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P
160938 - Paper ISOPE'93
AN EVALUATION AND EXTENSION
OF THE SHALLOW DRAFT
DIFFRACTION THEORY
By: B. Buchner
March 1993.
ABSTRACT
This paper presents an evaluation and extension of the shallow draft diffraction theory. In this theory it is assumed that offshore structures with a shallow draft can be idealized for the diffraction and radiation problem as a flat plate in the free surface. This allows simplified relationships for the boundary Integral equation and the Green's function In three dimensional diffraction theory. The method is meant to provide a fast and simple design tool for the preliminary design of shallow draft structures, the use of which is wide spread in offshore industry. The present shallow draft diffraction theory is developed by Wu and Pnce, based on the earlier work of MacCamy and Kim.
However, no extensive validation of the shallow draft model, or evaluation of the applicability of the method in engineering
problems, was published until now. This validation and evaluation is the purpose of the first part of the paper. Generally, three important parameters are recognized: the beam to draft ratio, the ratio between the vertical position of the centre of gravity above the centre of buoyancy and the beam and the ratio between the wave length and the beam.
In the second part of this paper the shallow draft theory Is extended with the calculation of mean second order wave drIft forces on this shallow draft structures. Use Is made of the
momentum relations of Maruo and Newman. The IdealizatIon of the structure as a flat plate, allows also In this case simplified formulations. Finally, some first resuRs of the shallow draft drift force calculations are presented.
KEY WORDS: hydrodynamics. diffraction theory. ocean engineering, drift forces, preliminary design, wave forces, hydrodynamic coefficients
AN EVALUATION AND EXTENSION OF THE SHALLOW DRAFT DIFFRACTION THEORY
Bas Buchner
Maritime Research Institute Netherlands (MAR IN)
INTRODUCTION
The basics of the present shallow draft diffraction theory can be found by MacCamy [9]. In 1960, he proposed a calculation procedure for the two-dimensional radiation problem of a heaving cylinder with zero draft. His work can be seen as a special application of the two dImensional (strip) theories developed In that period. He found that the zero draft assumptions allow simplified relationships for the Integral equations In the problem. With his two-dimensional strip theory, he also calculated the added mass
and damping of a heaving circular disc in the free surface. Kim [5-7] later converted MacCamy's theory to the three dimensional case. With a source distribution method, he calculated the added mass and damping for heaving, rolling and pitching elliptical plates. Due to the development of the fully three dimensional diffraction methods in later years, the shallow -or zero- draft method was not further extended.
However, in 1986 Wu and Price [13-17] proposed the application of the method for offshore structures with shallow draft, flat bottom and (nearly) vertical sides. The use of this sort of structures is very wide spread in offshore Industry. Wu and Price combined the insight of the fully three dimensional diffraction methods, with the benefits of the shallow draft formulations, like the disappearance of the normal derivative of the Green's function, the short
calculation time and the lack of irregular frequencies. Finally, they developed a prediction method for the hydrodynamic coefficients, wave forces and motions of shallow draft structures. A review of this method will be given in thispaper.
Infortunately Wu and Price did not publish an extensive validation of the zero draft model, or an evaluation of the applicability of the method In engineering problems. Therefore, in this paper the shallow draft theory will be validated andevaluated.
The purpose of this study was not to defend the shallow draft theory, but to study the possibilities for its applicationobjectively. This is important to notice, because one of the main conclusions of this report is, that the applicability of the theory Is quite limited. On the other hand the shallow draft diffraction theory presents a good and fast motion prediction method for (the preliminary design
of) real shallow draft structures. For this sort of structures usual three dImensional diffraction methods can cause significant numerical problems and require a lot of computation time. Therefore it is important to find the ranges of applicability of the shallow draft theory. Generally, three important parameters will be investigated:
The ratio between the beam (B) and the draft (T) of the structure: BIT
The ratio between the height of the centre of gravity above the centre of buoyancy (Zg') and the beam of the structure: Zg'/B
The ratio between the wave length and the beam of
the structure: AJB
In addition the shallow draft method will be extended with the calculation of mean second order wave drift forces. This is done
by using the momentum balance method proposed by Maruo [10] and Newman [11]. Due to the shallow draft approximation, It Is possible to simplify the formulation of the Kochin-function, In the same way as it was done with the boundary integral equation in the first order calculations.
A REVIEW OF THE SHALLOW DRAFT DIFFRACTION THEORY Because the first aim of this study is the evaluation and extension of the shallow draft diffraction theory, in this chapter a short review of this theory will be given. Special attention is paid to the shallow draft assumptions and their effect on the resulting formulations. In later chapters, this will be used for the explanation of the differences between the shallow draft and normal three dimensional diffraction results.
Assumptions
In the shallow draft diffraction theory it is assumed that:
The draft (T) of the structure is small compared to its length (L) and beam (B)
The structure has a flat bottom
- The sides of the structure are vertical
Based in these assumptions, Wu and Price [13-17] proposed to determine the wave-structure interaction as follows:
Like in normal linearized diffraction theory, the Froude-Krylov forces due to the undisturbed wave pressure are evaluated over the mean wetted surface of the structure.
The diffraction and radiation potentials however, are evaluated over an approximate plate lying in the undisturbed free surface. This allows simplified formulations for the Green's function and the boundary integral equation used in three dimensional diffraction theory.
The schematized method and the coordinate system are given in the figure below.
z -axiS
G(O.O,Z,.)
y-axis
axis
Because the idealized fiat plate in the free surface has no vertical elements, the horizontal diffraction force is neglected. This is also the case for the horizontal added mass, damping and the coupling between the horizontal motions and rotational displacements of roll and pitch.
The motion amplitudes are derived for each frequency as solutions of the general equation of motion, which can be writtenas:
;
[(_w2(M+A,)_(B,j)+c,4
Xaj = F1denotes the constant inertia matrix of the structure, Aq and are defined as the added mass and damping matrices
respectively and C,1 is the hydrostatic restoring matrix. F, denotes the external (wave) force or moment in the i-th direction.
The velocIty potentIal
To obtain the matrices, forces and moments in (1), inlinearized diffraction theory use is made of a velocity potential for sinusoidal fluid motion, defined as:
c1(x,y,z,) =(x,y,z) e_t
In this way the dependence on time is separated from the spatial component. Because of the linearization of the problem, the spatial component can be split up again in contributions from the
undisturbed wave field 41. the diffracted wave 41 and the radiated waves due to the motions of the body 41. Because the surge, sway and yaw motions do not cause radiating waves in the shallow draft approximation, the total velocity potential becomes:
5 cj [
°+Ø'+
Xaj '.3 2-X 1 c(O,c,ZG) y-axiSA
I 4__
A
k-ax isThe undisturbed incident wave can be expressed as:
= _.i9Fi evz '(xcOS+y5Th
Where ca is the wave amplitude, v the wave number and the
direction of wave propagation.
Once the radiated potentials are known, the hydromechanic coefficients can be calculated:
.-
J Im(f)ndS
= -Q
J
Re(ndS
Combination of the undisturbed wave potential and diffracted wave
potential will give the wave forces and moments, according to:
2 2t' ()
JJ.(Q)ãG(PQ)ds
=_Jvr(Q)G(PQ)dS
ôn (12
In this way the flow around the structure is described by a source-dipole distribution over the facets of the mean wetted surface of the body. The normal velocity v1 in the right hand side of the equation is defined by the boundary conditions for the diffraction and radiation problem.
The Greens function in the boundary integral equation, in fact the potential of a pulsating source under a free surface, can be expressed as:
G(PQ)= 1/r+1/r+I1+iI2
v(Z.l-) esds=_Vz)[HQ(V+
Y0(v]+2 I
2+j1/2j
Jo [v '2 = 2.ni?J(vR)
The diflerent parameters in this function are defined In the figure
below.
R
S
The boundary Integral equation and Green's function It can be proved that the boundary integral equation for a normal three-dimensional diffraction method can be formulated as:
P
The modified expressions
In this paragraph, the modified expressions for the three
dimensional shallow draft diffraction theory will be derived. This will result in a boundary integral equation without the normal derivative of the Green's function and in a simple expression for the Green's function itself.
Starting point of the derivation is the formulation of the deep water Green's function (9).
It can be proved [2,3] that the normal derivative (n1 = n2 = 0, n3 = -1) of the Green's function can be written as:
6G
--(1/r+ 1/,d)+v(2/t
+ I)+ n'12(10)
ôz
With this expression, it is possible to bypass the normal derivation of the Green's function. The first term In (10) Is zero in the shallow draft approximation, when the field (P) and source (Q) element are not the same. For P=Q, this normal derivation is included in the first term of the boundary Integral equation (8). This becomes 4ir in stead of 2it.
6G ôn
This results at last in the modified expression for boundary integral equation:
4r
( + v J(0) G (P, 0) dS =
-J G (P0) v
(0) dS
The Green's function itself (9) reduces without the depth dependent terms into:
(13)
G (P, 0) = 2/R - 2EV [H0 (vR) + Y0 (vR)]
+ i2wJ0 (vR)
Finally, the following boundary conditions in the boundary integral equation have to be satisfied to obtaInthe radiation and diffraction potentials in the shallow draft approximation.
For the diffraction potential:
(14)
v=!:±:= -a°l
on on
Iz.-draft
for j=7
For the radiation potentials:
for J=3,4,5
denotes the generalized direction cosine In direction J. It should be noted that the boundary condition for the diffraction problem. is taken at the real draft of the structure. The further evaluation of this potential, by means of the boundary integral equation, is done over the idealized plate in the free surface. Physically, (14) and (15) show that the diffraction and radiation potentials have to compensate the normal velocity due to the undisturbed wave or due to the motions of the body.
The horizontal forces and motions
Because the added mass, damping and diffraction forcesare
neglected for the horizontal motions, we will only have to consider the Froude-Krylov forces due to the undisturbed wave pressure, for the calculation of these motions.
Based on this, Wu and Price derived a number of analytical expressions for the motions of structures with special geometries, like rectangular barges and circular, or triangular, platforms. They are discussed extensively in [161.
Because the application of these expressions is limited, a different and more general approach is chosen for this study. The
undisturbed wave pressure Is for this purpose Integrated over a string of side elements forming the boundary of the structure. Because the sides of the structure are vertical and the draft constant, the Froude-Krylov force in the i-th direction is approximated with its value at half the draft of the structure:
F,=_Q9CaJ e O5vT,a(xcosfl+ysinfl) n,viS S
It should be noted, that this makes it possible to study the effect of the horizontal Froude-Krylov forces on the motions of rolland
pitch. when the centre of gravity is not In the centre ofbuoyancy. This is not the case with the analytical solutions of Wu and Price. Irregular frequencIes
it is a well known problem that the integral equation does not give unique solutions at certain frequencies. At this so called irregular frequencies, the coefficient matrix of the discretized integral equation is badly conditioned.
This problem has to be associated with the eigenfrequencies of the interior Dirichlet problem. In fact, at these frequencies the potentials are satisfying the Laplace equation, the boundary condition on the free surface and the Dirichlet boundary condition
= 0 on the inner side of the floatingbody.
For simple structures It is possible to formulate analytical soiutlons of this problem. In this way the irregular frequencies can be predicted. Eatock Taylor [1] published an expression for the irregular frequencies of rectangular barges with length (L), beam(B) and draft (T):
= (g}'cothyl)1"2
(ax)2
(fl)2
for a, /3= 1,2,3,...
It will be clear from this expression that the lowest irregular frequency tends to Infinity when the draft is going to zero. This means that the irregular frequency problem has vanished when the draft is assumed to be zero.
EVALUATION
Based on the shallow draft diffraction theory, the program 'PLATO' has been developed.
To evaluate this theory, the hydrodynamic properties of a series of rectangular barges was analyzed with PLATO and with the fully three dimensional diffraction program MATTHEW. These barges were derived from the barge Investigated by Brown, Eatock Taylor and Patel In [1], which has a BIT ratio of 7.6. The new barges were made with a draft 1/4. 1/2, 2 and 4 times the original draft, whereas the other dimensions were kept constant. This resulted in 5 different barges with a BIT ratio between 1 .9 and 30.5.
Also a number of different heights of the centre of gravity were Investigated, to evaluate the effect of this height on the applicability of the shallow draft approximation.
Effect of beam to draft (BIT) ratIo
For the evaluation of the effect of the BIT ratio on the applicability of the shallow draft method, the origin of the coordinate system was placed In the centre of buoyancy at half draft. In this way only the effect of the draft on the added mass, damping and wave forces Is determined for structures with one side element. The influence of the side elements on the added mass and damping in roll and pitch will be discussed In the next section, where the effect of the vertical position of the centre of gravity above the centre of buoyancy is studied.
In Figure 1 a comparison is presented between the fully three dimensional calculations and the shallow draft results. It compares the added mass and damping in heave from the shallow draft methods with the fully three dimensional results for barges with an increasing BIT ratio. Roll and pitch showed the same sort of
trends.
To evaluate the accuracy of the shallow draft approximation, an error function is defined as:
E=
R30R° represents the shallow draft results (added mass, damping or wave forces), R, represents the fully three dimensional results. This definition Is useful because It represents directly the relative error, and the nature of that error, due to the shallow draft approximation.
Generally, It Is observed that the shallow draft results forthe damping, can be seen as the limiting case of the usualthree dimensional calculations with increasing B/T. The higher damping in the shallow draft approximation, can be explained by the fact that the disturbance of the free surface reaches Its maximum when
the draft is zero. This is also the explanation for the higher added mass at the lower frequencies. But the reason for the lower added mass at the higher frequencies is less clear.
In the usual three dimensional diffraction calculations, the added mass reaches a limiting value, somewhat higher than its minimum, when the frequency tends to infinity. But in the shallow draft approximation, the minimum vanishes and the added mass seems to go to zero when the frequency is increasing. A same trend can be found in the two dimensional zero draft calculations of MacCamy [9]. The reason for this difference is discussed in [3]. According to Wu and Price [13], it is possible to obtain accurate shallow draft results with a small number of elements describing the approximate plate. However, it seems that this accuracy is rather artificial, because calculations showed that the number of elements has a large effect on the calculation of the damping. See the figure below, for 12 (.), 4.8(+), 96(), 160(0), 300(x) and 432(0) elements.
It should be noted that not only the horizontal coefficients are neglected in the shallow draft approximation, but also this coupling between the horizontal translations and vertical rotations.
As in fully three dimensional diffraction theory, the Froude-Krylov forces are evaluated over the mean wetted surface of the structure. ThIs Implicates, that the differences In the wave forces and moments between the shallow draft and three dimensional calculations, are due to the differences in the diffracted wave
pressure.
As an example, In Figure 4 a comparison is presented between the vertical wave forces for the barges with increasing BIT ratio. The centre of gravity is situated in the centre of buoyancy at half draft again. The error function is also presented in the figures. The idealization of the structure as a flat plate in the free surface, results in larger wave making effects. Therefore the diffracted wave pressure will be greater in the shallow draft approximation. The diffracted wave pressure generally has a counteracting effect on the vertical wave forces. This causes lower total vertical wave forces and moments In the shallow draft approximation.
In Figure 3 a comparison Is shown between the horizontal x-forces for different B/T ratios. From this figure It becomes clear that the undisturbed wave pressure produces very irregular horizontai wave forces on the structure in this frequency range. This is due to the fact that the wave forces on opposite vertical sides of the structure, are sometimes cancelling each other at certain frequencies.
It should also be noted that the total horizontal wave forces are generally larger In the three dimensional calculation. This is due to the fact that, contrary to vertical forces, the diffracted wave pressure has a positive effect on the total horizontal forces, see Hogben and Standing [18].
As was the case with the hydromechanic coefficients, the shallow draft results are better for the lower frequencies. But it should be noted that at the higher frequencies, the forces and moments are small for themselves. Therefore these differences will have little effect on the critical motions of the structure.
Generally speaking, this wave force results support the conclusion based on the results for the hydrodynamic coefficients, that the shallow draft method should not be used for structures with a BIT ratio less than about 7.6.
The effect of the position of the centre of gravity (Zg'/B) It is common practice in linearized hydromechanics to define the wave-structure interaction, with respect to a space fixed coordinate system with its origin In the undisturbed free surface. We will call this coordinate system O. The centre of gravity Is in that case
lying on the z-axis, with coordinates (O,O,Z0,). For motion calculations however, the origin Is translated to the centre of gravity. The new coordinate system is called G,,. This vertical translation has effect on the wave exitin'g moments and the hydromechanic coefficients.
It will be clear that the horizontal forces will Introduce additional moments, when the origin is translated to the centre of gravity. This extra moment can be expressed by:
F4'
fO
F1(19)
F5'I=
0 1x
F2F6'J IZGJ F3
0 0.5 1.5 2
2g
In fact, a small number of elements results In a smaller damping. This means that with a low number of elements, the shallow draft calculation seems to be more accurate for barges with a lower BIT ratio, than that is actually the case.
Based on the observation of the error functions for the damping, it should be concluded that the shallow draft method can not be used for structures with a B/T ratio less than 7.6, especially at the higher frequencies (o'>l.0)
In Figure 2 the horizontal damping and cross-coupling coefficients are shown, for sway and sway into roll respectively. The added mass coefficients were showing the same sort of trends. It will be clear from these figures, that the damping for the horizontal motions is very dependent on the draft of the structure. For structures with a BIT ratio larger than 7.6 these coefficients seem
negligible.
The presented cross-coupling coefficients are due to the suction-force on the bottom of the structure, when the structure is surging or swaying. This effect is clearly identified in the pressure distribution in the sway mode, shown in figure below.
Which results in the following corrected moments:
F= F+F2Z
F=
ZGThe effect of the translation on the hydromechanic coefficients is less clear. First we have to consider the fact, that these
coefficients introduce forces and moments with respect to the origin, when the body is moving. The correction for the hydromechanic coefficients can therefore be obtained with the following train of thought.
First the origin is translated to the centre of gravity. Then we assume a translation or rotation with respect to this new coordinate system G. This motions are translated to motions with respect to the old coordinate system With the old coefficient matrices, it is now possible to obtain the forces and moments, caused by this motion with respect to the old system
Afterwards these forces and moments have to be converted to forces and moment with respect to the new system again.
Such reasoning leads for the vertical translation of the centre of gravity (for a structure with vertical sides) to:
N15= N51
O15ZG011
N55= 0-2ZG015+2011
N24 = N = 024 + ZGQ7
N=
0-2ZG024+20-where N,, and 0,, denote the new and old hydrodynamlc coefficients respectively.
To determine the effect of the vertical position of the centre of gravity, a series of calculations Is done with a ratio Zg'/B between 0.0 and 0.4. Zg' denotes the vertical position of the centre of gravity above the centre of buoyancy. Because it was concluded in the previous section that the shallow draft theory can not be used for structures with a BIT ratio less than 7.6, we will restrict ourselves to the barges with a BIT of 7.6 and 30.5.
In Figure 5 a comparison is presented between the shallow draft and fully three dimensional diffraction wave exciting roll moments with an increasing height of the centre of gravity. This is done for both B/T=30.5 (left) and BIT=7.6 (right).
Generally, the same trend of a decreasing wave moment with an increasing height of the centre of gravity, is found in the shallow draft results and usual diffraction calculations. It will be clear from this figure that a higher centre of gravity can be allowed when the BIT ratio is small. For the barge with B/T=30.5 it possible to use a Zg/B ratio of 0.20. But for the other barge (with B/T=7.6) a ratio Zg/B=O.10 is the limit.
Because the approximate plate used for the radiation problem in the shallow draft theory has only horizontal elements, the vertical position of the centre of gravity will not affect the added mass and damping terms in roll and pitch. But In the non-zero draft situation, this is not the case. The coefficients for roll and pitch, and the cross-coupling coefficients for this motions with surge and sway,
are dependent on the position of the centre of gravity. This is due to the fact that when the centre of gravity is above the centre of buoyancy, a rotation results in a horizontal displacement of the structure. This is shown in the figure in the introduction and in
(21).
Figure 6 shows the added mass and damping in roll for a Zg'IB ratio between 0.0 and 0.40. The barge has a BIT of 7.6. It will be clear from these figures, that the position of the centre of gravity, has a large decreasing effect on the damping in roll. Because the shallow draft damping was too high already, this will have a further negative effect on the correlation between the shallow draft and normal three dimensional diffraction results.
The effect of the wave length on the results
As mentioned in the previous sections, the results of the shallow draft approximation are getting worse at Increasing frequencies. To explain this effect, In this paragraph we will take a closer look at the behaviour of the Green's function at the higher frequencies. When we consider the definition of the Green's function, we see that the draft has an Important effect on certain parts of the
formulation, especially at the higher frequencies. Generally, this is due to the following factor in both I and /2:
,c'3D(w)
ve2
= -e4
which reduces in the zero draft case to: F0(w)=
It will be clear, that the exponential decaying behaviour has vanished in (23). Physically, the factor F represents the fact that the disturbance of the free surface due to an underwater pulsating source, decreases rapidly when the frequency Is Increasing. But in the zero draft case, this decaying effect has vanished.
This is shown for the real part of the Green's function In the figure below. It should be noted that even for small values of the draft the amplitude of the maxima Is decreasing at the higher frequencies. This effect has totally vanished at zero draft.
S 10 is 20
-s- 1.0.50,,, 1.0.25
--
T.0.i2S,-I- T.0.0L2$,., .,- 1_on.
THE MOTION RESULTS
Finally, the resulting motion results of the shallow draft diffraction theory will be discussed. We will restrict ourselves again to the barges with a B/T ratio of 30.5 and 7.6.
Figure 8 shows the horizontal motions of surge and sway for barges with a BIT ratio of 7.6 for different positions of the centre of gravity. The comparison between the shallow draft and fully 3-D results is reasonably good, especially for the surge motion. When vertical position of the centre of gravity Is Increasing, the
hydromechanic coupling between the roll and sway motion becomes more important. Because this coupling is neglected in the shallow draft approximation, the differences are large near roll resonance.
The heave motion in head and beam seas for the barge with BIT_-7.6, is presented in Figure 7 . The shallow draft results in head sea are remarkably good. But in beam sea, there Is a considerable difference between the shallow draft and fully three dimensional result, especially for the barge with a B/T ratio of 7.6. This can be explained by the lower wave exciting force and the higher added mass and damping.
In [1] Brown, Eatock Taylor and Patel presented model tests of a transportation barge of 86.40 m length, 28.80 m beam and 3.78 m
draft (B/T=7.6) at scale 1:36. In [13] Wu and Price presented a
comparison between the shallow draft calculations and these model tests. The calculations and model tests show reasonable comparison, especially in heave, roll and pitch. In sway, the neglect of the coupling with the roll motion (Figure 10) Is identified
again.
The comparison between the shallow draft roll motion and the experimental values, is remarkably good. Especially whenwe consider the fact that usual three dimensional calculations generally over predict the roll motions, which are governed by viscous effects. Based on this observation, Wu and Price [13] conclude that the shallow draft theory is a better method for the prediction of the roll motion.
However, it is the question whether such a conclusion is really justified. The larger damping In the shallow draft theory Is namely an effect of the approximation in the model. This approximation results in an artificial sort of damping, which Is not due to real physical phenomena like viscous effects. The results are good for this particular barge and wave height, but this does not mean that this is also the case in other situations.
MEAN WAVE DRIFT FORCES ON SHALLOW DRAFT STRUCTURES
A body floating on waves in an ideal fluid, is subjected to a pressure that can be expressed with Bernoulli's equationas:
(24) P =
_QRe( we_Dt)_j-QI12
QgZ
In the first order dIffractIon calculations, only the oscIllatIng first term of (24) was used. This resulted In purely harmonic forces on, and oscillations of, the body, which are proportional to the wave amplitude of the incident waves. This wave amplitude Is assumed to be small.
But when we use the whole expression (24), Integrate up to the instantaneous position of the free surface and take terms up to the
second order in wave amplitude, we get a extra second order force, which Is quadratic In the IncIdent wave amplitude. This force consists of a slowly varying part and a steady component. This mean wave drift force will now be determined for shallow draft structures, based on a modification of the momentum relations proposed by Maruo [10J and Newman [11].
The far field potentials
The velocity field around a floating structure in an ideal fluid, can be described by the combination of the undisturbed wave potential, the diffraction potential and radiation potentials.
For an arbitrary point P(x,y,z) in the fluid, the diffraction or radiation potential can be obtained with:
(25) (P) =
_J_JIG(P
Q) ÔG(P,4;
[
ânon
S
G(P,Q) is the Green's function. The normal derivative in equation Is determined by the appropriate boundary condition for the diffraction or radiation problem.
Far away from the body, the combination of the radiation and diffraction potentials represents regular sinusoldafly outgoing waves. Based on this observation, a far field approximation of equation (25) can be obtaIned by substituting the Green's function (9) in (25) and assuming that the argument P is large. This yields, see for instance Newman [11]:
v \1/2
f/(p)b_.
(-)
H(x+9)R4)
4(P)° is the sum of the diffraction and radiation potentials.
H(9) is defined as the Kochln-function [8], which represents the far field potential due to the diffracted and radiated wave fields. For a unit normal pointing into the fluid this potential can be expressed for all the diffraction and radiation potentials as:
H(0)
=
J
$ Is the mean wetted surface of the structure. Because the radiation potentIals are linearly dependent on the motion amplitude in that particular mode, the total Kochln function becomes:
6
H(9) = H7(0) + 11(0)
i-i
in (26) and (27), use is made of the polar coordinates (r,O,z). In these new coordinates, the undisturbed wave potential can be expressed as:
= -
.f! vz+iRcos(fl+8)]
cvThe total potential In a point far away from the body, can now be described by the sum of (26) and (29). From this potential, the normal and tangential velocities at a large distance A from the origin, can be derived In polar coordinates according to:
(30)
8=
Re(e_t)
V6 =Re()_.
RÔ0
The momentum relations
To obtain the mean wave forces and moment on a freely floating structure, use is made of the rate of change of linear and angular momentum. Therefore, a control surface is defined by the mean wetted surface of the body (S), the free surface (SF) and a cylindrical control surface at infinity (Sj.
For the rate of change of linear and angular momentum, the following expressions can be given, see for instance Hong [4] and Newman [11]:
JJ [pfli+QVx(VnUn)JdS
S +S,*S.
=
-
J
J [xn - yn1) +
- yV)(Vn-
u)]ds
dt
S +Sc.s.S.
M,, M. K denote the rate of change of momentum In the x,y and v direction respectively. V is the normal velocity of the fluid on the boundary, whereas U expresses the normal velocity of the boundary itself. The unit normal n1 (n,ri2,n3) is pointing into the
fluid.
It should be noted that the normal velocity of the fluid (Va) and normal velocity of the boundary itself (Un), are the same at the ship's surface (S) and in the free surface (SF).
The contribution of the pressure term on the ship's surface (5), represents the desired wave forces and moment. When we now assume that the velocity of the closing cylinder itself is zero (LJ =
0). we obtain: dM F1 =
J
J[pni
+QVxVnJdS+-S. dM (35)F2=
JJ[Pn2+QvYvn]ds+--S.dK
F6 =J
J[xn2_
yni) + Q(xV- yV) Vn]dS +
S.The velocities in the x, y and normal direction can be converted to velocity expressions in the polar coordinate system according to:
V= Vjcos0- V9sinO
V,= VRSInO+ V8cosO
V, = - Vq
Because the average rate of change of momentum for a periodic motion is equal to zero, the desired mean wave forces and moment can be derived from equation (34-36) as:
=
- f
J[pcoso
+QVFVRCOS0 Vesirr8)]RWdz
S.
F2=
-
J
J[psine
+QVR( VRSInO + V9cosO)]R.iidZS.
F6 =- JJQVRVOR2cVdZ
The horizontal bars denote the tIme average.
The shallow draft approximation of the drift forces
Substitution of the velocities derived with the far field potentials in the momentum relations, will give the desired relations for the mean wave drift forces and moment. In this paper, the complete derivation of the formulations will not be given. See for an extensive treatment Newman [10] and Hong [4].
They prove that, when we take terms up to the second order in wave amplitude, the mean wave forces and moment can be expressed by:
QOa
=
0)jcos0
+ _cosflReI-r-)]
0 2r
F2
iiIx+ 0)12 sinO
-sinflR4Ivfi)]
8Xj
0
2x
=
_imJ H*(,r+8)H(x+9)
+9_!vfrrkH'(,r_fl)}
H is the complex conjugate of H, whereas H' denotes the
differentiation of H with respect to 8, accordIng to:
These expressions are somewhat different from those published by Newman [11]. This is due to the different definition of the undisturbed wave potential and the oppsite direction of the unit
normal.
We will now modify the Kochin function 1-1(0) for the shallow draft approximation. Following the same train of thought as for the first order calculations, we obtain:
=-I
I
P2 +QV( V,,-
U)]dS
S (44)H'(fl)
= dM dtdM
dt[8] Kochin, NE (1951). "On the Wave Resistance and Lift of Bodies Submerged in a Fluid", SNAME Technical and Research
RuI,IPtin Nçt ,1 .R
JJ[on
'j
This gives for the total Kochin function, because only the vertical motions are generating waves:
5
hO)=H(0)+Xaj
ii(o)
'-3
H7 denotes the diffraction potential due to the bottom ofthe
structure. The diffraction on the vertical sides Is neglected. It will be clear from (45) that the normal derivation of the far field Green's function have vanished. This is also the case for the draft dependent term.
High frequency limits of the wave drift forces
When the frequency Increases, the waves are becoming shorter. The fluid motion is in this case limited to a small layer near the free surface. Because the motions of the structure are very small at this frequencies, the incident wave is almost completely reflected on the sides of the structure. This causes a wave free 'shadow' on the other side of the structure.
It will be clear that this effect has not been taken into account in the shallow draft calculation of the drift forces. The wave drift forces will therefore be to small in the high frequency region. A different method has to be chosen to predict the drift forces and moment due to the totally reflected wave.
This effect can be studied with either the direct pressure integration (See Pinkster [12]), or with the momentum relations applied on the incident and totally reflected wave (Maruo [10]). Both considerations are based on the assumption that the reflected and incident wave directions havethe same, but opposite, angle with the normal on the side of the structure. This resuits in the following relation for the high frequency limits:
Q1
;E1=_____j cos2(8-9)na dl
LL denotes the non-shadow part of the structure for each wave direction, n the direction cosines and 0 the angle of the unit normal into the fluid with the x-axis.
Some first results
Because no experimental results for shallow draft structures were available, some first calculations of the mean wave drift forces were carried out on a shallow draft barge with B/T=10.0. For the barge in bow-quartering seas, Figure 9 presents a comparison between the shallow draft (PLATO) results and the results of two programs based on fully three dimensional theory. 'WAMIT' is based on a far field approach for the calculation of the drift forces, 'DELFRAC' makes use of a direct pressure Integration technique. The results are satisfactory for the lower frequencies and around the pitch and roll resonance peaks. At the higher frequencies the reflections on the sides of the structure become dominant, which are not incorporated In the shallow draft approximation.
NOMENCLATURE
Tte shallow draft diffraction theory is a simple and fast tool for the preliminary design of shallow draft offshore structures.However, the following conclusions about its applicability seem justified:
The ratio between the Beam (B) and the Draft (T) has an important effect on the accuracy of the shallow draft method. It is concluded that the application of the shallow draft theory is limited to structures with a BIT ratio larger than 7.6. The vertical position of the centre of gravity above the centre of buoyancy (Zg') is another important parameter In the applicability of the shallow draft method. For an acceptable accuracy the ratio between Zg' and the beam of the structure has to be limited by 0.20 for a barge with B/T=30.5, but by Zg'IB=0.1 0 for a barge with B/T=7.6.
The comparison between the usual three dimensional calculations and the shallow draft results, Is worsening when the frequency Increases (w'> 1.0). This Is due to the large effect of the draft on the Green's function in this region.
The calculation of mean wave drift forces for shallow draft structures needs to be validated with model tests before firm conclusions can be drawn.
ACKNOWLEDGEMENTS
The author wishes to thank Prof.Dr.lr. J.A. Pinkster for his enthusiastic support and advice during this project, which was a part of the Master's thesis of the author at the Deift University of Technology. Marine Structure Consultants (MSC) in Schiedam (The Netherlands) sponsored this study. Ir. Joost A. van Santen is acknowledged especially for his critical and inspiring supervision during the project.
REFERENCES
Eatock Taylor, R, Brown, D T and Patel, MH (1983). "Barge Motions in Random Seas - a Comparison of Theory and Experiment", Journal of Fluid Mechanics, Volume 129.
Buchner, B (1991). "An Evaluation and Extension of the Shallow Draft Diffraction Theory", TU Delft/MSC report PF 8914-1746, Schiedam.
Buchner, B (1993). "Hydrodynamic Properties of Shallow Draft Structures", To appear In: Ocean Engineering.
Hong, VS (1983). "improved Prediction of Drift Forces and Moment", David Taylor Naval Ship Research and Development Centre, Report No. DTNSRDC-83/69.
Kim, WD, MacCamy, AC and Stelson, TE (1962). "The Forced Oscillation of Shallow Draft Ships", Final Report Bureau of Ships
Fundamental Hydromechanic ResearctProgram, Project S-R009 01, 1962.
Kim, WD (1963). "On the Forced Oscillations of Shallow-Draft Ships", Journal of Ship Research, October 1963.
Kim, WD (1965). "On the Harmonic Oscillation of a Rigid Body on a Free Surface", Journal ofFluid Mechanics, Volume 21,
Part 3.
16 14 12 10 6 4 2 0 0.5 1 1.5 2 - WT-3C.5 - 8(1.1525 0(1.71 - 0(1.4.8 8(1.1.8
Figure 2: Some non-dimensional horizontal coefficients in the three dimensional calculations for increasing BIT ratio: damping in sway (above) and damping cross-coupling between sway and roll
(below). 25 B'24 25 -
I-2g F1' 12 10 8 6 4 2 0 12 10 8 6 4 2 0 1.4 12 0.8 0.6 0.4 02 0 0 - 8(140.5 '+ StT.15.25 -- 0(1.71 -8 8/T.3.$Figure 4: The non-dimensional z-forces in beam sea, calculated with three dimensional diffraction theory (above) and the shallow draft approximation (middle) for different values of the B/T ratio, The error due to the shallow draft approximation is presented in the lowest graph.
-F3' F3' E 2 25 8/1.1.5Figure 5: The k-moment on the barge with B/T=30.5 (left) and BIT=7.6 (right) in beam sea, with different positions of the centre of gravity: The usual diffraction theory calculation (above), the shallow draft result (middle) and the error function.
0 0.5 1.5 2 2.5
ri
- 8(1-30.5 4- 0(1.15.30 . OtT-iS '8- 8/1-3.8 -k- 8/1.1.8
Figure 3: The non-dimensional x-forces in head sea, calculated with three dimensional diffraction theory (above) and the shallow draft approximation (below) for different values of the BIT ratio.
1.5 0.5
0.8 0.6 04 0.2 0 0.25 0.2 0.15 0.1 0.05 0.8 0.6 04 0.2 1.2 0.8 06 04 0.2 0
- z_o.o
ZqSO2O (mm/m 1.2 0.5-
1.5 -- 2g'/8O.1OFigure 6: The three-dimensional added mass (above) arid damping (below) in roll for the different values of the Zg/B ratio with a three dimensional diffraction program.
2 A'
25/v
V-. I-
.j 2g 25rw
.-.. /-
2g Figure 7: The heave motion In head sea (above) and beam sea (below) for a barge B/T=7.6.(mm 1.2 0.8 0.6 0.4 02 0 1.2 0.8 0.6 0.4 0.2 - PLATO. ZUO.0 Zq-.1Q
- ze-0.0
Z-020- Z-0.O5
-+- Z.0.40Figure 8: The surge motion In head sea (above) and the sway motion in beam sea (below), for a barge 9/1=7.6.
150
100
WAMIT -- OELFRAC - PLATO
Figure 9: The X-drift forces on the transportation barge In quartering seas.
