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

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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: â 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]. â 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 âfully 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) the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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 Âµ between body and incident wave is defined such that Âµ=180Â° denotes head sea and Âµ=90Â° 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 Î±i, the following approximations are valid: sin(Î±i)=tan(Î±i)=Î±i, cos(Î±i)=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=Î¶(0): (6) The last term in (6) is explicitly written: (7) the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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 âshifting' 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 Ï>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, Ï the potential of an element of unit strength including all mirror images. Ï 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 30Â° 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 âcut-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=1â¦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) the authoritative version for attribution. (14)

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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) the authoritative version for attribution. We introduce the abbreviations: (31) (32) (33) (34) Recall that the instationary pressure contribution is: (35)

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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 âzero' line for the other âsection' 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 âsection' 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. the authoritative version for attribution. If we apply this to the second-order time-average longitudinal force on the ship, we obtain: (45) Combining (20) and (28), yields: (46)

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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 â 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 âsection' coordinate s can be approximated to first order in the vicinity of the steady wave profile by a tangential straight line: (53) zâ² is a vertical coordinate with origin at the height of the steady surface pointing downwards. Let nâ² be a modified normal: (54) Let N be the unit normal on the contour in the x-y-plane. Then (55) p(0)(zâ²) We develop in a Taylor series around zâ²=0 (average free surface=steady free surface): (56) zâ² As and z point both downwards, the derivation is interchangable. The âsteady' pressure is zero at the âsteady' free surface. Thus: (57) The p(1) term is simple. A Taylor series gives p(1)(zâ²)âp(1)(0). Then: Similarly the other first-order quantity is simply multiplied by Z in the integration over zâ². 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 the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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 âother' potential to ensure consistently symmetrical (i.e. f(y)=f(ây)) and antisymmetrical (f(y)=âf(ây)) behavior. The second derivatives of the harmonic potentials are neglected in the expression for for simplicity (âdesperation 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 Ï values. For these cases, good agreement with experiments for motions was demonstrated, [2], [3], [8]â[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 the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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 âcut-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 âcut-out' technique was employed and the steady results interpolated from the âsteady' grid to the âunsteady' 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= â0.078Lpp (23.95m behind amidships). The motions agree rather well for both head sea and oblique sea with Âµ=150Â°, 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 Ï<0.4â¦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 (90Â°). 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 Î»/L<1.25 for Âµ=180Â° and Î»/L<1.0 for Âµ=150Â°. These limits correspond for the investigated low Froude number to Ï-values around 0.35â¦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<Î»/L<0.75, measured and computed pressures at the ship bottom agree well. The strip method results for pressures are worse for short waves Î»/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 Ï>0.4. Unless techniques are developed to extend it to smaller Ï-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 Ï-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 the authoritative version for attribution.

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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 âfully three-dimensional' methods. ACKNOWLEDGMENT We are grateful for the support of our younger colleagues Shuji Mizokami and âCowboy' Tanaka in preparation of the results. REFERENCES 1. Bertram, V. and Yasukawa, H., âRankine source methods for seakeeping problems,â Jahrbuch der Schiffbautechnischen Gesellschaft, Springer, 1996, pp. 411â425 2. Bertram, V., âNumerical investigation of steady flow effects in 3-d seakeeping computationsâ, 22. Symp. Naval Hydrodyn., Washington, 1998 3. Bertram, V. and Thiart, G., âA Kutta condition for ship seakeeping computations with a Rankine panel methodâ, Ship Technology Research 45, 1998, pp. 54â63 4. Bertram, V., âFulfilling open-boundary and radiation condition in free-surface problems using Rankine sourcesâ, Ship Technology Research 37, 1990, pp. 47â52 5. Iwashita, H. and Ito, A., âSeakeeping computations of a blunt ship capturing the influence of the steady flowâ, Ship Technology Research 45, 1998, pp. 159â171 6. Jensen, G., âBerechnung der stationÃ¤ren PotentialstrÃ¶mung um ein Schiff unter BerÃ¼cksichtigung der nichtlinearen Randbedingung an der freien WasseroberflÃ¤cheâ, IfS Report 484, Univ. Hamburg, 1988 7. Nakatake, K. and Ando, J., âRankine source method using rectangular panels on water surfaceâ, 11. Workshop Water Waves and Floating Bodies, Hamburg, 1996 8. Bertram, V., âVergleich verschiedener 3D-Verfahren zur Berechnung des Seeverhaltens von Schiffenâ, Jahrbuch Schiffbautechnische Gesellschaft, Springer, 1997, pp. 594â600 9. Bertram, V. and Thiart, G., âA Rankine panel method for ships in oblique wavesâ, Euromech 374, Poitiers, 1998, pp. 221â229 10. Bertram, V. and Thiart, G., âFully three-dimensional ship seakeeping computations with a surge-corrected Rankine panel methodâ, J. Marine Science and Technology, 1998, pp. 94â101 11. Bertram, V. and Thiart, G., âFully 3-d seakeeping computations for real ship geometriesâ, Jahrbuch Schiffbautechnische Gesellschaft, Springer, 1998, pp. 244â249 12. Tanizawa, K., Taguchi, H., Saruta, T. and Watanabe, I., âExperimental study of wave pressure on VLCC running in short wavesâ, J. Soc. Nav. Arch. Japan 174, 1993, pp. 233â242 13. Mizoguchi, S., âExciting forces on a high speed container ship in regular oblique wavesâFrequency selections for calculating exciting forces by the strip methodââ, J. Kansai Soc. Nav. Arch. Japan 187, 1982, pp. 71â83 the authoritative version for attribution.

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Fig. 2: Like Fig. 1, but for Âµ=150Â° Fig. 1: Motions for VLCC, Fn=0.131, Âµ=180Â°; â¢ experiment, â RSM, Â· strip method INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 364

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Fig. 4: Like Fig. 3, but for Âµ=150Â° Fig. 3: VLCC, Fn=0.131, Âµ=180Â°; unsteady pressure angle; 90Â°=bottom, 0Â° starboard CWL; â¢ exp., â RSM, Â· strip method INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD at x=â0.078Lpp plotted over circumference 365

About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. Fig. 6: As Fig. 5, but for Fn=0.3 Fig. 5: RAOs for motions and added resistance for S175, Fn=0.25, Âµ=180Â°, â¢ exp., â RSM INVESTIGATION OF GLOBAL AND LOCAL FLOW DETAILS BY A FULLY THREE-DIMENSIONAL SEAKEEPING METHOD 366

lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the original typesetting files. Page breaks are true to the original; line 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 âcut-out' technique is used for a tanker and possibly S-175 container ship as well. Why do you think the âcut-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 âcut-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 âcut-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.