Rene H. M. Huijsmans
I
I 'CENTS LEW
UNIVERSI
$CheepshydromerhAnicaArchief
Mekelweg 2, 2628 CD Delft
II
Te1:015-2786873/Fax:2781836
Mathematical
MOdelling
of the Mean Wave Drift Force
in Current
A Numerical and Experimental Study
Rene H. M. Huijsmans
labrwatorium voor ScheepshydromechanIca Archler klekelweg 2, 2628 CD Delft Tab 015 - 786873 . Fax: 015.781338Mathematical Modelling of the Mean Wave
Drift Force in Current
Printed by:
Drift Force in Current
A Numerical and Experimental study
Proefschrift
ter verkrijging van de g-raad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus Prof. Jr. K.F. Wakker, in het openbaar te verdedigen ten overstaan van een commissie,
door het College van Dekanen aangewezen, op ma.andag 17 juni 1996 te 10.30 uur
door
Rene Herman Maria HUIJSMANS
Dit proefschrift is goedgekeurd door de
promotor-Prof. Dr. Jr..A.J. Hermans
Samenstelling promotiecomrnissie:
Rector Magnificus, voorzitter
Prof.drir. A.J.Hermans, TU DELET,promotor
Prof dr ir J A Pinkster, fac WbMt Prof.dr.ir.J.H.Vugts, fac Ct
Prof ir M van Hoist, fac WbMt
Prof.dr.B.Molin, Ecole Superieur Marseille,Pr Prof G.E.Hearn, U-New Castle, Groot Britannie
Prof.dr.ir. G.Kuiper, fac WbMt ISBN-nummer: 90-75757-02-6
Contents
1Abstract
3 2Introduction
5
3,Mathematical formulation
3.1 Problem formulation . . .. ... .
. . 11 113.1.1 Linearization of the free surface condition . . 14
3.1.2 Linearization of the body boundary 'condition 15
3.2 The potential function 16
3.3 The boundary condition on the free surface 19
3.4 The body boundary conditions,,,
.
20 3.5 The steady potential . . 214 Expansion of the potential
'254.1 The integral equation . ,
4.2 The amplitude distributions of the potentials .
25
30
5 the Green's function
335.1 The expansion of the 'Green's function 33 5.2 The zero order Green's function 00 37 ,5.3 The first-order Green's function
.... .
6, O. II 385.3.1 A transformation in the complex plane 38
5.3.2 An expression of derivatives of if', . . . 39
5.3.3 The agreement of both expressions . 40
5q4 The uniform expansion of the Green's function . 40
5.4.1 Large distance B. .. 41
5.5 Suppression of irregular frequencies 47
5.5.1 The Lid method: theory 47
5.5.2 Implementation of lid method 51 5.5.3 Discussion lid method 65
6 The forces on the body
676.1 Added mass and damping 67 6.2 The exciting forces and the motions 71
6.3 The mean wave drift forces 79
7 Model test experiments
857.1 Wave drift force measurements 85
7.1.1 Passive mooring 85
7.1.2 Active mooring 87
7.2 Extinction tests in regular waves 89
7.3 Model test conditions 93
8
Validation
998.1 200 kDWT Tanker 100% Loaded 99
8.1.1 First order responses 100
8.1.2 Wave drift forces in current T100 % 106
8.1.3 Wave drift damping T100 % 107
8.2 200 kDWT tanker 40% loaded 108
8.2.1 First order Responses 110
8.2.2 Wave drift forces in current T40 % 116 8.2.3 Wave drift damping T40 % 118
8.3 200 kDWT tanker 70% loaded 119
8.3.1 Wave drift damping T70 % 120
8.4 Time domain results 200 kDWT tanker 40% loaded . 121
9 An engineering view of wave drift damping
123CONTENTS
A Derivation of integral equation
139Integral equation irregular frequencies
143B.1 Integration of free surface panel 145
C Integration rankine source
149The fsc for the radiation potential
153The m3-terms 157
F The computations of 7,b1 159
F.1 A transformation in the complex plane 159
F.2 An expression of derivatives of Oo 161
F.3 The derivatives of the 01 163
The far field expansion
167G.1 The residue of //) 167
G.2 The method of stationary phase 168
G.3 Asymptotic behaviour of /ko 171
Error estimates from experiments
175I
Dynamic positioning at model scale
1771.1 Global set-up 177
1.2 Components in a DP system 178
1.3 The control loop 179
1.3.1 Mathematical model of the ship 179
1.3.2 The extended Kalman filter 180
List of Figures
2.1 A typical mooring layout of turretmoored tanker .
2.2 Undisturbed stationary flow .., 8,
2.3 Disturbed stationary flow . . , llo
3.1 System of coordinates. . .
. ... .
. 173.2 The coordinate system and the six modes of ship
mo-tion. .
...
, .
. . 183.3 Steady wave system . jh1A 22
4:1 Coordinates at waterline. . . 26
4.2 Stationary potential in free surface around a 200 kDWT
tanker in cross-flow conditions.. . . .
r.
,.. 31 5.1 'Contours of integration. ., ....
. . 355.2 Wave pattern of oscillating translating source r < 1/4. 35 5.3 Contours of integration. . 37
6.4 ?Po,. 1? is variable, the source in (0,0, 1), w = 1.4. .. , 38
5.5 01, R is variable, the source in (0, 0, 1), w = 1.4. . . 41
5.6 Large distance 2/4 , R is variable, the source in (0,, 6,-1),
.5.7 i and t/.7(= R is variable, the source in (0,41,,,-1),
44
5.8 and FRO, R is variable, the source in (0,0, 1),, ,ta
5,9 Ship. . . z. !J. r t;j) 7.; ; ";. 47
5.10 Ship section ; , 47
5.11 Panel distribution box .
,
.516.12 Surge added mass no lid.
,
53vi LIST OF FIGURES 5.13 Surge damping no lid 53
5.14 Surge-pitch added mass no lid. 53
5.15 Surge-pitch damping no lid. 53
5.16 Heave added mass no lid. 54
5.17 Heave damping no lid. 54
5.18 Roll-sway added mass no lid. 54
5.19 Roll-sway damping no lid. 54
5.20 Roll added mass no lid. 55
5.21 Roll damping no lid 55
5.22 Yaw added mass no lid 55
5.23 Yaw damping no lid 55
5.24 Surge added mass with lid. 56
5.25 Surge damping with lid. 56
5.26 Surge-pitch added mass with lid 56
5.27 Surge-pitch damping with lid. 56
5.28 Heave added mass with lid 57
5.29 Heave damping with lid 57
5.30 Roll-sway added mass with lid. 57
5.31 Roll-sway damping with lid 57
5.32 Roll added mass with lid. 58
5.33 Roll damping with lid. 58
5.34 Yaw added mass with lid. 58
5.35 Yaw damping with lid. 58
5.36 Surge added mass effect free surface panels. 59
5.37 Surge damping effect free surface panels. 59
5.38 Surge-pitch added mass effect free surface panels. 59
5.39 Surge-pitch damping effect free surface panels 59
5.40 Heave added mass effect free surface panels. 60
5.41 Heave damping effect free surface panels 60
5.42 Roll-sway added mass effect free surface panels. . 60
5.43 Roll-sway damping effect free surface panels 60
5.44 Roll added mass effect free surface panels. 61
5.45 Roll damping effect free surface panels 61
5.46 Yaw added mass effect free surface panels. 61
5.47 Yaw damping effect free surface panels 61
5.48 Surge added mass comparison original-lid 62
5.49 Surge damping comparison original-lid 62
5.50 Surge-pitch added mass comparison original-lid 63
5.51 Surge-pitch damping comparison original-lid. 63
5.52 Heave added mass comparison original-lid 63
5.53 Heave damping comparison original-lid. 63
5.54 Roll-sway added mass comparison original-lid 64
5.55 Roll-sway damping comparison original-lid 64
5.56 Roll added mass comparison original-lid 64
5.57 Roll damping comparison original-lid. 64
5.58 Yaw added mass comparison original-lid 65
5.59 Yaw damping comparison original-lid. 65
6.1 Panel description Sphere 70
6.2 Panel description free surface around Sphere 70
6.3 Surge-heave added mass coefficients of a half-immersed
sphere of radius a and Fr = 0.04. 71
6.4 Surge-heave added mass coefficients of a half-immersed
sphere of radius a and Fr = 0.04.
(No free surfacecontribution) 71
6.5 Surge-surge contribution to the added mass and the
damping coefficients of a half-immersed sphere of radius
a and Fr = 0.04. 72
6.6 Added mass coupling coefficients Fn=0.0 and Fn-0.05 72
6.7 Damping coupling coefficients Fn=0.0 and Fn-0.05 . 73
6.8 The surge exciting forces, without speed, computed by Nossen, the Haskind relation and pressure integration. 75
6.9 The heave exciting forces, without speed, computed by Nossen, the Haskind relation and pressure integration. 75
6.10 The surge exciting forces. 76
6.11 The heave exciting forces. 77
6.12 The surge motion 77
6.13 The heave motion 78
6.14 Drift force on sphere for Fn=0.0 and Fn=0.04 80
6.15 The drift forces on a restrained sphere. 83
616 The drift forces on a free sphere, without the stationary potential in the body boundary conditions 83
viii LIST OF FIGURES 6.17 The drift forces on a free sphere, with the stationary
potential in the body boundary conditions 84
6.18 The wave drift damping on a free sphere 84
7.1 Set-up of wave drift force measurement passive mooring 86
7.2 Spring characteristics mooring system. 87 7.3 Dynamic positioning set-up top-view 88
7.4 Dynamic positioning set-up cross-section 88
7.5 Set-up extinction test. 90 7.6 Decay curve surge in regular waves unfiltered. 91
7.7 Decay curve surge in regular waves filtered 91
7.8 Logarithmic decrement 92
8.1 Body plan of the 200 kDWT tanker. 99 8.2 Panel description of 200 kDWT tanker fully loaded. . 101
8.3 Panel description of free surface 200 kDWT tanker fully
loaded 101
8.4 Surge response in 180 degree waves, no current. . . . 102
8.5 Heave response in 180 degree waves, no current 102
8.6 Pitch response in 180 degree waves, no current. . . . 102
8.7 Surge response in 180 degree waves, 1.5 m/s current. 103
8.8 Heave response in 180 degree waves, 1.5 m/s current. 103
8.9 Pitch response in 180 degree waves,1.5 m/s current. . 103
8.10 Surge response in 135 degree waves, 1.5 m/s current. . 104
8.11 Sway response in 135 degree waves, 1.5 m/s current. . 104
8.12 Heave response in 135 degree waves, 1.5 m/s current. 104
8.13 Yaw response in 135 degree waves, 1.5 m/s current. . 104
8.14 Surge response in 135 degree waves, 135 degree current
at 1.5 m/s. 105
8.15 Sway response in 135 degree waves, 135 degree current
at 1.5m/s 105
8.16 Heave response in 135 degree waves, 135 degree current
at 1.5 m/s. 105
8.17 Yaw response in 135 degree waves, 135 degree current
at 1.5 m/s. 105
8.18 Wave drift force surge in 180 degree waves, 180 degree
8.19 Wave drift force surge in 180 degree waves, 180 degree
current at 1.5 m/s. 106
8.20 Wave drift force surge in 135 degree waves, 180 degree
current at 1.5 m/s. 106
8.21 Wave drift force sway in 135 degree waves, 180 degree
current at 1.5 m/s. 106
8.22 Wave drift force surge in 135 degree waves, 135 degree
current at 1.5 m/s. 107
8.23 Wave drift force sway in 135 degree waves, 135 degree
current at 1.5 m/s. 107
8.24 Wave drift damping surge in 180 degree waves. 108
8.25 Panel description of 200 kDWT tanker 40% loaded. . 109
8.26 Surge response in 180 degree waves, 1.2 m/s current. 110
8.27 Heave response in 180 degree waves, 1.2 m/s current. 110
8.28 Pitch response in 180 degree waves, 1.2 m/s current. 111
8.29 Surge response in 150 degree waves, no current. 111
8.30 Sway response in 150 degree waves, no current. 111
8.31 Heave response in 150 degree waves, no current 112
8.32 Roll response in 150 degree waves, no current 112
8.33 Pitch response in 150 degree waves, no current. 112
8.34 Yaw response in 150 degree waves, no current 112
8.35 Surge response in 150 degree waves, 1.2 m/s current. 113
8.36 Sway response in 150 degree waves, 1.2 m/s current. . 113
8.37 Heave response in 150 degree waves, 1.2 m/s current. 113
8.38 Roll response in 150 degree waves, 1.2 m/s current . . 113
8.39 Pitch Response in 150 degree waves, 1.2 m/s current. 114
8.40 Yaw response in 150 degree waves, 1.2 m/s current. . 114
8.41 Surge response in 150 degree waves, 150 degree current
at 1.2 m/s. 114
8.42 Sway response in 150 degree waves, 150 degree current
at 1.2 m/s. 114
8.43 Heave response in 150 degree waves, 150 degree current
at 1.2 m/s. 115
8.44 Roll response in 150 degree waves, 150 degree current
at 1.2 m/s. 115
x LIST OF FIGURES
8.45 Pitch response in 150 degree waves, 150 degree current
at 1.2m/s 115
8.46 Yaw response in 150 degree waves, 150 degree current
at 1.2 m/s 115
8.47 Wave drift force surge in 180 degree waves, 180 degree
current at 1.2 m/s. 116
8.48 Wave drift force surge in 150 degree waves, 180 degree
current at 0.0 m/s. 116
8.49 Wave drift force sway in 150 degree waves, 180 degree
current at 0.0 m/s.
8.50 Wave drift force surge in 150 degree waves, 180 degree
current at 1.2 m/s. 8.51 Wave drift force sway
current at 1.2 m/s. 8.52 Wave drift force surge
current at 1.2 m/s. 8.53 Wave drift force sway
current at 1.2 m/s.
8.54 Wave drift damping in surge in 180 degree waves. . . .
8.55 Panel description of 200 kDWT tanker 70% loaded. . .
8.56 Wave drift damping in surge in 180 degree waves, 180 degree current at 1.2 m/s.
8.57 Wave drift force surge in 150 degree waves; Fn=0.02 150 degree current, Fn=0.02.
8.58 Wave drift force sway in 150 degree waves; 150 degree
current, Fn=0.02.
9.1 Approximation wave drift damping surge using Aranha's
expression for a floating sphere. 126
9.2 Approximation wave drift damping surge using Aranha's
expression for a 200 kDWT in head waves and current. 126
10.1 Estimates of wave drift force using extinction tests. 128
in 150 degree waves, 180 degree in 150 degree waves, 150 degree in 150 degree waves, 150 degree
122
122
A.1 Computational domain integral equation 140
C.1 Integration over a Panel j. F. A A 150
C.2 Projection onto Panel j. . /EA CA A 150
List of Tables
511 Location irregular frequencies for a square box. . 52
7.1 Model test conditions for current speed OM m/s T100%
in 180 deg. waves. 93
7.2 Model test conditions for current speed 1.5 m/s T100%
in 180 deg. waves and 180 deg. current direction.. . 94
7:3 Model test conditions for current speed 1.5 m/s T100% in 135 deg. waves and 180 deg. current direction. . . 94
7.4 Model test conditions for current speed 1.5 m/s T100%
in 135 deg. waves and 135 deg. current direction. ,. . 95 715 Model test conditions for current speed 1.2 m/s T100%
in 180 deg. waves and 180 deg. current direction. . . 95
7.6 Model test conditions for current speed 1.2 m/s T40% in 180 deg. waves and 180 deg. current direction. . 96i
7.7 Model test conditions for current speed 1.2 m/s IT40 %
in 150 deg. waves. 96
7.8 Model test conditions T40/ 97'
7.9 Model test conditions T40% for current speed OM m/s
in 150 deg. waves. 97
7.10 Extinction test conditions for current speed OM m/s
T40% in 180 deg. waves. 98
7.11 Extinction test conditions for current speed OM m/s
T70% in 180 deg. waves'. 98
8.1 Particulars of the 200 k DWT tanker fully loaded.,
, .
100 8.2 Particulars of the 200 k DWT Tanker 40% loaded ., A ,. 109 8.3 Particulars of the 200 kDWT tanker 70% Loaded. . 120Chapter 1
Abstract
In this study results are reported on the modelling of the wave drift
forces on a vessel in regular deep water waves with forward speed. In chapters 3 and 5 the mathematical background of the boundary integral method for the computation of the first order and second order wave drift forces on floating bodies at low forward speeds is presented. By means of the Green's theorem a source distribution is derived. The Green's function (source function) and source strength are evaluated asymptotically for small values of the forward velocity. Also, the forward speed Green's function is linearized with respect to forward speed. The first two terms of the source strength over the mean wetted surface of the body is then computed from two sets of integral equations. The kernel of these sets of integral equations has the same form as the integral equation for the zero speed problem. In addition to the zero speed problem, a free surface integral enters the right hand side of the integral equation for the source strengths. In the
development of the linearized forward speed Green's function with
re-spect to forward speed, corrections on the asymptotic approximation are also given in order to arrive at a proper uniform expansion with respect to forward speed. Since the encounter frequencies are usually higher at forward speed than at zero speed for head on and bow
quar-tering waves, the effect of 'irregular frequencies' is also described. A robust lid method is put forward to solve the effects of this 'irregular frequency' problem. The mean wave drift forces are found by a far
field analysis. The results of the wave drift forces on a floating sphere in regular waves compare favourably to the results of the study of Zhao
and Faltinsen [82]. To validate the approach for the determination of
the wave drift forces, model test experiments were performed on a 200
kDWT tanker in fully loaded as well as balast condition. From the comparison with the results of model tests it is concluded that the
linearized forward speed description works well for head current cases
Chapter 2
Introduction
In the exploration and production of oil and gas in offshore locations more and more use is made of moored floating vessels. The
introduc-tion of the first floating vessels as producintroduc-tion platforms was motivated,
among other reasons, by an absence of a pipeline infrastructure in the vinicity of the oil wells. Nowadays, however, the increasing capital costs of a fixed platform for deep water oil production and the need for environmentally safe removal of the platform once production has
stopped provide further incentive for the development of moored
float-ing production systems. As the moorfloat-ing system has to withstand the
forces of wind, waves and current, a lot of emphasis has been placed in
the last twenty years on reliable assesment of the motions caused by environmental conditions. Given that the mooring system is likely to encounter severe environmental conditions in its service lifetime, the real need is for the assesment of the motions of the floating vessel in such conditions, which requires state-of-the-art numerical techniques. The full nonlinear treatment of the flow around the vessel in these severe wave conditions, including eg viscous effects, is still very far from practical application. Therefore there is still a need for a linear
approach tothe fluidbody interaction, in which the essential details of the fluidbody flow is maintained. The applicability of such linear approaches should however be validated against model test experi-ments, in which the environmental conditions can be controlled and
monitored more reliably than in real life. In the assesment of the loads
on the mooring system of the vessel essential work has been accom-plished in the past by van Oortmerssen [69], who calculated to first order the reaction of the vessel due to motions and waves using a lin-earized frequency domain potential flow theory. He showed that it is possible to compute in time domain, based on frequency domain re-sults, the nonlinear forces of the mooring system acting on the vessel.
The hydrodynamic reactions of the vessel were described by means of
convolution of the motion velocity with retardation functions. One of the properties of a catenary moored floating vessel is that the natural
period of the mooring system is very high (from 50 seconds up to
sev-eral minutes). Using a linear model of the fluidbody interaction one does not arrive at the correct excitation at the natural period. Several authors [54, 14, 48] have pointed out in the past that the weak non-linear fluid-body interaction is responsible for the excitation at the natural period of the system. A pioneering effort was performed by
Pinkster [54] in which he carefully derived a pressure integration tech-nique to arrive at the second order wave excitation (meaning quadratic
with respect to the incident wave height) in regular waves. He also
derived approximate expressions for the second order wave excitation
in bichromatic waves. The correctness of Pinkster's approximative approach for the low frequency excitation was demonstrated by Ben-schop et al. [5] and Yue [38]. As a consequence of the motions of the
vessel in the vinicity of the natural period of the system, not only the
wave excitation is of importance, but also the damping of the complete
mooringvessel system. In the case of a first order process we see that the amplitude of the surge motion of the moored vessel at the natural
frequency is given by
Flu(p,)
lx(1.1)1 = pBxx(p)
where it is the natural frequency of the mooring system in surge (x),
Hrs(p,) is the damping of the mooring-vessel system at the natural
fre-quency and Fritz) is the wave exciting force at the natural frefre-quency Since by nature the moored vessel itself has very low damping in the surge direction (minimum resistance) all other possible sources of
damping will influence the surge motion of the vessel as well.
7
internal turret external turret
Figure 2.1: A typical mooring layout of turret-moored tanker
An inventory of contributions to the damping has been given by Wich-ers [75]. He demonstrated, by means of carefully performed model test experiments, that the resonant low frequency motions were influenced
by wave drift damping, which is caused by the presence of high
fre-quency first order waves and first order motions. In earlier work Re-mery and Hermans [58] had already indicated that not only a correct description of the wave drift forces at resonance was necessary, but also an accurate description of the damping. This gave an indication of a complex interaction between first and second order motions. One of the findings in the work of Wichers et al. [75, 32, 77, 79] was that one could relate this low frequency damping, also called wave drift damping, to the resistance increase of a vessel sailing in waves. An-other important damping effect was reported by Huse [33]. He showed
that high frequency oscillations at the top of a catenary also lead to contributations to the low frequency damping on the moored vessel.
Experimental verification of this effect has been reported by Huse [33]
and Wichers et al. [76]. This effect shows that one cannot simply
decouple the low frequency motions from the high frequency chain dynamics (see Huijsmans et al. [32]).
surface and the body boundary at each time step. First attempts at
modelling the flow following a complete non-linear description of the free surface flow have been performed amongst others by Romate [60] and extended to large body motion by van Daalen [70] and Broeze [9]. Possible weak non-linear extensions from linearized potential flow
de-scriptions of the three-dimensional motions of a vessel in waves were
discussed by Beck [4]. Experience in the use of such nonlinear
.41
Figure 2.2: Undisturbed stationary flow
approaches as reported by Beck [4] has demonstrated that the
compu-tational effort required effort is still very large. In addition, problems were reported by Romate [59], Broeze, and Van Daalen [9, 70] with respect to the correct modelling of the evolution of the boundaries at intersection points. In this area a lot of progress is still required
to bring such non-linear flow models within reach of engineering prac-tice. Approximate forward speed linearized potential flow models have
been put forward to model the wave drift force at low forward speed [29], [23],[55],[82]. Forward speed diffraction effects have been
mod-elled using boundary element methods using the Green's function for a
translation oscillating source. The first effective results were reported
amongst others by Chang [10] , Bougis [6], Inglis [34]. The main
draw-backs of their approach were that the Green's function was (and still is) very cumbersome to compute and stationary potential flow effects
-Ux + (ps
.st
Figure 2.3: Disturbed stationary flow
were not accounted for. The linearization of the flow was taken around
the undisturbed incoming stationary flow Ux instead of around the disturbed stationary flow Ux , as visualized in figures 2.2 and
2.3.
First attempts at using a low forward speed formulation were given by Huijsmans and Hermans [30] and Grekas [16]. In order to model the wave drift forces in regular waves Huijsmans [27] used a pressure
distri-bution integration technique, including disturbed stationary flow field effects. This technique had been developed earlier by Pinkster [54] for zero speed wave drift forces. One of the problems associated with the pressure distribution integration technique that was encountered, especially for the forward speed pressure distribution integration, was
the correct treatment of the derivatives of the water velocities over the
body boundary. Using a constant source panel description for the
wa-ter velocities on the body boundary requires a numerical differentation of the water velocity over the body boundary. This may lead to
incon-sistent results. Therefore an alternative formulation of the wave drift forces was derived [23], [29]. This alternative formulation was based
on conservation of impulse and energy considerations as derived by Maruo [41] and Newman [44]. One of the advantages of this formu-lation of the wave drift forces was the absence of the derivatives with
respect to the water velocities. The formulation for low forward speed
was also independently derived by Nossen [52] and Grue et al. [17].
Grue [20] also gave an expression of the mean yaw moment in regu-lar waves. One drawback of using the alternative formulation of the wave drift forces is that it is then not possible to model the wave drift
forces in hi-chromatic waves. If such results are needed, especially for shallow water applications, one has to refer back to the pressure integration techniques. Using time domain type of methods for the linearized potential flow problem Prins [56] gave an accurate account of how wave drift forces could be calculated using a pressure
distri-bution technique including low forward speed effects. Sierevogel et al.
[64] extended this linearized time domain approach to higher forward
speeds. One disadvantage of the time domain algorithm is that it
creates a large computational burden, both on memory as well as on
CPU. One way of overcoming this problem is to solve the ship motion
problem in the frequency domain. In linearized theory the time do-main solution and the frequency dodo-main solution are equivalent (see Cummins [12] and Ogilvie [53]). We shall therefore reformulate the ship motion problem into the frequency domain.
This study is confined to the low forward speed in regular water waves
Chapter 3
Mathematical formulation
In order to calculate the hydrodynamic forces on the vessel, we
de-velop an expression for the pressures on the vessel. Assuming that the flow is irrotational and no viscous effects are present, we are able to describe the ship motion problem in a potential flow formulation. In this chapter the velocity potential is written as the summation of a steady and a non-steady part. Also, the integral equation and the free surface condition for the potential are derived.
The formulation of the ship motion problem is presented in thefirst section. In the second section the velocity potential is presented and the non-steady part of the velocity potential is described. The third section deals with the boundary condition on the free surface, the
fourth section presents the body boundary conditions. The last section
gives the general equations for the steady potential and also explains
how the derivatives of the steady potentialare obtained.
3.1
Problem formulation
The object of this study will be a floating vessel, sailing in deep water
in the presence of waves and current. We assume that the forward speed effects can be modelled analogously to the effects of current. This means that towing the vessel in waves or applying current and waves onto the vessel can be interchanged with an appropriate notion
of the wave frequencies involved.
We shall use a coordinate system fixed to the vessel, such that the undisturbed free surface coincides with the z=0 plane. The ver-tical zaxis is positive in upward direction. We distinguish between frequencies in the earth-fixed coordinate system denoted as coo and frequencies of encounter co in a ship-fixed system. One of the main purposes of this study is to model and compute the hydrodynamic interaction of the vessel with waves and current.
In earlier work of ship motion studies of eg Bougis et al. [6] and Inglis
[34], the basic linearization of the flow was around the ambient flow U, not taking into account the effect of the change of the stationary flow field due to the presence of the body. Basically this means that geometrical restrictions limit the applicability of such a linearization scheme. However, for ship type vessels with relatively large length over beam ratios ( larger than 3) sailing at zero drift angles, this ap-proach became a widely used approximation. Socalled strip theory type of ship motion theories have been developed based on these as-sumptions. In the wave resistance type of ship problem as formulated by Dawson [13] and discussed by eg Raven [57] and Jensen [35], the double body flow became the linearization point of the formulation of the wave making potential. In cross flow conditions, which are very frequently encountered by offshore moored vessels, the linearization around the double body flow is required. For floating bodies with sharp corners at the stern or bow of the ship, the basic flow around which the linearization is taken may also include viscous effects. The way the vorticity is generated at these sharp edges must be taken into
account.
In the present approach no viscous effects are included, which means that no vorticity will be present in the fluid, nor will it be generated due to boundaries in the fluid.
From Newton's law we derive the motions of the vessel. dM_Y_ F
Lit
3.1. PROBLEM FORMULATION 13
integration of the pressure on the wetted part of the vessel S.
F
t =- prkelSNeglecting all viscous effects in the fluidvessel interaction the
pres-sure distribution can then be calculated using Bernoulli's equation. We assume the fluid to be irrotational, therefore we can introduce a velocity potential 4, describing the local velocity in the fluid by V4. After integration of the Euler equations the Bernoulli equation reads:
1
P = Po pcDt
2pIV .1.12
pgz + U2
2In order to calculate the forces on the body we need an expression for the potential 4. The potential 4 satisfies the continuity equation in the fluid leading to Laplace's equation.
Ad, 0
The vessel is moving in a fluid bounded by the free surface and the sea floor. The free surface is an unknown quantity at first. At the
free surface the kinematic and dynamic conditions are satisfied, which
state that once a fluid particle is in the free surface it will not leave the free surface and the pressure at the free surface stays constant. The conditions at the free surface now read:
1\74>12 -FkW2 = o
= 0 At the free surface z = C(x,t)
6 + vcl.
(3.1)
The boundary conditions at the wetted part ( S ) of the vessel states that no flux of water is entering the vessel.
(V .72,)4 = 0 at S (3.2)
where the velocity U is taken with respect to the coordinate system
fixed to the vessel in the average position. Since we confine ourselves
to the ship moving in deep water we shall not impose a sea bottom
boundary condition. The problem defined so far, is still very
attempts have been made to include nonlinear effects such as
pre-sented by Broeze [911 and Van Daalen [70] for three-dimensional wave
problems and two-dimensional ship motion problems. (For a review
see Beck [4].) Inclusion of forwardspeed effects in the mathematical modelling of the three- dimensional ship motions is still a major task.
We shall therefore linearize the boundary conditions around small
am-plitude ship motions and small amam-plitude incoming waves. Prins [56] recently presented a time domain solution procedure for the linear three-dimensional ship motion problem with forward speed. He used rankine sources distributed over the free surface and the mean wetted part of the floating vessel to describe the evolution of the free surface
and the vessel motions with time.
Artificial boundary conditions at infinity are required to close the com-putational domain. Sierevogel [63] derived time independent artificial
radiation conditions. One disadvantage of the time domain algorithm 'is that it creates a large computational burden, both on memory as well as on CPU. One way of overcoming this problem is to solve the
ship motion problem in the frequency domain. In linearized theory the time domain solution and the frequency domain solution are equivalent
(see Cummins [12] and Ogilvie [53]). We shall therefore reformulate the ship motion problem into the frequency domain.
3,.1.1
Linearization of the free surface condition.
Combining the kinematic and dynamic boundary conditions (3.1) at
the free surface gives 1
+ (V$ V$)t
V(VS VS) + giSz = 0 at z= (3.3)al. PROBLEM FORMULATION 15 Using (b\ 1
ta24)\
= 4.(x,y,o,t) + c
(Tz)z.0
+az2)z.0
(3.4) For we write:=
+ -1-c74, vck) + (3.5) 2 Equation (3.3) becomes at z 0: 1+ oz+ (vt-vcp)t -
-1 + -1v(-
r12) 2(ct+ gz)z +
+V(I) V(Vcb 2V43)=0+ OW)
(3.6)For the purpose of linearization we decouple the potential into a steady and a non-steady part.
43(s, t) = 0(x) + q-5 ( ,t ) (3.7)
The free surface condition now reads, after retaining only the terms
linear in .(75 and quadratic in q5 :
(-1511 + 2(Vcb Vq-51) Vcb) + gc75.2
+c7i- v(v(
v-(75)-
vt,- u2)(qsz, + c;z)+
q5,(Vcb- VOH- c-bt) = Oat z=0 (3.8)
3.1.2
Linearization of the body boundary
condi-tion
The body boundary condition can be linearized as follows:
Assuming small oscillatory motions of the vessel we apply the body boundary conditions to the mean wetted surface of the vessel( S ) For the steady potential ç:
(V -m) = 0, on the body S
is1)(x,y,C,t)
g
and for the nonsteady part
(V n)(Tk = at [(IVO '1/47)cx
(a V)V0]
7_1_a denotes the displacement of the vessel at some point ( x ) on
3. In
the remainder we denote the mean wetted surface still as S or it will be clearly stated in the context.
In terms of translation and rotation with respect to the center of gravity a is defined as
a ---- X -I- c x (x
where X and Si are the translatory and rotational motions of the vessel in the center of gravity. These equations were first derived by Timman and Newman in 1962 [68].
3.2
The potential function
In this section the velocity potential is described and the time depen-dent part is split into a diffracted and a radiated part.
The following restrictions apply for the flow around the vessel:
The fluid is an ideal fluid, there is no viscosity.
The fluid is incompressible and homogeneous.
The fluid has an irrotational motion.
There is a gravity force field g.
The depth h is supposed to be infinite.
The fluid velocity u is expressed by the gradient of a velocity potential
3.2. THE POTENTIAL FUNCTION 17
The fluid is incompressible and homogeneous which states V it = 0. Now the potential function 4' satisfies Laplace's equation in the fluid
domain.
V243(x,t) = 0 for x E fluid domain (3.10)
The total velocity potential function will be split into a steady and a
non-steady part.
4'(_,t;U) = U + (/)8( .;ti) + (3.11)
In this equation U is the incoming ambient flow field, obtained by considering a coordinated system fixed to a ship moving under a drift angle ac with defined as:
= x cos a, y sin ac
Figure 3.1: System of co-ordinates.
Here 0.(x; U) is the steady disturbance defined as 08(x; U)
= Ux .
The time dependent part of the potential -(4 consist of an incoming and diffracted potential cD and radiated wave potential OR time har-monic with frequency co, where ce is the frequency in the coordinate system fixed to the ship, also called frequency of encounter.
It is convenient to separate OR into contributions from all the 6 modes
Waves
,t;:(1.)
Figure 3.2: The coordinate system and the six modes of ship motion.
of motion: surge, sway, heave, roll, pitch, yaw denoted as either
j=1,6 or as
The radiation potential due to the motions of the body may be written as
6
t; U) = et E ejcb;(x; U)
(3.12)j=1
where is the motion in the jth mode and, Oi is the corresponding
potential.
The wave potential ii)D will be split into a diffracted wave potential 07
and an incident plane wave potential q5.0 due to the incoming waves. We will assume the incident waves are harmonic in time.
t;1/) e -iwi(00(x) q57(_ ;U)) (3.13)
with 00()gCa ko[2-1-ix cos )3-4-iy sin .3] (3.14)
wo
where is the amplitude of the wave height of the incoming wave in direction fl. The frequency and wave number, wo respectively ko =
-`4 = -2--7 are in the ship-fixed coordinate system. The relation between
the earth-fixed and ship-fixed frequency c4)0 and w is as follows: The
x,y,z,0,0,0
3.3. THE BOUNDARY CONDITION ON THE FREE SURFACE19
frequency in the earth-fixed coordinate system is:
w = koU cos(/3 ac) (3.15)
3.3
The boundary condition on the free
surface
The vertical elevation of any point on the free surface may be defined by a function z = C(x,y,t). Newman[47, chapter 6] shows that the effects of the free surface must be expressed in terms of appropriate boundary conditions on this surface. In this section the free surface condition is derived in the frequency domain.
The free surface condition for the non-steady part of the velocity
po-tential can be computed by using equation (3.11) in (3.8). In appendix
D the derivation is presented, applied for head current.
The free surface condition now becomes formulated in the frequency
domain:
w2(75+2icoUx + U2 + = 0 at z =- 0 (3.16)
where D(x,-c-b) is a linear differential operator acting onq 5 as defined
in the appendix (D).
We assume -q-5(x,t;U) to be oscillatory (see section 3.2).
6
t; U) = (-tin R (00 + (1)7
E
ox.i)e " = (k(-;
2=-1
(3.17)
From the appendix (D) we have the following expression for D(x,q-5),
neglecting nonlinear terms and U2 terms:
D(x, )=-vxv-(x.,-+xyy)
(3.18)We apply the Green's theorem to a problem in D, inside S and to the problem in De outside S, where S is the ship's hull. The potential
+
function inside S obeys condition (3.16) with D".10, while far away from the body the free surface condition reads:
1u)20,-E 2ica/Ox u20,rx + = 0
(340
The 'derivation of the Green's function will be treated in the next
chapter. In chapter 4 equations (3.16) and (3.19) will be used to derive
a source and vortex distribution. The Green's function will satisfy the homogeneous adjoint far field free surface condition as defined in
equation (3.16).
The body boundary conditions for the diffraction problem have to be
treated carefully, since the incoming wave potential 00 does not satisfy
the free surface condition (3.8), while the incoming plus diffracted
wave potential does satisfy the free surface condition (3.8 ). Denoting
the linear homogeneous far field free surface operator as C, we then
write:
+ ibd)i p (x,40 + ("Aid) (3.20)
which then results in the following free surface condition for(-Ad,
cor-rected for the incoming wave potential, with
(04
=Icyj,d)
= 2iwUD
2iwU D (x,;(,) (3.21)3.4
The body boundary conditions
In this section the body boundary conditions are further defined. The body boundary conditions for the unknown radiation and diffraction
potential are (Newman[491):
acki(x;U)
iwn; Umi
= 1,(3,22)
ann an = 7
where
the Cartesian components of the normal vector n j -= 1,2,3 =
3.5. THE STEADY POTENTIAL 21
and
withx=ç.
The normal derivatives of each radiation potential consist of a part that represents the normal velocity at the mean position of the body and a part that shows the change in the local steady field due to the motion of the body.
The derivation of these body boundary conditions follows from the
work of Timman and Newman [68].
In appendix E the terms are written in terms of the derivatives of the steady potential and the normal vector. The m,-terms also consist of second derivatives of the steady potential.
The computation of these second derivatives is a difficult problem as
many authors (see Zhao and Faltinsen [82]) have outlined. To take the second derivative of a potential function defined as constant over a flat panel is inconsistent. One way to avoid numerical difficulties in
estab-lishing the second derivatives is to use eg higher order panel methods or just an approximative scheme, which calculates the flow in points at a certain distance from the vessel, and then perform the differenti-ation numerically and extrapolate the results to the body surface. In 2-D Zhao and Faltinsen [82] have shown that this approach will give correct results. Wu [81] adapted the integral equation for the steady
velocity, which he showed led to a higher accuracy of the derivative of
the steady velocity.
3.5
The steady potential
This section gives the conditions of the steady potential, as used by
Hess and Smith[25].
The steady part of the velocity potential is given by U(x cos ot,
y sin ac) Ux, where U (x cos ct, y sin °cc) is the ambient uniform
= (n, V) (V(x
(x cos a, y sin ac)))j =
1,2,3= (71 V ) (x x V(x (x cos a, y sin a)))
j = 4,5,6
=
amis. .... 'Wee
e....)::-...-:-....
,...e....:.,."...c..r. Z ...z.. e -. _..-41111.:e 4010...,:"Ltet'S4IeZre:/sa°.Cli.a..
apeteare....,,Crry,.../...."...0... 411/ crs, ..., ,...e...e...0. atereree:00 'P. 4.4101.41'. ..../".42.4110... ..". ea 4Paro....-1":"Ann. "GreCers dite Citami.
41Gror ..eAres. ...are eteraller 'OAPs
Figure 3.3: 'Steady wave system
current and Ux is the steady disturbance due to the body. Therefore
by definition:
= U (x + x cos ac + y sin ac)
The steady potential fulfills the body boundary condition.,
Ox(x)
art
_
'gni cos ct, + n2 sin GO on SIn the Hess and Smith[251 approach a source distribution is used. x(x)
may be written as:
x(4 =
1 a81(4)as,
47r s
_
0.23)
(3.24)
If the Froude number Fr = 1,, With / being the characteris-tic dimension of the body, the free surface condition for the steady potential is the classical double body flow condition, approximated
by:
(x) at z = 0 1(3.25)
az
In stationary flow problem (ship wave resistance type problem) a steady wave system from the vessel emerges significantly unless the
3.5. THE STEADY POTENTIAL 23
Fn is very small. As an example of such a wave system the wave pattern ,around a vessel is depicted in figure 3.3. Now we can write:
21rers(x)+11On
(-1)-r
ce(e)dSe = 41r(ni cos ac +n2 sin ac) (3.26)-with r From this expression it is evident that we only need two independent updates of the source strength with respect to the current direction. The source strength for any other current direction can then be readily determined. Denoting the source strength for ac = 0 as u. a, = 90 as crl, it follows that
crs(ac) = cr(1' cos ac + Sin an
To compute the body boundary condition we need an expression for the derivatives of the steady potential. The first derivatives of the steady potential or the velocities can be determined without any diffi-culty from the expression of the potential as a source distribution. In order to determine the second order derivatives of the steady poten-tial, use is made of a numerical differentiation scheme
The advantage of using a source distribution over a Morino formula-tion originates from the fact that the steady velocity is determined with the same accuracy as the source strength Here the numerical differentiation is taken of the steady velocity. In the work of Prins
[HI for the solution of the ship motion problem in time domain, use is made of a numerical scheme to determine the double derivative of the
steady potential. He reports that this must be done with great care.
-
.Chapter 4
Expansion of the potential
In this chapter the expansion of the potential is derived, using the integral formulation (the free surface condition, equation (3.16)) and the Green function of chapter 5.
The first section treats the way Hermans and Huijsmans[24] derived the potential using a source distribution. The last section deals with the amplitude of the potential for the far field.
4.1
The integral equation
First the source strength is computed by using the free surface con-ditions. An expansion of source strength is then derived. With the source strength we are able to compute an expansion of the potential. Combining the formulation inside and outside the ship, equations (3.16) and (3.19), we obtain a description of the potential function defined outside S by means of the source and vortex distribution. The formulation is an extension of the one found in Brard [8].
IL
s-y(e)-1G(x,)d, 5 + fiscr(e)G(x
,OdSe 2iwg ULL(0G(x
, )chi ++Ug2 fw
[y(0 a
aeG(x, at-MO aT7T(e)}
di +
U2
+ anc(e)G(x, + 2ical
f G(x
,0D(x, OcISc = 471-0(x_) (4.1)g wt, g FS
where at = cos(0x,t), aT = cos(0x, T)
, a, =
cos(0x,n) and where it is the normal and t the tangent to the waterline, and T=-t x it the hi-normal. Here G(x,) is the Green's function satisfying thehomo-geneous free surface boundary condition.
Figure 4.1: Coordinates at waterline.
It is clear that with the choice -y() = 0 the integral along the wa-terline gives no contribution up to order U. The source distribution we obtain this way is not a proper distribution, because it expresses
the function 0 in a source distribution along the free surface with a strength proportional to derivatives of the same function q5. In order to include the effect of the operator D on the integral equation, an iterative approach should be adopted. Based on updates of the poten-tial 0 and the velocity V0 in the free surface an iterative procedure could be formulated, but this approach has not been followed here. However, the formulation is linear in U and the integrand tends to zero rapidly for increasing distance R.
So finally we arrive at the formulation:
fiscr(0G(
)dSe + VLancr(OG(x,4")chi +
4.1. THE INTEGRAL EQUATION 27
Using the body boundary conditions, which are worked out in equation
(3.22), at the mean position of the hull
644)
= Vq5. n = V(x) at E S (4.3) Onand taking
at
of equation (4.2) and taking the limit of xED, to
X E S r
ac(T_,)
u2 f
ac(x,
)a(e)
dn anx 27r o-(0fsa(0
dS g WI, 2iwU ag IFS OniG(x_,OD(x, ck)dS 47rV(x) at E S (4.4)
D(x, 0) is the linear differential operator acting on 0 as defined in appendix (A). The quadratic terms in x are neglected. So D(x, 0) is
VX7745
Xyy).The normal derivative means the normal derivative with re-spect to x = (x,y, z).
The formulation obtained thus far does not give any new view on the integral equation with forward speed, except that the free sur-face term has been added. Apart from the steady potential influence equation (4.4) is equivalent to the formulations used by Inglis [34] or
Bougis [6]. The Green's function as it appears in equation (4.4) is still the translating oscillating source as formulated by Wehausen and Laitone [74] and subsequently used for the ship motion problem by Inglis, Bougis and others. One of the main drawbacks of the use of this Green's function formulation is the rather cumbersome way it is computed. To date little progress has been made in trying to com-pute this Green function as efficiently as, for example, in the zero speed Green's function demonstrated by Newman [46] or Telste and
Noblesse [66]. Therefore, we shall impose an additional restriction on
the use of this Green's function: it will only be applied for low speeds or more correctly for low Brard numbers (7 = < 1/4).
It is interesting to note that in classical forward speed formulation, in which the steady potential is neglected, a contribution over the
x
=
waterline is seen. Careful analysis by Nossen [52] did show that this term is cancelled once the steady potential is taken into account.
We consider small values of U, such that 7- =
<' The source
4strength and potential function will be expanded as follows:
crJ(e, U) = a-J0(4)+ Tail(e) + 0(72) (4.5)
U)
= 0j0() +
j1W + 0(72) (4.6)And for the Green's function we write:
G(x,) =
+ rq,b1(, ) + (4.7)where Go(x,e) is the zero speed Green's function.
Go(x,) =
+We now can write equation (4.4), at x E S, for j = 1, , 7:
OG(4,
-27rerio(0+llscrio(0
an,x°
47r0(x)
(4.8)and
azPi(x,e)
-27,731(0 f iscr3i(049G 00n(T.,_0ds,= f fso-,o(4.) an-x- dSe +
+2i
[1a
S 8nxG0(
o0D(x, )dse+ 47rVJI (x)
(4.9)
where Go(x,e) ,- ..+.7.1)0(x,O, with 'tiro is the zero speed pulsating wave
sources, and 17J(x; U) = 1'io(x)+7Vi1(x)d- 0(72) as in equation (3.22).
= = { -iwn3 000
- an
j= 1,
,6j = 7
{!
mij = 1,
...
,6 2i fFS -22- G° D(x'00)dSej = 7
an.. (4.10) (4.11)The subscripts Jo and ji mean respectively the zero- and first-order expansion in the jth mode of motion.
e)
e)D(X,
4.1. THE INTEGRAL EQUATION 29
In solving the integral equations (4.8) and (4.9) we encounter the
problem of the irregular frequency phenomena. The existence of these
irregular frequencies for the water wave problem dates back to the
publications of John [37, 36] in 1950.
Since the ship motion problem is formulated at forward speed, this
gives rise to encounter frequencies higher than the frequencies nor-mally used in the ship motion problem at zero speed for head and bow quartering waves. As will be made clear later on, the effect of " irregular frequencies" will influence the numerical solution of the problem. These socalled irregular frequencies tend to enter the ship motion problem at higher frequencies (this however actually also de-pends on the geometry of the vessel). Therefore attention will be paid to reduce the effect of the irregular frequencies on the ship motion
problem..
For lir we see that 00 depends of ca and coo, which in turn depends
on the speed of the vessel. Since 0,0 does not satisfy the free surface condition (3.8) we correct the 07 in order to let 07 -I- or, satisfy the free surface condition (3.8). From the analysis in chapter 3 we see that the last term in (4.11) gives the necessary correction. We have to take care that ta does not become too small, because then the factor becomes too large. When we use an asymptotic expansion, the terms
of the expansion have to be of the same order. So a small w makes the
first order term become much larger than the zero order term. In that case we are trying to make an asymptotic expansion for both small T as well as small co. A similar approach, but for small (ay has been given by Van de 'Stoep [74
The potential function equation (4.6) now becomes,:
,T71r crp(e)go(x,e)dSE
(.12)
417r crjo(e)?1(x,e)dSe :tits 0-,b(e)G.(;,ods,
it
Go(x,e)Dcx,ods,
(4.13)
we can evaluate 03 and g53 from the equations (4.12) and (4.13).
From the body boundary condition can be seen that the second
deriva-tive of the steady potential must be computed. Section 3.5 describes how we arrive at a numerical value of these quantities.
The integral equations for the source strength co and al are solved using a conventional panel method. We approximate the mean wet-ted surface of the body by quadrilateral panels for which we assume constant source strength over the panel. The integration of the rank-ine source part of the Green's function was discussed by Fang [15]. Because of some misprints in his original publication, the integration over a panel of 1/r and (1/r) is reiterated in the appendix C. The frequency dependent part of the Green's function is integrated using Euler. The algebraic equations that follow after the discretization of the integral equations can be solved either using classical LU
decom-position or by using an iterative solver. The latter is more useful if one
requires large number of panels. The iterative scheme used is based on a method published by Sonneveld [65]. His method pivots around the use of conjugated gradient type of methods for non-self adjoint operators. For a review of these type of methods one is referred to the
work of Van de Vorst [73, 72].
In the right hand side of the integral equation (4.9) for al, an
integration has to be made over the entire free surface around the body. The extent of this integration over the free surface is, however, limited due to the fact that the steady potential disturbance behaves like a dipole, the integrand decays like R-1 with R being the polar distance to the vessel. An example of the extent of the influence of the stationary potential over the free surface is given in figure 4.2.
4.2
The amplitude distributions of the
potentials
4.2. THE AMPLITUDE DISTRIBUTIONS OF THE POTENTIALS31
Free surface Velocity in 150 deg current
Figure 4.2: Stationary potential in free surface around a 200 kDWT
tanker in cross-flow conditions.
is used to compute the drift forces.
Far away from the body, we have a radiation condition stating that
03 must behave as outgoing waves:
1 - fel (0) iz+ifils/1-47-2 sin2
H;(0)e
The far field approximations for the Green's functions are given in equation (5.30).
CL U)
1
fle
for j 1, , 7 (4.14)
ict (0) {z±tiW1-47-2 sin2
With the amplitude 8
,h(e, 9).
k1(9)eki(e);[c+iev cos 5=2r sin2o)-Fin(sin 5-1-27 cos sin 6)11-i-'41It
where
/01(9) = 27- cos O)+0(7-2)1
0.10
The function H results from the asymptotic expansion of the far field potentials in equation (4.2).
4r0.14) f iscr(e)G(1,0dSe
+2irifG(x,,e)D(x, 450)dge
FSSo the 'amplitude H of the potentials becomes
Ha(0)
+-1
cra(e)h(e,O)dSe4r
s2ff h(e,O)D(x, cho)dSe
0.16)
r
FSwith h(e,03).as in equation (5.31).
The amplitude of potential His the result of the &Mowing equation:.
47r0i =
fis kJ!
dS+2irifFS kfraD(x, cfr4j dS (4.17)
So the amplitude H 'of the potentials becomes, with Tucks theorem::
Hi(9) =
J Jr k 'Oh
(h V(xni dS
4r
s an 2/Cjj
[031)(x,h)] dS' '(4:18) H7(9) = ff icbDahdSir f
D(x, h)d.S (4.19) 47r s an 27r FSWe need the sum of the Ha's to compute the drift forces. We define:
Chapter 5
The Green's function
To solve the integral equation (equation (4.4) ), we have to compute the Green's function. Once the expression for the Green's function is found we then can compute the source and the potential distributions.
The first section of this chapter' gives the asymptotic expansion of the
Green's function. In the second section the zero order Green's func-tion 7,bo is treated. The third secfunc-tion gives two ways to compute the first order Green function '01: a transformation in the complex plane and an expression based on the derivatives of the zero order Green's
function. The fourth section deals with the non-uniformity of the
first-order Green function. In the last section we derived the derivatives of the first-order Green's function. We need these derivatives for the potential expansions in the next chapter.
5.1
The expansion of the Green's
func-tion
In this section we present an asymptotic expansion of the Green's function. The Green's function has to satisfy the conditions on the velocity potential 4)(x, y, z,t) = G( ,e;U)exp(iwt):
'The subscripts and / in this chapter are the terms of the asymptotic
expan-sion and not the modes of motion as in the preceeding chapter.
1. '724, =- 0,,
z < 0, '(,y, z)
(4,Th 0 (Lapface's equation)''2. Ste + 2E14. xt E124.xr y, t). Oat mean water level
4:1)(x, y, z, t)
=
cocwt;
ial;(x,y,z,t),harmonic everywhere in z <
limz_,_,V42, = 0 for all x,y,t (no' flux on the sea-floor)
5 lim
= 0 for all t,
112
= (y
6. 4qx,y,0,0) = itt(x,y,,0,0)1= 0
In accordance with Wehausen [74] we introduce:
G(x,e;U)=
1,e;ti
15.0r[
where r=
el and r1=-e' is the image of e with respect to the free surface. This Means with e = (e, C), = (x e)2 +1(y + (z+ C)2.
The Green's function follows from the oscillatory translating source function presented in Wehausen and Laitone[74]. In the case T <
where 7 = I, the function 1P(x,e; U) is written as follows:
0(x, e;U) = 2irgfo de die F(01,,k)H- fd9f dk F(0,k) (5:0)
71"
LI
where
k ek1z-ECI-0--e) cos01cos [k(y
v)' sin 0]
FO 9,k) =
gk (co kU cos Or
The 'contours L1 and L2 are given as in figure 5.1.
These contours are chosen in such a way that when 11 --) oo then
-t,b > &and the 'radiation' conditions are satisfied. The radiated waves
are outgoing and the Kelvin pattern is behind the ship.
-5.1. THE EXPANSION OF THE GREEN'S FUNCTION 35
k
2ii
nek 3 k
°C2
Figure 5.1:1 Contour& of integration.
Figure 5.2: Wave pattern of oscillating translating :source < 1/4.
,Sclavounos [42], it is assumed that the ship does not producewaves in
front of the vessel ( r > 1/4) . This condition is required due to the
upwind difference scheme they used to dampen out upstream waves.
Therefore these methods are less suited for slow speed ship motion theories (
r < 1/4). We only take 7 <
with that speed and wave frequency the vessel will not overtake the radiated waves. 7, Brazes number, is a non-dimensional parameter defined as "21.9
k1
The values kt are the poles of F(9,4 So: gk, f(ca+ k,U cos 9)2 =D.
We have to pay attention to the value of cos 9 in both the integrals in equation (5.2): in the first integral 0 < 9 < so LosG is negative and
in the other integral cos 9 is positive.
The values of lc, behave as follows:
Wct, Vgks
4/02,
_ 1VI
4T cos Oco 2T cos 9I + VI
4T cos 0 = CO 2T cos 9For small values of r these poles behave as follows:,
Vgek3
0(1)
T 0 (5'5) N/g/c2, Vgica rol
A careful analysis of the asymptotic behaviour of 11,(x,eU) for small values of T leads to a regular and an irregular part.
7,/,(x,epU)
= 0
0(z, e)
211-01( 7t) +
Ug2"LE )+
:6 (5.6)In Hermans and Huijsmans [24] (see appendix F) it is shown that due to the highly oscillatory behaviour the influence of % may be
neglected in our first order correction for small values of T.4
The behaviour of k1 and k3 gives rise to a regular perturbation series with respect to r. In contrast, k2 and k4 originates a highly oscillating contribution which gives rise to a non-uniform expansion. However, the position of the last two poles moves to infinity, therefore it can be treated separately. Hr 10, the contours L1 and L2 become the same (figure 5.3):. With it = it follows:
kek(.+CI
00(x,e)
=
2t
Jo(kR)dkik2ek(z+0
1,14z,iej = 4i cos 9,
(k n)2Ji(kR)dk
where 9t= arctan wz-g or in an other way R cos 9''
-{2kek(x+C)J0(kR)}k=n + 2
JL kic
kekvi-C)Jo(kR)dk =kek(z+c)
Jo(kR)dki W9)
= 2rucen(z-E0j0(nR) +2. PVIL
k
The zero-order Green's function, without the Rankine 'singularity is
computed in the algorithm FINGREEN, derived by Newman [45]. So in FINGREEN (1 + 00) is computed. Figure 5.4 shows the
be-haviour of the amplitude of -7.1i + //Jo, the real and imaginary part (in
The figure respectively 00, Re* and Im00)..
k = K k
Figure 5.3,: Contours of integration.
5.2
The zero order Green's function 00,
In this section the zero-order Green's function of the asymptoticex,
pansion is given.,
The equation (5.7) for Ibo can beisplit into the residue and the principal value integral.
5..2. THE ZERO ORDER GREEN'S' FUNCTION 00 37"
5.3
The first-order Green's function /Pi
In this section two methods to compute 7/Ji are given. The first sub-section gives an equivalent expression for ?p, by a transformation in the complex field. In the second subsection 0, is transformed into an
expression which only contains derivatives of 00. The plot in the third subsection shows the agreement of both expressions.
2.4
2.0
0.8
0.4
= 4i cosOf
Figure 5.4: -00, R is variable, the source in (0,0, 1), w = 1.4.
5.3.1
A transformation in the complex plane
An equivalent expression for ?ki is given by a transformation in the
complex field (Abramowitz [1]).
Using Ji(kR) = :,12{1111)(kR)+ H?)(kR)] equation (5.8) becomes:
7P0(x, ) = + 2/cf ek(z+C)Jo(kR)dk,
k ts (5.13)
The Green's function now becomes:
1 1
G(T_,e;U) //)*(x, U) (5.14)
where
r
r,tp.(x., ;u) =- 7,4(_;_,4) + (5.15)
5.3. THE FIRST-ORDER GREEN'S FUNCTION 7,G1 39
In appendix F.1 it is shown that we can rewrite 7,L) , with Re as the
real part and :1m as the imaginary part. These parts can be written
as follows:
Re{t1i} = 47r cos B'e't(z+C)tc [(1 tc(z +())J1(tcR)H- nR.Io(KR)15.11)
m{ } =
47r
costrek(z+C) [(1 + K(z ())Yi(kR) nRY0(nR)]8
coserr k2
(k [2nk cos k(z-f7r (n2 k2)2
+ (n2 k2)sin k(z+C)idk (5.12)
5.3.2
An expression of derivatives of 1,bo
In this subsection we transform the expression for '01 in equation (5.8)
into an expression which only contains derivatives of 00. From the
adapted version of the algorithm FINGREEN, we compute thezero
or-der Green function, without the Rankine singularity and its deriva-tives. Once the expression for 1,bj is recast into expressions of /Po and derivatives of tko , we adapted the original FINGREEN algorithm to incorporate the values of ?pi. The extra computational burden is very
minimal with respect to the CPU time. However, storage of this function will require extra memory capacity.
The computer time will be slightly increased, because the time for the calculation of the forward speed influence is negligible compared with the zero speed computations.
The expression for 7/70 according to equation (5.7) can be rewritten
as:
-and where
ek(z+c)
i/Vx,e) = 2K,L Jo(kR)dk (5.16)
Appendix F.2 presents the following expressions for respectively the real Re and imaginary Elm part of iki(x,
Re{ VII }= 47r cos tr eK(z+c) KR]. n(z C)).4(KR) tcliJo(IcR)] (517),
Elm{
2 cos ft{
[1 + K(z 0115 111 IC'where
7,1)6*(_,E) = 2' PlIf ek(z+C) Jo(kR)dk
lc it
0.19
(5.19)1
5.3.3
The agreement of both expressions
The plot in this section, figure 5.5 shows the agreement of the
expres-sions derived in section 5.3.1 and 5.3.2. In the plot the expresexpres-sions for
in section 5.3.1 (a transformation in the complex plane) are given as Reiki(1) and /m1(1). Results derived from the 'derivatives of lbo)
are denoted as Rob]. (2) and Inytki(2).
As appears from figure 5.5, the two 'expressions of the potential
are completely equivalent.
The algorithm which describes the computation of based on an
expression of 1,b0 was first reported by Huijsmans and Hermans [30].
The same approach was also used by others (Nossen [51] ,
One and
Palm [18]) to compute their slow speed Green's function..
5.4
The uniform expansion of the Green's
function
The plot in figure 5.5 makes it clear that has a non-uniform be-haviour for large R. This means that = iko mbl also behaves.
+ +
+
7,bi
5.4. THE UNIFORM EXPANSION OF THE GREEN'S FUNCTION41 -6 n n Irnqii (1), (2) Rettii (1). (2) SI 5 10 15 20 25 30 kR
Figure 5.5: 01, R is variable, the source in (0, 0, 1), w = 1.4.
non-uniform. In this section we try to write 7,b as an uniform asymp-totic expansion.
In the first subsection a large horizontal distance R, between r and e will be dealt with. The size of the vessel becomes large with respect to 7, 7-R = 0(L).
If the size of the ship is order L with respect to 7 R, it is not sufficient
to use the Green's function (equation (5.1)), with V, +77k1. Using
equation (5.2) for gives a non-uniform expression for large R. This is treated in the second subsection.
5.4.1
Large distance It
We look at equations (5.11) and (5.12) to find the origin of the non-uniformity. First we write
=
We then define 01 as 01 corrected for the non-uniform part. Finally we give an expression for that is defined accordingly.
The first-order Green's function can be written as
= Re{V;1}
i :son-W. This gives:
47r cos ifiek(2+C) [tc11(.1)(KR) ts2(z C)1111)(KR)
n2RI41)(KR)]
-8icos 8' fcc lc' Ki(kR) {2nk cos k(z+C)
7r o (n2 k2)2
(k2
K2)sink(z-1-01 dk (5.20)According to Hermans and Huijsmans [29], it turns out that the in-tegral behaves like 0(R), hence it leads to an uniform expansion
with respect to T. The integral is 0(1) as T + 0, VR E [0, oo). We have to consider the term between the brackets [...] in equation
(5.20). Using the asymptotic expansions of the Bessel functions:
(la)
visRexp[+i(cR 7r/4)]
HI1)(n 1R) exp 3r/4)]
By inspection we see that the first term between the brackets of(5.20)
does not show a non-uniform behaviour. The second term of (5.20) does give rise to non-uniform behaviour for large values of z+ C.
How-ever, we restrict ourselves to finite values of z C. Our main concern
is the third term of (5.20).
47r cos 6'eK(z+C)K2 (nR) (5.21)
For large values of R we can use the asymptotic expansion of HA'. 1P1( ,0 =
4r cos Oie'(z+c)tc2R 2
71-KR
(5.22)
Because of the term VT?, this part of 11)1 causes the non-uniform
be-haviour. If we define 0-1 as 01 minus the non-uniform part, we have an uniform expression for
,
= 01(x, + 47r cos 0/e".(z+C)K2R 2 eikR-1 (5.23)
irkR
