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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 368
Prediction of Wave Pressure and Loads on Actual Ships by the
Enhanced Unifed Theory
M.Kashiwagi (Kyushu University, Japan)
S.Mizokami, H.Yasukawa, Y.Fukushima (Mitsubishi Heavy Industries, Japan)
ABSTRACT
To establish a new practical calculation method in place of the conventional strip method, performance of the
enhanced unified theory is investigated through the comparison of computed results with a large number of experiments
conducted with VLCC tanker and container ship models. In this paper, compared are the ship motions, the pressure
distribution, and the wave loads. The enhanced unified theory is essentially based on 2-D computations but takes account
of 3-D and forward-speed effects. Furthermore the effects of wave diffraction from the bow part near the waterline are
taken into account in a rational way. Despite these theoretical improvements, the results of comparison for the wave loads
are not so good as expected. Since the pressure and wave loads are strongly influenced by the accuracy of ship motions,
more improvement is needed for precise prediction of the ship motions particularly near the resonance of heave, roll, and
pitch.
INTRODUCTION
In the design stage of actual ships, the strip theory is still in routine use for computing the ship motions, added
resistance in waves, pressure distribution, and so on. Recently, 3-D computation methods based on the free-surface
Rankine panel method have been developed, but they are time-consuming from a practical viewpoint, and validity for
various ship shapes is not confirmed.
On the other hand, the strip method is versatile and its prediction is relatively good, considering that the computation
time is short and the theory is simple. However, several shortcomings in the strip method have been recognised; for
instance, the pressure distribution near the ship bow and stern and the added resistance in short waves are not in good
agreement with experiments. These shortcomings are related to improper treatment of 3-D and forward-speed effects.
To account for these effects in the framework of slender-ship theory, many theoretical works have been made.
Among them, the unified theory, originally developed by Newman (1978) and extended to the diffraction problem by
Sclavounos (1984), is recognised as one of the successful slender-ship theories. The unified theory could bring in a
certain amount of 3-D effects to a strip-theory type solution in a rational manner. However, it was still not satisfactory.
For instance, the wave diffraction from the bow part near the waterline could not be taken into account, and thus the wave-
exciting force in surge and the added resistance in short waves were usually underestimated.
To incorporate the effects of the wave diffraction near the bow and other effects dismissed as higher order in the
slender-ship theory, the original unified theory was enhanced by Kashiwagi (1995); in which the radiation problem of
surge is solved in the same fashion as the heave and pitch modes, and the effects of wave diffraction from the bow part
near the waterline are taken into account by retaining the x-component of the normal vector in the body boundary
condition of the diffraction problem. Furthermore, 3-D and forward-speed effects on lateral modes of motion are
incorporated as well. Validity of this enhanced unified theory (abbreviated as EUT in the present paper) has been
confirmed only for mathematical ship models like a prolate spheroid and a modified Wigley model (Kashiwagi et al.
2000). The unified theory is essentially based on 2-D computations and thus the computation time is very short compared
to that needed in 3-D Rankine panel codes; this feature is promising as a practical design tool in the design stage of actual
ships.
For the purpose of establishing a new practical calculation method in place of the strip method, we have investigated
usefulness and applicability of the enhanced unified theory, through comparison of computed results with a large number
of experi
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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 369
ments using actual ship models. In the present paper, some results of comparison are shown for models of a VLCC tanker
and a recent container ship. The experiments were conducted in regular waves, and the incidence angle of the wave was
changed rather densely. Although there are many experimental data, shown in this paper are the ship motions and
pressure distribution of a VLCC tanker and the wave loads (vertical bending and torsional moments) of a container ship.
We can see some noticeable improvements over the strip method particularly in the pressure distribution and wave
loads, but predictions of the enhanced unified theory are still not perfect in some cases when compared closely with
experiments. Discussion is made on possible reasons of disagreement with experiments.
ENHANCED UNIFIED THEORY
Mathematical formulation
We consider a ship advancing with constant speed U and undergoing oscillatory motions with circular frequency ω
in deep water. The analyses will be performed using a Cartesian coordinate system, which moves steadily with the same
constant speed as a ship. The x-axis is directed to the ship's bow and the z-axis is directed downward (see Fig. 1).
Fig. 1 Coordinate system and notations
Assuming the inviscid fluid with irrotational motion, the flow can be described with the velocity potential, which is
expressed as
(1)
(2)
(3)
(4)
where denotes the incident-wave potential; A, ω0, k0, χ are the amplitude, the circular frequency, the wavenumber,
and the incidence angle of incoming wave, respectively; g is the gravitational acceleration. in (1) denotes the steady
disturbance potential due to the forward motion of a ship. in (2) denotes the scattering potential and the radiation
potential of the j-th mode with complex amplitude Xj, where in particular j=1 for surge, j=3 for heave, and j=3 for pitch.
To obtain a solution for the purpose of practical calculations, the enhanced unified theory (hereafter abbreviated as
EUT) is applied in this paper. In the subsections below, the outline of the theory will be given. For more details, we refer
the readers to Kashiwagi (1995, 1997).
Radiation problem
In the inner region close to the ship hull, variation of the flow along the x-axis is small compared to that in the
transverse section and the wave radiation at infinity is out of concern. Therefore, the velocity potential in the inner region
satisfies
(5)
(6)
(7)
where K=ω2/g. nj and mj in (7) denote the j-th component of the unit normal directing into the flu0id and of the so-
called m-term representing interactions with the steady flow; these are considered on the contour (CH) of the transverse
section at statio0n x along the ship's length.
The general inner solution satisfying (5)–(7) takes the following form:
(8)
where φj and are the particular solutions, corresponding to the first and second terms on the right-hand side of (7),
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respectively. denotes a homogeneous solution, which can be explicitly given by for the symmetric modes
(j=1, 3, 5) and by for the antisymmetric modes (j=2, 4, 6), where the asterisk means the complex conjugate.

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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 370
In the present paper, contributions of are neglected in computing the m-term and thus mj=0 for j=1~4, m5=n3, and m6=
−n2. Moreover, with slenderness assumption, n5=−x n3 and n6=x n2.
In accordance with these approximations, we can obtain the followings:
(9)
The unknown coefficient Cj(x) in (8) can be determined by the matching with the outer solution. In the outer region
far away from a ship, the solution can be represented by a line distribution of singularities along the x-axis. Considering
only the symmetric modes (j=1, 3, 5), the source distribution may be used, in which its strength Qj(x) is unknown due to
lack of the body boundary condition.
The method of matched asymptotic expansions gives the following results:
(10)
(11)
where σj and are the 2-D Kochin functions to be computed from φj and respectively. The kernel function f(x
−ξ) in (10) represents the 3-D and forward-speed effects; the Fourier transform of which is expressed as
(12)
where
(13)
and the upper and lower expressions in the brackets are taken according as κ>|k| and κ<|k|, respectively.
Once the integral equation (10) for Qj(x) is solved, it is straightforward to compute Cj(x) from (11), thereby
completing the inner solution, which will be used for computing the pressure, the added-mass and damping coefficients,
and the wave loads.
Diffraction problem
Unlike the radiation problem, we assume that the rapidly-varying part with respect to x is of the same form as that
is, eiℓx. Thus the scattering potential may be sought in the form of
The governing equation and boundary conditions for the slowly-varying part ψ7 are given as
(14)
(15)
(16)
It is noteworthy that the governing equation is the 2-D modified Helmholtz equation and the wavenumber in (15) is
not K but k0. The effects of wave diffraction from the bow part near the waterline are taken into account by retaining n1-
term on the body boundary condition.
Considering only the symmetric component with respect to y=0, the inner solution can be constructed in the form
(17)
where and ψ2D denotes a numerical solution of (14)–(16).
Here is the unknown coefficient of a homogeneous solution, which can be determined by matching (17)
with the outer solution.
The results of the matching are expressed as
(18)
(19)
cosh−1(|sec
where hS(χ)=csc χ χ|)−ln(2|sec χ|) and σ7 is the 2-D Kochin function to be computed from ψ2D. The kernel
function f(x−ξ) is identical to that used in the radiation problem.
The numerical method adopted here for solving the integral equations, (10) and (18), is essentially the same as that
shown in Sclavounos (1985), using Chebyshev polynomials for representing the unknown source strength and using a
Galerkin scheme with orthogonal properties.
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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 371
Hydrodynamic and hydrostatic pressure
Retaining only the first-order linear terms in Bernoulli's pressure equation, the spatial part of the unsteady pressure is
given by
(20)
Here the first term on the right-hand side is the hydrodynamic part, with V defined as
and the second term represents the change in the hydrostatic pressure due to ship motions from the equilibrium position.
In accordance with neglect of in computing the m-term, an approximation of is employed in the present
paper.
Substituting (2) as in (20), the total oscillatory pressure can be divided into three components; those are
written as
(21)
where pD, pR, and pS denote the diffraction pressure, the radiation pressure, and the change in the hydrostatic
pressure, respectively.
In the diffraction problem, differentiation with respect to x may be applied only to the rapidly-varying term, eiℓx, and
thus pD is given by
(22)
R S
Likewise, p and p are given in the nondimensional form as
(23)
(24)
The symmetric part of by EUT can be expressed by the homogeneous component (the second line) in
(17). The same is true for the asymmetric part, although its explicit form is not shown here.
The radiation potential by EUT is given by (8). Consistent with approximations for the m-term and the
hydrodynamic forces (which will be explained next), differentiation with respect to x in (23) is performed only for j=5
and 6.
The complex amplitude, Xj/A, of the j-th mode of motion will be given as a solution of the ship-motion equations.
Hydrodynamic forces
In the radiation problem, the force acting in the i-th direction is computed in terms of pR and the results can be
summarised in the form
(25)
(26)
where Aij and Bij are the added-mass and damping coefficients in the i-th direction due to the j-th mode of motion.
In this paper, as shown in (9), mi and can be expressed with ni and φj. Therefore, all integrals in (26) along the
contour (CH) of the transverse section at station x are evaluated using the following 2-D results:
(27)
The wave-exciting force in the i-th direction can be computed by integrating pD multiplied by ni over the ship hull.
Using (22) and (17), the symmetric components (i=1, 3, 5) are expressed as
(28)
We note that hydrodynamic forces related to surge (i=1) are computed by EUT, with 3-D and forward-speed effects
taken into account.
Ship motions
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In the linear theory, the symmetric modes (i=1, 3, 5) of motion can be computed independent of the antisymmetric
modes (i=2, 4, 6) for a ship symmetric with respect to y=0. Therefore, the longitudinal motions (surge, heave and pitch)
can be computed from the coupled motion equations:
(29)

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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 372
where Mij is the mass matrix and Cij is the restoring-force matrix to be computed from the hydrostatic pressure pS
given in (24). Nonzero elements in these matrices are
(30)
where ∇ is the displacement volume; Iyy is the moment of inertia about the y-axis and κyy is the corresponding radius
of gyration; AW is the waterplane area and xW is the x-coordinate of the center of waterplane area; and is the
distance between the center of gravity and the longitudinal metacenter.
The transverse motions (sway, roll and yaw) are computed in the same way. However, the damping coefficient in
roll is modified to take account of viscous effects, which are crucial especially near the resonance frequency. Namely
(31)
where denotes the value computed by EUT and kW is a correction coefficient, represents nonlinear
components due to the vortex shedding and the shearing force on the wetted surface of a ship, and represents the lift
component in the presence of the forward speed.
Wave loads
We consider first the vertical shearing force on the transverse section at x=x0 along the ship's length. As shown in
Fig. 2, the vertical shearing force is defined as positive when acting in the negative z-direction.
Fig. 2 Positive directions of the wave loads
Since the force is the sum of the pressure force and the inertia force due to the body acceleration, the vertical
shearing force can be given by
(32)
where xA denotes the aft end of a ship and w(x) is the weight distribution along the x-axis.
When computing the pressure force from pR given by (23), may be discarded, which is consistent with the
treatment in computing the added-mass and damping coefficients in heave.
The result after substituting the pressure can be expressed in the nondimensional form as follows:
(33)
where the nondimension is made in terms of a=L/2 (L being the overall length) for the x-axis, and b=B/2 (B being the
breadth) for the y- and z-axes. Therefore xA/a=−1 and other quantities are defined as
(34)
The hogging moment is defined as positive for the vertical bending moment (see Fig. 2). With this definition, the
vertical bending moment on the transverse section at x=x0 is computed by
(35)
Here, to be consistent with the pitch restoring moment, C55, in the motion equation, the contribution of n1-term is
included in computing the hydrostatic pressure force. (ℓG is the z-coordinate of the center of gravity.)
It should be also noted that the integral associated with the radiation pressure including differentiation of with
respect to x may be treated
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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 373
as
(36)
This transformation is consistent with the treatment in computing the added-mass and damping coefficients in pitch.
With these taken into account, the nondimensional calculation formula for the vertical bending moment takes the form
(37)
where S(x) denotes the area of the transverse section and ℓB(x) is the z-coordinate of the center of transverse-section
area.
The expressions for other components of the wave loads can be obtained in an analogous manner.
For the subsequent comparison with experiments, let us describe the expression for the torsional moment. Defining
the torsional moment acting counterclockwise about the x-axis to be positive (see Fig. 2), the torsional moment on the
transverse section at x=x0 is computed by
(38)
where and is the distribution of the moment of inertia in roll.
As is the same as the vertical shearing force, we note that in the radiation pressure may be discarded from a
viewpoint of consistency with the computation of the added-mass and damping coefficients in roll. Furthermore, since
only the antisymmetric components of the pressure contribute to (38), the nondimensional calculation formula for the
torsional moment is given as follows:
(39)
where is the transverse metacentric height and is the gyrational radius of roll in the transverse
section; both are nondimensionalized in terms of b.
It should be noted that the 3-D and forward-speed effects are taken into account in EUT even for the antisymmetric
part of and the lateral modes of the radiation potential (Kashiwagi, 1997).
RESULTS AND DISCUSSION
Outline of the strip method
In this paper, the results of the strip method established by Salvesen, Tuck and Faltinsen (1970) (which is
abbreviated hereafter as STFM) are shown and compared with the results of EUT and model experiments.
In STFM, the contour of the transverse section is approximated by the Lewis form, and 2-D hydrodynamic
computations are implemented by Ursell-Tasai's method. Surge is treated as an independent mode, with only the Froude-
Krylov force and the inertia force due to the ship's mass taken into account. The computer code used in this study solves
the diffraction problem directly, in which the freesurface condition of (15) is satisfied; that is, the wavenumber in the free-
surface condition is not K but k0.
Wave-induced ship motions
The experimental data of ship motions and the pressure distribution used for comparison in this paper are for a
VLCC tanker model. The principal particulars of this tanker model are shown in Table 1. The experiments were carried
out at Ship Research Institute and their results were reported by Tanizawa et al. (1993).
Although many experimental data exist, only the amplitudes of heave and pitch are shown in Fig. 3 for various
angles of the wave incidence, together with corresponding results by EUT and STFM. The Froude number was set equal
to Fn= 0.131. EUT takes account of 3-D and forward-speed effects in the radiation and diffraction forces.
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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 374
Therefore, EUT is expected to give a better prediction for the ship motions. However, the numerical results by EUT and
STFM are almost the same in the degree of agreement with experimental data. EUT tends to underpredict the pitch
motion in the longer wavelength region at χ=120~180°, which is due to underprediction of the pitch exciting moment and
overprediction of the pitch damping coefficeint, according to a recent study (Kashiwagi et al. 2000).
Fig. 3 Amplitudes of heave (left) and pitch (right) of a VLCC tanker in waves (Fn=0.131)
Table 1 Principal particulars of a VLCC tanker model
Lpp (m) 4.500 0.100
B (m) 0.793 κyy/Lpp 0.241
d (m) 0.285 κxx/B 0.355
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Cb 0.807 Fn 0.131
Pressure distribution
Comparison of the pressure distribution is shown from Fig. 4 to Fig. 6 for a VLCC tanker model

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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 375
at Fn=0.131. The pressure was measured at some points on the contour of the transverse section at S.S. 4.22 (0.1 m ahead
of S.S. 4). The abscissa of each figure is the position along the contour and θ=−90°, 0°, and 90° correspond to the weather
side, the centerline, and the lee side, respectively.
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Fig. 4 Pressure distribution at S.S. 4.22 of a VLCC tanker (λ/Lpp=0.3, Fn=0.131)
Figure 4 shows the results of λ/Lpp=0.3, at which the ship motions are almost zero except for the roll motion around
χ=30°. Therefore the pressure distribution in Fig. 4 may be regarded as the pressure induced by the wave diffraction only.
The overall agreement between computed results by

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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 376
EUT and the measurements is satisfactory. Compared with the results of STFM, EUT gives a noticeable improvement
near the ship bottom at χ= 180°. However, EUT tends to overestimate the pressure on the weather side in oblique waves.
Fig. 5 Pressure distribution at S.S. 4.22 of a VLCC tanker (λ/Lpp=0.75, Fn=0.131)
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Figure 5 shows the results of λ/Lpp=0.75, at which the pressure is influenced by the ship motion. Because of the
paucity of measured points on the lee side, the definitive judgement on the superiority of EUT cannot be made. However,
EUT seems to give better results than STFM, particularly in the following wave (χ=0°).

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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 377
Fig. 6 Pressure distribution at S.S. 4.22 of a VLCC tanker (λ/Lpp=1.25, Fn=0.131)
We can see that the wave pressure at λ/Lpp= 1.25 shown in Fig. 6 is relatively small in amplitude except for χ=90°. In
the present case, the roll motion becomes large around χ=90° due to the roll resonance. In fact, the change in the
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hydrostatic pressure due to the roll motion, the second term on the right-hand side of (24), becomes dominant near the roll
resonance. Therefore precise prediction of the roll motion is crucial in estimating the pressure. For that purpose, as shown
in (31), inclusion of the nonlinear viscous damping force in the ship-motion equation is very important. It should be noted
that

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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 378
the same modification for the roll damping coefficient using (31) is made both in EUT and STFM. (Unless the nonlinear
viscous damping is taken into account in roll, the pressure at χ=90° in Fig. 6 becomes tremendously large.)
Fig. 7 Vertical bending moment at S.S. 5 of a container ship (Fn=0.215)
Wave loads
In order to make a thorough investigation on the wave loads, measurements of the vertical bending moment and the
torsional moment have been carried out using a container ship model at Nagasaki R&D Center of Mitsubishi Heavy
Industries. The experiments were conducted for various angles of the wave incidence and at five different Froude
numbers. Furthermore, the wave loads were measured at seven stations along the ship's length.
The length-to-breadth ratio, L/B, and the block coefficient, Cb, of the tested ship model are 6.45, and 0.59,
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respectively. The nondimensional metacentric height in roll, was set to 0.03 and the gyrational radius in pitch,
κyy/L, was 0.25 in the experimental setup.
Figures 7 and 8 show the vertical bending moment and the torsional moment, respectively. These

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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 379
were measured at S.S. 5 and Fn=0.215. Looking at closely, computed results of EUT differ from the measured results in
some respects. Firstly for longer wavelengths in following and quartering waves, EUT tends to overestimate the vertical
bending moment, and secondly in beam wave (χ=90°), EUT is different from the measured results even in the variation
tendency. These discrepancies may be attributed to imperfect agreement of ship motions, because the wave loads are
strongly dependent on the results of ship motions.
Fig. 8 Torsional moment at S.S. 5 of a container ship (Fn=0.215)
Regarding the torsional moment at S.S. 5 (Fig. 8), the overall agreement between computed and measured results is
favarable, considering the value itself is small compared to the vertical bending moment. However, we can see a
difference in the variation tendency for oblique waves (χ=60° and 120°).
Figures 9 and 10 show the same items of the vertical bending and torsional moments, respectively, but the measured
section along the ship's length is S.S. 7. Compared to the results at S.S. 5 (Fig. 7), the amplitude of the vertical bending
moment decreases but the variation tendency with respect to λ/L and χ is more or less the same.
Looking at the torsional moment at S.S. 7 (Fig. 10), a noticeable improvement by EUT over STFM can be seen in
the shorter wavelength region for the case of χ=30°. However, in other angles of the wave incidence, there are still
discrepancies between computed and measured results. Since nonlinear and forward-speed effects on the damping force
are important in the roll mode, those effects might be a reason of discrepancy in the torsional moment.
Figure 11 shows the dependence of the Froude number on the vertical bending moment at S.S. 5 in head waves
(χ=180°). We can see that the value increases slightly as the ship's speed increases, but the overall variation tendency with
respect to λ/L is the same.
Regarding the degree of agreement, we can point out that EUT tends to underestimate around λ/L=0.7 and this
tendency becomes prominent
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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 380
with increasing the Froude number. In fact, the forward-speed effects are not properly taken into account in EUT
especially for higher Froude numbers. To improve in this respect, the forward-speed terms must be incorporated into the
free-surface condition even in the inner problem of the slender-ship theory.
Fig. 9 Vertical bending moment at S.S. 7 of a container ship (Fn=0.215)
CONCLUDING REMARKS
The enhanced unified theory (EUT) encompasses the strip method and takes account of the 3-D and forward-speed
effects in a rational way. Moreover, EUT can compute the surge-related hydrodynamic forces in the same manner as for
heave and pitch, and thus the longitudinal ship motions (surge, heave and pitch) are computed from fully coupled motion
equations among these three modes. In the diffraction problem, EUT can also account for the wave diffraction from the
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bow part near the free surface, because contributions of the n1-term are retained in the body boundary condition. We
expected that these theoretical improvements over the strip method would result in good prediction of the distribution of
wave pressure and wave loads even for actual ships.

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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 381
Fig. 10 Torsional moment at S.S. 7 of a container ship (Fn=0.215)
With this expectation, EUT was applied to a VLCC tanker model and a recent container ship model, and computed
results were compared with experiments to confirm applicability of the theory. However, the results were not so good as
expected before starting this study. One important reason of this is that the prediction of ship motions is not improved so
much when compared to the results of STFM, since the ship motions are influential in the prediction of the wave pressure
and resulting wave loads.
Of course, the effects of nonlinearity and three-dimensionality of the flow may be important particularly around the
bow part, and viscous effects are also important near the stern. These effects must be accounted for by more sophisticated
3-D computation methods. However, EUT is still advantageous from a practical viewpoint, because it is efficient in
computations. Further study is needed for precise prediction of the ship motions, for which the forward-speed effects on
the free-surface condition must be taken into account in a more rigorous way.
ACKNOWLEDGMENTS
The authors would like to thank Mr. H.Sueoka and Dr. T.Kuroiwa of Mitsubishi Heavy Industries for their help in
the course of the present study. Mr. Y.Tozaki in assisting numerical computations is also greatly acknowledged.
REFERENCES
Kashiwagi, M., “Prediction of Surge and Its Effects on Added Resistance by Means of the Enhanced Unified Theory,” Transactions of West-Japan
Society of Naval Architects, No. 89, 1995, pp. 77–89.
Kashiwagi, M., “Numerical Seakeeping Calculations Based on the Slender Ship Theory,” Ship Technology Research, Vol. 4, No. 4, 1997, pp. 167–192.
Kashiwagi, M., Kawasoe, K. and Inada, M., “A Study on Ship Motion and Added Resistance in Waves (in Japanese),” Journal of Kansai Society of N.
A., Japan, No. 234, 2000, in press.
the authoritative version for attribution.

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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 382
Newman, J.N., “The Theory of Ship Motions,” Advances of Applied Mechanics, Vol. 18, 1978, pp. 221–283.
Salvesen, N., Tuck, E.O. and Faltinsen, O.M., “Ship Motions and Sea Loads,” Transactions of SNAME, Vol. 78, 1970, pp. 1–30.
Sclavounos, P.D., “The Diffraction of Free-Surface Waves by a Slender Ship,” Journal of Ship Research, Vol. 28, No. 1, 1984, pp. 29–47.
Sclavounos, P.D., “The Unified Slender-Body Theory: Ship Motions in Waves,” Proceedings of the 15th Symposium on Naval Hydrodynamics,
Hamburg, 1985, pp. 177–192.
Tanizawa, K., Taguchi, H.Saruta, T. and Watanabe, I., “Experimental Study of Wave Pressure on VLCC Running in Short Waves (in Japanese),”
Journal of Society of Naval Architects, Japan, No. 174, 1993, pp. 233–242.
Fig. 11 Vertical bending moment at S.S. 5 of a container ship (χ=180°)
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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 383
DISCUSSION
P.Sclavounos
Massachusetts Institute of Technology, USA
I would like to congratulate the authors for a series of thorough studies, including the present one, refining the
unified slender body theory for the prediction of motions and wave induced loads of realistic ship hulls. I would like to
commend the authors' diligent refinement of the theory and its validation against careful experimental measurements. It is
the experience the authors are reporting that has led me to switch my energies since the mid 80's to the development of
the 3D Rankine Panel Method SWAN. I am however very pleased to see that several of the shortcomings of the unified
slender body theory with forward speed have been removed by the authors.
I have a few comments and questions to which I welcome the authors' response:
The good correlation of STFM and EUT for the motions of the tanker is probably due to her low Froude number. It
would be interesting to see how both methods perform at U=0. We have seen some very good performance of unified
theory in that limit which seems to be corroborated by the results the authors present.
In our experience with SWAN, one of the most important forward speed effect arises from the careful and complete
treatment of the m-terms on the body boundary condition. These terms may actually be computed from the double
body flow without the need to model surface wave disturbances. Their evaluation would require the use of a 3D panel
method, yet their values can be input into EUT as forcing terms in the body boundary condition. It would be
interesting to see the effect of this “experiment” on the performance on the EUT. It has been our experience that the
m-terms in the body boundary condition contribute much more significant forward speed effects than their
counterpart on the free surface condition, as the authors appear to suggest.
What is the behavior of the EUT near the τ=1/4 singularity in quartering waves. It is my recollection that the unified
theory solution developed in the late 70's and early 80's by Nick and myself was predicting a singularity in the
motion predictions at τ=1/4. We have since seen that this singularity is not really present in SWAN, yet it remains
difficult to resolve. How does EUT deal with this delicate regime when it arises in your computations and
experiments?
AUTHOR'S REPLY
1) Concerning good performance of EUT in the limiting case of U=0, we have shown many supporting results
not only for a single body in open sea but also the tank-wall interference and catamaran problems (see
reference [A1]). From theoretical viewpoint, the unified theory is a perfect one in the framework of slender-
body theory.
2) Regarding the effect of the steady disturbance in the body boundary condition, our recognition seems to be
different from that of discussor. Firstly, our purpose is to develop a practical calculation method which must
be easy to implement but reliable in the design stage. Secondly, the results of the radiation forces (especially
the diagonal coefficients) by the present method are in good agreement with experiments. Recent results for
modified Wigley model with L/B=6.67 are shown in reference [A2]. The degree of agreement by means of
EUT is almost perfect in A33, B33, and A55. The results of High-Speed Slender-Ship Theory (HSSST) are also
shown in reference [A2] and HSSST gives obvious improvement in B55. This tendency is much more
prominent in the cross-coupling terms. HSSST uses also the uniform flow assumption in the body boundary
condition, but the forward-speed effects in the free-surface condition are fully taken into account. In fact, we
have learned from Rational Strip Theory (RST) of Ogilvie & Tuck that the forward-speed effect in the free-
surface boundary condition is important in the cross-coupling terms, which improves clearly over the strip
theory. HSSST encompasses RST, and thus it is natural to see good agreement with experiments. From
experiences mentioned above, we suggest that the forward-speed effects in the free-surface condition may be
more significant than the effects of disturbance potential in the
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PREDICTION OF WAVE PRESSURE AND LOADS ON ACTUAL SHIPS BY THE ENHANCED UNIFED THEORY 384
body boundary condition.
3) As shown in some figures in references [A1] and [A2], EUT shows singularity at the frequency equal to
τ=1/4, and experimental data also show rapid variation near τ=1/4. I suppose the results of SWAN are
imperfect near τ=1/4, and we cannot discuss this sensitive behavior with questionable numerical results.
[A1] M.Kashiwagi (1997); Numerical Seakeeping Calculations Based on the Slender Ship Theory, Ship Technology Research (Schiffstechnik), Vol. 4,
No. 4, pp. 167–192.
[A2] M.Kashiwagi (2000); The State of the Art on Slender-Ship Theories of Seakeeping, Proceedings of the 4th Osaka Colloquium on Seakeeping
Performance of Ships, October, 2000.
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