yields
(9)
We substitute the Green function G0eiωt, which satisfies the free surface condition (9) and the radiation condition, for G in the equation (8). Then the equation (8) for on SB is transformed into
(10)
The first case is chosen when the incident wave of the wave number k
exists and otherwise the second case. CB is the intersection of SB and SF. K0=g/U2 and τ=ωU/g.
A key point tor implementing this approach is an efficient evaluation of the Green function G0. Iwashita and Ohkusu (1992) developed an efficient scheme to evaluate it numerically using a special mathematical expression by Bessho (1977).
(11)
where r″=(x′, y′,−z′) and
The expression (11) of the Green function has several extraordinary features: it is straightforward

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
to control the accuracy of its numerical evaluation because (11) is a genuine single integral; it is analytically integrated over a panel over which is uniformly distributed (Iwashita and Ohkusu (1992)).
The linear part of the unsteady hydrodynamic pressure Peiωt on the body wetted surface SB* at the exact instantaneous ship potion is evaluated by
(12)
All the functions in this equation are to be evaluated on the surface SB at the mean position of the ship. When computing the hydrodynamic force on the ship, Ρ must be multiplied by n+Δn and integrated on SB. The second parenthesis of (12) represents the restoring force in the generalized sense which is caused by the ship’s displacement in the nonuniform steady pressure field. It is independent of the unsteady flow because we know is not affected by . Therefore experimentally this term will be measured by prescribing a small steady displacement or rotation on a ship model during it runs on otherwise a calm water. Naturally the second line does not exist when the ship motion is suppressed. We notice that the first term in the second parenthesis, when we assume the wetted surface of SB is below z=0, will give the restoring force on calm water at zero forward speed.
The followings are a few points in question which arise when we implement this approach.
Choice of in (6) and (12) to evaluate V. Iwashita et al (1994) employed the double-model flow whose definition is given in the next section 3.
The singularity associated with the intersection of SF and SB. Iwashita et al (1992, 1993, 1994) assumed on CB; is continuous at the intersection. They do not enforce the boundary condition exactly at the intersection but collocate at two points, one on SB and the other on SF which are very close to the intersection.
Convergence of the solution as the size of the panels on SB approaches zero. SB is discretized into numerous quadrilateral panels over which and its derivatives are approximated by a superposition of quadratic spline functions. Over each panel the product of the Green function and or its derivatives are numerically integrated. Convergence of their solutions is tested on increasing the panel numbers up to 2,000 (Iwashita et al. (1994)). Their conclusion is that 1,000 panels on SB will be sufficient on all the practical parameter values they are concerned and for ordinary hull forms; the computation will be extremely economical without sacrificing the accuracy by replacing the distribution with a single isolated G0 over each panel. They observed no instabilities in their numerical computations.
Finally we comment that similar approach to solve for in the time domain under the linear free surface conditions (4) and (5) is possible by employing the Green function in the time domain (King et al. (1989))
Validation by Experiment
Validation of physical model assumed in a theoretical approach must be tested on ‘hydrodynamic’ experiment rather than ‘practical’ experiment. Even such a sophisticated theoretical approach as described in this section is often tested by comparing the prediction with the global hydrodynamic force and wave-induced ship motion measured at the towing tank. Such experiments are indeed ‘practically’ useful. However the global force is an integrated effect of hydrodynamic pressure on the ship and the ship motion is a result of a combined effect of many hydrodynamic factors. Fine agreement in the prediction of the ship motions does not necessarily allow us decide our theoretical approach is hydrodynamically correct; bad agreement does not let us know where the theory is wrong. More analytical experiment such as measurement of fluid pressure distribution and flow visualization will be necessary to prove their actual advantage or to pinpoint their inaccurate modelling.
Results of extensive test for pressure distribution on a VLCC ship model (Cb=0.81, L/B=5.1) running in waves of various wave headings with its motion suppressed (diffraction problem) are reported (Iwashita et al. (1993)). Fig. 1, 2 and 3 are the results at the wave length-to-ship length ratio λ/L=0.5, Froude number Fn=0.2 and the wave height 0.02 of the ship length. Fig. 1 is amplitude of the pressure normalized by the amplitude of the incident wave at a section at St. No. 2, Fig. 2 at St. No. 5 and Fig. 3 at St. No. 9. The wave heading angle from 0° to 180° at upper to lower figures. The pressure is plotted vs azimuth angle measured anticlockwise from portside to starboard (−90° represents the the portside water line).
From the following to the head waves the agreement of the predicted (solid lines) with the measured (white circles) is excellent except for near the water surface at the bow section. It means the Green function method with uniform steady basis flow predicts well the wave pressure distribution on relatively blunt hull forms. This result is more than

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
we expected.
The discrepancy is very clear at the bow section. First of all, we remark that this would not be clear if we employed traditional way of experimental validation such as the comparison of the global wave exciting force on the ship. One obvious feature of the discrepancy is that the wave pressure at the bow section is extremely small on the water line at the ship’s mean position. Perhaps the pressure gauges at the water line will go out from the water periodically and it will record the zero pressure during this period. When the time series of the pressure thus recorded is Fourier analyzed, the amplitude of the fundamental frequency component must be very small compared with the amplitude obtained assuming a real sinusoidal temporal variation of the pressure as in the linear theory. It is not so serious problem because it depends on the definition of the pressure amplitude rather than the inaccuracy of the theory; if we compare the amplitude of the wave elevation instead of the pressure, we can avoid this ambiguity.
More serious discrepancy is that the measured pressure at the position a little below the water line is much larger than the predicted. If we are concerned with the vertical force on the ship, this inaccuracy in the theoretical prediction does not lead to serious practical problem: one of the reasons why a heuristic theory like the strip theory often appears to be valid. But from more hydrodynamical view point it will be a significant problem. Moreover the discrepancy will be practically serious when we are concerned with the local wave loads or added resistance of the ship.
No theoretical explanation is available at moment for the discrepancy. But the following guess is not unreasonable. Our physical model assumes the uniform flow as a basis flow. It is obviously not correct at the bow part; the flow must be deformed largely from the uniform flow at the bow part of blunt hull forms. So one remedy for improving theoretical prediction which occurs to us first is to introduce more accurate basis flow in the free surface condition (it will be described in the next section). At the station No. 9, where we observe the largest discrepancy, the steady wave surface is depressed lower than the mean water surface. On this steady surface the unsteady wave is superposed. Therefore the depth of a pressure gauge located close to the water line that is measured downward from the actual water surface is much shallower than the depth of the identical pressure gauge we assume in the linear theory, which is the depth measured from the surface of the unsteady
Fig. 1 Hydrodynamic pressure Fn=0.2, λ/L=0.5

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
Fig. 2 Hydrodynamic pressure Fn=0.2, λ/L=0.5
Fig. 3 Hydrodynamic pressure Fn=0.2, λ/L=0.5

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
wave elevation superposed on z=0. The pressure gauge located actually shallower than we assume will record larger pressure than we predict in the linear theory. It implies that the effect of the steady wave elevation ηS must be taken into account in the prediction of the wave pressure at the bow part.
3. Linear Theory II
Inclusion of in the steady flow is more plausible when accounting for the steady flow effect on the free surface conditions for if only we are able to find an appropriate . We first write the complete free surface conditions:
(13)
(14)
We are concerned with the unsteady part of the free surface conditions. We introduce an assumption for the steady flow explained later and linearize the free surface condition with respect to and η; we ignore the terms of Ο(η2) and the higher order. Then we have linearized free surface conditions for
(15)
(16)
, the steady flow which we employ in deriving (15) and (16), is the double body flow that satisfies the free surface condition
(17)
Other assumptions are:
ηS is of higher order than and the free surface condition is transferred from z=ηS to z=0
The assumption (i) implies that the steady flow and the steady wave elevation are much larger than the unsteady counterparts. The second (ii) will be valid if we assume the low forward speed of the ship. The steady wave elevation ηS of the flow satisfying (17) is given by
(18)
An assumption of very small U certainly leads to the higher order ηS which can be ignored. But now we do not employ such formalism on the consistency of the assumption. We just understand that the double model flow must be physically more plausible in accounting partly for the nonuniform steady flow dominating around ordinary hull forms which are not slender despite the assumption of ηS of the higher.
Linearization with respect to η admits transferring the body condition from the exact instantaneous hull surface under the water to the one at the mean position SB; the body condition is identical to (6). The velocity potential is written in the same form as the equation (8).
Mathematical expression of the Green function satisfying the free surface conditions (15) and (16), and the radiation condition will be extremely complicated even if it exists. So the so-called Rankine panel method employs a simple Rankine source function for G in the equation (8):
(19)
General idea of solving the integral equation (8) with the free surface conditions (15) and (16) is a time marching scheme starting with an appropriate initial condition. The free surface conditions (15) and (16) are used to update the velocity potential and the wave elevation η on SF (z=0). Solution of the ship motion equation provides the flux over the hull surface SB.
The integral equation (8) with G given by (19) is solved numerically with respect to the unknowns over SF and on SB with introduction of an appropriate radiation condition on S0. The radiation condition for the unsteady flow is never given in an explicit form when the ship has a forward speed and implementation of this condition is, unless G itself satisfies it, not straightforward. We will discuss this problem later.
Rankine panel method solving , particularly its numerical consistency and stability, has been systematically studied in both the frequency domain and

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
the time domain by Sclavounos and Nakos (1988), Nakos and Sclavounos (1990), Nakos et al. (1993), Vada and Nakos (1993), Kring (1994) and Kring et al. (1996). A fine review is given in Sclavounos (1996). In virtue of their investigation Rankine panel method is now one of the most reliable approach to find numerically based on the free surface conditions (15) and (16) or more general free surface conditions. We summarize their achievement and discuss some questions in the following:
The boundary domain is discretized into plane quadrilateral panels, over which are approximated by a summation of bi-quadratic spline base functions.
They used the so-called ‘empliciť Euler scheme (explicit for the integration of (15) and implicit for (16)) for time-marching integration of the free surface conditions. They studied systematically stability of this time marching scheme and reached the conclusion that the index Δx/Δt2 for lower U and Δx2/Δt3 for higher U must be larger than some critical value for the stable time marching integration. Here Δx is a spatial scale of the panels.
They found that discretizatin of the free surface, no matter how fine it is, distorts dispersive relation governing the wave propagation on the surface; the dispersive relation on the discretized free surface is different from that on the continuous free surface. It lets some wave components propagate into wrong direction. This analysis gives a consistency criterion which assures the convergence of the solution as each panel size approaches zero. The criterion is clearly stated in a relation of the panel aspect ratio, the panel Froude number and the reduced frequency in the frequency domain analysis.
A finite size of the panel causes another problem: aliasing of the energy of the waves with higher wave number than the Nyquist wave number π/Δx. It leads to spurious oscillation of the wave surface. They proposed a numerical filter to avoid the oscillation.
Radiation condition in their approach is enforced as a numerical wave absorbing beach away from the ship; the numerical wave absorbing beach is a layer of the free surface panels surrounding the free surface mesh and located distant from the ship. On this layer Rayleigh’s artificial viscosity type damping is introduced in the free surface condition. This idea was apparently successful after some numerical experiments. Yet it seems to require some know-how for this technique to succeed in developing robust scheme.
Condition of zero disturbance and zero wave slope in front of the ship works as a radiation condition for the case of Uω/g>0.25 if the analysis is in the frequency domain.
They do not mention how they treated with the singularity at intersection of the free surface and the hull surface. Our guess is that both the free surface condition and the hull surface condition are prescribed at the intersection as in the case at no forward speed (Dommermuth and Yue (1987)).
Integration of the equation of ship motions under the effect of hydrodynamic force must be done as a part of the time marching scheme of in the time domain: the ship motions are determined by evaluated at a time step; the boundary value problem is solved with the updated ship motions to determine new and the resulting hydrodynamic force. The highest derivative of the equation of the ship motions is naturally the ship inertia term on the left side of it. The forcing terms (hydrodynamic force) too have implicitly acceleration-dependent part. Instability will occur in the numerical integration of such system. They avoided it by separating the hydrodynamic force into two parts: the instantaneous added mass part proportional to the instantaneous acceleration of the ship and the memory part dependent on the time history of the ship motion and velocity in the form of the convolution. The former part is transferred to the left of the equation to be combined into the inertia term; the remaining forcing term does not contain the acceleration dependence any more. Integration of this system is generally stable. It increases, however, the computing time because we have to evaluate two components of the hydrodynamic force separately.
Instead of G of (19) we may use the Green function G0 (11). Hereafter we consider the problem in the frequency domain. We write the equation (8) with the Green function G0
(20)
where the choice of the last case depends upon if the incident wave exists or not. C0 is the intersection of SF and S0 at z=0. The integration on C0 is derived from the original integration on S0 by assuming vanishes on C0 because it is sufficiently distant from the ship and the double body flow decays rapidly. We notice that the of (20) satisfies the radiation condition because G0 satisfies it.

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
The boundary domain is discretized into plane quadrilateral panels as in the case of Rankine panel method, over which are approximated by a superposition of bi-quadratic spline base functions. The integral equation (20), the body condition (6) prescribed on SB and the free surface conditions (15) and (16) with ∂/∂t replaced by iω will provide a system of linear equations on the unknowns.
In this formulation we do not need a special technique to numerically enforce the radiation condition. Instead a number of the unknowns on SF is more than that in Rankine panel method. Success of this approach will be dependent on the integration of the product of the Green function and on each panel of SF. A mathematical expression of the Green function proposed by Iwashita and Ohkusu (1991) may facilitate this integration.
Kashiwagi (1994) proceeds to an analytical transformation of the part of (19) to be integrated on SF. He reduces the number of the unknowns on SF by utilizing the free surface condition obtained after eliminating η from (15) and (16). This transformation increases the order of the derivatives of the functions to be evaluated on SF and the consequence is computational difficulty. His result is, however, mathematically significant. He proved a modified Haskind relation and the reciprocal relations on the hydrodynamic force which hold for on the double model flow .
Other approaches to solve on the double model flow are: Yasukawa and Sakamoto (1991) introduces a special source function satisfying the free surface condition on the local flow for G in the equation (8). Takagi (1990) extends the radiation layer with damping, as used for the wave absorbing beach in Rankine panel method (for example Kring et al. (1994)), to the whole free surface but reduces the damping to the tuned limit. The former’s source function is mathematically complicated and use of G0 instead of their source function will be more plausible. In the latter the tuning of the damping has to be determined rather empirically and no theoretical method is available.
Validation by Experiment
In section 2 we demonstrated that comparison of hydrodynamic pressure predicted and measured would be indeed the best way to investigate validity of the theories for hydrodynamic as well as practical viewpoint. Reliable experimental data of hydrodynamic pressure on the hull surface is not easy to obtain. Moreover how the measured pressure is to be compared with the predicted is sometimes ambiguous, particularly for the hydrodynamic pressure at a location which goes in and out the water during a period of the wave-induced ship motion. Observation of the wave elevation and comparison with the theoretical is easier in terms of experimental instrumentation and its accuracy is much more reliable than that of the measured fluid pressure. The wave elevation has close relationship with the pressure in the vicinity of the water surface and there is no ambiguity on what is to be compared with the predicted wave elevation.
We concentrate on the unsteady wave generated by a ship in the frequency domain. Actually the unsteady wave field generated by a ship translating at forward speed in waves is invisible at tank test because other waves such as the steady waves and the incident waves coexist. We need a technique to separate each of those waves. Another problem is that instantaneous distribution of the wave elevation around a ship model is not perfect data for the case of the unsteady wave η. A complete picture of η is obtained only after the spatial distribution of the amplitude and the phase is accurately measured. A technique was developed (Ohkusu and Wen (1996)) to overcome those difficulties and obtain the distribution of η around a ship model without a large number of wave probes installed.
Figs. 4, 5, 6 and 7 compare the theoretical wave elevation with the experimental one at ωt=0( cos component) and ωt=π/2 (sin component) at several x positions. Wave elevation is normalized by the amplitude of the incident wave and plotted vs y. A ship model is a Series-60, Cb=0.80 at the ballast condition. The model’s motions are restrained in head seas of the wave length to ship length ratio λ/L=0.5 and Froude number Fn is 0.20. The data shown in Figs. 4 to 7 are obtained by excluding the steady wave and the incident wave from η. The measured wave elevation plotted the most left in each of the figures is the wave elevation almost on the hull surface at the widest section of the ship; at the bow part the wave elevation is at the location little away from the hull surface (this is owing to the system of the wave measurement, see Ohkusu and Wen (1996)). Reason why we selected the ballast condition for the comparison is that we know the diffraction waves are higher than those at the full load condition particularly near the bow.
η is computed from , which is obtained by Rankine panel method assuming the double model flow in the free surface conditions, with
(21)

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
Fig. 4 Wave elevation at St. No. 9
Fig. 5 Wave elevation at St. No. 8
Fig. 6 Wave elevation at St.No. 7
Fig. 7 Wave elevation at St. No. 6

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
Fig. 8 Wave elevation along the hull surface
where stands for the gradient in the horizontal plane. The comparison of the computed η with the measured reveals the significant disagreement at St. No. 9 and 8. The measured wave height is considerably higher than the theoretical. However as we go away from the bow to downstream, the agreement is improved. At the midship zone the agreement is rather excellent. Conclusion is that taking of the double model flow into our modelling does not improve the inaccuracy of the theory though the inaccuracy was reasoned due to the uniform steady flow assumption in section 2. A disappointing fact is that this three dimensional computation does not improve the prediction with a simpler ‘2.5 dimensional’ approach of the high speed slender body theory with the double model flow taken in the free surface condition (the results are not shown here).
In order to confirm this conclusion we measured the wave elevation along the hull surface forward of St. No. 9 section and compare it with the theoretically predicted. Time series of the wave elevation is recorded by the capacitance type wave probe. Amplitude of the fundamental harmonics of the wave elevation thus recorded is plotted in Fig. 8. This value includes the effect of the incident waves. Reliability of this result is confirmed by other way (analysis of the video image). Discrepancy between the measured and the predicted is extreme but is expected from the difference we observe in Fig. 4.
4. Semi-Nonlinear Slender Body Theory
The steady wave elevation at the bow part is supposed to be high with high speed vessels and the effect of such high steady wave elevation must be significant on the unsteady flow at the bow part. We have shown already by two examples in section 2 and 3 that the steady wave elevation appears to dominate the accuracy of the wave pressure prediction at the bow part of even not high speed and not slender ships.
Full nonlinear theory must be our final goal but simpler approach capable to account for some of nonlinear features efficiently will be practically useful. Faltinsen and Zhao (1991) proposed an approach useful for high speed vessel and able to account for the effect of the steady wave elevation consistently. It is derived following Ogilvie’s bow flow model of the steady flow (Ogilvie (1972)).
Since high speed vessels are supposed to be of slender hull form, it is legitimate to introduce the slenderness parameter ε≪1. But we assume the slope of the hull form into the x direction is O(ε1/2) instead of Ο(ε). It sounds rather artificial but it is conceivable that the hull slope and therefore the flow quantity vary more rapidly of the order Ο(ε1/2) into the x direction at a part of the ship length, for example, at the bow part even if the hull form is globally slender of the order Ο(ε). Naturally x component of the unit normal n to the hull surface is Ο(ε1/2) and it follows that and ηS=Ο(ε). The former results from the body boundary condition and the latter is deduced from the steady wave elevation computed from . Variation of the flow will be
in the fluid domain close to the hull surface. Here f represents a flow quantity we are concerned.
Under such assumptions we must not transfer the free surface condition from z=ηS to z=0 because they yield ηS∂/∂z=O(1). A model with the free surface condition to be satisfied on the steady wave surface will be likely to account for what the linear theories could not in the previous sections. Retaining the first order terms with respect to and η we write the free surface condition

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
(13)
And the radiation part is expressed as the same manner as its first-order counterparts. Since we primarily concern with the second-order wave exciting force, we skip it.
3. Application of HOBEM
A particular characteristic of HOBEM compared to CPM is the introduction of shape functions. The geometric shape and physical variables are interpolated by shape functions in higher-order elements. Therefore boundary-value problem is solved more accurately by using a HOBEM (11).
3.1 8-node Boundary Element
In this paper, 8-node bi-quadratic elements are adapted. Fig. 2 illustrates this element and shape functions.
Fig. 2: 8-node bi-quadratic element and shape function
By using shape functions, the velocity potential and geometry are expressed as follows;
(14)
3.2 Integral Equation
The typical integral equation for velocity potential is well known. It contains dipole integral and solid angle, which cause numerical difficulties and require additional calculations. Eatock Taylor and Chau (8) derived an improved form of integral equation by adding an interior flow model, which removes dipole singularity and solid angle.
(15)
where,
However, the wave Green function causes unfavorable interior resonant flow, known as irregular frequency problem. To prevent this difficulty, the condition for zero potential in the interior water plane is put into the eq. (15) (16). The resulting integral equation becomes
(16)
For the purpose of numerical evaluation, the body surface and interior water plane are discretized into higher-order elements. Then the integral equation turns to be algebraic equation.
(17)
In the above equation, NT is the number of nodes including that of interior surface. Detailed descriptions for influence matrix, Dji and Aji, are shown in Choi (17).

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
The source and dipole integrals on the singular element are performed with the help of polar-triangular transformation (11).
3.3 Derivatives by Using Higher-Order Element
The evaluation of the second-order force requires derivatives. In CPM, it is performed by differentiating given integral equation. However, its numerical work is awkward due to the increased strength of singularity on singular elements. On the contrary, it is simple in HOBEM, because the velocity potential and geometrical shape are expressed in terms of mathematical shape functions. Derivatives of them are also expressed in terms of the derivatives of the shape functions.
For first-order derivatives, the transformation of coordinates is to be
(18)
For second-order derivatives, 6 values of second-order derivatives must be known in the mapped coordinate system. But can not be defined. If the Laplace equation is satisfied on boundary surfaces, the unknown can be calculated. For example, transformed second-order derivatives are like below;
(19)
where,
(20)
4. Second-Order Wave Forces
Nonlinear wave forces are attributed to the nonlinearity of the free surface and the body motion. To the second-order of wave amplitude, this effect is represented by mutual interaction of the first-order quantities. Thus characteristic frequency of the second-order force is either difference- or sum-frequency of component wave frequencies of ωk and ωl.
(21)
(22)
The difference part, denoted by Fkl(2)−, may cause large slowly-varying motions of a moored vessel, and the sum part, denoted by Fkl(2)+, may cause high-frequency vertical vibrations of a tension-leg platform, so called springing. In the light of physical origins, the second-order force can be divided into two parts; quadratic products of the first-order properties and the second-order potential. The former, denoted by Fqkl(2), was derived systematically by Pinkster (5), and it can be calculated by using the first-order solutions only. But the evaluation of the latter, denoted by Fpkl(2), is very difficult due to the complicated second-order potential.
Previous studies and experiments sofar clarify that the Fqkl(2)− part is dominant in the differnce-frequency components (18). However, in the sum-frequency component, the Fpkl(2)+ part can not be neglected in comparison with Fqkl(2)+. Recent studies on this force due to the second-order potential are activated in phase with the appearance of tension-leg platforms. The last component in eq. (22), Frkl(2)±, is the hydrostatic restoring force due to the second-order motions.

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
4.1 Second-Order Potential Force
The second-order potentials, which satisfy the Laplace equation and boundary conditions from eq. (11) to eq. (13), can be expressed as the sum of difference- and sum-frequency parts.
(23)
(24)
In the above equations, * denotes complex conjugate. The incident wave potential in eq. (23) is known by Bowers (19).
As stated previously, we concentrate herein on the second-order wave exciting force due to the incident and scattered waves. The formulation of the second-order excitation is quite similar for both differnce- and sum-frequency components. The wave exciting force is normalized by the square of wave amplitude to take
(25)
where the subscript j denotes the direction of the force and moment. The symmetry relation is well known so that one can evaluate it in the lower half region of the frequency the combination diagram.
(26)
To evaluate the force, the second-order scattered potential must be sought, as shown in eq. (12) and eq. (13). Molin (7) solved this problem by using the Haskind relation. His result contains the first-order quantities only.
(27)
where F7kl± is the complex form of the free surface forcing term in eq. (12), corresponding to sum- and difference-frequencies. Auxiliary potential ψjkl± is exactly the same as the first-order radiation potential with sum- and difference-frequencies. The first and second terms in the right-hand side of eq. (27) are the contribution of incident waves and the third one comes from the first-order motions. These terms can be evaluated by using solutions of the first-order problem and the second-order incident waves. However, the last integral is to be performed over the entire free surface. Most previous researches on this topic have focussed on developing numerical schemes, in order to evaluate accurately over a finite region. This direct integration requires long computation time and huge computer memory. It was known that the convergence of the integral according to the size of region is rather poor (9). Another difficulty is that, as shown in eq. (8) and eq. (9), the kernel contains higher-order derivatives and accurate first-order quantities must be available. It can be tied over by using HOBEM.
To overcome the difficulty of integral over entire free surface, it is divided into 3 subregions; inner, intermediate and outer ones (See Fig. 3).
Fig. 3: Subdivision of Free Surface
4.2 Inner Region
In the inner region, the surface integral is evaluated accurately by using higher-order elements. This method requires less number of elements than CPM. The values of the potential in this region are calculated by using integral equation. Second-order

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
derivatives are evaluated by the scheme as explained previously, after some modifications.
4.3 Intermediate Region
In the intermediate region, all the integrands are expanded in terms of eigen functions in cylindrical coordinates.
(28)
En and Bem are eigen functions in the vertical and radial directions, which are expressed by hyperbolic functions and Bessel functions, respectively. kn are eigen values, called wave numbers. As shown in eq. (9) and eq. (27), the integrands consist of products of 3 first-order potentials. The integrals of the circumferential direction can be evaluated analytically with the help of the orthogonality of sinusoidal functions. The integrals in the radial direction are performed numerically. This semi-analytic approach saves numerical burden considerably. More detailed descriptions on eigen functions and coefficients are shown in Choi (17).
4.4 Outer Region
In the outer region, asymptotic expansions of integrands are utilized. After the integral is approximated in the circumferential direction by using the method of stationary phase, the integral from the radial distance ρ2 to infinity is carried out analytically. This yields to as follow;
(29)
where
Ρ± and Q± are expressed in terms of Fresnel functions, which correspond to the stationary phase of β and β+π, respectively. Since the numerical domain can not cover the whole region, it serves as correction to numerical results. Its value oscillates rapidly and decays slowly as shown in Fig. 4. Therefore the convergence of the result as a function of the size of numerical region, S1 and S2, is poor. To obtain an accurate result, the contribution of the outer region should be considered.
Fig. 4: Fresnel Functions
5. Numerical Results and Discussions
5.1 HOBEM vs. CPM
To demonstrate the advantage of HOBEM, a comparison study for HOBEM and CPM is made. For this purpose, hydrodynamic properties of a sphere and a rectangular barge are calculated.
■ Sphere
Since some analytic results are available, sphere is one of popular objects for validation study. To test the convergence, three different discretizations are adopted for CPM and HOBEM, respectively.
Heave added mass in deep water is shown in Fig. 5. Compared with the analytic result of Hulme (20), the convergence rate of HOBEM is faster than that of CPM. Hence, accurate results can be obtained with less number of elements by using HOBEM. Also, it can be seen that the scheme of eq. (16) guarantees

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
the successful removal of irregular frequencies in a wide frequency range.
Fig. 5: Heave Added Mass of Floating Hemisphere
Fig. 6: Convergence diagram of Floating Hemisphere
A convergence diagram is represented in Fig. 6. In this figure, the number of unknowns means the number of elements for CPM and the number of nodes for HOBEM. The relative error is calculated by using analytic value at va=1.0, and the convergence rate is found as follow.
surge added mass
∝ 1/N1.0
∝ 1/N2.1
for CPM
for HOBEM
heave added mass
∝ 1/N1.0
∝ 1/N1.9
for CPM
for HOBEM
surge damping
∝ 1/N1.0
∝ 1/N2.2
for CPM
for HOBEM
heave damping
∝ 1/N1.0
∝ 1/N1.5
for CPM
for HOBEM
The convergence of HOBEM with quadratic-order is approximately double of CPM, except for heave damping. Therefore we can conclude that HOBEM is a very accurate and efficient scheme.
Fig. 7: Time Mean Surge Drift Force of Floating Hemisphere

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
The comparison is also earned out for the time mean drift force. The results are shown in Fig. 7, where analytic result of Kudoh (21) is included. Considering first-order quantities, the CPM gives the force overestimated and the convergence is very slow, especially in high frequency range. However, the accuracy and convergence of HOBEM is excellent. Only 48 elements are sufficient. To calculate the time mean drift force, first-order derivatives of the velocity potential should be evaluated. Hence, the derivative scheme, eq. (18), is highly recommended. It can be concluded that HOBEM is very useful for the evaluation of the second-order force.
■ Rectangular Barge
This geometry is chosen to understand the effect of shapes with sharp edges. The size of barge is 30 m(L)×22 m(B)×1.5 m(T), and displacement is 990 m3. It floats freely in water of depth 15 m. In order to test the convergence, four different discretizations are used for CPM and HOBEM, as shown in Fig. 8 and Fig. 9. Panel 4 and element 4 discretize the sharp edge densely.
Fig. 8: Constant Panels (quadrant)
Fig. 9: Higher-Order Elements (quadrant)
Fig. 10: Surge and Heave Added Mass of Barge
Fig. 11: Time Mean Surge Drift Force of Barge at Head Sea

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
It is seen from Fig. 10 that the convergence of HOBEM for hydrodynamic coefficients is faster than that of CPM. The added masses of surge and heave by CPM are larger than HOBEM. This is caused by neglecting solid angle at the edges. Since the solid angle is included in HOBEM, the effect of shape edge is taken into account more precisely.
In the case of the time mean drift force, noticeable differences are found between CPM and HOBEM. There exist apparent peaks in the CPM results and the peak magnitudes continue to decrease as the number of panels increases. In contrast, the convergence of the HOBEM results is very fast and the sharp peaks do not appear even for coarse discretizations. In the case of CPM, the result seems to be sensitive to the panel resolution near sharp edges. In the CPM results, the slightly negative drift is seen in the short range before the curve rises. This physically unacceptable result was reported in other works, especially for the drift of sharp edged bodies. However, the negative drift force is negligibly small in the result of HOBEM. It seems to be the effect of panel resolution near the sharp edge in CPM, also.
5.2 Second-Order Potential Force
The second-order potential force is known to be negligible in the difference-frequency excitation. Therefore only the sum-frequency excitation is considered. To verify the method developed herein, the second-order force on a sphere is calculated. And then, ISSC TLP is taken for a practical application.
■ Sphere
For sphere, the analytic values for the sum-frequency force are available in Ref. (22). They calculated it using the ring source integral equation method for water depth of 3×(radius of sphere).
Fig. 12 represents the discretization of the sphere and numerical free surface region, S1. The number of higher-order elements is 192 for the body and 256 for the free surface. The radius of S1 (=ρ1) is 3a.
In Fig. 13, surge and heave forces of double frequency are shown for a stationary sphere. The quadratic part of the force agrees with the analytic value. The evaluation of heave force is successful, also. However, some deviation in surge force is observed in the range of high frequencies. It seems to be caused by oscillations of auxiliary surge potential near the body surface in high frequency range. It can be seen that the second-order potential force makes a great contribution to the sum-frequency excitation.
Fig. 12: Discretization of Sphere and Free Surface
Fig. 13: Surge and Heave Force of Double Frequency on Stationary Sphere
In the case of a freely floating sphere, the results are shown in Fig. 14. In this figure, Fbb means the force due to the first-order motion as shown in eq. (27), which contains second-order derivatives. As shown in this figure, the second-order derivatives are performed accurately on the body surface. Hence, it can be concluded that the derivation scheme of eq. (19) and eq. (20) is very effective. The peak value near va=1 is induced by the heave motion at the natural frequency.

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
Fig. 14: Surge and Heave Force of Double Frequency on Freely Floating Sphere
In Fig. 15, the surge force caused by the free surface integral is illustrated. The results are compared with different size of inner and intermediate regions. The radius of the boundary between S2 and S3, is the same. It should be decided where the asymptotic expansion or the method of stationary phase are applied. In this paper, ρ2 is determined by the following equation.
In the above equation, k0 is the wave number of the longest wave among two component waves and auxiliary radiation wave. The value for small numerical region is the same as that for wider one. Therefore, it is demonstrated that the introduction of the intermediate region is successful. However, the results for a subregion (S1 or S2) are quite different dependent on the size. It implies that the integration is fluctuating largely according to the size of domain. The contribution from S3 is not negligible. It is mainly due to the interaction between the first-order incident wave and disturbed wave. Hence, one should take outer region integral into account.
Fig. 15: Contribution of Free Surface Integrals on Freely Floating Sphere
■ ISSC TLP
As mentioned previously, high frequency forces like the sum-frequency excitation can excite the stiff vertical modes (heave, pitch and roll) of a tension-leg platform. For example, springing can cause the fatigue failure of tensioned tethers. Researches on the sum-frequency force are therefore focused on the application to the TLP. In this paper, the ISSC TLP is chosen as an example calculation. Its main dimensions are summarized as below;
Spacing between Column Centers (L): 86.25 m
Diameter of Column: 16.87 m
Breadth of Pontoon: 7.5 m
Height of Pontoons: 10.5 m
Draft: 35 m
Displacement: 54500 ton
Weight: 40500 ton
Water Depth: 450 m
More detail information is given in Eatock Taylor and Jefferys (23).
Fig. 16 represents the discretizations of the ISSC TLP and free surface region. The number of

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
higher-order elements is 140 and 512 for the body and free surface, respectively. The radius of S1 (=ρ1) is 90 m.
Fig. 16: Discretization of ISSC TLP and Free Surface
Double frequency heave and pitch excitations of the stationary TLP in head sea are shown in Fig. 17. The results are compared with the numerical results of Lee, et al. (24) and Liu, et al. (9) and also with the experimental result of Matsui, et al. (25). The qualitative trend of results is similar to each other. At high frequency, a good agreement with that of Lee, et al. (23) is found. They calculated with 4048 and 4928 constant panels for the body and free surface.
Fig. 18 shows the contributions of quadratic term and the second-order potential force in the pitch moment. It is seen that the contribution of quadratic term is small.
The results of the compliant TLP are represented in Fig. 19. They are very similar to the case of the stationary TLP. In the figure, Ff and Fb denote the component of free surface integral and body surface integral as given in eq. (27). The contribution of the free surface integral is dominant in the high frequency range. Hence, it should be noted that an accurate evaluation of the free surface integral is necessary to the sum-frequency excitation of TLP.
Fig. 20 shows the force due to the free surface integral and each subregion integral. The total value for small numerical region is the same as that with wide one. It means that an introduction of a wider intermediate region may save the numerical work for the inner region. In the pitch moment, the contribution from the outer region is remarkable.
Fig. 17: Heave Force and Pitch Moment of Double Frequency on Stationary TLP
Fig. 18: The Components of Pitch Moment on Stationary TLP

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
Fig. 19: Heave Force and Pitch Moment of Double Frequency on Compliant TLP
6. Conclusion
An accurate and efficient numerical scheme is constructed based on higher-order boundary elements. The convergence and accuracy are improved compared to the constant panel method. As shown in numerical examples, this scheme is particularly effective for the body with sharp edges.
The second-order potential force is formulated with the help of Molin’s approach. The difficulty of infinite free surface integral is overcome by introducing the intermediate and outer regions. It reduces the numerical burden remarkably. The wave force contribution from the far field is derived analytically in terms of Fresnel functions, which enhance the convergence of numerical results. Numerical examples shown herein clearly indicate the efficiency of the present method.
Fig. 20: Contribution of Free Surface Integrals on Compliant TLP
References
1. Havelock, Τ.Η., “The Pressure of Water Waves upon a Fixed Obstacle,” Proc. Royal Soc. London, Ser. A, Vol. 175, No. 963, 1940.
2. Havelock, T.H., “The Drifting Force on a Ship among Waves,” Phil. Magazine, Ser 7, Vol. 33, 1942.
3. Maruo, H., “The Drift of a Body Floating on Waves,” J. Ship Res., Vol. 4, No. 3, 1960.
4. Newman, J.N., “The Drift Force and Moment on Ships in Waves,” J. Ship Res., Vol. 11, 1967.
5. Pinkster, J.A., “Low Frequency Second Order Wave Forces on Vessels Moored at Sea,” Proc. 11th Symp. Naval Hydrodyn., 1976.
6. Pinkster. J.A., “Low Frequency Second Order Wave Exciting Forces on Floating Structures,” N.S.M.B., Pub. No. 650, 1980.
7. Molin, B., “Second Order Diffraction Loads upon Three-Dimensional Bodies,” Appl. Ocean Res., Vol. 1, 1979.

OCR for page 341

Twenty-Second Symposium on Naval Hydrodynamics
8. Eatock Taylor, R. and Chau, F.P., “Wave Diffraction—Some Development in Linear and Non-Linear Theory,” J. Offshore Mech. and Arctic Engng., No. 114, 1992.
9. Liu, Y.H., Kim, M.H. and Kim, C.H., “Double-Frequency Wave Loads on a Compliant TLP,” Proc. 3rd International Offshore and Polar Engng., 1993.
10. Hess, J.L. and Smith, A.M.O., “Calculation of Nonlifting Potential Flow about Arbitrary Three Dimensional Bodies,” J. Ship Res., Vol. 8, No. 3, 1964.
11. Liu, Y.H., Kim, C.H. and Lu, X.S., “Comparison of Higher-Order Boundary Element and Constant Panel Methods for Hydrodynamic Loadings,” J. Offshore and Polar Engng., Vol. 1, No. 1, 1990.
12. Boo, S.Y., “Application of Higher Order Boundary Element Method to Steady Ship Wave Problem and Time Domain Simulation of Nonlinear Gravity Waves,” Ph. D. dissertation, Texas A&M Univ., 1993.
13. Hong, S.Y., “Analysis of Steady and Unsteady Flow Around a Ship Using a Higher-Order Boundary Element Method,” Ph. D. Thesis, Seoul National Univ., 1994.
14. Newman, J.N., “Marine Hydrodynamics,” The ΜIT press, 1976.
15. Ogilvie, T.F., “Second-Order Hydrodynamic Effects on Ocean Platforms,” Proc. Intl. Workshop Ship and Platform Motions., 1983.
16. Hong, D.C., “On the Improved Green Integral Equation Applied to the Water Wave Radiation-Diffraction Problem,” J. Soc. of Naval Architects of Korea, Vol. 24, No. 1, 1987.
17. Choi, Y.R., “An Analysis of Second-Order Wave Forces by Using a Higher-Order Boundary Element Method,” Ph. D. Thesis, Seoul National Univ., 1997.
18. Choi. Y.R., Won, Y.S. and Choi, H.S., “The Motion Behavior of a Shuttle Tanker Connected to a Submerged Turret Loading System,” Proc. 4th International Offshore and Polar Engng., 1994.
19. Bowers, E.C., “Long Period Oscillations of Moored Ships Subject to Short Wave Seas,” Trans. R. Inst. Naval Archit., 1976.
20. Hulme, A., “The Wave Forces Acting on a Floating Hemisphere undergoing Forced Periodic Oscillations,” J. Fluid Mech., Vol. 121, 1982.
21. Kudoh, K., “The Drifting Force Acting on a Three-Dimensional Body in Waves,” J. of Soc. Nav. Arch. Japan, Vol. 141, 1977.
22. Kim, M.H. and Yue, D.K.P., “The Complete Second-Order Diffraction Solution for an Axisymmetric Body. Part 2. Bichromatic Incident Waves and Body Motions,” J. Fluid Mech., Vol. 211, 1990.
23. Eatock Taylor, R. and Jefferys, E.R., “Variability of Hydrodynamic Load Predictions for a Tension Leg Platform,” Ocean Engineering, Vol. 13, No. 5, 1986.
24. Lee, C.H., Newman, J.N., Kim, M.H. and Yue, D.K.P., “The Computation of Second-Order Wave Loads,” Proc. Offshore Mech. and Arctic Engng., 1991.
25. Matsui, T., Suzuki, T. and Sakoh, Y., “Second-Order Diffraction Forces on Floating Three-Dimensional Bodies in Regular Waves,” J. Offshore and Polar Engng., Vol. 2, No. 3, 1992.