MaTS
MARIEN TECHNOLOGISCH SPEURWERK
Netherlands Marine Technological Research IA COMPARISON BETWEEN TWO-DIMENSIONAL
AND THREE-DIMENSIONAL CALCULATIONS FOR THE BEHAVIOUR OF TRANSPORT BARGES
AND A STUDY ON THE FORWARD SPEED EFFECT
VM-II-3 August 1981
L i
P1981-2
Industrièle Raad
voor de Oceanologie
Netherlands Industrial Council for Oceanology
-A COMP-ARISON BETWEEN TWO-DIMENSION-AL -AND
THREE-DIMENSIONAL CALCULATIONS FOR THE BEHAVIOUR OF TRANSPORT BARGES AND A STUDY ON THE FORWARD SPEED EFFECT
N.S.M.B. Order No. Z 02586
Ordered by: Netherlands Industrial Council for Oceanology (I.R.0.)
P.O. Box 215 2600 AE DELFT The Netherlands
Reported by: A.B. Aalbers
MANAGEMENT SUMMARY OF 'MOTIONS OF TRANSPORT BARGES'
When preparing an ocean tow of a loaded cargo barge, an estimate of the barge motions is needed in order to assure the safety of the cargo.
Possible aspects that can be assessed are the loads on the seafastenings, the accelerations to which the cargo is subjected and the likelihood that overhanging parts of the cargo may hit the water surface.
Current computational methods for the determination of barge motions which are at the disposal of engineers are based on the strip theory and the
three-dimensional diffraction theory. In strip theory calculations the wetted surface of a floating body is schematized by means of a number of
strips. In the case of the three-dimensional diffraction theory the hull surface is represented by a number of plane elements. The latter theory is applicable to floating objects of arbitrary shape, while the applicabi-lity of the strip method is restricted to slender bodies, such as ships. Since typical towed barges are not slender, the applicability of the strip method is not at all obVious. Advantage of the strip method is the lower computer cost of the calculations when compared to the three-dimensional diffraction method.
Both methods are based on the linear potential theory and compute the motion amplitudes for a regular sinusoidal wave of given frequency. By repeating the calculations for various frequencies and applying spectral techniques the motions in a random sea can be estimated.
As partof the Marine Technological Research Program (MaTS) a study has been carried out by the Netherlands Ship Model Basin concerning the behaviour of towed transport barges in waves. The study was conducted in the following two phases:
- In phase I the diffraction theory was used to calculate the hydrodynamic
coefficients and motions of a barge with dimensions of length x breadth x draught - 150 x 150 x 10m. The calculations were performed using both a coarse and a fine distribution of elements and the results were compared to available model test data.
In phase I it was found that in general good agreement existed between the results calculated using diffraction theory and the modeltest data. Increasing the number of surface elements used to represent the barge
did not in general lead to a significant change in the analysed coefficients and motions.
In phase II it was found that, although some significant differences
existed in the hydrodynamic forces, in general strip theory cannot be rejected as being inadequate or inapplicable.
The comparison between the two theoretical methods strongly suggests that the application and the manner of application of either one should be left to the experienced specialists who is aware of the limitations in their validity and can assess their appropriateness for each specific application.
Finally it should be emphasized that both these theoretical methods are based on linear potential theory which neglects the influence of viscosity. Since the roll motion of a barge close to its natural frequency is strong-ly influenced by non-linear viscous forces, neither method can accuratestrong-ly predict this motion.
The complete results of the study are given in the following two reports: phase I : NSMB report No. 02586-I-ZT 'The effect of element size in
diffraction theory calculations for transport barges', July 1980
phase II : NSMB report No. 02586-2-ZT: 'A comparison between
two-dimen-sional and three-dimentwo-dimen-sional calculations for the behaviour of transport barges and a study on the forward speed effect', August 1981.
CONTENTS
Page
INTRODUCTION 1
1.1. General 1
1.2. The computer programs 2
DESCRIPTION OF THE BARGE AND REVIEW OF THE
CALCU-LATIONS 5
2.1. The barge 5
2.2. Schematization of the barge geometry 5
2.3. Review of the calculations 6
2.4. Definitions and presentation of results 7
DESCRIPTION OF THE STRIP THEORY AND DIFFRACTION
THEORY AND A COMPARISON OF THEIR RESULTS 9
3.1. Short description of the programs, including the
basic assumptions 9
3.1.1. General 9
3.1.2. Three-dimensional diffraction theory: program
DIFFRAC 10
3.1.3. The two-dimensional strip theory: program SHIPMO 13
3.1.4. Review of assumptions and simpZifications 15
3.2. Comparison of the results of strip theory and
dif-fraction theory calculations 17
3.2.1. General 17
3.2.2. The wave exciting forces and moments 17
3.2.3. The hydrodynamic coefficients 20
3.2.4. The barge motions 23
3.2.5. Summary 26
DISCUSSION ON THE INFLUENCE OF THE BASIC ASSUMP-TIONS AND APPROXIMAASSUMP-TIONS ON THE ACCURACY OF THE
CALCULATIONS 29
4.1. The accuracy in the calculation of the wave loads 29
4.1.1. General 29
4.1.2. The total wave loads on the barge 29
4.1.3. Discussion on the longitudinal force distribution 31
NETHERLANDS SHIP MODEL BASIN PAGE
CONTENTS (continued)
Page 4.2. The accuracy in the calculated hydrodynamic
coefficients 35
4.2.1. General 35
4.2.2. Differences in the total hydrodynamic coefficients 36
4.2.3. The longitudinal distribution of the hydrodynamic
coefficients 38
4.3. Summary 44
THE FORWARD SPEED EFFECT 46
5.1. General 46
5.2. The wave exciting forces and moments 49
5.3. The motions 51
5.4. Conclusive remarks as to the influence of forward
speed 54
CONCLUSIONS 56
LIST OF REFERENCES 58
LIST OF SYMBOLS 59
APPENDIX I : DESCRIPTION OF THE DIFFRACTION THEORY 61
APPENDIX II: SHIP MOTIONS CALCULATED BY STRIP THEORY .... 69
Tables ( 2)
1. INTRODUCTION
1.1. General
In April 1979 the Netherlands Industrial Council for Oceanology commissioned the Netherlands Ship Model Basin to carry out a study concerning the behaviour of transport barges under the influence of waves. The purpose of this study is to gain in-sight in physical phenomena governing the behaviour of this type of vessel and the applicability of existing methods for computing the behaviour. In this study, the behaviour of the barge encompasses not only the motions under the influence of waves, but also the wave loads which induce motions and the hydrodynamic reaction forces to motions, known as added mass
and damping.
This report concerns the second of the three phases in which the study has been subdivided. A brief summary of the three phases is given below:
Phase I : In this part of the study results of computations,
based on three-dimensional linear potential theory concerning a barge at zero forward speed in regular waves, were compared to establish what degree of
refinement is necessary to give satisfactory results. The results were also compared with existing experi-mental results to establish confidence in the
compu-tation method.
Phase II : This part of the study will be mainly devoted to a comparison of results obtained using computation methods based on three-dimensional linear potential theory on one hand and "strip" theory methods which
approximate the three-dimensional case using two-dimensional linear potential theory results on the
NETHERLANDS SHIP MODEL BASIN PAGE
other hand. The purpose of this part of the study is to increase insight in the applicability of strip theory methods for the determination of the behaviour of barges at zero speed. The results of three-dimen-sional potential theory computations will be consid-ered "true" results.
In this phase the influence of forward speed on the behaviour of a barge will be determined, using re-sults of computations based on strip theory. The
three-dimensional potential theory computation method cannot be applied for this case, since at present -the influence of forward speed is not included in -the
computer program used for this study.
Phase III: This part of the study will be mainly focussed on the influence of effects which are not accounted for in computation methods based on linear potential theory. This phase will be devoted to the effects of barge motions amplitudes, viscous effects and any non-lin-ear effects which may be considered to have a
signif-icant influence on the behaviour of barges. The ap-plicability of the principle of linear superposition will also be considered.
More details concerning Phase I are given in Report No. 02586-1-ZT (ref. [1]). Phase III will be the subject of a future report.
1.2. The computer programs
The two computer programs, available at N.S.M.B. for calculating the motions of floating structures, which are considered in this report are:
SHIPMO, based on the strip theory;
DIFFRAC, based on three-dimensional linear potential theory.
The so-called strip theory is a method to solve the three-dimen-sional problem of the motions in waves of a floating body, by reducing it to a two-dimensional problem, which is much simpler to solve. The basis of the method is that the mean wetted sur-face of a ship .(or other floating structure) is divided into a number of strips as shown in the figure below:
Each cross-section of the ship is considered to be part of an infinitely long cylinder. Each two-dimensional problem so con-structed is solved separately and after that the solutions are combined to yield a solution for the ship as a whole. A condi-tion for the applicability of the strip method is, that the
floating body must be slender, which means that any change in a longitudinal direction is small with respect to changes in a transverse or vertical direction. In the strip theory, the effect of a forward speed of the floating body can be taken into account in an approximative way. SHIPMO has been applied successfully for ships in waves in the frequency range of prac-tical interest.
NETHERLANDS SHIP MODEL BASIN PAGE
The computer program DIFFRAC is based on the linear three-dimen-sional potential theory. For the computation of the velocity potential, the mean wetted part of the hull of the vessel is approximated by a number of plane elements, representing a dis-tribution of source singularities each of which contributes to the velocity potential describing the fluid flow. The computer program is applicable for vessels of arbitrary shape at zero forward speed. The water depth is restricted to approximately one wave length of the wave corresponding to the frequency under consideration. DIFFRAC has been applied successfully for various offshore structures and ships in shallow water in the range of wave frequencies of practical interest.
2. DESCRIPTION OF THE BARGE AND REVIEW OF THE CALCULATIONS
2.1. The barge
The calculations were carried out for a typical offshore barge with a length, beam and depth of approximately 90 m, 27.5 m and 6.0 m respectively. The draft was 2.75 m and the displace-ment 6430 m3. The exact dimensions are given in Table 1, in which the characteristic quantities for the weight distribution are also given. The bow of the barge intersects the water sur-face at an angle of 45 degrees, which differs somewhat from the
30 degree angle which is found for the "standard" barge. The main reason for adopting such a bow angle was to attune the DIFFRAC and SHIPMO schematizations (see also Section 2.2.).
2.2. Schematization of the barge geometry
The schematization of the wetted surface of the barge for the computations with the program DIFFRAC is given in Figure 1.
The results of the first phase of the project [1] showed that it was advisable to have relatively small elements in the vi-cinity of the bilges of the barge. Therefore, an element
dis-tribution was selected in which the dimension of the elements was 2.711 lc 2.743 m2 around the bilges. In the middle part ofthe bottom the elements were somewhat larger: 5.422 * 5.486 m2 which, however, is still small compared with the wave length at
the highest frequency in the calculations. The total number of elements was 287.
The schematization of the barge hull for the strip theory cal-culations is given in Figure 2. These calcal-culations were carried out with 17 sections, of which 16 were identical (27.43 m wide
NETHERLANDS SHIP MODEL BASIN PAGE
and 2.743 m deep) while the shape of the bow of the barge was simulated by using a section having the full width but a depth of only 0.001 m.
In SHIPMO, the velocity potential for the sectional shape of interest is found by conformal transformation of Ursell's exact solution for a circular cylinder. In the present case a close-fit conformal mapping procedure was used. As shown in Figure 3, a transformation polynomial containing ten coefficients proved to give an optimal representation of the actual sectional shape of the barge.
The schematizations of the hull shape for the two computer pro-grams are attuned to each other in such a way, that in the three-dimensional calculations the barge was subdivided into several parts with their respective centres on the sections of the strip theory schematization (see Figure 2). For these parts the hydro-dynamic coefficients and wave forces were determined separately.
2.3. Review of the calculations
In Table 2 a review is given of all the cases considered in the computations.
The water depth in the SHIPMO calculations is infinite; for the DIFFRAC computations the water depth was chosen maximal within the limitations of the program. This resulted in a water depth varying with the frequency corresponding to the relation kh =9.5
(in which k is the wave number = 270, and h is the water depth) in case of higher frequencies. For wave frequencies below w =0.7 rad./sec. the water depth was held constant at a value of 200 m.
For all cases the water depth may be considered as being rela-tively deep (except the depth of 85 m in the additional
2.4. Definitions and presentation of results
All results are given with respect to the system of axes with the origin in the centre of gravity as shown in Figure 1.
The results of the computations of the first order oscillatory wave loads and motions in regular waves are presented in the form of frequency response functions of the amplitude and phase angle to a base of the full scale wave frequency w in rad./sec. Frequency response functions for the amplitude of the first order wave forces are expressed in kN per metre wave amplitude. Frequency response functions for the amplitude of the first order wave moments are expressed in kN.m per metre wave ampli-tude. The frequency response functions for the amplitudes of the linear motions are expressed in metres per metre wave am-plitude. The frequency response functions for the amplitudes of angular motions are expressed in degrees per metre wave
amplitude. Phase angles are expressed in degrees. The phase angle is positive when the quantity under consideration reaches its positive maximum value before the crest of the undisturbed incoming wave passes the mean position of the centre of gravity of the body. This means that if the wave elevation c(t) of the undisturbed incoming regular wave at the mean position of the
centre of gravity is:
360
C(t) = Ca sin 7TT . wt
then the first order oscillatory wave load is presented by:
F(t) =
Fa sin (360 .
wt +FC)
27
where: E = phase angle between force and wave in degrees
= wave frequency in rad./sec. Fa = amplitude of the force.
NETHERLANDS SHIP MODEL BASIN PAGE
The positive directions of the wave forces and linear motions coincide with the positive directions of the body axes shown in Figure 1. The positive directions of the moments are also shown in Figure 1. The angular motions fit the directions of the moments according to the right-hand screw rule. Added mass and damping coefficients may be expressed in kN (forces) or kN.m (moments) per unit acceleration and velocity respectively. Accelerations and velocities for linear motions are expressed
-2
in m.sec. and m.sec.-1 respectively. For angular motions these
-are expressed in rad.sec72 and rad.sec1. respectively. Since 1000 kN = 1 Gg.m.sec72 (1 Gg = 106 kg), added mass coefficients of forces due to linear accelerations are expressed in Gg, while added mass coefficients for moments due to angular accelerations are expressed in Gg.m2.rad71. Damping coefficients of forces
-due to linear velocities are expressed in Gg.sec1. , while damp-ing coefficients for moments due to angular velocities are
expressed in Gg.m2.sec71rad71. The units in which the added mass and damping coupling coefficients are presented, are com-mensurate with the aforegoing.
3. DESCRIPTION OF THE STRIP THEORY AND DIFFRACTION THEORY AND A COMPARISON OF THEIR RESULTS
3.1. Short descri2tion of the 2rogramsL including the basic assum2-tions
1.1. General
Both the strip theory and the three-dimensional diffraction theory are based on the linear potential theory.
The fluid is assumed to be inviscid, homogeneous, irrotational and incompressible. The fluid flow may then be described by a velocity potential. The velocity potential is a scalar function of the co-ordinates and of time. The fluid velocity in any di-rection is equal to the gradient of the velocity potential in that direction. Knowledge of the velocity potential of the flow around the vessel is sufficient for the computation of fluid pressures, wave loads, etc.
The vessel is considered as a rigid body, oscillating sinusoidal-ly about a state of rest, in response to excitation by a long-crested regular wave. The amplitudes of the wave as well as of the ship motions are supposed to be small. In both computer pro-grams the motions are calculated by using linear equations of motion.
In both computer programs the hydrodynamic problem of the float-ing body, movfloat-ing in waves is divided into two more easily solv-able problems, i.e. the problem of calculating the wave forces on a fixed body and the calculation of the reaction forces on a body, which is subject to forced oscillation in still water.
The results of both calculations are combined to yield the equa-tion of moequa-tion for the floating body in waves.
NETHERLANDS SHIP MODEL BASIN PAGE
3.1.2.
Three-dimensional diffraction theory: program DIFFRACBesides the afore-mentioned assumptions, no further basic as-sumptions are required for the three-dimensional diffraction theory.
For the computation by the program DIFFRAC of the velocity po-tentials describing the fluid flow, the mean wetted part of the hull of the vessel is approximated by a number of plane elements, representing a distribution of source singularities each of which contributes to the velocity potential describing the fluid flow. The choice of the number of elements used to approximate the form of the hull is influenced by the follow-ing factors:
The complexity of the shape of the hull:
If the hull is irregular in form, more elements will be used than in the case of the hull forms which are smooth or of a simple form, consisting of a number of flat surfaces.
The length of the shortest wave for which computations are to be carried out:
In general the sizes of the elements are chosen, so that the distance between the centres of consecutive elements will be between 1/6 and 1/10 of the wave length.
The aim in both cases is to ensure that the flow characteris-tics and its spatial variations are accurately described.
The way in which the elements are distributed over the hull surface is mainly based on past experience. The problem as to whether a given distribution will yield acceptable results, is usually investigated by repeating computations using a differ-ent distribution and comparing the results. It is assumed that increasing the number of elements, will yield results which will tend to the limit, valid for the true hull form.
The mathematical formulation of the source singularities has been chosen so that the boundary conditions valid at the free
surface, the sea floor, at great distances from the body and within the fluid are automatically satisfied. These boundary
conditions are:
- Outside the hull, the amount of fluid is constant (equation of continuity).
The (moving) free surface of the fluid is a surface of con-stant pressure equal to the atmospheric pressure.
No fluid particles pass through the (moving) free surface. No fluid particles pass through the sea floor.
Except for the undisturbed incoming waves, all waves i.e. those generated by the presence of the body, travel outwards from the body.
For a given distribution of elements, the strengths of the sources are determined based on the requirement, or boundary condition, that the velocity component of the fluid in a direc-tion normal to the hull satisfies certain criteria. When the wave loads on the captive body are considered the normal veloc-ity of the fluid including the effect of the incoming waves must be zero. In computing the added mass and damping
coeffi-cients it is required that the normal velocity of the fluid be equal to the velocity of the oscillating body in the direction of the normal. Both requirements simply imply the watertight-ness of the hull. In the diffraction theory program the calcu-lation of the source strengths requires these boundary condi-tions to be fulfilled only at the centre points of the plane elements.
Having solved for the unknown source strengths, the velocity potential describing the flow and related quantities such as fluid pressures and forces can be computed. Basically, in the
NETHERLANDS SHIP MODEL BASIN PAGE
program in-phase and out-of-phase components of wave pressures, wave loads and motions are computed relative to the positive maximum of the undisturbed incoming regular wave potential
at the mean position of the centre of gravity (C.G.) of the vessel. The fluid pressures which yield the added mass and
damping are computed as in-phase and out-of-phase components relative to the positive maximum accelerations and velocity respectively of the motions of, or in the case of angular mo-tions, about the C.G.
The fluid pressures are computed at the centre of each plane element, by adding the contributions of all single elements. The contribution of a surface element to the pressure in its own centre is dependent upon the shape of the element. The
in-fluence of this effect is not taken into account in the program DIFFRAC. In fact, all surface elements are supposed ,to have a circular circumference. In case of a square element, the error in the pressure contribution of the own element is only one percent, and therefore it is as a rule attempted to keep the
shape of the elements as square as possible.
In order to compute the total fluid forces or moments, which are given relative to the C.G., it is assumed that the fluid pressure is constant over each plane element and equal to the pressure computed at the centre of the element. The process of integration of the pressure force acting on each element is a simple summation process which, besides the fluid pressure and area of the plane element under consideration, also takes into account the orientation of the plane element and the co-ordi-nates of the centre of the element, relative to a rectangular system of body axes with the origin in the C.G. of the body. It will be clear that if more elements are used, the potential accuracy of this integration procedure will be increased.
A detailed description of the method of computation is given in ref. [8]. Appendix I contains a brief description of the mathematical theory basic to DIFFRAC.
. The two-dimensional strip theory: program SHIPM0
In the strip theory, the velocity potential around the floating body is approximated by using two-dimensional linear potential theory results for the separate strips.
Besides the general assumptions of linear potential theory as mentioned in Section 3.1.1. the following additional assumptions are needed in the strip theory.
Any change in shape in a longitudinal direction
iS
small with respect to changes in a transverse or vertical direction. In practice, this means that the floating body should be slender(L/B >> 1 and L/T >> 1).
The actual three-dimensional potential for heave, sway and roll is sufficiently approximated by the successive two-dimensional potentials for the various sections.
In other words: mutual interaction of sections is not accounted for. moreover, because each section is supposed to be part of an infinitely long cylinder, the fluid flow is supposed to pass entirely underneath the body; fluid flow around the ends of the body is not taken into consideration. It will be clear that this assumption is violated especially near the vessel's ends. Ogilvie [4] has shown that for a body in forced oscillation the three-dimensional effects decrease at increasing frequencies: the second assumption is reasonably satisfied if the correspond-ing wave length is in the order of magnitude of the body's
length or less.
NETHERLANDS SHIP MODEL BASIN PAGE
3. In hydrodynamic respect pitching for any section is identi-cal with loidenti-cal heaving at the same velocity and yawing is identical with local swaying at the same velocity.
This assumption is needed because in two dimensions a body can perform only sway, heave and roll motions. The surge motion can-not be handled by the strip method.
The computer program SHIPMO starts with the computation of the hydrodynamic coefficients. The added mass and damping
coeffi-cients for heave, sway and roll are determined for a prescribed sectional shape at a range of frequencies. This is done by a transformation of Ursell's exact solution for a circular cyl-inder to a solution for the sectional shape of interest. The exact solution of Ursell contains a converging series of multi-poles. For heave six multipoles and for sway and roll eight multipoles are included, which is generally considered to be sufficient to give accurate values. The transformation of these exact solutions to the required sectional shape is performed by a "close-fit" conformal mapping procedure. The added mass and damping coefficients for the whole body are then obtained by integrating the sectional values.
In the second part of SHIPMO the wave forces on the whole body are calculated. The total wave force consists of two parts, the so-called Froude-Kriloff force and the diffraction force. The Froude-Kriloff force is the force exerted by the pressure in the wave without taking account of the disturbance by the presence of the vessel. This component can be obtained by sim-ply integrating the undisturbed wave pressure over the surface of the floating body. In the computer program, however, a sim-plification is introduced which requires that the waves are
long with respect to the beam and the draft of the vessel.
The diffraction force is the force component due to the distur-bance of the wave. It is computed according to
Korvin-Kroukov-ski's relative motion hypothesis, in which it is stated that with respect to the effect of the fluid velocity on the hull
no difference can be made between the case of diffracted waves on a fixed hull and the case of the same hull oscillating in calm water. This means that the diffraction force can be com-puted using the added mass and damping coefficients. Vugts [2] has shown that this is theoretically correct as long as local values of the velocities and local values of the added mass and damping coefficients are considered.
In the program SHIPM0, however, sectional values are used for the hydrodynamic coefficients, while also a mean value is used for the orbital velocities. These simplifications require again that the waves are long with respect to the beam and the draft of the vessel.
A further source for possible inaccuracies is the neglect of force contributions from local surge due to rotational motions. After computing the diffraction force per section, the total
force is obtained by integration along the length of the body.
1.4.
Review of assumptions and simplifications
The table on the next page gives a comprehensive review of the main assumptions and simplifications in the two methods and their effects.
It was expected that the diffraction theory would yield the better results for the present calculations, because the shape of the barge cannot be considered as "slender". So in the strip theory calculations inaccuracy might occur due to this non-slenderness, due to end-effects from bow and stern and due to other three-dimensional effects. On the other hand, however, the bilges of the barge were described more accurate in SHIPM0
NETHERLANDS SHIP MODEL BASIN PAGE
than in DIFFRAC, so for instance the value for roll damping is probably more accurate than the result from diffraction theory. Since both theories assume the fluid to be ideal, a discussion on the effects of e.g. viscosity will be given in a later phase of this study. So, the major items of interest are the three-dimensional effects which are not incorporated in strip theory, and the effects due to the barge being not slender.
Method Assumption/Simplification Acceptable if: Effect on:
BOTH METHODS
- fluid is ideal
- small amplitudes of waves and motions
- rigid body
- long-crested harmonic waves
_
in general acceptable in general acceptable for large volume float-ing bodies roll STRIP METHOD - body is slender - three-dimensional effect neglected- pitching = local heaving, yawing = local swaying - Froude-Kriloff simplification - Korvin-Kroukovski
simplifi-cation
- approximation of geometry by strips, integration procedures
L/B >> 1, L/T >> 1 A < L X » B A >> T ) >> length of strips
_
..._
_
coefficients and wave forces wave forces coefficients and wave forces THREE-DIMENSIONAL DIFFRACTION - no influence of shape of facets - approximation of geometry by facets; pressure constant over facet facet almost square A >> facet size; no sharp edges_
coefficients and wave forces roll3.2. Comparison of the results of strip theory and diffraction theory calculations
3.2.1. General
In the following paragraphs the results of the calculations with diffraction theory and strip theory for the complete barge will be compared. Since it is assumed that the three-dimensional
calculations yield accurate results, (see for instance Phase I of this study), a discussion of the results of the program SHIPMO with respect to the program DIFFRAC will be given in the sum-mary at the end of Section 3.2.
The sequential order will be a comparison of the wave forces and moments first, then of the hydrodynamic coefficients and finally of the motion responses. In the comparison of the motion responses it will thus be possible to discuss certain differ-ences by referring to previous paragraphs. The discussion of the various items will be subdivided with respect to the six degrees of freedom of the barge.
In this section the differences in the discussed quantities as calculated by the two computer programs will be denoted as neg-ligible when they are less than 3% of the highest level of the diffraction theory value in the frequency range under consider-ation in this study.
Differences in phase of less than 10 degrees will also be de-noted as negligible.
The wave exciting forces and moments
Surge force (Figures 4 and 5)
SHIPMO is based on two-dimensional theory, so the surge forces cannot be calculated. The results from diffraction theory show
NETHERLANDS SHIP MODEL BASIN PAGE
the effect of the inclination of the bow: at high frequency the surge force for waves coming from 180 and 135 degrees is about
30% lower than for waves coming from astern.
In bow and stern quartering waves at the highest frequencies the surge forces are lower than the surge forces for head and stern waves respectively.
Sway force (Figures 6 and 7)
In bow and stern quartering waves the differences between the results of the strip theory and diffraction theory calculations are very small: the maximum differences are found at 0.5 . w 0.75 rad./sec. where the strip theory values are approximately 80 kN/m (or 5%) lower than those from diffraction theory.
In beam waves some difference is found at high frequency: at w = 1.05 rad./sec. the strip theory value is 600 kN/m (about 15%) higher than the value from diffraction theory, While at w = 1.25 rad./sec. the strip theory value lies below the result of diffraction theory. The differences in phase angle are prac-tically zero, except at the highest frequency where a difference of about 40 degrees is found.
Heave force (Figures 8, 9 and 10)
In the frequency range 0.5 w 0.75 rad./sec. the strip theory values are somewhat higher than those from diffraction theory. The maximum difference is about 2000 kN/m at w = 0.5 rad./sec. These differences are found for all five wave directions.
For higher frequencies the strip theory results are somewhat smaller than those from diffraction theory and since the force level is low the relative differences are rather large: about 50%. The differences in phase are very small at low frequency; at high frequencies (w 0.8 rad./sec.) they may be considerable
Roll moment (Figures 11 and 12)
In the wave directions under consideration, the differences are very small at low frequencies but increase with increasing fre-quency; especially in beam waves. In that case the value from
strip theory at w = 1.25 rad./sec. is about twice as high as the value from diffraction theory. In general, the phase differ-ences are small; the only exception is a difference of almost
170 degrees, found at w = 1.25 rad./sec. in bow and stern quar-tering waves. (See also Section 4.1.).
Pitch moment (Figures 13 and 14)
In the mid-frequency range the pitch moment, calculated with strip theory is somewhat higher than that calculated with dif-fraction theory, for all the wave directions considered. The
largest difference is about 60 MN.m/m (or 25%) at w0.6 rad./sec. The phase differences are small (about 10 to 30 degrees) except in head and following waves at w = 1.05 and 1.25 rad./sec. and
in bow and stern quartering waves at w = 1.25 rad./sec., where these differences amount to some 100 degrees.
A very small effect - due to the longitudinal asymmetry of the vessel - can be observed in the diffraction theory results: for head waves the pitch moment is slightly larger than for waves from astern.
Yaw moment (Figure 15)
The yaw moments as calculated by strip or diffraction theory show hardly any difference; only at w = 1.25 rad./sec. a small - but relatively large - change in amplitude is found (6000 kN.m/m).
The phase differences are small. (The maximum difference is 30 degrees at w = 1.25 rad./sec.).
NETHERLANDS SHIP MODEL BASIN PAGE
.2.3.
The hydrodynamic coefficients
Added mass in surge (Figure 16)
As mentioned before, SHIPMO gives no results for this quantity, so no comparison can be made.
Added mass in sway (Figure 17)
The results from the two calculation methods are in reasonable agreement. In the mid-frequency range, however, at 0.5 < w
0.85 rad./sec., the values from strip theory are somewhat lower than those from diffraction theory. (Maximum difference: 0.38 Gg or 14% at w = 0.7 rad./sec.).
Added mass in heave (Figure 18)
Though the calculations with both theories show the same quali-tative behaviour, the strip theory values for w > 0.7 rad./sec. are somewhat higher than those from diffraction theory. Below this frequency the strip theory results are lower than those from diffraction theory, which difference increases for lower frequencies.
Added roll inertia (Figure 19)
The results from both computer programs agree well; for w < 0.8 rad./sec. the strip theory values are only slightly higher
(about 10% at maximum).
Added inertia for pitch (Figure 20)
The strip theory results are higher than the results from the diffraction theory. In the mid-frequency range the results agree rather well, but for decreasing frequency the two-dimensional calculations yield steeply increasing values whereas the three-dimensional results stay at a lower, constant level. (Although the result at 0.285 rad./sec. may be affected by the smaller water depth, Vughts' [2] results for an ellipsoid show a similar tendency for deep water).
Added inertia for yaw (Figure 21)
The added yaw inertia as calculated by diffraction theory shows a maximum at a frequency of about 0.75 rad./sec., while the strip theory results show this maximum at 0.65 rad./sec. So, at frequencies below 0.7 rad./sec. the strip theory yields too high values. For the frequency range above 0.75 rad./sec. the results from strip theory are somewhat lower than the diffrac-tion calculadiffrac-tions; the trend, however, is the same. The maximum differences are found at the highest and lowest frequencies con-sidered in the calculations and amount to about 250 Gg.m2/rad., which is about 20% at w = 0.45 rad./sec. and 30% at w = 1.25 rad./sec.
The sway-roll inertia coupling coefficients (Figure 22)
According to the linear potential theory, the hydrodynamic cou-pling between forces and motions of different mode is symmetric. Therefore SHIPMO only calculates a24 and b24 and assumes a42 and b42 to be equal to these quantities. DIFFRAC, however, calcu-lates the coupling coefficients individually. The results given in Figure 22 show that a42 from the diffraction theory agrees well with the strip theory values, but a24 from the diffraction
theory shows a rather large deviation at frequencies between 0.4 and 0.8 rad./sec.
Surge damping (Figure 23)
Also here, only diffraction theory results are available.
Sway damping (Figure 24)
The two-dimensional and three-dimensional results are in good agreement; the largest difference is 0.12 Gg/sec. at w = 0.5 rad./sec. Since the level of the damping is low, this difference means that the strip theory value is about 70% too high.
NETHERLANDS SHIP MODEL BASIN PAGE
Heave damping (Figure 25)
The strip theory values are higher than the diffraction theory values; at low frequencies the difference amounts up to 80%.
Damping coefficient for roll (Figure 26)
A considerable difference between the results from strip theory and diffraction theory is found over practically the whole fre-quency range considered. The strip theory values are higher than those from diffraction theory: at w = 0.75 rad./sec. this difference is 25 Gg.m2/sec. or 80%.
Damping coefficient for pitch (Figure 27)
The values from strip theory are considerably higher than the results of the three-dimensional calculations. At low frequency
this is most pronounced, since the three-dimensional results tend to zero while the strip theory values are very large.
Damping coefficient for yaw (Figure 28)
The strip theory results are somewhat higher than the diffrac-tion theory results. Large propordiffrac-tional differences in the mid-frequency range occur because there the values from diffraction theory are very low (at w = 0.6 rad./sec. the difference is 250 Gg.m2/sec., or a factor 5). Above w = 0.6 rad./sec. the yaw damping increases strongly and though the absolute difference stays about the same, the discrepancy reduces to some 20% at w = 1.05 rad./sec.
The sway-roll damping coupling coefficient (Figure 29)
The damping coupling coefficients agree reasonably well in the frequency range considered. Yet, at increasing frequency the differences between the results of the two theories increase to values of about 2.5 Gg.m (which is about 40% of the magnitude of the strip theory results).
3.2.4. The barge motions
In the discussion below the motion response amplitude operators which were calculated all apply to the case with zero forward
speed.
Surge (Figures 30 and 31)
Here only results from diffraction theory are available and so no comparison can be made.
Sway (Figures 32 and 33)
In bow and stern quartering waves the agreement is very good, except in a narrow frequency range 0.05 rad./sec. wide
-around the natural roll frequency, where the strip theory re-sults show a much stronger influence of the roll motion than the three-dimensional results. Though the roll response from
strip theory is lower at these frequencies, the effect on sway is more pronounced due to the larger absolute value of the added mass coupling coefficient a24 (see Figure 22). In beam waves the same behaviour is observed, but the differences around the nat-ural roll frequency are somewhat larger than in the case of quartering waves: at w = 0.65 rad./sec. the strip theory value is about 0.37 m/m smaller than the value from diffraction theory. Both in quartering waves and in beam waves the calculated phases agree very well, except in the narrow frequency range around the natural roll frequency and at the highest frequency. The maximum differences are about 45 degrees.
Heave (Figures 34, 35 and 36)
In head waves and following waves, the differences between the strip and diffraction theory results are negligible, except at w = 0.8 rad./sec., where the strip theory value is practically zero and the diffraction theory values are about 0.2 m/m.
NETHERLANDS SHIP MODEL BASIN PAGE
In bow and stern quartering waves the differences are negligible or very small in the frequency range below w = 0.8 rad./sec. At w = 0.8 and 1.05 rad./sec. the strip theory values are some-what lower (about 0.1 m/m) than the results from diffraction theory. Yet, one may consider the differences in these four wave directions to be small.
In beam waves the results from strip theory are lower than those from the diffraction theory for wave frequencies above
0.6 rad./sec. The difference is largest at w = 1.05 rad./sec., where it amounts to 0.3 m/m or 40%. Also, the qualitative be-haviour of the two response curves is different: the DIFFRAC results show a hump near the natural heave frequency, while the SHIPMO curve is smoothly decreasing for increasing frequen-cies. This difference between strip method and three-dimensional theory for heave in beam seas has been observed in several other cases of barge-like floating structures, also with other comput-er programs. The typical behaviour of the three-dimensional re-sults has been confirmed by model tests (see ref. [1]).
At frequencies around w = 0.8 rad./sec. the lower values for strip theory can be traced back to lower wave exciting forces and smaller damping. At higher frequencies, the lower wave
forces are considered to cause the smaller values of the motion response from strip theory.
The differences in phase are very small at wave frequencies below 0.8 rad./sec. At w = 0.85 rad./sec. in head and following waves the amplitudes from strip theory are practically zero and so the phases are less accurate. (Differences of about 180 de-grees for following waves are observed). In beam waves the phases agree very well.
Roll (Figures 37 and 38)
In bow and stern quartering waves the differences between the two and three-dimensional calculations are small, except at the frequencies close to the natural roll frequency. Those differ-ences can be related to the difference in the roll damping, which is calculated in the two programs. This, together with the difference in added inertia results in a small shift of the peak in the roll response from strip theory to a slightly lower frequency and in a considerable reduction in the height of the peak at w = 0.625 rad./sec., which frequency is very close to the natural roll frequency. In beam waves the same behaviour is observed, although at w = 0.625 the reduction due to larger roll damping in strip theory is more pronounced. As will be further discussed in Section 4.2.2., the damping as calculated by SHIPMO is considered more reliable. Therefore, it should be stated here that with regard to the roll motion response the strip theory results must be regarded as more realistic compared to the results from the diffraction program. It should be re-marked, however, that due to the neglect of viscous effects the
roll damping is still much too small. This subject will be dis-cussed amply in the report of Phase III of this study.
The phases agree well for the wave directions considered, ex-cept at w = 1.25 rad./sec., where the roll amplitude is very small and consequently the accuracy in the calculation of the phases is not very high.
Also
the difference in the phase of the roll moment contributes to the observed differences in phase.Pitch (Figures 39 and 40)
In head and following waves the differences between strip the-ory and diffraction thethe-ory calculations are negligible for wave frequencies below 0.75 rad./sec.; above that frequency the val-ues from two-dimensional theory are lower than those from the three-dimensional calculations. At w = 1.05 rad./sec. the
two-NETHERLANDS SHIP MODEL BASIN PAGE
dimensional calculations give a value for the motion response operator which is 0.5 deg./m below the value from diffraction and is practically zero. This can be explained from the smaller pitch moment and larger added inertia and damping which are
calculated by SHIPMO. At this frequency the calculated phases also differ considerably: about 90 degrees.
In bow and stern quartering waves the agreement is good, but at the higher frequencies (w 0.7 rad./sec.) the strip theory results are lower than those from diffraction theory. The dif-ference is at most 0.6 deg./m at w = 1.05 rad./sec. The differ-ences in phase are small, but increase with increasing frequency.
Yaw (Figure 41)
The yaw response functions in bow and stern quartering waves as calculated by the two computer programs are in good agree-ment, yet at the wave frequency range above w = 0.5 rad./sec. the strip theory values are slightly higher: about 0.2 deg./m. The phases agree well too. This agreement may be somewhat
sur-prising at first sight, remembering the differences which were found in the yaw added inertia coefficients. This can be under-stood from the fact that the yaw added inertia is much smaller than the barge's own inertia and so the differences in yaw added inertia (which are at most 250 Gg.m2) are less than 5% of the total inertia coefficient in the equation of motion. Yet, the lower added inertia from the strip theory is seen to yield some-what larger motions in this case, since the yaw moment from strip theory and diffraction theory is practically identical.
3.2.5. Summary
The wave exciting forces generally agree in a qualitative as well as a quantitative way. Large discrepancies, however, are
range. Also some difference is found for the pitch moment in the mid-frequency range and in the sway force at high frequen-cies. As far as the phases are concerned, the agreement is good, except at frequencies where the level of the forces or moments is low and at higher frequencies. See Section 4.1. for a more detailed discussion on the discrepancies in the wave loads.
The agreement in added mass coefficients is generally good. For the damping coefficients, however, only a reasonable agree-ment is observed for sway. In Section 4.2. these discrepancies will be extensively discussed.
It is somewhat surprising to see that, though the differences in the wave forces and moments as well as those in added mass and damping coefficients were considerable in various cases, the differences in the calculated motions are generally small. In this respect, it should be noted that in most cases the dif-ferences in added mass (or inertia) are small compared with the total of the barge's own mass (or inertia) and the added mass
(or inertia). Only in case the added mass is large with respect to the barge's own mass, as in heave, these differences do have considerable influence on the calculated motions. Further, in the calculated motion responses, the influence of the damping coefficients is generally small, except in small frequency ranges near the natural frequency of heave, pitch and - especially -roll. A last remark should be that the diffraction forces in SHIPMO (see also Section 4.1.3.) are calculated from the added mass and damping coefficients, so higher coefficients may yield higher forces, and these effects partly counterbalance each other
in the equation of motion, leaving the motion amplitudes rela-tively unaffected.
From the basic assumptions in the strip theory as described in Section 3.1. the expectation arose that the results would be inaccurate at low frequencies and at high frequencies. The first deviations would be caused by the high frequency assumption in
NETHERLANDS SHIP MODEL BASIN PAGE
the calculation of the hydrodynamic coefficients and the latter by the long wave length assumption for the calculation of, for instance, the wave loads. In SHIPMO, the simplifications in the calculation of the Froude-Kriloff forces and the diffraction forces according to Korvin-Kroukovski are most accurate for long waves. In all calculations with SHIPMO the interaction be-tween the sections is considered negligible.
In the given comparison of the results from SHIPMO and DIFFRAC for the total barge, the consequences of the above requirements and assumptions in SHIPMO are not easy to isolate. Therefore, the next chapter will contain a detailed discussion of the ef-fects of the assumptions on the accuracy of the results.
4. DISCUSSION ON THE INFLUENCE OF THE BASIC ASSUMPTIONS AND APPROX-IMATIONS ON THE ACCURACY OF THE CALCULATIONS
4.1. The accuracy in the calculation of the wave loads
4.1.1. General
In the following paragraphs the differences which were observed between the strip theory and the diffraction theory calculations will be discussed and related to deficiencies in the methods, wherever such a discussion is considered worthwhile. Distinction has been made between the results for the whole body and the results of the calculation of the longitudinal distribution of the wave loads.
4.1.2. The total wave loads on the barge
The wave exciting forces and moments consist of the Froude-Kriloff undisturbed wave forces and the diffraction forces. In SHIPMO the latter are determined using Korvin-Kroukovski's rela-tive motion hypothesis. Therefore, the added mass and damping coefficients per section are used and the resulting forces are integrated over the whole body. In the calculation of the total added mass and damping the frequency is assumed to be high in order to obtain accurate results for a body of restricted length. It may be concluded that the diffraction forces are also subject to this restriction. The relatively small magnitude of the dif-fraction forces at low frequencies, however, reduces the effect of inaccuracies due to this restriction.
On the other hand, the Froude-Kriloff forces and the diffraction forces according to Korvin-Kroukovski are both subject to sim-plifications in the strip theory program, which require long wave lengths.
NETHERLANDS SHIP MODEL BASIN PAGE
These simplifications are:
- the sectional wave forces are obtained by using an approxi-mation of the local wave pressures;
- the sectional values for the added mass and damping coeffi-cients are taken instead of local values.
The relative importance of the two conflicting requirements de-scribed in the above two paragraphs is not known. So, inaccura-cies can be expected at all frequeninaccura-cies. It is assumed that if the wave length is about the same as the length of the barge, which is at about 0.85 rad./sec., these inaccuracies will be at their minimum.
The comparison between SHIPMO and DIFFRAC with respect to the wave loads on the total body shows the deviations for the fre-quency range considered in this study. (See Section '3.2. and Figures 6 through 15). In the table below the agreement between strip theory and diffraction theory results in three frequency ranges is indicated. Force or momen t Wave direction (deg. )
Frequency range (rad./sec.)
0.3 <w< 0.75 0.75 <w < 1.0 w . 1.0
F 135/45 reasonable good good
Y
90 good reasonable reasonable
Fz 180/0 reasonable reasonable reasonable
135/45 reasonable reasonable reasonable
90 reasonable reasonable reasonable
M (i)
135/45 good reasonable poor
90 good reasonable poor
M0 180/0 reasonable reasonable reasonable
135/45 reasonable reasonable reasonable
M 11)
The most serious discrepancies are found in the roll moment at w > 1.0 rad./sec. At these high frequencies the simplifications in SHIPMO as to the use of an approximation for the wave pres-sures and to the use of sectional values for added mass and damping for the calculation of the diffraction forces are no longer justified. Apparently, the roll mode is more sensitive to these simplifications than the other modes of motion.
4.1.3. Discussion on the longitudinal force distribution
The standard barge is rather short with respect to its beam (L 3.3 B); it has a small draft (B = 10 T) and a rectangular shape, so it may be concluded that the barge is all but slender. The requirement for the strip theory that any change in shape in a longitudinal direction must be small with respect to changes in a transverse or vertical direction is met quite well over
almost the entire length of the barge, but it is strongly vio-lated at the bow and the stern. Consequently, it may be expected that three-dimensional effects, i.e. end-effects, have a con-siderable influence on the accuracy of the strip theory calcu-lations for the wave loads.
It may be useful to distinguish specific end-effects from the more general three-dimensional effects. Though both effects are related to the restricted length of the barge, in the programs the bow and stern are simulated in a specific way:
for DIFFRAC by a series of inclined plane elements and verti-cal elements respectively;
for SHIPMO by a section with full width and a very small draft and by a truncation respectively.
NETHERLANDS SHIP MODEL BASIN PAGE
Especially at the bow the DIFFRAC and SHIPMO representations may lead to characteristic effects, which are compared in the
discussions on the next pages. At the stern, the differences are considered less important and therefore, no special atten-tion is payed to that.
End-effects show clearly in the plots of the longitudinal force distribution as changes in magnitude near the bow. Sometimes, such changes in magnitude extend over a major part of the barge
and sometimes there exist differences in level between the strip theory results and the diffraction theory results. These differences may be related more generally to three-dimensional effects.
The end-effects and the variation of the forces over the length of the barge in beam waves are studied by calculating the forces per unit length (and unit wave amplitude) on the sectional bod-ies with the diffraction program. These values are compared with those from strip theory and those obtained from the damping co-efficients of SHIPMO by application of the Newman relations:
2 1/2
1F11
The Newman relations [5] are based on a two-dimensionalization of the theory of Haskind [6] in which a relationship was estab-lished between the exciting force in waves and the far field velocity potential for forced oscillations in calm water.
= F ya = b' I w YY eq. (1a) 2 1/2 = F
zaa
= [Pgw b' I eq. (lb) zz 2 1/2 = M /C (1)a a =[22
w b' (tql)eq (lc)
This procedure has the advantage of being independent of the requirement that À >> B, T but on the other hand no phase
in-formation can be obtained and since the outcome is strictly related to the origin in the water plane, the OG transformation
for the roll moment may cause some problems.
The results of the comparison of the forces per unit length on the sectional bodies are given in Figures 42, 43 and 44 as a function of the location.
For the transverse horizontal force 1F11 the results from strip
Y
theory below w = 1.05 rad./sec. agree rather well with the val-ues obtained by applying Newman's relations on the damping co-efficient for sway from strip theory. Between diffraction theory and the Newman relations the agreement is good in the whole
frequency range. The results also show that the three-dimension-al effects do not play an important role in the cthree-dimension-alculation of the sway force at the frequencies below 1.05 rad./sec.
The end-effects cause a large reduction of the force close to the bow.
The comparison for the vertical force
IF;1
shows that the re-sults from strip theory agree well with those from the Newman relations at wave frequencies below 0.85 rad./sec. A reasonable agreement between the results from diffraction theory and New-man's relations is obtained as far as the mean level is con-cerned. Three-dimensional effects appear to be important, as shown by the difference in level and their influence on the magnitude of the local force over a considerable part of the barge. It may be concluded that at frequencies below 0.85rad./sec. the heave force on the total body will be calculated fairly accurate in strip theory, due to the agreement in mean level, although there exist large discrepancies in longitudinal direction. The pitch moment, however, will show some deviations if compared with three-dimensional calculations because it may
NETHERLANDS SHIP MODEL BASIN PAGE
be expected that the three-dimensional effects - here only vis-ualized for beam waves - are even more pronounced in bow quar-tering or head waves. (See for instance the study of Ogilvie on end-effects [7]). A typical end-effect is the steep increase of the force at the bow section, which seems superimposed upon the tendency of the three-dimensional effect.
The level of the roll moment per unit length
NI
from strip theory agrees well with the results of the Newman calculations, except for w = 1.25 rad./sec. At this frequency it also lies considerably (400%) above the values from diffraction theory. The level ofNI
calculated with the Newman relations shows agood agreement with the results from diffraction theory, except with respect to the end-effects and at w = 1.25 rad./sec. At the most forward sections, the end-effects cause a sharp in-crease in the local value of the roll moment to a level which is about twice as high as at the midships for low wave frequen-cies and about eight times as high for the highest frequenfrequen-cies. The differences between the DIFFRAC and SHIPMO representations result in an opposite behaviour of the end-effects at the bow.
As to the roll moment calculated with the Newman relations, the OG transformation has been carried out in an approximate way. It was stated earlier that no phase information can be obtained from the calculation of the wave forces according to Newman. The correct OG transformation for the roll moment uses the fol-lowing formula: --2 2 , + 2(C.G.) = 2(O.) + OG .F (O.) Mdp M4) Y + 2 OG.F (0.).M (0.).cos(cm. - EF ) Y ci) Y eq. (2)
Here, this formula was reduced to:
M (C.G.)
= M(0.) ± OG.F (0.)Y
+ in case (E
-EF
) = 0 deg. Y- in case (EN
-EF
) = ±180 deg.Y
eq.
(3a,
3b) In these formulas (C.G.) and (O.) denote that the quantity is given with respect to the Centre of Gravity and the Origin re-spectively. The results from strip theory show that in beam waves the phase difference is practically 180 degrees, so the approximation is considered valid.From the discussion of the longitudinal distribution of the sway and heave force and the roll moment in beam waves, it can be concluded that the agreement between the values from SHIPMO, from DIFFRAC and from the Newman relations is generally good below w = 1.05 rad./sec. Above this frequency, the forces and moment from SHIPMO deviate from the DIFFRAC and Newman results, probably due to the approximations in the strip theory program which require A >> B, T. Three-dimensional effects play an im-portant role in
Fz and M0 at all frequencies, and for FY and M(I)
at the frequencies above 1.0 rad./sec.
4.2. The accuracy in the calculated hydrodynamic coefficients
4.2.1. General
It was already pointed out in Section 3.1. that the problem of the oscillating body in calm water is in strip theory accurate-ly solved for high frequencies onaccurate-ly, because the approximation of the three-dimensional potential by successive two-dimensional
NETHERLANDS SHIP MODEL BASIN PAGE
potentials for the oscillating body was, according to Ogilvie [4], only acceptable at frequencies for which X L. The most important three-dimensional problems that may be expected in the calculation of the hydrodynamic coefficients with strip theory are the end-effects due to bow and stern configuration and the inaccuracy in added mass and damping at the frequencies corresponding to wave lengths larger than the length of the barge (X > L).
4.2.2. Differences in the total hydrodynamic coefficients
The comparison between the strip theory and diffraction theory calculations for the hydrodynamic coefficients of the whole barge, as given in Figures 17 through 22 and 24 thraugh 29, show the differences in the frequency range considered in this study. In the table below, the agreement between those results is indicated for three frequency ranges.
Hydrodynamic coefficient
Frequency range (rad./sec.)
0 . 3 < w 0 . 75 0 . 75 < w -; 1.0 1 . 0 .. w 1.25 a22 a33 a44 a55 a66 b22 b33 b44 b55
b
00 reasonable reasonable reasonable reasonable reasonable good poor poor poor poor good good good poor reasonable good reasonable poor poor poor good good good poor reasonable good reasonable reasonable poor reasonableThough the results for the added mass coefficients are generally better than those for the damping, the overall picture is that for the higher frequencies the agreement improves. Exceptions to this are the pitch added mass and damping coefficients, the reasons of which are to be sought in the end-effects, which in-fluence on pitch will be discussed in Section 4.2.3.
The differences observed in the heave and pitch added mass and damping coefficients for the frequencies below 0.7 rad./sec. can be denoted as three-dimensional effects in the way Vugts [2] showed by means of a comparison of calculations for an ellipsoid
(with exact three-dimensional theory) and a "two-dimensional" cylinder (strip theory).
So, it can be concluded that the results confirm the expecta-tions with respect to the high frequency assumption as stated in Section 4.2.1.
With respect to the calculated roll damping for the whole barge, a poor agreement is observed between the strip theory results and those from diffraction theory. In the previous phase of this study [1] it has been shown that the roll damping values from diffraction theory are sensitive for the element size, especial-ly near the bilges. In this case with the standard barge, the rounding of the bilges is more closely described in the close-fit procedure of strip theory than in the element distribution of the diffrcation theory. (See Figure 3 for the close-fit re-sult). From the Phase I results and from experience with other projects it is known that in case of sharp bilges SHIPMO yields better results for the roll damping than DIFFRAC.
Apparently, for the roll damping a proper description of the section contour is more important than the influence of three-dimensional effects in the case of barges.
It should be emphasized here, however, that the roll damping as predicted by SHIPMO is still much too low due to the neglect of viscous effects.
NETHERLANDS SHIP MODEL BASIN PAGE
Considering the coupling coefficients, Phase I of this study [I] revealed that the values for ajk and aki or bik and bki from diffraction theory show better agreement with an increasing number of elements. Here, it is observed that the values from DIFFRAC for
a42 and b42 agree well with the SHIPMO results. For
a24 and b24' however, a considerable difference is found, especially for the inertia coefficient in the frequency range of 0.45 to 1.0 rad./sec. This discrepancy may reduce if a finer element distribution near the bilges of the barge is chosen. Although this explanation is supported by the results of Phase I of this study, some caution should be taken as to such a
comparison, since the barge in the first phase differed consid-erably from the present one.
4.2.3. The longitudinal distribution of the hydrodynamic coefficients
The added mass and damping coefficients per unit length have been calculated from the diffraction theory results for the ten sectional bodies. In Figures 45 through 50 these values are presented for heave, sway and roll respectively.
In both computer programs the linear concept implies that pitch and yaw can be considered as local heave and sway respectively.
In
the strip theory, however, the interaction effect between the various sections is not taken into account, whereas this effect is included in the diffraction theory. This interaction effect in pitch differs from that in heave, because for the pitching barge the various sections heave at different amplitudes.There-fore, also a comparison has been made for the added inertia and damping moment for these quantities. Note that they are present-ed in the plots, being dividpresent-ed by the lever squarpresent-ed. It should be noted here also, that the influence of force components in surge direction at bow and stern on the pitch and yaw moments are included in the diffraction program, but not in the strip theory.
Sway (Figures 45 and 46)
The longitudinal distribution for the added mass in sway from strip theory calculations differs locally from the diffraction theory results. For w < 1.05 rad./sec. the three-dimensional calculations yield near amidships higher values and near the bow lower values. So, three-dimensional effects are of some im-portance. The SHIPMO representation of the bow shows the same qualitative behaviour as the DIFFRAC representation, which may be explained from the reduction in draft which is simulated in both representations. At the frequencies of 1.05 and 1.25 rad./
sec., however, the diffraction theory results show a peculiar behaviour, viz, a minimum in the magnitude close to amidships. This cannot be explained satisfactorily. The longitudinal dis-tribution of the sway damping behaves qualitatively similar to the sway added mass and therefore we refer to the above discus-sion.
Heave (Figures 47 and 48)
For heave, the longitudinal distribution of the added mass from SHIPMO differs considerably from the DIFFRAC results, the latter showing an important reduction in magnitude close to the bow. Due to the fact that the level of the strip theory results is such, that the deviations from the diffraction theory values amidships and at the ends compensate each other, the integrated results for the whole barge agree reasonably well. The damping coefficients for heave as calculated from strip theory and dif-fraction theory agree well at the highest frequencies; at lower frequencies the level differs considerably, in which range the high frequency assumption for the calculation of the hydrodynam-ic coeffhydrodynam-icients in strip theory is most strongly violated. The end-effects appear as a pronounced increase of the damping near the bow, which is also observed in the strip theory results due to the adoption of a bow section with a very small draft.
NETHERLANDS SHIP MODEL BASIN PAGE
Further, in the three-dimensional calculation it is observed that the damping reduces in the positive x-direction over the
forward third part of the barge, especially at the mid-frequency range. The typical bow effect described above is superimposed upon this trend.
Roll (Figures 49 and 50)
For roll, the added inertia coefficient corresponding to strip theory has a longitudinal distribution which agrees reasonably with the results from diffraction theory. It should be noted that the three-dimensional effect is not very pronounced, con-trary to the observations for heave and sway.
In the roll damping as function of the longitudinal location large differences exist between the strip theory results and the diffraction theory results. The differences in level have
al-ready been discussed on page 25, Section 3.2.4. The bow effect in the DIFFRAC results is pronounced and shows a steep increase of the roll damping per unit length at the bow, especially at the highest frequencies. It is thought that these effects can mainly be related to the realistic representation of the bow in DIFFRAC by means of a series of elements, which are inclined with respect to the water surface. When heaving, these elements
act like a plunger type wave generator and the outgoing wave energy can be related to damping. Note that the outgoing waves in case of the DIFFRAC calculations are travelling in all direc-tions, while in SHIPMO only transversely outgoing waves are con-sidered. Since it is expected that the keel of the barge also sends out waves, this effect is also present at the vertical stern, but to a much lesser extent. Therefore, as a check, the DIFFRAC values of the roll damping per unit length at the bow and stern have been plotted in the figure on the next page.
o 1600 w 1200 C.) _CI H 4-1 U cl) 4) 0 CS, En U w E 800 o CD-, I
Eco
R11 400 d, O 1 Body 1 (bow) Body 10 (stern) Body 8 (amidships) ...,Wave frequency (w) in rad./sec.
The plot also shows the unit length roll damping for the mid-ships section and it is observed that the roll damping at the bow and stern increase proportionally with the value amidships, except at the highest frequencies above w = 0.85 rad./sec.
Note that this plot serves as an illustration only and since the sectional body at the bow is shorter than the sectional body at the stern, the end-effect for body 1 is somewhat over-emphasized. Yet, it may be concluded that the damping at bow and - less pro-nounced - at the stern increases at the higher frequencies due to the increase of outgoing wave energy.
The strip theory calculations for the hydrodynamic coefficients per unit length at the bow section do not include the effect of the bow shape, apart from the reduction in draft. It is observed that SHIPMO calculates very low values for the roll damping at
O 0.5 1.0
NETHERLANDS SHIP MODEL BASIN PAGE
