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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 355
Investigation of Global and Local Flow Details by a Fully Three-
dimensional Seakeeping Method
V.Bertram (HSVA, Germany)
H.Yasukawa (Mitsubishi Heavy Industries, Japan)
ABSTRACT
A fully-three-dimensional Rankine panel method in the frequency domain is validated for local pressures, motions,
and added resistance. Previous formulae for added resistance contained errors resulting in large differences to
experiments. This has now been remedied. The method is linearized with respect to wave height. The steady flow
contribution is captured completely by solving the fully nonlinear wave-resistance problem first and linearizing the
seakeeping problem around this solution. The same grids on the hull are taken for both steady and seakeeping
computation. On the free surface different grids are used, either following quasi-streamlined grids or rectangular grids
with cut-outs for the hull. The results from the steady solution are interpolated on the new free-surface grid. The method
is applied to various test cases. Motions are in good agreement with experiments, but this is also the case for strip method
results. Local pressures, especially for shorter waves, are much better predicted than by strip method. The added
resistance is sensitive to higher derivatives of the potential and a numerical differentiation of these terms may be
preferable to using higher-order panels.
1. INTRODUCTION
The most commonly used tools to determine seakeeping properties are based on strip theory. The strip method
approach is cheap, fast, and for most cases also quite accurate. However, strip methods do not perform so well for high-
speed ships, full hullforms (tankers), ships with strong flare, and generally for low encounter frequencies which typically
occur in following seas. They are also questionable with regard to local pressures which are needed as input for finite-
element analyses.
Approaches to improve predictions of seakeeping properties should capture:
– 3-D effects of the flow
3-D effects are important for low encounter frequencies and full hull forms. 3-D diffraction at the bow region
of tankers contributes considerably to added resistance, [1].
– Forward-speed effects
Strip methods include forward speed by the change in encounter frequency. But forward speed enters the ship
motion problem in additional ways: the local steady flow field, the steady wave pattern of the ship, and the
change of the hull form and wetted surface due to squat (dynamic sinkage and trim).
We will present here a 3-d Rankine singularity method (RSM) which captures all forward-speed effects. The method
is ‘fully three-dimensional', i.e. both steady and unsteady flow contributions are captured three-dimensionally. For a
recent survey of Rankine singularity methods for forward-speed seakeeping, we refer to [1], [2].
2. THEORY
2.1. Physical model
We consider a ship moving with mean speed U in a harmonic wave of small amplitude h. We assume an ideal flow.
Then the fundamental field equation is Laplace's equation. In addition, boundary conditions are postulated:
1. No water flows through the ship's surface.
2. At the trailing edge of the ship, the pressures are equal on both sides. (Kutta condition)
3. No water flows through the free surface. (Kinematic free-surface condition)
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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 356
4. There is atmospheric pressure at the free surface. (Dynamic free-surface condition)
5. Far away from the ship, the disturbance caused by the ship vanishes.
6. Waves created by the ship move away from the ship. For certain combinations of frequency of incident wave
and speed of the ship, waves created by the ship propagate only downstream. (Radiation condition)
7. Waves created by the ship should leave artificial boundaries of the computational domain without reflection.
They may not reach the ship again. (Open-boundary condition)
8. Forces on the ship result in motions. (Average longitudinal forces are assumed to be counteracted by
corresponding propulsive forces, i.e. the average speed U remains constant.)
2.2. Mathematical model
All coordinate systems here are right-handed Cartesian systems. The inertial Oxyz system moves uniformly with
velocity U. x points in the direction of the body's mean velocity U, z points vertically downward. The Oxyz system is
fixed at the body and follows its motions. When the body is at rest position, x, y, z coincide with x, y, z. The angle of
encounter µ between body and incident wave is defined such that µ=180° denotes head sea and µ=90° beam sea.
The body has 6 degrees of freedom for rigid body motion. We denote corresponding to the degrees of freedom:
u1 surge motion of O in x-direction, relative to O
u2 sway motion of O in y-direction, relative to O
u3 heave motion of O in z-direction, relative to O
u4 angle of roll=angle of rotation around x-axis
u5 angle of pitch=angle of rotation around y-axis
u6 angle of yaw=angle of rotation around z-axis
The motion vector is and the rotational motion vector are given by:
(1)
(2)
All motions are assumed to be small of order O(h). Then for the 3 angles αi, the following approximations are valid:
sin(αi)=tan(αi)=αi, cos(αi)=1. The theory has been described rather extensively by [2]. In the following, we will therefore
only briefly review the theory except in cases where changes to [2] justify a more detailed discussion.
We decompose potential and free surface elevation into steady and time-harmonic parts:
(3)
(4)
The superposition principle can be used within a linearised theory. Therefore the radiation problems for all 6 degrees
of freedom of the rigid-body motions and the diffraction problem are solved separately. The total solution is a linear
combination of the solutions for each independent problem.
The harmonic potential is divided into the potential of the incident wave the diffraction potential and 6
radiation potentials. It is convenient to divide and into symmetrical and antisymmetrical parts to
take advantage of the (usual) geometrical symmetry:
(5)
The conditions satisfied by the steady flow potential are:
The particle acceleration in the steady flow is:
We define an acceleration vector
For convenience we introduce an abbreviation:
At the steady free surface:
On the body surface:
The combined, linearized free-surface condition is at z=ζ(0):
(6)
The last term in (6) is explicitly written:
(7)
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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 357
With the abbreviation the boundary condition at is:
(8)
The Kutta condition requires that at the trailing edge the pressures are equal on both sides. For monohulls, this is
automatically fulfilled on the centreplane for the symmetric contributions. Then only the antisymmetric pressures have to
vanish, compare (17):
(9)
This yields on points at the trailing edge:
(10)
For catamarans, the Kutta condition requires for both symmetric and antisymmetric contributions, that the pressures
on both sides of the trailing edge are the same. This is enforced by selecting pairs of collocation points at the trailing edge
and matching the pressures.
The 2 unknown diffraction potentials and the 6 unknown radiation potentials are determined by approximating the
unknown potentials by a superposition of a finite number of Rankine higher-order panels on the ship and above the free
surface. For the antisymmetric cases, in addition Thiart elements (semi-infinite dipole strips on the plane y=0), [2], [3],
are arranged and a Kutta condition is imposed on collocation points at the last column of collocation points on the stern.
The l.h.s. of the four systems of equations for the symmetrical cases and the l.h.s. for the four systems of equations for the
antisymmetrical cases share the same coefficients each. Thus four systems of equations can be solved simultaneously
using Gauss elimination.
Radiation and open-boundary conditions are fulfilled by the ‘shifting' technique (adding one row of collocation
points at the upstream end of the free-surface grid and one row of source elements at the downstream end of the free-
surface grid), [4]. This technique works only well for τ>0.4, as also demonstrated by [5]. Elements use mirror images at
y=0. For the symmetrical cases, all mirror images have same strength. For the antisymmetrical case, the mirror images on
the negative y-sector have negative element strength of same absolute magnitude.
Each unknown potential is then written as:
(11)
mi is the strength of the ith element, φ the potential of an element of unit strength including all mirror images. φ is
real for the Rankine elements and complex for the Thiart elements.
The same grid on the hull is used as for the steady problem. The grid on the free surface is created new. The
quantities on the new grid are linearly interpolated within the new grid from the values on the old grid. Outside the old
grid in the far field, all quantities are set to uniform flow on the new grid. The interpolation of results introduces only
small differences as observed in various test cases.
Structured grids on the free surface are generated by one of the following techniques:
1. The longitudinal grid lines follow quasi streamlines around the hull. The transverse grid lines are
equidistantly spaced on lines y=const. A maximum entrance angle of 30° is kept which results in zones not
covered by the grid near the bow and stern of blunt ships.
2. A rectangular grid is created consisting of lines x=const. and y=const. Panels within the waterline are deleted.
The first technique is well suited for slender ships, the second technique better for blunt ships. The second technique,
called ‘cut-out' technique was proposed for the steady wave-resistance problem by [6] and [7]. Both grid generation
options are available for both steady and seakeeping free-surface grid generation. While it is in principle possible to
switch from one grid type to the other between steady and seakeeping computations, it is recommended to use
consistently cut-out grids for full hulls like tankers and the standard grid option (quasi streamlines) for slender ships.
After the potential (i=1…8) have been determined, only the motions ui remain as unknowns. The forces and
moments acting on the body result from the body's weight and from integrating the pressure over the instantaneous
wetted surface S. The body's weight is:
(12)
m is the body's mass. and are expressed in the inertial system ( is the inward unit normal vector):
(13)
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(14)
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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 358
is the center of gravity. The pressure is given by Bernoulli's equation:
(15)
(16)
(17)
Eqs. (13) and (14) yield:
(18)
(19)
The ship is in equilibrium for steady flow. Therefore the steady forces and moments are all zero. The first-order parts
give (r.h.s. quantities are now all functions of):
(20)
(21)
where and Eqs. (18) and (19) (steady equilibrium) have
been used. Note: The difference between instantaneous wetted surface and average wetted surface still
has not to be considered as the steady pressure p(0) is small in the region of difference.
The instationary pressure is divided into parts due to the incident wave, radiation and diffraction:
(22)
Again the incident wave and diffraction contributions can be decomposed into symmetrical and antisymmetrical parts:
(23)
(24)
Using the unit motion potentials and the pressure equation (17) the pressure parts pi are derived:
(25)
The individual terms in the integrals (20) and (21) are expressed in terms of the motions ui, using the vector identity
(26)
(27)
The relation between forces, moments and motion acceleration is:
(28)
(29)
Mass distribution symmetrical in y is assumed. etc. are the moments of inertia and the centrifugal moments with
respect to the origin of the body-fixed Oxyz-system:
(30)
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We introduce the abbreviations:
(31)
(32)
(33)
(34)
Recall that the instationary pressure contribution is:
(35)
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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 359
(36)
Then we can rewrite (26), (27) and (28):
(37)
The weight terms and contribute with W=mg:
(38)
The mass terms and contribute:
(39)
with M being
(40)
where the radii of inertia have been introduced: etc.
Combining Eqs. (36) and (37) yields a linear system of 6 equations in the unknown ui that is quickly solved using
Gauss elimination.
2.3. Added resistance
Following a similar approach as for the first-order forces, a formula for the added resistance can be derived that uses
only quantities computed so far. The added resistance is the negative time-averaged value of the x-component of the
second-order force. If t1 and t2 are time-harmonic quantities, the time-average of t1t2 is where is the
conjugate complex of
(41)
The force in x-direction is given by:
The integral over the wetted surface can be expressed as a double integral over a body-fitted curvilinear coordinate
system. One coordinate follows rather longitudinal lines from stern to bow, the other coordinate follows the hull contour
from the free surface down to the keel. One of the longitudinal coordinate lines follows the contour of the steady wave
profile and this is the ‘zero' line for the other ‘section' coordinate. This modified waterline contour C accounts also for
steady trim and sinkage and differs usually the still waterline contour. The contour line C splits at the stern and both sides
run from stern to bow. The ‘section' coordinate runs from the actual free surface Z to the keel K. Then we can re-write any
integral over the wetted surface as:
(42)
(43)
(44)
We can thus split the integration into one integral over the average wetted surface S(0) and a correction double integral.
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If we apply this to the second-order time-average longitudinal force on the ship, we obtain:
(45)
Combining (20) and (28), yields:
(46)
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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 360
Thus:
(47)
The second-order pressure p(2) is:
(48)
The term containing the time-derivative of the second-order potential vanishes in the time-average:
(49)
The term ∇ p(1) involves again second derivatives of the potential on the hull:
(50)
Z is the first-order difference between average (steady) and instantaneous wave profile on the hull:
(51)
with:
(52)
The curvilinear ‘section' coordinate s can be approximated to first order in the vicinity of the steady wave profile by
a tangential straight line:
(53)
z′ is a vertical coordinate with origin at the height of the steady surface pointing downwards.
Let n′ be a modified normal:
(54)
Let N be the unit normal on the contour in the x-y-plane. Then
(55)
p(0)(z′)
We develop in a Taylor series around z′=0 (average free surface=steady free surface):
(56)
z′
As and z point both downwards, the derivation is interchangable. The ‘steady' pressure is zero at the ‘steady' free
surface. Thus:
(57)
The p(1) term is simple. A Taylor series gives p(1)(z′)≈p(1)(0). Then:
Similarly the other first-order quantity is simply multiplied by Z in the integration over z′.
Thus eventually we get for the time-averaged second-order longitudinal force:
(58)
The added resistance is:
(59)
The integrals in the Ti are evaluated numerically over the starboard half only and multiplied by 2 for symmetrical/
symmetrical and antisymmetrical/antisymmetrical pressure-normal combinations only. (Antisymmetrical/symmetrical
combinations yield zero contributions.) The decomposition into
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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 361
symmetrical and antisymmetrical parts complicates evaluating the added resistance. The individual terms are decomposed
into symmetrical and antisymmetrical parts:
(60)
(61)
(62)
(63)
(65)
(64)
Note that the second component of each of these vectors contains y-derivatives of the ‘other' potential to ensure
consistently symmetrical (i.e. f(y)=f(−y)) and antisymmetrical (f(y)=−f(−y)) behavior. The second derivatives of the
harmonic potentials are neglected in the expression for for simplicity (‘desperation rather than physical insight').
We introduce the abbreviation
(66)
(67)
(68)
with
(69)
(70)
Retaining only symmetrical terms in T1 and T2 yields:
(71)
Due to symmetry, the above integrals are twice the value of the integrals over the starboard half only.
3. APPLICATIONS
3.1. Local pressures
So far, applications of the present RSM were shown only for relatively high Froude numbers, which for most angles
of encounters and wave lengths of interest result in sufficiently high τ values. For these cases, good agreement with
experiments for motions was demonstrated, [2], [3], [8]–[11]. Numerical studies showed that the influence of the steady
flow on the results for motions is significant, for moderate wave lengths, but negligible for short and long waves. This
was explained by purely numerical investigations of local pressures. A research cooperation allowed now to investigate
local pressures for a VLCC, Table I, at Fn=0.131. The exact geometry of the test case is confidential. The pressures are
the amplitudes of the pressure fluctuation, i.e. pressures without hydrostatic and steady hydrodynamic pressures.
The tanker was discretised using 495 ele
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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 362
ments. First the steady fully nonlinear wave resistance problem was solved. The grid on the free-surface was generated
using the ‘cut-out' technique. This technique generates a structured grid consisting of rectangular elements. Elements
which are partially or totally inside the hull are then eliminated and then the shifting technique is applied. This technique
is known to give better results for full hulls than streamlining a grid around the hull. The fully nonlinear method used 3
iterations which reduced the error at the free surface by 4 orders of magnitude.
The same grid for the hull was employed for the seakeeping computations (at the dynamic trim and sinkage).
Pressure integrations considered only the area submerged in the steady case. The free surface in the seakeeping
computations was discretised with typically 1400 elements. Again the ‘cut-out' technique was employed and the steady
results interpolated from the ‘steady' grid to the ‘unsteady' grid. Test computations for two wave lengths with free-surface
grids involving approximately 4200 elements yielded results that were only 5% different. This may be interpreted as that
the coarser discretisation is sufficient.
The computational results are compared to measurements of [12] and MHI strip method results. The strip method is
based on standard STF method with Lewis section representation, but includes an empirical correction [13]. The results
include motions and pressures on the hull at a location x= −0.078Lpp (23.95m behind amidships). The motions agree
rather well for both head sea and oblique sea with µ=150°, Figs. 1 and 2. However, strip method also predicts heave and
pitch motions well. In fact, for long waves strip methods gives better results than the RSM. This is not surprising. Strip
methods are known to predict heave and pitch motions well for usual ships and ship speeds. The present RSM uses the
shifting technique which deteriorates in performance for τ<0.4…0.5. Sway and yaw are also well predicted, the maximum
of the roll motion is underpredicted. This may be due to the deterioration of the shifting technique, as for a fast
containership with Fn=0.275 Bertram [2] obtained significant overprediction for roll resonance as expected for a method
that does not include empirical corrections for nonlinear roll damping.
Figs. 3 and 4 compare pressures. Starboard is the weather side. For head waves the computed pressures are of course
symmetrical to the midship plane (90°). One point on the port side was then plotted on its corresponding position on the
starboard side. Pressures computed by the RSM agree well with measured pressures for λ/L<1.25 for µ=180° and λ/L<1.0
for µ=150°. These limits correspond for the investigated low Froude number to τ-values around 0.35…0.4. For short
waves, the computations underpredict the pressures at the bottom of the ship compared to measurements. However, as the
pressures should decay exponentially with depth like all wave effects, for short waves the near-zero values of the
computation appear to be more plausible and we assume that they reflect in this case reality better than the measured
values. For waves of moderate length 0.5<λ/L<0.75, measured and computed pressures at the ship bottom agree well. The
strip method results for pressures are worse for short waves λ/L=0.2, 0.3 where diffraction effects are stronger than
radiation effects.
In summary, the RSM predicted pressures and motions well, the strip method predicted pressures in short waves
badly, but motions well. The RSM is currently limited in practice to approximately τ>0.4. Unless techniques are
developed to extend it to smaller τ-Values, the RSM will remain a research tool of limited functionality. We see hybrid
methods matching an inner RSM solution to an outer Green function method or Fourier-Kochin solution as most
promising approach to extend the method to low τ-values, but at present no such research is planned due to lack of funds.
Table I: Test case VLCC
Lpp 307.00 m zg 4.333 m
B 54.00 m kx 19.193 m
T 19.50 m ky 73.987 m
CB 0.813 kz 76.750 m
KG 15.17 m kxz 0m
xg 10.045 m
3.2. Added resistance
We show here applications to the ITTC standard test case S-175 containership in head seas. Computations are
compared to experiments of Mitsubishi Heavy Industries. Fig. 5 shows results for Fn=0.25 and Fig. 6 for Fn=0.3. For all
three motions the agreement is good. In fact, the computational results for heave for long waves appear to be more
plausible, as they tend as expected monotonously to 1. The discrepancies between experiments and computations for the
higher Froude number Fn=0.3 are most likely due to nonlinear damping effects in the experiments. For Fn=0.275, [2]
showed that the phase information is also correctly captured, at least
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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 363
for wave lengths where the RAOs are not approximately zero.
The added resistance is similarly well captured. The differences for the higher Froude number are explained by the
differences in capturing the motions. The derived formula seems to be debugged now and can be recommended for other
‘fully three-dimensional' methods.
ACKNOWLEDGMENT
We are grateful for the support of our younger colleagues Shuji Mizokami and ‘Cowboy' Tanaka in preparation of
the results.
REFERENCES
1. Bertram, V. and Yasukawa, H., “Rankine source methods for seakeeping problems,” Jahrbuch der Schiffbautechnischen Gesellschaft, Springer, 1996,
pp. 411–425
2. Bertram, V., “Numerical investigation of steady flow effects in 3-d seakeeping computations”, 22. Symp. Naval Hydrodyn., Washington, 1998
3. Bertram, V. and Thiart, G., “A Kutta condition for ship seakeeping computations with a Rankine panel method”, Ship Technology Research 45, 1998,
pp. 54–63
4. Bertram, V., “Fulfilling open-boundary and radiation condition in free-surface problems using Rankine sources”, Ship Technology Research 37,
1990, pp. 47–52
5. Iwashita, H. and Ito, A., “Seakeeping computations of a blunt ship capturing the influence of the steady flow”, Ship Technology Research 45, 1998,
pp. 159–171
6. Jensen, G., “Berechnung der stationären Potentialströmung um ein Schiff unter Berücksichtigung der nichtlinearen Randbedingung an der freien
Wasseroberfläche”, IfS Report 484, Univ. Hamburg, 1988
7. Nakatake, K. and Ando, J., “Rankine source method using rectangular panels on water surface”, 11. Workshop Water Waves and Floating Bodies,
Hamburg, 1996
8. Bertram, V., “Vergleich verschiedener 3D-Verfahren zur Berechnung des Seeverhaltens von Schiffen”, Jahrbuch Schiffbautechnische Gesellschaft,
Springer, 1997, pp. 594–600
9. Bertram, V. and Thiart, G., “A Rankine panel method for ships in oblique waves”, Euromech 374, Poitiers, 1998, pp. 221–229
10. Bertram, V. and Thiart, G., “Fully three-dimensional ship seakeeping computations with a surge-corrected Rankine panel method”, J. Marine
Science and Technology, 1998, pp. 94–101
11. Bertram, V. and Thiart, G., “Fully 3-d seakeeping computations for real ship geometries”, Jahrbuch Schiffbautechnische Gesellschaft, Springer,
1998, pp. 244–249
12. Tanizawa, K., Taguchi, H., Saruta, T. and Watanabe, I., “Experimental study of wave pressure on VLCC running in short waves”, J. Soc. Nav. Arch.
Japan 174, 1993, pp. 233–242
13. Mizoguchi, S., “Exciting forces on a high speed container ship in regular oblique waves—Frequency selections for calculating exciting forces by the
strip method—”, J. Kansai Soc. Nav. Arch. Japan 187, 1982, pp. 71–83
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Fig. 2: Like Fig. 1, but for µ=150°
Fig. 1: Motions for VLCC, Fn=0.131, µ=180°; • experiment, ○ RSM, · strip method
INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD
364
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Fig. 4: Like Fig. 3, but for µ=150°
Fig. 3: VLCC, Fn=0.131, µ=180°; unsteady pressure
angle; 90°=bottom, 0° starboard CWL; • exp., ○ RSM, · strip method
INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD
at x=−0.078Lpp plotted over circumference
365
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Fig. 6: As Fig. 5, but for Fn=0.3
Fig. 5: RAOs for motions and added resistance for S175, Fn=0.25, µ=180°, • exp., ○ RSM
INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD
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INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 367
DISCUSSION
H.Chun
Pusan National University, Korea
1) I think that you solved iteratively the nonlinear free surface boundary condition. In that case, how did you treat the
coefficients (such as added mass, damping, restoring force) of the motion equation?
2) You mentioned that the panel shift method can not meet the radiation condition for some cases. Do you have any
alternative idea to satisfy the radiation condition for such cases?
AUTHORS' REPLY
1) We treated the linearized (unsteady) ship motion problem based on fully nonlinear steady flow[2]. Then, since the
ship motion problem is linearized, we can define all coefficients such as added mass, damping, restoring force and
exciting forces. Of course, the effect of steady wave elevation and flow on the coefficients is included in the computations.
2) We see so-called hybrid methods matching an inner RSM solution to an outer Green function method or Fourier-
Kochin solution as most promising approach to extend the method to low values.
DISCUSSION
M.Kashiwagi
Kyushu University, Japan
Firstly I wish to commend you for completing complicated calculations of the added resistance with the pressure
integration method. I have a couple of questions concerning the grids. I understand that what you call ‘cut-out' technique
is used for a tanker and possibly S-175 container ship as well. Why do you think the ‘cut-out' technique goes well? If you
use the alternative quasi-streamline grid, how is the result going to be? Does the computation break down or are obtained
results much different from experiments? Are there some criteria of which grid should be used for given values of the
block coefficient and the Froude number?
AUTHORS' REPLY
We used ‘cut-out' method for the computation of VLCC. The reason is that quasi-streamlined grids either give very
distorted grids near the bow and stern or do not locate collocation points on the water surface near the ship ends. Distorted
cells lead to problems with the radiation condition. The ‘cut-out' method is robuster in this respect for full waterline
forms. We recommend the cut-out method for ships with block-coefficients above 0.7.
DISCUSSION
M.Ohkusu
Kyushu University, Japan
Apparently, a pressure transducer is located on the water line in your experiment. This transducer is naturally out of
water some duration during one period of the motion. Then the time history of the pressure measured will be of not a
sinusoidal but a truncated sinusoidal curve.
So I wonder how you treated with the truncated curve to derive your value of pressure. If you take the first harmonic
components of this curve, you will obtain much smaller amplitude of the pressure. Nevertheless, your pressure at the
water line looks consistent with the pressure at other locations.
AUTHORS' REPLY
As you pointed out, we observed the time history of a truncated sinusoidal curve in the pressure measurement in the
vicinity of the water line. From the time history data, amplitude was defined as variation between zero-level and the
positive peak value[12]. So the experimental data plotted in Figs. 3 and 4 is not the first harmonic components of the
curve. In the computations, we do not take the truncated effect into account. The reason why the calculated accuracy is
insufficient in the vicinity of water line may be due to treatment of the truncated effect.
the authoritative version for attribution.
Representative terms from entire chapter:
added resistance
