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A SIMULATION MODEL FOR A SINGLE POINT MOORED TANKER
Dr.Ir. J.E.W. Wichers
Publication No. 797 TR diss j Maritime Research Institute Netherlands 1637 Wageningen, The Netherlands
A SIMULATION MODEL FOR A SINGLE POINT MOORED TANKER
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus,
Prof. Dr. J.M. Dirken, in het openbaar te verdedigen ten overstaan van een commissie
door het College van Dekanen daartoe aangewezen, op dinsdag 7 juni 1988
te 14.00 uur
door
Johannes Everardus Wicher Wichers
■ . ^ ' " ' ^ ^ N .
,- . , i l 1 '-V)\
ÜL-U-I
geboren te Groningen
civiel ingenieur
STELLINGEN
I Het computermodel voor de s i m u l a t i e van een SPM-systeem b l o o t g e s t e l d aan s t room ( wind en o n r e g e l m a t i g e golven kan gecompl iceerd z i j n . De m o e i l i j k h e i d s g r a a d van h e t model i s e c h t e r vaak omgekeerd even red ig met de s c h a a l van B e a u f o r t .
I I Voor he t bepa len van de k r a c h t in de boegdraad van een t a n k e r afgemeerd in stroom a l l e e n moeten beha lve de gemiddelde s t roomsne lhe id ook de r i c h t i n g s - f l u c t u a t i e s v e r o o r z a a k t door bv. macro-wervels bekend z i j n .
I I I Op de huid van een afgemeerde t a n k e r worden vele z inkanoden a a n g e b r a c h t . D i t maakt de s c h a l i n g van de R e y n o l d s - a f h a n k e l i j k h e i d van de model- n a a r de p r o t o t y p e - w a a r d e n een s tuk g e m a k k e l i j k e r .
IV Door de bodem van de v o o r s t e en a c h t e r s t e tankcompar t imenten van een t anke r afgemeerd in golven t e ve rwi jde ren en de compart imenten aan te s l u i t e n op w i n d t u r b i n e s kan e n e r g i e opgewekt worden." Deze e n e r g i e kan aangewend worden om de g r o t e l a a g f r e q u e n t e schr ikbewegingen t i j d e n s s to rm t egen te werken.
V Men kan de natuur slechts overwinnen door zich naar haar te schikken (Francis Bacon/ 1561-1626). VI Het gebruik van de resultaten van vroegere experimentele onderzoeken naar de neerwerking rondom een zandribbel te zamen met de recente vortex blob theorieën kan leiden tot nieuwe inzichten in zand transportberekeningen. VII Het toepassen van het oude p r i n c i p e van de spudpaa l -a fmer ing voor s n i j k o p z u i g e r s b u i t e n g a a t s d u i d t op he t o n d e r s c h a t t e n van de k rach t en van de z e e .
VIII Zolang een t e c h n i c u s 10 kN b l i j f t voe len a l s een tonf i s voor hem de overgang van he t t e c h n i s c h m a a t s t e l s e l naa r het p r a k t i s c h m a a t s t e l s e l ofwel het S i - s t e l s e l een wel z e e r o n p r a k t i s c h e s t a p .
IX Op de ringwegen van de g r o t e Amerikaanse s t e d e n houden de a u t o m o b i l i s t e n z ich a l j a r e n k e u r i g aan de s n e l h e i d s l i m i e t . D i t hoef t n i e t het gevolg te z i j n van de vermoede d i s c i p l i n e van de Amerikanen.
A SIMULATION MODEL FOR A SINGLE POINT MOORED TANKER
Dr. Ir. J.E.W. WICHERS
Publication No. 797 Maritime Research Institute Netherlands
Wageningen, The Netherlands
SHELL Tunirex - Tazerka Field - FPSO Terminal , Tunisia
(a SHELL Photograph)
1
EXXON - OS & T Terminal, Santa Barbara Channel, California (Courtesy of IMODCO Inc., Los Angeles, California, USA)
ELF I t a l i a n a - Rospo Mare Fie ld - FSO Terminal, Adriat ic Sea, I t a l y (Courtesy of S ingle Buoy Moorings Inc.)
Louis iana Offshore Oi l Port (LOOP), Gulf of Mexico, USA
(Courtesy of SOFEC I n c . , Houston, USA)
3
Hudbay Oil - Lalang Field FPSO terminal, Malacca Strait (Courtesy of Bluewater Terminals S.A. Switzerland)
PEMEX - CAYO ARCAS, FSO Terminal Baya de Campeche Gulf of MEXICO, MEXICO
(Courtesy of Enterprise d'Équipement Mechanique et Hydraulique, Paris, France)
4
CONTENTS Page
1. INTRODUCTION 11
REFERENCES (CHAPTER 1) 21
2. LOW VELOCITY DEPENDENT WAVE DRIFT FORCES 23 2.1. Introduction 23 2.2. Equations of motion for a tanker in head waves 24 2.3. Displacement and velocity dependency of the hydrodynamlo
forces 33 2.4. Experimental verification of the velocity dependency of the
mean wave drift force in regular waves 37 2.4.1. Test set-up and measurements 37 2.4.2. Extinction tests in still water and in waves 40 2.4.3. Towing tests 45 2.4.4. Evaluation of results of extinction tests and towing
tests 47 2.4.5. Deviation from linearity at higher forward speeds .... 50
2.. 5. The mean wave drift force in regular waves combined with current 55 2.5.1. Towing speed versus current speed 55 2.5.2. Regular waves traveling from an area without current
into an area with current 59 2.6. Computation of the low velocity dependent wave drift forces ••• 64
2.6.1. Introduction — 64 2.6.2. Theory ; 65 2.6.2.1. Linear- ship motions at forward speed 66 2.6.2.2. Wave drift force at low forward speed 68 2.6.3. Results of computations and model tests 71 2.6.4. Evaluation of results 74
2.7. The low frequency components of the wave drift forces and the wave drift damping coefficient 75
- to be continued -
5
- continued -
2.7.1. Introduction 75 2.7.2. Wave drift forces at zero speed 76 2.7.3. The approximation of the low frequency components .... 79 2.7.4. Total wave drift force in irregular waves without
current 80 2.7.5. Stability of the solution and contribution of the
oscillating wave drift damping coefficient 82 2.7.6. Total wave drift force in irregular waves combined with
current 85 2.7.7. Evaluation of results in irregular waves 87
REFERENCES (CHAPTER 2) 88
3. HYDRODYNAMIC VISCOUS DAMPING FORCES CAUSED BY THE LOW FREQUENCY MOTIONS OF A TANKER IN THE HORIZONTAL PLANE 91
3.1. Introduction 91 3.2. Equations of the low frequency motions 95 3.3. Hydrodynamic viscous damping forces in still water 100
3.3.1. Equations of motion in still water 100 3.3.2. Test set-up and measurements 101 3.3.3. Viscous damping in the surge mode of motion 103 3.3.4. Viscous damping due to sway and yaw motions 109
3.4. Hydrodynamic viscous damping forces in current 115 3.4.1. Equations of motion in current 115 3.4.2. Test set-up and measurements 119 3.4.3. Current force/moment coefficients 122 3.4.4. Relative current velocity concept for the surge mode
of motion 123 3.4.5. Relative current velocity concept for the sway mode
of motion 128 3.4.6. The dynamic current contribution 129
- to be continued -
6
- continued
i 3.4.7. Evaluation of the semi-empirical mathematical models in current 138
REFERENCES (CHAPTER 3) 145 \
4. EVALUATION OF THE LOW FREQUENCY SURGE MOTIONS IN IRREGULAR HEAD WAVES .' 147
4.1., Introduction 147 4.2. Frequency domain computations in irregular head waves without
current 148 4.2.1. Theory 148 4.2.2. Computations 152 4.2.3. Model tests 153 4.2.4. Evaluation of results 157
4.3- Time domain computations in irregular head waves with and without current 161 4.3.1. Theory 161 4.3.2. Computed wave drift forces and mean wave drift
damping coefficient 163 4.3.3. Computed motions 166 4.3.4. Model tests 167 4.3.5. Evaluation of results ' 170
REFERENCES (CHAPTER 4) 173
5. EVALUATION OF THE LOW FREQUENCY HYDRODYNAMIC VISCOUS DAMPING FORCES AND LOW FREQUENCY MOTIONS IN THE HORIZONTAL PLANE •• 175
5.1. Introduction 175 5.2. Tanker moored by a bow hawser exposed to regular waves 176
5.2.1. Introduction 176 5.2.2. Computations '•• 177
- to be continued -
7
- continued -
5.2.3. Model tests 179 5.2.4. Evaluations of results 180
5.3. Tanker moored by a bow hawser exposed to current 180 5.3.1. Introduction 180 5.3.2. Computations 181 5.3.3. Model tests , 182 5.3.4. Evaluation of results 182
5.4. Tanker moored by a bow hawser exposed to current and wind 185 5.4.1. Dynamic stability of a tanker moored by a bow hawser .. 185 5.4.2. Determination of the stability criterion 190 5.4.3. Computations 192 5.4.4. Model tests r 193 5.4.5. Evaluation of results. 194
REFERENCES (CHAPTER 5) 197
6. SIMULATION OF THE LOW FREQUENCY MOTIONS OF A TANKER MOORED BY A BOW HAWSER IN IRREGULAR WAVES, WIND AND, CURRENT 199
6.1. Introduction 199 6.2. Equations of motion 200 6.3. Computations 204 6.4. Model tests 208 6.5. Evaluation of results 209
REFERENCES (CHAPTER 6) 217
7. CONCLUSIONS 219
APPENDIX , .. 223
REFERENCES (APPENDIX) . . 232
- to be continued -
8
NOMENCLATURE 233
SUMMARY 239
SAMENVATTING 241
9
CHAPTER 1 INTRODUCTION
Systems consisting of jackets with process platforms and seabed pipelines to produce and transport crude are normally used for large offshore fields. For medium sized and marginal oil fields more and more tanker-shaped vessels moored to a single point are used. To this end the processing equipment is placed on the deck of the tanker, serving as a loading terminal. Transportation of crude is then accomplished by mostly special purpose tankers shuttling back and forth. In case a tanker moored to a single point is used as a storage unit the tanker serves as loading terminal only. For this type of system the tankers are kept on station by using one mooring point. This solution allows the tanker to weathervane according to the prevailing weather conditions and to stay on location with minimum mooring loads.
Single point mooring (SPM) systems have been installed in areas with moderate to severe weather conditions.
1 A.\"<<W^V*SyA\\
///W///W//A&//AV'//AV'///ÏX( y/^//^vy/£y//j4>y/# y / w ^ M x w x i w ^
Figure 1.1 Examples of mooring systems of single-point moored tankei
11
An example of a permanently moored process and storage tanker under moderate weather conditions is Weizhou, People's Republic of China [l-l]. In this case the tanker has been moored by means of a bow hawser to a fixed pile. In areas with more severe weather conditions the mooring systems can vary from chain/turret systems (Rospo Mare [l-2]) to rigid articulated systems (Tazerka [l-3]) and hybrid-type structures (Jabiru [1-4]). Some examples of SPM systems are shown in Figure 1.1.
SPM moored vessels are subjected in irregular waves to large, so-called first order wave forces and moments, which are linearly proportional to the wave height and contain the same frequencies as the waves. They are also subjected to small, so-called second order, mean and low frequency wave forces and moments, which are proportional to the square of the wave height. The frequencies of the second order low frequency components are associated with the frequencies of the wave groups occurring in irregular waves as indicated in Figure 1.2.
20
0
WAVE SPECTRUM
MEASURED : 4^S"„=12.6 m; T , = 1 4 . 0 s THEORETICAL: c 0 » 1 3 . 0 m ; =12 .0s (P .M. )
^L
A '
ft
\ \
- - » . -
re c 2000
o
•z. o
}—
a. i / l
0
TEST NO. 7499 DERIVED FROM LOW FREQUENCY PART OF SQUARED WAVE RECORD DERIVED THEORETICALLY BASED ON SPECTRUM OF MEASURED WAVE
\ \ \\ \\ V
\\ \\
0 .5
WAVE FREQUENCY in rad.s
1.0 -1
0.25 GROUP FREQUENCY in rad.s"
0.50
Figure 1.2 Spectra of waves and wave groups (wave registration lasted 12 hours prototype time)
12
The first order wave forces and moments are the cause of the well-known first order motions- Due to the importance of the first order wave forces and motions they have been subject to investigation for several decades. As a result of these investigations, prediction methods have evolved with a reasonable degree of accuracy for many different vessel shapes, see for instance [1-5], [l-6] and [1-7].
Typical features of SPM moored tankers are the very low natural frequencies of the modes of motion in the horizontal plane. At low frequencies the hydrodynamic damping values are small. Excited by the second order wave forces and moments large amplitude low frequency motions may be induced in the horizontal plane. The origin and characteristics of the second order wave drift forces and moments in irregular waves have been the subject of study for some time, see for instance [l-8], [l~9] and [1-10]. /
The result is that it could be established that the motions of a vessel moored to a single point not only consists of high frequency motions (with wave frequency) but also of low frequency motions. These motions induce mainly the mooring forces.
For the design of mooring systems it is still common practice to carry out physical model tests to obtain the design loads. In the last ten years, however, several computer simulation programs for vessels moored to a single point have been developed. At present the application of such programs, if at all, is limited to preliminary calculations. The reasons for the reluctance to apply such computation methods are due to failures in describing the governing physical phenomena and a lack of reliable input data.
In this thesis a theoretical study will be described and experimental results will be presented for the input and the methodologies involved in the computer simulations of the low frequency motion behaviour of a tanker moored to a single point-
13
Concerning the low frequency motions in the horizontal plane, distinctions and restrictions will be made for the 1- and the 3- degrees-of-freedom (DOF) case.
The 1-DOF case concerning SPM tanker systems exposed to severe weather conditions, in which the waves, wind and current are co-linear, is considered to be one of the most important design conditions. For the 1-DOF case of the moored tanker the study will deal with: - the total drift forces in head waves with and without current; - viscous surge damping in still water and in current; - solution of the equations of the low frequency surge motion in the
frequency and time domain.
The 3-DOF case considers SPM tanker systems in moderate weather conditions. For this kind of system a tanker moored by a bow hawser is chosen. Such a system can, due to unstabilities of the system combined with the environmental conditions, perform large amplitude, low frequency motions in the horizontal plane. For the 3-DOF case the following research has been carried out: - formulation of the coupled equations of the low frequency tanker mo
tions in the horizontal plane for non-current (still water) and current condi t ions;
- solution of the equations of the low frequency motions in the horizontal plane in the time domain for a tanker exposed to waves only;
- solution of the equations of the low frequency motions in the horizontal plane in the time domain for a tanker exposed to wind, waves and current.
These SPM simulations are based on studies performed in the past and are indicated in Figure 1.3.
Of the present developments, the theory and the experimental results will be given in the following chapters. In Chapter 2 attention is paid to the wave drift excitation as a function of low speed of the vessel.
14
DRI FTP 1980 [1-10]
DIFFRAC 1976 [1-7]
LOW FREQUENCY MOTIONS
LOW FREQUENCY HYDRODYNAMIC VISCOUS FORCES
LOW VELOCITY DEPENDENCY ON - HIGH FREQUENCY FORCES - HIGH FREQUENCY MOTIONS - WAVE DRIFT FORCES
WAVE DRIFT FORCES
HIGH FREQUENCY FORCES HIGH FREQUENCY MOTIONS
Figure 1.3 Historical review and present developments of SPM simulations
Experimental research showed that the introduction of the low velocity in the hydrodynamic theory is necessary in order to obtain the complete expression for the wave drift excitation. As a basic principle it was experimentally found that the total wave drift excitation can be assumed to be of potential origin and can be expressed as a' linear expansion to small values of the speed of the vessel. As a result of the expansion of the dependency of the low frequency velocity of the vessel on the quadratic transfer function of the wave drift force in non-current condition the transfer function can be split in two parts. One part of the quadratic transfer function is the low frequency velocity independent wave drift force (zero speed) while the other concerns the low frequency velocity dependent part of the wave drift force. Because of the low frequency velocity dependency that part of the wave drift force will act as a damping force. The damping force, linearly proportional to the low
15
frequency velocity, is called the wave drift damping force. Based on the wave drift force for small values of forward speed, transformations to the current condition can be carried out to obtain the quadratic transfer functions of the wave drift force in the steady current speed and of the associated wave drift damping coefficient.
The reason for the speed dependency of the wave drift excitation must be found in the first order hydrodynamic theory. Computations by means of 3-dimensional potential theory including linear expansion to small values of forward speed confirmed the velocity dependency of the first order hydrodynamic theory and that the low velocity dependency on the second order wave forces can be reasonably approximated [1—11 ], [l-12]. In this study the experiments and theoretical calculations have been restricted to vessels moored in head waves.
Considering the hydrodynamic reaction forces of potential nature besides the wave drift damping also the low frequency (first order) added mass and damping coefficients exist. The latter, however, is negligibly small. Because the tanker is surging in a real fluid the total damping consists of both the wave drift damping and a damping contribution caused by viscosity, see Figure 1.4.
For a sinusoidal excitation the transfer function of the low frequency surge motion of a tanker, moored in a linear system can be written as:
J± (u) = l (1.i)
la \ F 2,2 .22 y(Cll-muu ) + bnu
in which: Xi (u ) = amplitude of low frequency excitation force u = low frequency c,, = spring coefficient mll = v i r t u al mass coefficient b-ti = damping coefficient
16
Since the total damping is relatively small resonance motions can take place. Because in an irregular sea low frequency wave drift force components at the resonance frequency will occur, the magnitude of the transfer function will be determined by the value of the damping coefficient. To simulate the low frequency surge motion not only the wave drift damping but also the damping from viscous origin has to be known. The forces caused by viscosity cannot be fully solved by mathematical models. In Chapter 3 the experimentally derived damping coefficients for both the non-current and current condition are presented.
HYDRODYNAMICS SPM SYSTEM
POTENTIAL THEORY VISCOSITY
COMPUTER SIMULATION
Figure 1.4 Origins of the important parts of the hydrodynamics of SPM systems
In Chapter 4 results of computations of the low frequency motions of a tanker for the 1-DOF case are given. To elucidate the effect of 'the quadratic transfer function of the wave drift damping on the low frequency surge motions for the non-current condition frequency domain computations have been carried out. Therefore the tanker was moored in a linear mooring system and exposed to waves with increasing significant wave heights. The results of the computations have been verified by means of physical model tests. Exposed to a survival sea both without and with a co-linear directed current time-domain simulations of the low frequency
17
motions of the moored tanker were carried out. As a result of the speed dependency of the wave drift forces the excitation in waves combined with current will increase. The computed wave drift forces with and without current have been compared with results of measurements. Using the theoretical data as input the tanker motions have been simulated and the results compared with model measurements. So far the SPM simulations concern the computations of the low frequency surge motions only. The system involved is a permanently moored tanker exposed to survival conditions.
In this thesis on the one hand a system under severe weather conditions is considered while on the other hand a system will be studied which will be exposed to more moderate weather conditions. To this end a tanker moored by a bow hawser is chosen. A feature of such a system is that the tanker can perform low frequency motions in the horizontal plane with relatively large amplitudes. In absence of wind and waves the determination of the equations of motion of the low frequency motions in the horizontal plane give rise to difficulties in the description of the low frequency • hydrodynamic reaction force/moment components. As mentioned already for the viscous damping for the surge mode of motion also the damping force/moment components in the sway and yaw mode of motion can not be attributed to forces of potential nature only; they are for a dominant part determined by viscosity, see Figure 1-4. The force/moment components caused by viscosity can be determined by means of physical model tests.
In addition to the determination of the viscous damping coefficients in surge direction, in Chapter 3 the resistance forces and moments caused by the sway and yaw mode of motion have been determined by means of physical oscillation tests. Because it may be assumed that oscillations at low frequencies will induce different flow patterns along the vessel in still water or current a clear distinction is made between the non-current and the current condition for the formulation of the resistance components. For the non-current condition no formulation was found in literature. By means of the results of oscillation tests a formulation
18
of the resistance force/moment components has been established. For the current field case, however, several investigations have been carried out in the past to formulate the low frequency hydrodynamic damping force/moment components. By means of the formulation derived in this thesis the description as proposed by Wichers [l-13], Molin [l-14] and Obokata [l-15] has been evaluated.
In Chapter 5 the low frequency hydrodynamic viscous damping force/moment components have been validated by means of the low frequency motions in the horizontal plane. For the evaluation the results of the computations are compared with the results of physical model tests. For the non-current condition time domain computations for a bow hawser moored tanker exposed to long crested waves only were carried out. Large amplitude unstable low frequency motions occur in the horizontal plane. In the general case, however, a tanker moored by a bow hawser will be exposed to irregular waves, wind and current. Each of the weather components can have an arbitrary direction. To evaluate the large amplitude unstable low frequency motions the condition has to be considered in a current (and wind) field only. By means of the theory of dynamic instability the unstable conditions have been determined and used for the evaluation.
In Chapter 6 the simulations of the moored tanker under the influence of wind, current and a moderate, long crested sea state are discussed. In the equations of motion of the low frequency motions a distinction will be made between mathematical models. The distinction in the models concerns the relatively small or large low frequency motion amplitudes. The differences will be found in the treatment of the wave drift forces.
Because the large amplitude model consumes considerably more preparation and computer time for the simulation than the small amplitude model the dynamic stability program facilitates the choice of the model beforehand, as is shown in the flow diagram in Figure 1.5. The results of the computations have been compared with the results of model tests.
Finally, a review of the main conclusions is given in Chapter 7.
19
/ /
1 1 EXCITATION FORCES |
1 MEAN CURRENT I
| MEAN WIND |
| LOW FREQUENCY FORCES |
DAMPING FORCES
HYDRODYNAMIC VISCOUS DAMPING
WIND DAMPING
P | INERTIA FORCES |
| ADDED MASS |
DYNAMIC
STABILITY -UNSTABLE-
>> LARGE LF AMPLITUDE ) TIME DOMAIN 'DEGREE OF UNSTABILITY^
| HIGH FREQUENCY FORCES*
EXCITATION FORCES DAMPING FORCES INERTIA FORCES
FIRST ORDER WAVE POTENTIAL DAMPING
VISCOUS ROLL DAMPING
ADDED MASS
)HIGH FREQUENCY RESPONSE)
| LOW FREQUENCY FORCES
TRANSFER FUNCTION OF THE
TOTAL WAVE DRIFT FORCE
SMALL AMPLITUDE J LARGE AMPLITUDE J
Figure 1.5 Review of the hydro- and aerodynamic input for the SPM simulation program
20
REFERENCES (CHAPTER 1)
1-1 Mathieu, P. and Bandement, M.A.: "Weizhou SPM: a process and storage tanker mooring system for China", OTC Paper No. 5251, Houston, 1986.
1-2 Boom, W.C. de: "Turret moorings for tanker based FPSO's", Workshop on Floating Structures and Offshore Operations, Wageningen, November 1987.
1-3 Carter, J.H.T., Ballard, P.G. and Remery, G.F.M.: "Tazerka floating production system: the first 400 days", OTC Paper No. 4788, Houston, 1984.
1-4 Mace, A.J. and Hunter, K.C.: "Disconnectable riser turret mooring system for Jabiru's tanker-based floating production system", OTC Paper No. 5490, Houston, 1987.
1-5 Korvin-Kroukovsky, B.V. and Jacobs, W.R.: "Pitching and heaving motions of a ship in regular waves", Trans. S.N.A.M.E. 65, New York, 1957.
1-6 Hooft, J.P.: "Hydrodynamic aspects of semi-submersible platforms", MARIN publication No. 400, Wageningen, 1972.
1-7 Oortmerssen, G. van: "The motions of a moored ship in waves", Marin Publication No. 510, Wageningen, 1976.
1-8 Remery, G.F.M. and Hermans, A.J.: "The slow drift oscillations of a moored object in random seas", OTC Paper No. 1500, Houston, 1971-SPE Paper No. 3423, June 1972.
1-9 Molin, B.: "Computation of drift forces" OTC Paper No. 3627, Houston, 1979.
21
10 Pinkster, J.A.: "Low frequency second order wave exciting forces on floating structures", Marin Publication No. 600, Wageningen, 1980.
11 Hermans, A.J. and Huijsmans, R.H.M.: "The effect of moderate speed on the motions of floating bodies", Schiffstechnik, Band 34, Heft 3, pp. 132-148, 1987.
12 Huijsmans, R.H.M. and Wichers, J.E.W.: "Considerations on wave drift damping of a moored tanker for zero and non-zero drift angle", Prads, Trondheim, June 1987.
13 Wichers, J.E.W.: "Slowly oscillating mooring forces in single point mooring systems", BOSS 1979, London, August, 1979.
14 Molin, B. and Bureau, G.: "A simulation model for the dynamic behaviour of tankers moored to SPM", International Symposium on Ocean Engineering and Ship Handling, Gothenburg, 1980.
15 Obokata, J.: "Mathematical approximation of the slow oscillation of a ship moored to single point moorings", Marintec Offshore China Conference, Shanghai, October 23-26, 1983.
CHAPTER 2 LOW VELOCITY DEPENDENT WAVE DRIFT FORCES
21l^__Int reduction
To solve the low frequency surge motions of a moored tanker exposed to irregular head waves, the hydrodynamic input for the equation of the motion, being the low frequency reaction and excitation forces, have to be known.
By means of linear three-dimensional diffraction potential theory making use of a source distribution along the actual hull surface the reaction forces at the low frequencies can be computed, see Figure 2.1. The theory behind these reaction forces has been reported by van Oortmerssen [2-l]. The values of the component of the hydrodynamic reaction forces, which is in phase with the surge velocity becomes zero for the low frequencies. By means of extinction model tests Wichers and van Sluijs [2-2] showed, however, that for the low frequencies damping exists. This damping, as is indicated in Figure 2.1, is assumed to be of viscous origin.
Figure 2.1 Measured and computed low frequency surge damping and non-dimensional added mass coefficients in still water [2-2]
23
The excitation, inducing the low frequency motion, is supposed to be caused by the wave drift forces. Based on the output of the diffraction program and the transfer function of the first order motions, the direct pressure integration technique as proposed by Pinkster [2-3] delivers the quadratic transfer function of the wave drift force.
Applying the mentioned results as input to the equation of motion the low frequency surge motions can be computed. On base of the results of model tests Wichers [2-4] showed, however, that the predicted motions were overestimated. For a similar problem we have to go back to the work of Remery and Hermans [2-5] in 1971. In their experimental investigation and validation they had to use a surprisingly large damping coefficient for a correct prediction of the low frequency surge response.
In the last decade research has been carried out to understand the nature of the damping mechanism. Results of model experiments in regular waves followed by implementing low forward speed in the 3-dimensional diffraction potential theory showed that a large part of the damping could be attributed to the velocity dependency of the wave drift forces, [2-6] and [2-7]. In the next sections first the physical explanation will be given of the features associated with the velocity dependency of the wave drift forces followed by the computation procedures.
2.2._Ec[uations of_m°tion_for a tanker in_head waves
The motions of a moored tanker in irregular head waves consist of small amplitude high (= wave) frequency surge, heave and pitch motions and large amplitude low frequency surge motions. The frequencies of the low frequency surge motions are concentrated around the natural frequency of the system, see Figure 2.2.
To study the motions use has been made of two different systems of axes as indicated in Figure 2.3; the system of axes 0x(l)x(3) is fixed in space, with the Ox(l) in the still water surface and the 0x(3) axis coinciding with the vertical axis Gx3 of t h e ship-fixed system of axes 6x^x3 at rest.
24
(deg)
TIME (s)
Figure 2.2 A registration of the motions of a moored tanker model in head waves
Figure 2.3 System of co-ordinates
25
We shall assume that the surge, heave and pitch motions can be decoupled into the following form:
*i = «[^.o + Mll^+ 411^ x3 - « ^ . t ) +e2[x521)(t)+x^)(2i(t)]
x 5 = e*<l)(£,t) + e 2[xg(t) + x(2)(lL>t)] (2.2.1)
with e and n being small parameters, viz.: - e is related to the wave steepness; - T| considers the ratio between the two time scales of the motions: the
\i frequency range around the natural frequency of the system and the 10 frequency range of the wave spectrum frequencies.
And further: - xi' , X3' ' and xc' ' are related to the wave frequency surge, heave
and pitch motions; - xjif « x3lf anc* x51f stand for the large amplitude low fre
quency second order surge, heave and pitch motions; - xij,f* , x3hf an(^ x5hf represent the second order motions of
which the frequency range is twice the wave frequency range.
Of the second order motions only the low frequency part will be considered and will be denoted x *• '. For a simple sinusoidal excitation with wave frequency the equations of motion can be written as follows:
for k = 1,3,5 (2.2.2)
in which M, . is the inertia matrix of the vessel. Since the origin of the system of axes coincides with the centre of gravity of the vessel the inertia matrix can be written as follows:
26
Mkj =
M 0
0 M
0 0
0
0
X5-
(2.2.3)
while further: ai j(io) = matrix of added mass coefficients b^^u) = matrix of damping coefficients c^j = matrix of force restoring coefficients X^a ' = amplitude of the first order wave w = wave frequency exC = P n a s e angle between the first order wave force and the wave M = mass of the vessel Ic = moment of inertia of the vessel
The indices kj indicate the direction of the force in the k-th mode due to a motion in j-direction.
Besides the hydrostatic restoring forces, c ^ may also include restoring forces due to the mooring system, as long as this mooring system has linear load-excursion characteristics.
Since the hydrodynamic reaction coefficients a -j and bj. are frequency dependent, equation (2.2.2) can only be used to describe steady oscillatory motions for a purely linear response in the frequency domain. In irregular head waves the first order wave forces/moment will present all kinds of frequencies. In this case equation (2.2.2) cannot describe the motions in irregular waves. To describe the equations of motion one has to return to the time domain description using memory functions to represent the frequency dependent added mass and damping terms. This memory function or impulse response function is given by the Duhamel, Fal-tung or convolution integral. This function states that if for a linear system the response K(t) to an unit impulse is known then the response of the system to an arbitrary forcing function X(t) can be determined. The formulation is as follows:
27
x(t) = ƒ K( T ) X(t-x) dt (2.2.4) O
The impulse response theory has been used by Cummins [2-8] to formulate the equations of motions for floating structures. According to Cummins the reaction forces due to the water velocity potential may be derived by the impulse response theory by considering the vessel's velocity as input of the system.
Applied to equation (2.2.2) for a tanker moored in irregular head waves the time domain equations of motion can be formulated as follows:
J.l'3,5 K^ +^J ) S5 1 >V" K kJ ( X ) 4J 1 ) ( t" T ) ^ + CkJXJ1>} = X^)(t) for k = 1,3,5 (2.2.5)
where: Mu-j = inertia matrix of vessel mkj = matrix of constant (frequency independent) added mass coeffi
cients Kk. = matrix of impulse response function x = time shift c^j = matrix of force restoring coefficients X, ' ' = time varying first order wave exciting forces
Ogilvie [2-9] showed that the function Kk.,(t) is given by:
K, .(t) = - ƒ b, .(00) cos(ut) du (2.2.6) kj n Q KJ
where th,-(a>) are the first order potential damping coefficients of the vessel at the frequencies 00. The constant inertia coefficients were determined by:
m k j = akj(u) + ±r ƒ ^.(t) sinCu't) dt (2.2.7)
28
where aj.(w') is the frequency dependent added mass coefficient:corre-' sponding to an arbitrary chosen frequency u'.
Considering the complete equations of motion the total wave exciting force has to be taken into account. The total wave exciting force as present in irregular head waves consists of the following parts:
Xk(t) = x£1}(t) + x£2)(t) .. for k = 1,3,5 (2.2.8)
where X^ '(t) is the first order wave exciting force with the wave frequencies and X ^ '(t) represents the mean and the slowly oscillating parts of the second order wave drift force. The result will be that the equations of motion comprise a second order mean displacement and low and high frequency motions.
The natural frequencies in heave and pitch direction for mono-hull type structures are in the wave frequency range. In this range the induced mean and low frequency heave and pitch motions will be negligibly small. For the surge direction, however, the natural frequencies of the considered systems are in the low frequency range. The damping at these low frequencies is small. The result will be that in surge direction large amplitude low frequency motions combined with high frequency motions will occur.
The total fluid damping in surge direction is caused by the combined high and low frequency motions. Since for the low frequencies in surge direction negligibly small damping due to wave radiation exists (to < 0.08 rad.s ; see Wichers and van Sluijs [2-2]). The fluid damping force is assumed to be of viscous origin. Because the origin of the damping mechanisms are completely different (wave radiation versus viscosity) it is assumed that the damping forces will not mutually interfere. Therefore we assume that the wave radiation damping is excited through the first order motions, while the viscous damping forces are generated by the first and second order motions.
29
Following the afore-mentioned assumptions the equations of motion can be written as follows:
(M+a11(u1))-42)+Bu(u1)(x(12)+41))+BU2(^1)(i(
12)+41))
l i ^ M ^ - h u * ^ - *[2)<t> (2.2.9) 00
r {(M +m )x(1)+ ƒ K (x)i(1)(t-u)dT + c x(I)} = X(1)(t) j=l,3,5 J J J 0 J J iJ J L
(2.2.10) and
for k = 3,5 (2.2.11)
in which: aii(|ii) = added mass coefficient in surge direction at frequency u, B^(u^) = linear viscous damping coefficient in surge direction at fre
quency u^ Biiod^l) = quadratic viscous damping coefficient in surge direction at
frequency u, Ui = natural frequency of the system in surge direction X, ( '(t) = first order wave forces in surge direction Xi^ '(t) = second order wave forces in surge direction X^(t) = total wave forces in the k-direction x.j ' = wave frequency motion in j-direction x.^ ' = second order motion in j-direction x. = total motion in j-direction
Neglecting the influence of X-j' '(t) and X5( '(t) the moored tanker will oscillate with high frequencies in surge, heave and pitch directions and simultaneously perform low frequency large amplitude oscillations in surge direction. The hydrodynamic reaction and wave forces may be effected by the slowly varying velocity.
30
As an introduction to the problem of the velocity dependency of the hy-drodynamlc forces a simplified mathematical model of a linearly moored tanker will be considered: - the tanker will be exposed to a regular head wave with frequency u; - the linear spring constant of the mooring system will be C-Q; - a low frequency oscillating external force acting in surge direction will be applied to the tanker:
x\(t) = Xlacos t^t (2.2.12)
in which: u- = natural surge frequency of the system.
The total wave exciting forces in regular head waves consist of the following parts:
Xfc(t) = x£X)(t) + x£ 2 ) for k = 1,3,5 (2.2.13)
where Xk (t) is the first order wave exciting force and Xk^ ' is the mean wave drift force.
For the simplified model the equations of motion can be written as:
(M+a1 1(u1))x{2 )+B1 1( l i l)(xJ2 )^1 ))+B1 1 2(u1)(x52 )-Hi51 ))
| x p ) + x ^ 1 ) | + c n x p ) = x£2)+XL(t) (2.2.14)
;<1>4.„ r,,^lK, JV\ = vW, S ((H +a ( » ) ) x i ; + b M i 1 J + c X1 J | = X ^ ' ( t
j = l , 3 , 5 J J J J J J J (2.2.15)
and
_ 2 {(Mkj+akj(oo))x^bkj(üOi..+ck.x.} = Xk(t) for k = 3,5 (2.2.16)
Due to the wave forces the tanker will perform high frequency motions around a mean displacement. Due to the external force X (t) the result
31
will be that in surge direction large amplitude low frequency motions combined with high frequency motions will occur.
The coefficients ak-(u)) and b^iC") are the coefficients of the hydrody-namic reaction forces when the vessel oscillates with wave frequency u. Computed by means of the 3-D potential theory the coefficients are only dependent on the wave frequency, the water depth and the geometry of the underwater hull. Therefore the hydrodynamic reaction coefficients should be written as:
akj< u« il ( 2 ) = °>
b k j ( u ' h < 2 ) " ,°> (2.2.17)
The first order wave forces can be calculated with the 3-D potential theory.. The computed first order wave forces X, ' are only dependent on the amplitude and period of the incoming wave, water depth and the geometry of the, underwater hull of the body. The second order wave forces Xvv ' on a stationary floating body exposed to regular waves may be calculated by the direct pressure integration technique. In the theory of the direct pressure integration technique it .is assumed that the floating body only performs small amplitude high frequency motions around the mean position. Following the conditions of the mentioned computations the first order wave forces and the second order wave drift forces should be written as follows:
X^( X< 2?=0, X<2>=0,t)
x^V'M2^0' ii -0) (2-2-18)
As mentioned earlier, in reality the moored vessel in irregular waves performs small amplitude high frequency motions while traveling with large amplitude low frequency surge oscillations. In our simplified model with the tanker moored in regular waves it performs small amplitude high frequency oscillations while traveling with large amplitude
32
low frequency surge oscillations.
These observations imply that not only the hydrodynamic reaction forces but also the wave exciting loads may be influenced by the low frequency displacement and velocity of the vessel. By using the simplified model these effects on the motions in surge direction will be discussed in the next section.
213^_Disglacement_and_velocit2 dep_endency_ of the_hydrod2namic forces
Oscillating at high frequencies and simultaneously performing the low frequency large amplitude oscillations the hydrodynamic reaction forces of a structure will be affected by the slowly varying speed. Further, due to the low frequency large amplitude oscillations through the regular wave field, the wave forces will be affected by both the displacement and the speed. To study the displacement and velocity dependency we shall restrict ourselves to the equations of motion in surge direction, which are given by equations (2.2.14) and (2.2.15). The actual high frequency hydrodynamic reaction coefficients and the first order and second order wave forces should be written as follows:
*l;](».il(2))
bj^Oo.i/2)) for j = 1,3,5
X^Hx^^W.t) X^txd)^2),^2)) (2.3.1)
By applying the Taylor expansion of the reaction coefficients and the wave forces to the low frequency displacement and velocity up to the first order variations we obtain for the hydrodynamic reaction coefficient:
33
öa^co.O) j »lj(«-.il(2)) = a1;]((ü,0) + l i , xj
öx^
, . C ) \ ^ Öb (ü),0) b^u),*^') = b^Oo.O) + x3 x ^ ; for j-1,3,5 (2.3.2)
for the first order wave forces:
X1(D(x1(2))il(2))t) . Xl(l)(0,0,t) + Z L _ ^ I 1 . X ( 2 ) + ax,(1)(o,o,t)
bx)
ax.(1)(o,o,t) " 1 <■->->"' , ( 2 ) + -T7(2) - ^ ^.3.3) öx
and for the second order wave drift forces: dX(2)fx(1) 0 O)
X1(2)(x(D)x1(2),x1(2)) - X ^ H x ^ . O . O ) + l V-{2)' ' ;.X;2> +
ox^x^.o.o) + - ^ i}2) (2-3.4)
öx^
in which a^co.O), bj.((ü,0) and X1(1)(0,0,t), X1(2)(x(1) ,0,0) correspond to the coefficients and the wave forces as specified in equation (2.2.17) and equation (2.2.18). Substitution of equation (2.3.4) into equation (2.2.14) and equations (2.3.2) and (2.3.3) into equation (2.2.15) leads to an approximation of the assumed general equations of motion in surge direction of the vessel moored in regular head waves:
(M+au(^1))x(2)+B11(ul)(x(2)+x(1))+Bu2(,1)(x(2)+x(1))|x(2)4i(1)| +
(2) (2), (1) , °X<2)(x(1).°>°) (2) + c n.x(^ = x { ^ V .0,0) +— (Ij xl +
34
ax^feW.o.o) m „ — -jj- iJz;+X(t) (2.3.5) öx
and
n ■* öa (u,0) Z { M x 1 ) + (a «„.O) + ^ .x<2>)x^ +
j=i,3,5 iJ J iJ ai^ ; l J
db (o),0) + fb Oo 0) + ^ .x(2)].x( ) + c x ( l ) =
l lj^' ; .-(2) -X1 j-Xj + cljXj öx)
n. ax(1)(o,o,t) ax(1)(o,o,t) X^;(0,0,t) + — ^ -Xl + . ( 2 ) *i (2.3.6)
1
In equation (2.3.5) and equation (2.3.6) both the high and low frequency surge motion components are incorporated. The displacement and/or speed effects on the force components will be studied. Therefore the wave force components and the hydrodynamic reaction forces will be considered in more detail.
A regular1 wave can be described by:
C(t) = Ca.cosü)t
Relative to the slowly oscillating vessel this regular wave can be written as:
C(t) = Ca cos(wet + K Xl<2>) (2.3.7)
where u = frequency of encounter = io + K ii' ' < = wave number = 2n/X. X = wave length
35
The associated first order wave force in surge direction will yield:
X ^ H x / 2 ) , ^ 2 ) ^ ) = Xla^>((üe) c o s ( V + < x1(2)+exC(o,e)) (2.3.8)
in which: Xi ' '(u> ) = amplitude of the first order wave force e >-(io ) = phase angle between the first order wave force and the wave
For small values of x^ ' and x. equation (2.3.8) should actually correspond to equation (2.3.3). Equation (2.3.8) shows that the amplitude of the first order wave force will be low frequency modulated by the frequency of encounter. Further, the frequency of the wave force will result in a high frequency oscillation modulated by the frequency of encounter and the low frequency phase shift. The result is that the frequency of the first order wave force will vary but within the wave frequency range. Because the frequency is in the high frequency range the first order wave does not contribute to the low frequency damping. Considering the hydrodynamic reaction force components in equation (2.3.6) a similar explanation and conclusion can be drawn.
Of the second order wave drift force in a regular wave, as is indicated by equation (2.3.4), the first term is the mean wave drift force and will be a constant. Since the mean wave drift force is independent of the position of the tanker in the regular wave the derivative to the displacement will be zero. After inspection of the terms of equation (2.3.5) the equation of the low frequency motion in regular waves can be reduced as follows:
s(2) _ _ „ ,,. VA(2) _ „ ,., ,-(2) (M+a u(u 1))x^ ; = - B u ( u 1 ) x ^ ; - B u 2( li 1)^ 1 •(2) xl
+ ^2)(^/0,0).i[2>-c11.,<2)+t{2)(x<2>,0,0)+X1(t) 5x^
(2.3.9)
36
In the right hand side of equation (2.3.9) three low frequency damping coefficients can be recognized. The first two terms are assumed to be of viscous nature, while the last term relates to the low frequency velocity of the mean wave drift force.
In order to analyse and verify the separate terms of equation (2.3.9) model tests were carried out: 1. Motion decay tests.
- in still water - in regular head waves with various heights and periods
2. Towing tests at low speed. - in still water - in regular head waves with various heights and periods
2.4. Experimental verification of_the_velocity_ dependency of_the_mean wave drift force in regular waves
2.4.1. Test set-up and measurements
To verify the terms in equation (2.3.9) extinction and towing tests have been carried out. Use was made of a model of a loaded 200 kDWT tanker (scale 1:82.5). The particulars of the vessel for different loading conditions as will be used in this work are given in Table 2.1. The body plan and the general arrangement are given in Figure 2.4. For the extinction tests a linear mooring system was employed. The test set-up for the mooring arrangement is shown in Figure 2.5. The spring constant was 16 tf/m. The extinction tests were carried out in the Wave and Current Laboratory of MARIN measuring 60 * 40 m. The tests were performed in a water depth of 1 m. The low speed towing tests were carried out in the Seakeeping Laboratory of MARIN, having a water depth of 2.5 m, a length of 100 m and a width of 24 m. For the towing tests the mooring system, consisting of linear springs, was connected to the towing carriage. The spring constant amounted to 257.4 tf/m. During the towing tests the model was kept in
37
longitudinal direction by means of a light weight trim device connected at its forward and aft perpendicular.
Designation
Loading condition Draft in per cent of loaded draft Length between perpendiculars Breadth Depth Draft Wetted area Displacement volume Mass Centre of buoyancy forward of section 10 Centre of gravity above keel Metacentric height transverse Metacentric height longitudinal Transverse radius of gyration in air Longitudinal radius of gyration in air Yaw radius of gyration in air
Wind area of superstructure (a - lateral area - transverse area
Added mass a) » 0 rad/s (water depth 82-5 m)
Symbol
L B H T S V M FB~ KG GMt
CM,
kll k22 k66
ft): ALS ATS all a22 a26 a62 a66
Unit
m m m
Ü* m tfs2/m m m m m
m
m m
"2 ID'
tfs2/m ■ t£s2/m tfs2
t£s2 tfms2
Magnitude
Loaded
100% 100X
310.00 47.17 29.70 18.90
22,804 234,994 24,553
6.6 13.32 5.78
403.83
14.77
77.47 79.30
922 853
1,594 25,092
-83,618 -83,618
123,510,000
Intermediate
602 70% 310.00 47.17 29.70 13.23
18,670 159,698 16,686
9.04 11.55 8.66
15.02
77.52 83.81
922 853
755 10,940
-30,400 -30,400
59,607,700
Ballasted
25% 40%
310.00 47.17 29.70 7.56
13,902 88,956 9,295
10.46 13.32 13.94
15.30
82.15 83.90
922 853
250 5,375
-16,132 -16,132
23,200,000
Table 2.1 Particulars of the tanker
During the tests the surge, heave and pitch motions and the longitudinal mooring forces were measured. The surge and heave motions were measured in the centre of gravity (G) by an optical tracking device. The pitch motion was measured by means of a gyroscope. The sign convention is given in Figure 2.5. The mooring lines were connected to force transducers. All measurements were recorded on magnetic tape to facilitate the data reduction. All data were scaled to prototype values according to Froude's law of similitude.
38
31
fe ^
AP STATION 10 FP
i^-16-10
Figure 2.4 General arrangement and body plan
-
1 /
-
3 '*—l
" j4
J t k . G
G
+ x3
~ ^ x 6
l «—— +x,
_^
-
" * '
"
. ,, > •'•'
7F-
1 / J -C
:
AP
Figure 2.5 Test set-up
39
2.4.2. Extinction tests in still water and in waves
Applying equation (2.3.9) to the condition of extinction in still water the equation of motion reduces to:
(M+an(u1))x1(2) = -B u(u 1)xJ 2 )-B u 2(u 1)x5 2 )|x{ 2 )|-c 1 1x[ 2 ) (2.4.2.1)
The results of the extinction tests are shown in Figure 2.6 and 2.7. It appears that for the large amplitude surge motions the viscous damping force is approximately linearly proportional to the low frequency velocity (6112(^1) ~ 0 tf.s.m ). The theory and the procedure of determining the linear damping coefficient will be explained below.
Equation (2.4.2.1) can be written in a linear differential equation with constant coefficients:
(M+au(u1))xJ2)+B11(a1)xJ2)+c11xJ2) = 0 (2.4.2.2)
The solution of equation (2.4.2.2) is:
x ( 2 ) = e 2(M+a u ) ( C i C 0 S t i i t + ^ s i n ^ t ) ( 2 .4 .2 .3 )
in which:
, , C l l r B l l -.2 " l \ / ( M + a u ) L2(M+an)-1
= natural frequency of the system
and C, and Co are constants dependent on the initial position of the vessel.
Following equation (2.4.2.3) the decrease of the amplitudes of the decay curve x and x will be:
N N+l
40
hl'* A = _ ! ? L = e < M + all^l = e 6 (2.4.2.4)
*N+1
in which 6 is named the logarithmic decrement.
Because of the low damping of the considered system.i.e.
R 2 c böH^y] « (M^T < 2 - 4 - 2 - 5 >
the natural frequency will approximately correspond to the natural frequency of the undamped system. Because of the linearity of the damping for the large surge amplitudes the logarithmic decrement is constant and the value of the decrement can be determined from:
in xx - In x N + 1 o = ^
in which: N = number of oscillations
The damping coefficient becomes:
B l l , 6V c l l ( M f aU>'
5 c n B,, = —— ( 2 . 4 . 2 . 6 )
11 itjj. v J
For a detailed description reference is made to Hooft [2-10 ]. From Figure 2.6 and 2.7 the natural frequency and the damping coefficient can be determined. They amount to Uj = 0.0238 rad.s"1 and B n ( u ) = 18.2 tf.s.m respectively. As is indicated in Figure 2.1 the still water damping coefficient is caused by viscosity. The potential damping due to radiated waves is negligibly small at low frequencies.
41
SURGE (m)
40
20
n
-2.0
-40
"1 1 \l lf\ \
1 i 1 i 11 i 1 i
1 I \ I 5°°
STILL WATER
WAVES sa
1 / ' 'V if 1000 ^
-- 3.11 m ; T
' / ' 1' ' l\ \ 1500
11.8 s
». TIMF fs l
Figure 2.6 Registration of extinction test in s t i l l water and in regular waves
o CREST VALUES . TROUGH VALUES
50
20
10
^s2 ^^**,s^.
N> L.
r = 3.11 m ; T = 11.8 s \
C = 0.0 m ; T = 11.8 s * ~ » ^ ' ( s t i l l water)
k c <<u J l , , " \ i £ = 1.88 m ; " ^ ^ *
X T = 11.8 s
10 .20 N (NUMBER OF OSCILLATIONS)
30
Figure 2.7 Determination of the damping coefficients in. still water and in regular waves
42
Equation (2.4.2.2) applied to the condition of extinction in regular waves gives:
(M+ a i l(^))x[ 2 ) = - B u(u 1)i; 2 ) - c n X ; 2 ) + X<2)(x(1),0,0) +
ax}2>(5Ci),o,o) + 77(2) •*! <2-4-2-7)
ox| Further, if it is assumed that the damping coefficient in the last term on the right hand side of equation (2.4.2.7) is constant in the regular wave and denoted -B^» equation (2.4.2.7) reduces to:
(M+au(u))xJ2) = -(Bn+B1)x1(2)-c11x{2)+X1
(2)(XJ1),0,0) (2.4.2.8)
Results of the extinction tests in still water and in regular waves are given in the Figures 2.6 and 2.7.
Figure 2.7 shows that for both wave amplitudes used in the tests (constant wave frequency u) the total low frequency damping force is linearly proportional to the low frequency surge velocity. This leads to the conclusion that the contribution of the quadratic viscous damping to the total damping is negligibly small (B]_]_2~0)-
Based on the linearity of damping coefficients the viscous damping coefficient Bii can be separated from the total damping coefficient. The remaining damping coefficient is assumed to be caused by the waves. Therefore extinction tests were carried out in different wave heights and various wave frequencies. The separated damping coefficient Bi caused by the waves as function of the wave height squared is shown in Figure 2.8.
The damping coefficient appears to be linearly proportional to the square of the wave height. Since the wave drift force is linearly proportional to the square of the wave height the damping coefficient B- is assumed to be related to the wave drift force. For this reason B- is assumed to be of potential nature. The coefficient B^ is called the wave drift damping coefficient. The wave drift damping quadratic transfer
43
function as a function of the wave frequency can be written as follows:
B1(w) ox{2)(x{1),0,0)
< * ! 2 ) (2.4.2.9)
50
2b
(1
A u = 0.36 r a d . s "
• u = 0.38 r a d . s "
X u = 0.532 r a d . s "
D u = 0.56 r a d . s "
A u = 0.628 r a d . s " 1
O u = 0.80 r a d . s "
^<Zz^
x ^ ^ _ ^ . -
o—
^ x ^
. - — • —
10 2 . 2
C in m
Figure 2.8 Wave drift damping related to the wave height squared
Following equation (2.3.9) it appears that in a regular wave the wave drift damping coefficient represents the derivative with respect to the low frequency vessel velocity of the mean longitudinal wave drift force at zero speed. Based on the foregoing results the hypothesis can be made that for small values of the vessel's velocity the total or velocity dependent mean wave drift force can be written as:
X1(2)(2
(1))iJ2),u))=xWLx(1),0>(l))-B1((l)).iJ2) (2.4.2.10)
To prove this hypothesis the dependency of the mean wave drift force on the vessel speed has to be known. For this purpose towing tests were carried out in a range around zero speed.
44
2.4.3. Towing tests
Prior to the towing tests in regular waves towing tests in still water were carried out at various speeds. The towing directions were both backward and forward. Following equation (2.3.9) and taking for the low
.(2) frequency oscillating speed the steady speed x. = U the mean resistance force 3L, can be described as:
X T - - B i l ü - B 1 1 2 ü ü *pLTClc(U,4.cr)U (2.4.3.1)
in which: C^c(U,(|;cr) = longitudinal resistance coefficient
P L T
= relative current angle = mass density sea water = length between perpendiculars = draft of the vessel
The measured resistance coefficients C^(U,4>cr) as a function of the vessel's velocity and towing direction are shown in Figure 2.9.
* =0° us
0
ns
©
O 0
-3
(
-2
>
-1 0
<
+ 1
•>
U in m.s" +2
g 0
+3
* = 180 . Figure 2.9 Resistance coefficient measured during towing tests in still
water
The towing tests in regular waves were carried out under the same speed conditions as the still water towing tests (except for the 5 knot
45
speed). Following equation (2.3.1) and equation (2.4.3.1) and assuming that 5L, represents the total mean resistance force for the steady state condition, the total mean resistance force will be:
Xj = fcLTC^U.^tl^+Ki^)2] + x{2>(x(1),x[2),u) (2.4.3.2)
Since in a regular wave X^' '(x. ,x^ ' ,\j) is independent of x-^ ' and for the viscous resistance force formulation U » h[k\ ) equation (2.4.3.2) can be simplified into:
X,. = ^pLTClc(U,4,cr)U2 + x[2)(x(1),u) (2.4.3.3)
The force Xj^ '(x* ',U) actually represents the velocity dependent mean wave drift force or the added resistance force at a speed U of the vessel. From the experiments carried out for various wave heights, wave periods and speeds the mean wave drift force can be established as a function of the vessel's speed. In Figure 2.10 the measured quadratic transfer function of the mean wave drift force as function of the vessel's speed based on the earth-bound wave frequencies is shown. The results clearly indicate the dependency of the mean wave drift force on the speed of the vessel. It can be concluded that the mean wave drift force or added resistance seems to be a linear function of the (low) speed of the vessel.
Since the mean wave drift force is approximately linearly proportional to the low values of the vessel's speed U the gradient of the added resistance will be constant by approximation. The gradient of the transfer function derived from Figure 2.10 can be written as:
oX^dJ.x* 1)) — — , (2.4.3.4)
C ÖU a Similar conclusions were derived from the results of experiments carried out by Saito et al. [2-ll] and Nakamura et al. [2-12]. •
46
(tf.m"')
2 0 , 2 (m.s"1)
»=0.457 rad.s"1
O 2 c = 4.0 m 3
• 2 t = 6.0 il Ü
-20-,-
-15--
. / /
/> -5-"
L
-10
-2 0 2 (m.s"1) .
0.625 rad.s"1
-20.
-15--
-5"
-2 0 2 (m.s-1) - U
0.765 rad.s"1
Figure 2.10 The quadratic transfer function of the mean wave drift force as function of the towing speed for three earth-bound wave frequencies
2.4.4. Evaluation of results of extinction tests and towing tests
In terms of the quadratic transfer function of the wave drift damping coefficient the results as obtained from the extinction and towing tests have been plotted in Figure 2.11.
47
LOADED 200 kTDW TANKER IN HEAD WAVES
TOWING TESTS 2C, 4.0 m
X 2c = 6.0 m 9
EXTINCTION TESTS
/*
/
\ A.
v.
0 0.5 1.0 u in rad.s
Figure 2.11 Experimentally derived values of the wave drift damping quadratic transfer function
The trend of the experimentally determined transfer function is supported by the results of the experiments carried out by Faltinsen et al. [2-13]. From the experimental findings one may conclude that the expansions used in equation (2.3.4) hold for the added resistance gradient for low forward speed. The gradient corresponds to the wave drift damping coefficient, or:
B.<») SxP'tx'1'^!2'.») cl *<» a*u
öx[2)(x(1),U,U)) C2 au a
U=0
(2.4.4.1)
As a consequence of equation (2.4.4.1) the transfer function of the total wave drift force in a regular wave with frequency u can be written as:
x^CiJ2'.») x{2)(0,u) B1(io).i{2) (2.4.4.2)
48
From the experiments it was found that the wave drift force increases approximately linearly for low forward speeds. Based on the gradient the quadratic transfer function of the wave drift force as function of low vessel speed U in a regular wave with frequency w can be approximated by:
X(2)(U,io) X(2)(0,oo) B (u).U -^—5 —j i-, (2.4.4.3)
C £ «I a a a
The total transfer function of the wave drift force in a regular wave acting on a tanker, which performs low frequency oscillations superimposed on the steady toWing speed U can be approximated by:
X<2>(lHi<2),<-) X<2><0,eo) B.CoO.tu+x^) _i _è i i *: (2.4.4.4)
c r c a a a
This procedure will further be referenced to as the gradient method. Following the gradient method the quadratic transfer function of the wave drift force for various forward and backward steady speeds can be approximated. The values of the quadratic transfer function of the wave drift damping are taken from Figure 2.11, while the quadratic transfer function of the wave drift forces for zero speed have been computed [2-3]. The result is given in Figure 2.12.
From Figure 2.12 it can be concluded that the quadratic transfer functions of the wave drift force with low forward speed increase significantly. The knowledge of the gradient of the added resistance for zero speed or the wave drift damping coefficient is of importance.
From the experiments it was concluded that the wave drift force linearly increased for low forward speeds. In the next section a study is made at what speeds the increase of the wave drift force deviates from linearity-
49
COMPUTED U=0: WATER DEPTH =206.0 m [2-6]
« -WATER DEPTH = 82.5 m[2-3]
u> in rad.s
Figure 2.12 Quadratic transfer function of the wave drift force as function of the towing speed in regular head waves (earth-bound wave frequency) [2-6]
2.4.5. Deviation from linearity at higher forward speeds
The prediction of the wave drift force with low frequency velocity or constant speed is based on the gradient method for the wave drift force at zero speed. The gradient method assumes a linear increment of the wave drift force or added resistance for low forward speed (= order of the current speed).
50
In order to check the afore-mentioned condition the added resistance for low and higher forward speeds has been studied. In this study the vessel concerns a 125,000 m LNG carrier sailing in head waves at relatively deep water (175 m). The particulars of the LNG carrier are given in Table 2.2, while the body plan is shown in Figure 2.13.
For the zero speed case the transfer function of the wave drift force has been determined by means of computations, while the wave drift damping coefficient has been derived from decay tests as described by Wlchers and van Sluijs [2-2]. The values for the added resistance for higher forward speeds have been determined by means of model tests [2-14]. For the computation of the transfer function of the wave drift force the facet distribution is shown in Figure 2.14. The results of the computations are presented in Figure 2.15. The wave drift damping coefficients as derived from decay tests have been plotted in Figure 2.16.
M5336 scale 1:70
Designation
Length between perpendiculars Breadth Draft, even keel Displacement volume Metacentric height Centre of gravity above keel Centre of buoyancy forward of section 10 Longitudinal radius of gyration Block coefficient Midship section coefficient Waterline coefficient Pitch period Heave period
Symbol
L B T V GM KG
FB
c9" Tz
Unit
m m
m3
m m m m m
sec sec
125,000 m3
LNG carrier
273.00 42.00 11.50 98,740 4.00 13.70
2.16 62.52 0.750 0.991 0.805 8.8 9.8
Table 2.2 The particulars of the LNG carrier
51
Figure 2.13 Body plan of the LNG carrier
Figure 2.14 Facet distribution LNG carrier (symmetrical starboard side)
52
The extinction tests in still water and in regular waves to derive the wave damping coefficients were carried out in the Seakeeping Laboratory of MARIN. In the same basin the towing tests to determine the added resistance RAW for the higher speeds (Fn = 0.14, 0.17 and 0.20) were carried out. The description of the laboratory and the test set-up is given in Section 2.4.1. The results of the measured transfer functions of the added resistance for the higher speed values are given as function of the forward speed in Figure 2.17 for 6 wave frequencies. The wave frequencies are defined in an earth-bound system of co-ordinates. In the same Figure the transfer functions of the computed added resistance for zero speed and of the estimated values of the wave damping coefficients are plotted. Using these data the curves of the transfer functions of the added resistance as function of the forward speed have been faired.
O DERIVED FROM DECAY TESTS [2-2] + DERIVED FROM FIG. 2-17
'f 1
/ 1
/
r\ ■ \
■k—-r
Figure 2.15 The computed transfer Figure 2.16 The measured quadratic function of the wave transfer function of drift force for zero the wave drift coeffi-speed of the LNG cient of the LNG carrier carrier
53
RAW ( U )
X ( ,2 )(U)
( t f . n i )
n=0.400 rad.s 0.433 rad.s
o 0.476 rad.s
□ COMPUTED
=0.532 rad . s " 0.616 rad .s " 0.785 rad .s "
o COMPUTED
MEASURED [ 2 - 1 4 ]
U in m.s U in m.s Figure 2.17 The quadratic transfer function of the added resistance
curve as function of forward speed.
The results from Figure 2.17 indicate that the gradient method may be applied to predict the wave drift forces or added resistance for small values of forward speed being in the range of current speeds. For the forward speeds in the order of current speeds the added resistance will be approximately linear with the speed. At higher speeds, however, the added resistance becomes a strongly non-linear function of the speed. To approximate the total wave drift force of a vessel, which performs low frequency oscillations superimposed on the higher forward speeds U both the wave drift force and its derivative at speed U has to be known, which can be expressed as:
t<2Vx<2),u,) X[2)(U,U) B1(U,0)).xJ2) (2.4.5.1)
So far the quadratic transfer functions of the wave drift forces in reg-
54
ular waves were considered under towed conditions. In reality the tanker is moored in a current. The consequences for the transfer functions will be dicussed in the next sections.
2.5. The mean wave_drift_force in_regular waves combined with_current
2.5.1. Towing speed versus current speed
In the previous sections the quadratic transfer function of the mean wave drift force as function of towing speed U was dealt with. In reality, however, the tanker is stationary moored in current. From a theoretical point of view the transfer function of the wave drift force acting on a tanker towed with speed U or a stationary tanker moored in current with velocity V (=U) is the same if the frequency of encounter and the earth-bound frequency, respectively, are the same.
According to the linear wave theory the wave relations are defined with regard to the system of coordinates bound to the fluid, in which the wave propagates [2-15]. Using the system of axis ZQOQXQ, as indicated in Figure 2.18, for the regular wave with the potential <|>Q the following relations for the wave characteristics can be determined:
wave height: CQ = C&Q cos(ü)Qt + K Q X 0 )
and from the dispersion relation:
Figure 2.18 Earth-bound system of co-ordinates
55
u0 = K0g t a n h ( K O h )
with the wave velocity: CQ = \)/Tn = \ / — 2 ~ t a n n (Kr>'1) * (2.5.1.1)
in which: h = water depth K 0 = 2%/\Q
X = wave length
■ V
Figure 2.19 System of co-ordinates Z1O1X1 moving with speed U
For a towing speed U the regular wave should be related to a system of co-ordinates ZiOiXi moving with regard to system ZQOQXQ with velocity U in the direction of the positive XQ axis as is indicated in Figure 2.19.
With respect to the wave characteristics the following relationship exists between both systems of co-ordinates:
w l c l
h C a l
= = = =
io0 +
co +
\ C a0
*ou
U
(2.5.1.2)
For a tanker moored stationary in current the regular wave exists in combination with current. In this case it is normally assumed that both for prototype and model tests the wave frequencies and wave heights are" defined relative to an earth-bound system of co-ordinates. The linear wave theory, however, always defines the relations for wave characteristics relative to systems of co-ordinates bound to the fluid in which the wave propagates. In case of current the wave characteristics can be des-
56
cribed r e l a t i v e to a system of co-ord ina tes Z0O2X2 moving with the current speed as indica ted in Figure 2.20.
SYSTEM OF CO-ORDINATES
z 3 0 3 x 3 FIXED TO EARTH
hz
«z
^ \ «? ~'\J
V c ■
^ / ^ x ^ w / W i W ! ; SYSTEM OF CO-ORDINATES ■
z 2 0 2 x 2 FIXED TO CURRENT
Figure 2.20 System of co-ord ina tes r e l a t e d to current
Based on the fluid-bound system of co-ord ina tes the r e l a t i o n s descr ibed by the wave p o t e n t i a l $2 w i l l be s i m i l a r to the one descr ibed by the wave p o t e n t i a l <1>Q. Therefore the wave c h a r a c t e r i s t i c s in the system of co-ordinates moving with current speed V w i l l be analogous to equation ( 2 . 5 . 1 . 1 ) by changing the subscr ip t 0 i n t o 2:
wave he igh t : C, = C 0 c o s ( " ) 2 t + K2X2^
and
io„ = K2S t anh (K„h)
with the wave velocity: C2 = 2^T2 = \/~T~ t a n h (K2h) (2.5.1.3)
With regard to the system of co-ordinates fixed to the earth Z3O3X3 the following relations can be obtained:
= (02 + K2Vc
= C + V 2 c
Cal Ca0 (2.5.1.4)
57
Comparing equation (2.5.1.2) with equation (2.5.1.4) analogous wave characteristics exist if the vessel will be either towed with speed U in regular waves with frequency UQ or stationary moored in a current with speed V (=U) in regular waves with frequency uio if both frequencies are related to the fluid.
If current is considered the values of the quadratic transfer function of the velocity dependent wave drift forces as shown in Figure 2.12 and Figure 2.17 can be considered to be related to o -
Due tó the current speed V the wave frequency Uj will be transformed into the frequency of encounter 103 according to equation (2.5.1.4). Based on the gradient method and using the relation for the frequency transformation, the quadratic transfer function of the wave drift force and the wave drift damping coefficient as function of the current speed V can be determined according to:
Xl 2 ) ( Vc , U3 ) xi 2 ) ( 0 , w2 ) Bi(0.«>2>-Vc 2 = 2 2
C C C ^a a a B1(Vc,u3) B^O,^) 2 = 2 a a
in which: (1)3 = 102 + K2*Vc
(2.5.1.5)
Applied to the loaded 200 kTDW tanker moored in 82.5 m water depth the quadratic transfer function have been approximated for 2 kn head current; the results are presented in Figure 4.7.
58
2.5.2. Regular waves traveling from an area without current into an area with current
In the previous section the quadratic transfer functions of the second order forces were considered when acting on a vessel both towed and stationary moored in a current field. In this section the transfer functions of a tanker moored in an area without current and an area with current will be considered. It is assumed that the current is directed in the same direction as the propagation of the wave. To determine the relation between both regular waves the following conditions have to be fulfilled [2-15]:
- the relations between the wave characteristics are given with regard to the fluid, in which the wave propagates;
- the wave period in an earth-bound system of co-ordinates does not change when the wave travels from the area without current into the area with current.
To study the wave relations use can be made of earth-bound systems of co-ordinates, viz. Z-JOJX-J and ZQOQXQ for the areas with and without current respectively and the system zft-yK-) moving with the current as is shown in Figure 2.21.
Using the system of co-ordinates 2^02X2 moving with the current the wave relations can be determined. With regard to the wave frequencies Ü)Q the frequencies 002 will shift to smaller values by the term <2'Vc' s e e
Figure 2.21. The value <^ as function of the relations for the still water case can be numerically solved. For deep water the values <^ can be derived from:
59
^ 3 . yS^ ^
V c
+z3
0 3 S
• r^ ' • ' \^/
! x i '•aO,
"0= U J3
•<\f
WITH CURRENT
SYSTEM OF EARTH-BOUND CO-ORDINATES
WITHOUT CURRENT
WITH CURRENT
SYSTEM OF FLUID-BOUND CO-ORDINATES
Figure 2.21 The system of co-ordinates
the wave length relations
X2 = X3 = c\ 2n/g and \ Q = C2Q 2n/g
the celerity of the wave in current
c3 = c2 + vc
and further
60
X3 \ = 0
or
C2 =
and
V
C3 = -p— and
co
1 ^ l + \
the wave
L
«f 4V — ) co J
length
/>♦ 4V — )
C2+Vc co
(2.5.2.1)
(2.5.2.2)
In this situation the frequencies of the transfer functions of the wave drift force and the damping coefficient for zero speed as presented in Figure 2.22 have to be considered on base of frequency (Oo-
In order to arrive at the transfer functions belonging to the wave frequency Wo in the earth bound system of co-ordinates z,0-,x, the frequencies of the transfer function have to be shifted to:
U3 = U 2 + V V c ■
The appropriate values of the transfer function of the wave drift force can be determined by means of the gradient method, see Figure 2.22.
If the regular wave travels from an area without current into an area with current not only attention must be paid to the frequency transformation but also to the wave amplitude. For sake of completeness the theory on the change of the amplitude of waves running from still water into a current area will be shown below. Assuming continuity in transport of wave energy through zQ00 and z,0, we find:
V0 = V3 (2.5.2.3)
in which:
61
vo - Eo,cgO V3 = E3.Cg3
where the wave energy in each system of co-ordinates amounts to:
Eo = ** 4 E3 = hps Ca3 (2.5.2.4)
and the celerity of the wave energy will be:
cgo ° noco C g 3 = n 2 C 2 + V c ( 2 . 5 . 2 . 5 )
in which the transmission coefficient provided with the appropriate subscripts will be:
n = + iïnffeh) <2-5-2'6> By means of the energy relations the wave amplitude in the current field becomes:
Ca3 = C a o V n 2 c f ? V c <2'5-2-7>
From the result it can be concluded that the wave height will decrease if a regular wave will travel from an area without current into an area with current. Applied to irregular waves the following conclusion can be drawn: - since for a regular wave in the earth-bound system of co-ordinates the
wave frequency wili not change, the same will hold true for the frequency range of a wave spectrum;
- the spectral density of the waves will decrease when the waves are running from the area without current into the area with current.
62
QUADRATIC TRANSFER FUNCTIONS
V =2.06 m.s
-15-
- 1 0 -
ID in rad.s
PIERSON-MOSKOWITZ WAVE SPECTRUM
£ , , . , = 8.0 m ; T, = 11.0 s w1/3 1 V = 0 m.s"1
L C
<w1/3 = 5 . 9 m ; T , = 11.0 s V£ = 2.06 m.s"1
in .rad.s
Figure 2.22 Effect of an i r r e g u l a r sea running in to a current f i e l d with a current ve loc i ty of 2.06 m/s on the wave spectrum and the second order t r ans fe r functions
63
To elucidate the theory on the transfer functions and the wave spectrum an example is given. The wave spectrum concerns a Pierson-Moskowitz spectrum, of which the characteristics for no current amount to ? . ._ = 8.00 m and T. = 11 s. The transfer functions of the wave drift force and the second order fluid damping for the 200 kTDW tanker in zero speed condition are assumed to be known. The water depth is considered to be deep. If the waves travel from the area without current into the area with 4 knot current the effects on the wave spectrum and the quadratic transfer functions of the wave drift force and the wave drift damping coefficient are presented in Figure 2.22.
2i^i_2°5EHË2Ëi22_2£_ËlïS_i2ï_Y£i2£iï.Z_ëSEÊ2É£nt waYË_ÉEi££_£°E£Ê5
2.6.1. Introduction
The transfer function of the wave drift force at zero speed in regular waves can be computed by the direct pressure integration method [2-3]. The input of the direct pressure integration method may be based on the output of the diffraction model as reported by van Oortmerssen [2-1]. The diffraction model treats the ship motions for the zero speed case without any geometrical simplification of the underwater hull. The program is based on the solution of integral equations, where the potential function is written as a source distribution along the hull.
In order to determine the gradient or the wave drift damping coefficient at zero speed computations for small values of forward speed are necessary. For the computation of the velocity dependent wave drift forces the diffraction program has to be adapted for the speed effects.
A direct approach is reported by Inglis [2-16], Chang [2-17] and Bougis [2-18]. They introduced the forward speed effect by using the pulsating translating wave source function and certain line integrals. The present computation procedure is restricted to regular waves in deep water and is based on small values of the forward speed.
64
The low velocity dependent wave drift forces actually originate from the low speed dependency of the first order wave loads and hydrodynamic reaction forces. Taking the velocity dependence in mind Hermans and Huijsmans [2-7] pointed out that the original diffraction model based on zero speed [2-l] can be adapted for small values of the forward speed. Therefore the potential function written as a source distribution along the underwater hull and water-line was expanded with respect to small values of the forward speed U. Solving the fluid pressures along the hull and the fluid forces acting on the wetted surface the ship motions can be determined. By applying the direct integration method the transfer function of the wave drift force for small values of forward speed U can be computed.
Plotting the values of the quadratic transfer function on base of zero speed and small values of forward speed (for the same wave length), the wave drift damping coefficient can be determined. In order to determine the transfer functions for other speeds the gradient method can be applied.
2.6.2. Theory
For the theory on the computations reference is made to [2-7], [2-19], [2-20] and [2-2l]. For the introduction of the forward speed the total potential function can be split up in a steady and a unsteady part in a well-known way:
»(ï(J),t) = -Ux(l) +»(x(j);U) + $(x(j),t;U)
for j=l,2,3 (2.6.2.1)
in which: x(j) = system of co-ordinates as indicated in Figure 2.3 moving
with speed U in the positive x(l) direction U = incoming unperturbed velocity field obtained by consi
dering the system of coordinates x(j)
65 '
$ ( x ( j ) ; U ) = steady p o t e n t i a l funct ion <t>(x( j ) , t ;U) = o s c i l l a t i n g p o t e n t i a l function
-lw t = <Kx(j),u)e e
ID = frequency of wave encounter
The steady part does not contribute to the unsteady part directly. It plays a role in the free surface condition. Because of the considered very low Froude numbers (Fn = U//g.L « 0.1) the effect of the free surface is not taken into account and the contribution of $(x(j);U) is completely neglected. The time dependent oscillatory potential <t>(x( j) ,t;U) will be written as a source distribution along the hull and the water-line and will be expanded with respect to small values of U. By solving the potential the part linear with speed will lead to the speed effects in the ship motions and to the wave drift damping in the computations of the second order drift forces. The diffraction theory on the low speed dependent potential as derived in [2-7] is discussed in the Appendix.
2^6.2.1. _Linear_ship_motions_at_forward_s£eed
The flow field characterized by the low speed dependent potential can be computed with the diffraction program:
$(x(j),t;U) = *0(x(j),t) + t^CxCJ),!) (2.6.2.2)
in which: -iw t
*0(x(j),t) = 4>0(x(j)) e e
-iw t ♦jdUJ.t) = (xCj)) e e
u U e g
The oscillating fluid pressure as derived from the linearized Bernoulli equati'pn will be:
66
P(2(j),t) = -P<t>t(x(j),t;ü) or P(x(j),t) = p0(x(j),t) + xp^xCj)^) (2.6.2.3)
in which:
P0(ï(J).t) = " PaT * 0(- ( j ) , t )
Pl(x(J),t) --pgl^K^.t) -pi._»rTy0(ï(j),t)
Integration of the pressure along the mean wetted surface results in the hydrodynamic reaction forces in the system of co-ordinates fixed to the vessel:
X = - ƒƒ p.n.dS S0
in which:
n = generalized direction cosine on S (pointing outside is positive) SQ = mean wetted surface of the vessel X = X, for k=l,2,3 -k
Substitution of the pressure expansion (2.6.2.3) gives:
f x1
—
=
- ƒƒ so
- IS so
V
p l '
■ n.
-n.
.dS
.dS
with:
X = X° + tX1 (2.6.2.5)
For the moments analogous expressions can be derived.
For the unit motion in the j-mode one is now able to write the added
67
mass and damping coefficients as:
2 0 , 0 - to a. .= real X. . e kj kj
" iweh°ki = i m a8 X^j (2.6.2.6)
with similar definitions for a, . and b, . • kj kj
X, . is the reaction force in the k-mode due to a unit oscillatory motion in the j-mode. From the computed wave loads and added mass and damping coefficients the motion of the vessel can be determined using Newton's law of inertia.
2^6.2.2. Wave_drift_force_at low f.2£W££d_sgeed
For the derivation of the second order wave drift forces the fluid pressure as given by the unsteady Bernoulli equation has to be considered:
+ PQ + C(t) (2.6.2.7) P(x(j),t) = -pgx(3) - p$t - ^p|V$
where: Pn = atmospheric pressure x(3) = vertical distance below the mean free surface $ = velocity potential C(t) = constant independent of co-ordinates p = mass density of fluid
The second order (with respect to the wave height) wave forces can be computed now. In Bernoulli's equation P~ and C(t) can be taken zero without loss of generality. Assuming that a point on the hull is carrying out a first order wave frequency motion x, (j) about a mean
(0) position x, (j) and applying a Taylor's expansion to the pressure in the mean position, the following expression is found:
p ,vP<°> + ep(1) + s V 2 ) + 0(e3) (2.6.2.8)
68
where: e = a measure related to wave steepness p^ ' = the hydrostatic pressure
= " Pgxn0)(3)
p^ ' = the first order pressure = - PgxJ^O) - p*
(2) pv ' = the second order pressure = " ^p|v*|2 - p(x^1)(j) -V*t) (2.6.2.9)
The derivatives of the potential <j> are taken at the mean position of the point. The material derivative, D/Dt, results in a ö/öt, and a con-vective term -U.ö/ox operating on the potential $ . The potential 4> is regarded as a first order velocity potential ($ ). In order to determine the second order pressure more exactly a second order
~(2) potential $ has to be added to equation (2.6.2.9). The influence of ~(2) the second order velocity potential <|> , however, will be neglected
since this term does not contribute to the wave drift force in a regular wave.
The total force acting on the ship is:
X = - /ƒ p N dS (2.6.2.10) S
where: N = the instantaneous normal vector S = the instantaneous wetted surface. X = X(j) for j=l,2,3
Using a similar perturbation scheme for the wave loads as for the fluid pressure, we can write:
X = X ( 0 ) + eX ( 1 ) + e 2X ( 2 ) + 0(e3) (2.6.2.11)
69
in which: x(°' = the hydrostatic force obtained from integration of p'ü' along the
t(l) = mean wetted surface SQ the first order wave loads
After some algebraic manipulations the final expressions for the wave drift force becomes:
iw - - M pg WL
,(1) gr 2 n dl + a(1)x(M.x^X)(j)) +
ƒ ƒ -^p|v$|2 n dS - ƒ ƒ -p(x£ >(j) .V*t) n dS (2.6.2.12) S0 S0
in which:
a™ - (xf>, x™. x^))1 ... H J O
xi '(j) = first order motions of CG with regard to 0x(l)x(2)x(3) rl) Xy" (j) = first order motions of a point on the hull with regard to
0x(l)x(2)x(3)
For the moments analogous expressions can be derived.
Since the direct pressure integration method was applied to the case with small values of forward speed the final expression will be analogous to the expression for zero speed, see Pinkster [2-3]. Four contributions to the wave drift force can be distinguished. The terms in equation (2.6.2.12) are caused by: 1. The relative wave height at the mean water line; 2. Product of first order angular motions and inertia forces; 3. Bernoulli pressure drop due to first order velocities; 4. Pressure due to the product of first order motion and gradient of
first order pressure.
In equation (2.6.2.12) the forward speed dependent potentials and derivatives of these potentials have to be evaluated at the mean waterline and the mean wetted surface. The expressions we obtain are of similar nature as those obtained by Hearn and Tong [2-22J. However, their method
70
is based on 2-D strip theory with adaptations for the incorporation of diffraction effects. For the numerical scheme of the calculations of the wave drift forces at small values of forward speed reference is made to [2-20'].
2.6.3. Results of computations and model tests
The computations of the quadratic transfer function of the second order wave drift force and the wave drift damping coefficient were carried out for the loaded 200 kTDW tanker sailing at small values of forward speed in deep water and in regular head waves. For the computations for zero speed and small values of forward speed the tanker hull was schematized by a facet distribution as is shown in Figure 2.23. The number of plane elements amounted to 238 and the number of waterline elements was 60.
In section 2.4.4. the results of computations of the wave drift forces for zero speed were used. The computations concern the transfer functions of the wave drift force for a water depth of 82.5 m and 206 m. For the computation 302 facets and 74 waterline elements were used. The facet schematization is shown in Figure 2.24. For sake of completeness the results in numerical form are presented in Table 2.3.
Figure 2.23 Facet distribution of the tanker hull for the computation at zero and low forward speed in deep water.
71
For the hull as shown in Figure 2.23 the transfer function of the wave drift forces for zero speed and 1 kn and 2 kn forward speed have been computed. In Figure 2.25 the results of the computations of the transfer function for zero speed and 2 kn forward speed are given. Based on the transfer function for zero, 1 kn and 2 kn forward speed the gradients at zero speed have been determined in order to obtain the transfer function of the wave drift damping coefficient. The results of the computation and the experimental data are plotted in Figure 2.26.
A^m
Figure 2.24 Facet distribution tanker hull for the computation at zero speed in 82.5 m and 206 m water depth
-20 r
M)
COMPUTED
U=2 kn
MEASURED O •
/ / / / 1 II
il jfr
/ °V
\ °
0 -0.5 1-0 u in rad.s"
Figure 2.25 Quadratic transfer function of the wave drift force for a 200 kTDW tanker in head waves at zero and 2 kn forward speed (earth-bound wave frequency)
72
T( Cil^u^)
w l \
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96 1.04
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96 1.04
0 0
0.1 0.8
3.5 8.7
12.9 11.9
8.6 9.2
302 facets - 74 waterline elements 8.7 frequencies in rad/s 8.7 Water depth 206 m [2-6] 8.8
^ 0.189 0.266 0.354 0.444 0.523 0.560 0.600 0.630 0.713 0.803 0.887 0.978
0.189 0.266 0.354 0.444 0.523 0.560 0.600
0.134 0.597
1.912 6.947
12.36
13.89
302 facets - 74 waterline elements Frequencies in rad/s Water depth 82.5 m [2-3]
0.630 0.713
8.35
0.803
9.26
0.887
8.66
0.978
8.82
üO\2
0.253 0.354 0.444 0.523 0.560 0.600 0.630 0.713
0.253 0.354 0.444 0.523 0.560 0.600 0.630 0.713
0.462 1.7
5.68 12.79
14.17 13.34
14.50 8.28
238 facets - 60 waterline elements Frequencies in rad/s Deep water
Table 2.3 Computed transfer function of the wave drift force in regular
73
X<2)(U)-X(,2)(0)
(tf.nf')
0.560 rad.s
2 0 (in kn.)
DETERMINATION OF GRADIENT (COMPUTED)
oj=0.600 rad.s
1
(tf.s.m )
COMPUTED O x • EXPERIMENT
0 0.5 1.0 u in rad.s"
Figure 2.26 Quadratic transfer function of the wave drift damping coefficient for the 200 kTDW tanker in head waves (earth-bound wave frequencies)
2.6.4. Evaluation of results
Comparing the results of the model tests and the computations it can be concluded that a reasonable approximation of the wave drift damping can be achieved by means of the potential theory with low forward speed. A peculiar deviation occurs at wave frequency io = 0.523 rad.s . The results clearly indicate that the damping caused by the waves is dominated by the velocity potential. Viscous drag due to the orbital motions of the fluid particles in combination with the low frequency tanker motions as suggested by Lungren et al. [2-23J and Aage et al. [2-24] can be neglected.
74
Comparing the results of the computations of the wave drift force for zero speed for deep water and 206 m water depth some deviations occur. The tanker hulls were approximated with 238 and 302 facet elements, while 60 and 74 waterline element were used respectively. In the frequency range, where for both configurations deep water is valid, the results are to some extent different. It seems that the results are sensitive to the schematization. This might explain the deviation between measured and computed wave drift damping coefficients. It is recommended that more computations be carried out to study the sensitivity of the schematization.
2^7. The_low frec uency_ com£onents_of_the wave drift forces and the wave drift_damging_coefficient
2.7.1 Introduction
The foregoing sections dealt with the transfer functions of the wave drift forces and the wave drift damping coefficient for regular waves only. In irregular waves, however, for both the wave drift forces and the wave drift damping coefficients mean and low frequency components may occur. The frequencies of the low frequency components are associated with the frequencies of the wave groups.
It is assumed that both the transfer functions of the wave drift force for zero speed and the wave drift damping coefficients as obtained in 'regular waves are known. Based on these data approximations will be made to compute the low frequency components of the wave drift forces of the total wave drift forces including low frequency tanker motions with and without current. These approximations are allowed for deep water and small values of the natural frequencies of the system. The procedures will be presented in this section.
75
2.7.2. Wave drift forces at zero speed
In order to arrive at the theory of the approximations to compute the mean and low frequency components for the total wave drift force first the derivation will be given for the wave drift forces for zero speed as treated in [2-3].
The behaviour of the drift forces in waves can be elucidated by first looking at the general expression for the drift forces in a wave train consisting of two regular sinusoidal waves with frequencies u^ and CO2 and amplitudes Ci and C2*
The wave elevation is written as:
2 C(t) = E C. sin(o).t + E )
i=l x
= Cl sinCoo-t + e ^ + C2 s i n ( u 2 t + e ) ( 2 . 7 . 2 . 1 )
?!
£!
Figure 2.27 Regular wave group
For small differences between o> and u>2 a schematic representation of the wave train is shown in Figure 2.27. Such a wave train will be called a regular wave group. This type of wave train is characterized by a periodic variation of the wave envelope. The frequency associated with the
76
envelope is equal to Au = u - u>„ being the difference frequency of the regular wave components.
We will write the wave elevation in amplitude modulated form:
C(t) = A(t) sin(wt + ê) (2.7.2.2)
in which:
u = (w^ + u2)/2 E = (e]_ + e2)/2
It can be shown that the envelope becomes: 2 2
A(t) = [ E T. C.C, cos((io -w,)t + (e -e,))]* (2.7.2.3) i=li=l 1 J J 1 J
The square of the envelope is:
2 2 A2(t) = E E C.C. cos((w -w.)t + (E,-£.)) (2.7.2.4)
1=1 j=i 1 J J x J
A quantity which is a quadratic function of the wave amplitude, in this case the wave drift force, will be:
2 2 X,(2V) = Z E C.G.P, . cos((ü>.-u.)t + (e.-e.)) +
1=1 j=l J J J J
2 2 + £ 2 C.C.Q, . sinf(u -oo.)t + (E - E . ) ) (2.7.2.5)
1=1 J = 1 i J iJ i J i J
in which PJ, and Qj. are quadratic transfer functions dependent on two frequencies Bj and U)J. Generally P. , and Q. . are computed so that the following relations exist:
77
p i j - p j i Q i j - - Qji
P^J is that part of the quadratic transfer function which expresses the component of the drift force which is in-phase with the square of the wave envelope and Q ., expresses the quadrature part of the drift force. For the regular wave group the wave drift force is:
xj }(t) = C J P U + C 2P 2 2 + C1C2(P12+P21).cos((u)1-u2)t + (e^e2)) +
+ ^1C2(Q12-Q21) sin((i^-co^t + ( e ^ ) ) (2.7.2.6)
The formulation shows that the drift force contains several components. The first two are constant parts corresponding to the mean drift force in each of the regular wave components separately. The third and fourth parts are low frequency varying components which arise through the combined presence of the two regular wave components in the wave group.
The quadratic transfer function of the wave drift force in terms of amplitudes and phase angles are defined as:
T = TO^.u) ) = (P2<u)1,uj) + Q V ^ W J ) ) *
= quadratic transfer function of the amplitude of the wave drift force
E. . = arctan - -=-; = - (2.7.2.7) ij P(iüi,co )
= phase angle between the low frequency part of the second order force relative to the low frequency part of the square of the wave elevation.
Using the mentioned definition of the quadratic transfer function the wave drift force for the regular wave group can be written as:
78
2 2 X<2> } ;(t) = E E C.C.T cos((u -oo )t + (e,-e.) + e. .) (2.7.2.8)
1=1 ï=i J J J ■*■ J XJ
In irregular waves the wave drift force is:
,(2) N N Xj J(t) = E E q t l cos((w -u> >t + (e±-e.) + e ) (2.7.2.9)
1=1 j=i J J J J . J
The quadratic transfer function P^J and Q.. for zero speed can be computed by means of the direct pressure integration method.
2.7.3. The approximation of the low frequency components
The computations of the wave drift forces with low forward speed, however, have been developed for regular waves. Only the values for P(U,üJi,ooi) can be computed. Therefore for all computations the following assumption is made:
xJV) "i,wiy 2
a and
CKo^.wp = 0 (2.7.3.1)
In order to estimate the unknown low frequency varying components of the drift forces an approximation will be made. The approximation is possible if the water is deep and the natural frequency of the system is very low. In [2-3] it is shown that the influence of the second order potential on the low frequency parts is negligibly small for deep water. Neglecting the effect of the second order potential the low frequency varying part arises through the combined presence of two regular wave components with frequencies GOJ and OJ.. The frequency difference of im-
79
portance will be IO^-ÜJJ = u, where u is very small. If u is small then 0.( ,(0 .)~0 (for monohull type of structures).
Neglecting the quadrature part of the transfer function the low frequency component can be estimated as proposed by Newman [2-25]:
P(o) ,Q> ) + P((0 ,U) ) PO^.w..) i 5
J - J - (2.7.3.2)
Because the frequency difference is assumed small equation (2.7.3.2) will approximately correspond to:
(o.+u). to.-ho . P^.O.J) = p(^rJ-.-irJ-^
10.+(0 . (O.+ü) . T(Ul,io ) = p("i2-J'''JTJ"^ (2.7.3.3)
This approximation will be used for all computations.
2.7.4. Total wave drift force in irregular waves without current
Following the gradient method the total wave drift force in a regular wave group can be written as:
4?M . x < 2 > ( t ) + f e i = ox.
= C^T n + C%I22 + 2C1C2T12 cos(Aoo12t + Ae12) +
+ C?Dni1 + ^ D 2 2 X l +
+ 2CLC2D12 cos(Ao)12t + A E 1 2 ) X 1 ( 2 .7 .4 .1 )
80
in which:
*1
D l l
D22
D12
=
=
=
=3
i ( 2 )
ÖT11 ÖX.
5T22 öx.
ÖT12 öx.
The total wave drift force can be split up in a wave drift force and a wave drift damping part. The wave drift damping force contains several components. The coefficients of the first two terms correspond to the mean wave drift damping in each of the regular wave components seperate-ly. The third term stands for the low frequency varying part of the wave drift damping force. As mentioned for the derivative of 1(00 ,0).) to the low frequency velocity no data exists. To estimate the oscillating part of the damping the same procedure is proposed as applied to the oscillating part of the wave drift force:
u).-ko. to.+w . ÖTO^.Uj) ÖT(-irJ-,-ir
J-) D J (2.7.4.2)
J öxx bi
The total wave drift force in irregular waves without current will be:
x ! t } - ±ix .ix cicjTij ««««wt + ( e i - e j > + e i j ) +
N N + Z E ZX.O..X (2.7.4.3)
i=l j=l 3 J
81
2.7.5. Stability of the solution and contribution of the oscillating wave drift damping coefficient
The effect of the damping will play an important role for the condition that the natural frequency of the moored vessel will correspond to the frequency of the wave group:
Awi2 " » =\JmI^) Assuming a linear viscous damping the equation of low frequency motion can be written as:
(M + a u(u))x 1 + ( B u - C*D n - C2D22 +
- 2C 1C 2D1 2 cos(ut + E 1 2 ) ) X 1 + CJJXJ^ =
= C J T U + C 2T 2 2 + 2CXC2T12 cos(ut + e12) (2.7.5.1)
The damping term contains linear coefficients and a low frequency oscillating coefficient. Due to the low frequency oscillating coefficient the value of the damping coefficient can be positive or negative. Since a negative damping in the equation of motion can cause an unstable solution attention has to be paid to the magnitude of the oscillating damping coefficient with regard to the linear damping coefficients. Moreover, attention will be paid to the contribution of the damping force with the oscillating coefficient to the motions.
In order to judge the influence of the low frequency oscillating coefficient on the stability to the solution the equation of motion is simplified:
mx^ + f(t)x. + ex. = a.cosjit (2.7.5.2)
82
in which: f(t) = b, + D2 cosut a.cosu.t = oscillating part of the wave drift force.
Starting from the equation of motion for the tanker
mxj + f(t)ix + cxx = 0 (2.7.5.3)
and multiplying this equation by x.. , we obtain:
.2 mx1x1 + f(t)x + ex ij = 0 (2.7.5.4)
or in terms of energy:
■£ ( W j + *cxj) = - f(t)xj (2.7.5.5)
s2 This means that the decrease of the total energy corresponds to f(t).x... The system is called instable if the term f(t) is negative. In order to judge the stability the sign of f(t) will be studied.
The damping function f(t) consists of still water damping and the derivatives of wave drift force components to the low frequency velocity.
Because the natural frequency of the moored vessel is usually very low, the frequency difference of importance will be 10 -to. = u with u « 1.
Caused by the small value of the frequency difference the magnitudes of the damping coefficients will have approximately the same value:
oT(w ,10 ) oT((o.,<o ) ÖT(u> co.) ±—— = i-J- = — i - J - (2.7.5.6)
öi1 ox. ÖJL
83
Since:
Ci + Cj > 2 Ci Cj (2.7.5.7)
and because the still water damping has to be added to the linear parts of the wave damping coefficients it can be concluded that the low frequency oscillating damping coefficient will be smaller than the linear damping coefficient. Therefore the sign of f(t) will be positive for all values of t.
Besides the stability of the solution also attention will be paid to the contribution of the oscillating damping to the motion.
For the considered equation of motion
mx^ + (b^ + b2 cosut)x + ex = a cosut (2.7.5.8)
the following solution has been assumed:
xx = pQ + E (pn cos(nut) + qn sin(nut)) (2.7.5.9) n=l
Substituting the solution in the equation of motion it can be proven that the oscillating damping coefficient hardly contributes to the motions. Because the low frequency oscillating coefficient has to be multiplied by the velocity, of which the dominant components consists of cosut, the product results in a double frequency 2\x. This double frequency is beyond the resonance frequency of the low frequency surge motion and so the contribution will be negligible. It must be mentioned that in principle a constant part will remain, which will affect the mean wave drift force slightly.
Neglecting the term with the oscillating coefficient the total wave drift force in a regular wave group will be:
84
2 2
x^2)( t ) = i=i A CiCjTiJ "" vV + ( v e j> + eij) +
2 2 .
+ Z C D x (2.7.5.10) 1=1
Following the mentioned procedure the total wave drift force in irregular waves without current will be:
4? = .fx ^ ciSTij « ^ v v ' + ( e i-e j>+ e i j ) + '
N 2 . + 2 C.D x (2.7.5.11) 1=1
2.7.6. Total wave drift force in irregular waves combined with current
Assuming that the vessel is sailing at a low speed U and performing low frequency oscillations x1 in a regular wave group the total wave drift force according to the gradient method is:
Xt 2 > ( ul , w2 , l H il , t ) = C1 TU + C2T22 +
+ 2C1C2T*2 cos(Au12 + Ae12 + e12) + + C ^ ^ + C ^ ^ (2.7.6.1)
in which:
T* = T + D .U 11 11 11
T* = T + D .U 22 22 22 U
T* = T + D .Ü 12 12 12
The formulation shows an increase of the constant parts and the oscillating part of the wave drift force caused by the low forward speed U.
85
The formulations mentioned so far were based on a vessel sailing with low forward speed U in combination with low frequency oscillations, while the wave frequencies (o and (02 are considered with regard to earth. If the vessel is moored in a current with speed V (bow directed into the current) the earth related wave frequencies should be transformed into the wave frequencies io„ (= wave frequencies as measured at a fixed point in the Wave and Current Laboratory) at:
<0 . = (i), + T V
el 1 A., c
U) , = u, + I 2 1 V (2.7.6.2) The total wave drift force in irregular waves combined with current with a velocity V is:
.9, N N X^z;(t) = S Z C1CjT*(co1,(oj) cos((U.-a)j)t + ( e ^ ) + e^) +
N + £ ^ i ^ V 0 0 ! ^ ! (2.7.6.3)
in which: 10+10. (0.+10.
u.-hi). u -ho. oT( 1- J-, 1 ■ J) T*(U ,(0 ) = T(^I-jL.-JtJ") + — -Vc
ö ii
while co and w. stand for the frequency transformations according to:
w. = co.» + T V i 10 \i0 c
10, = U-n + T V j JO X j 0 c
86
in which o^g and W.Q are the frequencies in still water and \^Q and \JQ are the corresponding wave lengths.
2.7.7. Evaluation of results in irregular waves
Knowing the quadratic transfer functions of the wave drift force and the wave drift damping coefficient in regular waves by means of the approximation method the total wave drift force in irregular waves with or without current can be determined. In solving the equation of motion, however, the low frequency hydrodynamic reactive forces have to be known. Therefore in Chapter 3 the low frequency viscous reactive forces will be dealt with in general terms. The evaluation of the calculated wave drift force and the resulting motions in irregular waves with or without current will be done by means of model tests. The validation will be presented in Chapter 4.
87
REFERENCES (CHAPTER 2)
2-1 Oortmerssen, G. van: "The motions of a moored ship in waves", MARIN Publication No.510, Wageningen, 1976.
2-2 Wichers, J.E.W. and Sluijs, M.F. van: "The influence of waves oh the low frequency hydrodynamic coefficients of moored vessels", OTC Paper No. 3625, Houston, 1979.
2-3 Pinkster, J.A.: "Low frequency second order wave exciting forces on floating structures", MARIN Publication No. 600, Wageningen, 1980.
2-4 Wichers, J.E.W.: "On the low frequency motions of vessels moored in high seas", OTC Paper No. 4437, Houston, 1982.
2-5 Remery, G.F.M. and Hermans, A.J.: "The slow drift oscillations of a moored object in random seas", OTC Paper No. 1500, Houston, 1971.
2-6 Wichers, J.E.W. and Huijsmans, R.H.M.: "On the low frequency hydrodynamic damping forces acting on offshore moored vessels", OTC Paper No. 4813, Houston, 1984.
2-7 Hermans, A.J. and Huijsmans, R.H.M.: "The effect of moderate speed on the motions of floating bodies", Schiffstechnik, Band 34, Heft 3, September 1987, pp. 132-148.
2-8 Cummins, W.E.: "The impulse response function and ship motions", Department, of the Navy, David Taylor Model Basin, Washington D.C, Report 1661, October 19.62 / Schif f stechnik, Vol. 47, No. 9, Jan. 1962, pp. 101-109.
88
9 Ogilvie, T.F.: "Recent progress towards the understanding and prediction of ship motions", Fifth Symposium on Naval Hydrodynamics, Bergen, 1964.
10 Hooft, J.P.: "Advanced dynamics of marine structures", Wiley-Interscience Inc., New York, 1982.
11 Saito, K., Takagi, M., Okubo, H. and Hirashima, M.: "On the low-frequency damping forces acting on a moored body in waves", J. Kansai Soc- N.A., Japan, No. 195, 1984 (in Japanese).
12 Nakamura, S., Saito, K. and Takagi, M.: "On the increased damping of a moored body during low-frequency motions in waves", Proc. 3rd Offshore Mechanics and Artie Engineering (OMAE) Conference, Tokyo, April 1986.
13 Faltinsen, O.M., Dahle, L.A. and Sortland, B.: "Slow drift damping and response of a moored ship in irregular waves", Proc. 3rd OMAE Conference, Tokyo, April 1986.
14 "Added resistance in waves", MARIN Report No. 50332-1-OE, December 1983.
15 Ippen, A.T., Editor: "Estuary and Coastline Hydrodynamics", Engineering Societies Monographs, McGraw-Hill Book Company, 1966.
16 Inglis, R.B.: "A three dimensional analysis of the motion of a rigid ship in waves", PhD Thesis, University College, London, 1980.
17 Chang, M.S.: "Computation of three dimensional ship motions with forward speed", Proc. 2nd International Conference on Numerical Ship Hydrodynamics, Berkeley, 1977.
89
2-18 Bougis, J.: "Etude de diffraction-radiation dans Ie cas d'un flotteur indeformable animé par une houle sinusoidale de faible amplitude", PhD Thesis, Université de Nantes, 1980.
2-19 Huijsmans, R.H.M, and Hermans, A.J.: "A fast algorithm for the computation of 3-D ship motions at moderate forward speed" 4th International Conference on Numerical Ship Hydrodynamics, Washington, 1985.
2-20 Huijsmans, R.H.M.: "Wave drift forces in current", 16th Conference on Naval Hydrodynamics, Berkeley, 1986.
2-21 Huijsmans, R.H.M. and Wichers, J.E.W.: "Considerations on wave drift damping of a moored tanker for zero and non-zero drift angle", Prads, Trondheim, June 1987.
2-22 Hearn, G.E. and Tong, K.C.: "Evaluation on low frequency wave damping", OTC paper No. 5176, Houston, 1986.
2-23 Lungren, H. Sand, S.E. and Kirkegaard, J.: "Drift forces and damping in natural sea states", International Symposium on Ocean Engineering and Ship Handling, Gothenburg, 1982.
2-24 Aage, C , Lungren, H., Jensen, 0. and Velk, P.: "Scale effects in model testing of floating offshore structures", Proc 3rd OMAE Conference, Tokyo, April 1986.
2-25 Newman, J.N.: "Second order, slowly varying forces on vessels in irregular waves", International Symposium on the Dynamics of Marine Vehicles and Structures in Waves, London, 1974.
90
CHAPTER 3 HYDRODYNAMIC VISCOUS DAMPING FORCES CAUSED BY THE LOW FREQUENCY
MOTIONS OF A TANKER IN THE HORIZONTAL PLANE
3Il^_Introduction
A single point moored tanker exposed to irregular waves, wind and current will undergo not only low frequency surge motions, but in general will perform low frequency motions in the horizontal plane. The equations of motion will be governed by the low frequency force components as is shown in Figure 3.1.
EXCITATION FORCES
mean current
mean drift force
slowly varying drift forces
mean wind
DAMPING FORCES
hydrodynamic viscous damping
wave drift damping
wind damping
INERTIA FORCES
added mass
Figure 3.1 The mean and low frequency force components
In establishing the equations of motion difficulties arise in the description of the low frequency hydrodynamic reaction forces and moment acting on the hull. In the low frequency range the damping parts of the hydro-dynamic reaction forces and moments cannot be attributed to forces of potential nature only, but are for an important part determined by viscosity.
The forces and moments caused by viscosity cannot be fully solved by mathematical models but have to be determined by means of model tests. As a result the formulations for the horizontal motions are called semi-theoretical empirical mathematical models. In the past many investigations have been carried out to describe the manoeuvring of ships. Since the physical aspects of the low frequency motions of a moored vessel are similar to those of a vessel manoeuvring
91
at low speed, some of the formulations will be briefly reviewed here. Most of the manoeuvring models are based on the model presented by Abkowitz [3-l], see Section 3.2. In this model the manoeuvrability is described by means of linear and non-linear derivatives of the hydrody-namic forces and moments by perturbation models. Examples of these models are given by: - Inoue, Hirano and Kijima [3-2], 1981; - Hirano and Takashina [3-3], 1980.
In general these kinds of models are used for relatively high sailing speeds and small drift angles.
Another category of models describes the physical behaviour as a result of the three flow fields: - ideal incompressible flow; - viscous flow generating lift forces; - viscous cross flow in planes perpendicular to the longitudinal axis of
the ship.
Examples of these models are given by: - Gerritsma, Beukelman and Glansdorp [3-4], 1974; - Glansdorp [3-5], 1975; - Sharma and Zimmermann [3-6J, 1981; - Sharma [3-7], 1982.
An advantage of the proposed models is the increased insight of the contributions of the force and moment components in the physical process. Furthermore the drift angle can vary 360 degrees. A disadvantage as a consequence of the modeling is, however, that at zero rate of turn the remaining resistance force and moment components do not correspond to the actual steady current force and moment. This is of importance for the mean position of the tanker moored to an SPM in a combined weather condition.
92
A model describing the physics by taking into account only the ideal incompressible flow and the viscous cross flow In the plane perpendicular to the longitudinal axis of the vessel was proposed by: - Faltinsen, Kjaerland, Liapis and Walderhaug [3-8], 1979.
As a consequence of the modeling the same disadvantage for the mean position of the tanker can be mentioned as before.
An improvement for the mean position of the tanker was applied by: - Ractliffe and Clarke [3-9], 1980.
In accordance with [3-8] they replaced in the formulation the non-stabilizing Munk moment, which originates from the ideal incompressible flow (see Section 3.4.1), by the current moment formulation.
For a tanker moored in a current field the hydrodynamic forces on the hull consist of inertia parts caused by the ideal incompressible flow and resistance parts induced by viscosity. The resistance forces and moments (including the Munk moment) on a steady tanker in a real flow are dominated (or modified) by viscosity and are called the current force and moment components. For tanker-shaped bodies a considerable amount of data on the current force and moment components is available, see for instance [3-10] and [3—11J. Based on this knowledge a category of models has been developed of which the descriptions of the physics are based on the relative current concept:
- Wichers [3-12], 1979; , - Molin and Bureau [3-13], 1980; - Obokata [3-14], 1983.
These models ensure that in current and at zero rate of turn the tanker will take up the correct mean position in a combined weather condition. The total relative current force formulation consists of a quasi-steady relative current part and a dynamic current contribution. The dynamic current contribution involves the part of the hydrodynamic forces caused
93
by yaw of the tanker in the current field, see Section 3.4.
In order to determine the dynamic current force contribution model tests may be carried out. The tanker is rotated about a vertical axis through the centre of gravity for a number of steady low yaw velocities in a number of steady current velocities, while measuring the force/moment components in surge, sway and yaw direction [3-13].
STEADY YAW ROTATION IN CURRENT
(Vc=l.03 m.s
OSCILLATING YAK MOTION IN CURRENT: 1984 _ - » 1985 — o
(V =1.03 m.s"' ; |*6a|= .003 rad.s"') POTENTIAL PART
100
" I t fyn
A2dyn. . „. In t f
20,000
^A . -x-j
^
1 ' /
90 180 (♦c-x6) in deg
Figure 3.2 The dynamic current force and moment contribution due to steady yaw and an oscillating yaw motion
The tanker moored to an SPM, however, performs slowly varying oscillating yaw motions in the current field. Figure 3.2 shows that in order to obtain the correct information on the forces in a current field low frequency yaw oscillating tests have to be carried out instead of the steady yaw rotation tests.
It 'may be assumed that the oscillations at low frequencies will induce different flow patterns around the vessel in still water and in a current field. Therefore a clear distinction will be made between the damping forces and moment in still water and in current. For the determination of the low frequency damping forces a series of model tests were carried out as is indicated in Table 3.1.
94
surge mode sway mode yaw mode
oscillatory motions in calm water
steady linear motions / steady current forces
oscillatory motions in current
Table 3.1 Review of experiments
Based on the results of the experiments a description of the damping forces and moments in still water and in current can be derived. Using the experimental results the category of models based on the relative current concept [3-12], [3-13] and [3-14] will be checked with respect to their reliability. For the still water case no formulation was found in literature. A formulation for the still water case will be proposed in this chapter.
5i^ • _lSH££i°5Ë_°l the_l°w_!Ee3uËI?£v._9°£i:2!ïs.
To study the motions of the vessel in 3 degrees of freedom (in the horizontal plane) use is made of two systems of co-ordinates as is indicated in Figure 3.3: - the system of axes 0x(l)x(2) is earth-fixed; - the frame Gx-^ is linked to the vessel with its origin in the centre
of gravity. Based on the earth-fixed system of co-ordinates the differential equations of motion according to Newton's second law are:
Mx(l) = X(l)
Mx(2) = X(2)
Ix(6) = X(6) (3.2.1)
95
in which: M = mass of the tanker I = moment of inertia of the tanker.
x(1)
Figure 3.3 System of co-ordinates of a tanker moored to a SPM and sign convention of weather direction
In literature the ship's manoeuvrability is mostly described by a set of differential equations of motion relative to a ship-fixed system of coordinates. The transformation to the ship-fixed system of co-ordinates has the following consequences:
+x(2)
«(1)
X(j) = T Xj for j = 1,2,6 (3.2.2)
and
2( j ) = T i j
where:
T =
COS X. 0
sin x,
0
( 3 . 2 .
- s i n x 0 6 cos x, 0
0 0 1
while for the acceleration the following transformation
Figure 3.4 The origin of the centrifugal is found: effects
96
5(J) = T 5j + T ïj - (3-2.4)
Substitution of equation (3.2.2) through equation (3.2.4) in equation (3.2.1) yields the equations of motion for the ship-fixed system of coordinates:
M.Cx-1 ~ x2x6) = Xx
M(x2 + i:x6) = X2
Iï6 = X6 (3.2.5)
The acceleration components are modified by the so-called centrifugal effects as shown in Figure 3.4.
Consequently the equations of motion expressed in the absolute ship's accelerations along the instantaneous directions of the ship-fixed axes are as follows:
M.x1E = Xj_
M.x2E = X2
l/x6 = X6 (3.2.6)
in which x^E and x 2 E are the accelerations in the earth-fixed system of co-ordinates along x^ and x2~axis respectively.
For the low frequency motion components x, , x2 and x,- the complete equations of motion for the ship-bound system of axes are:
M(x+Dx) = X H + X w + X m + X D + X T ( 3 . 2 . 7 )
w h e r e :
97
X = <
M O O
O M O
O O I ,
O O -x„
O O +i,
O O O
X^ = hydrodynamic reaction and current forces X = wind force -w X = mooring force —m
wave drift force X = thrust of main and auxiliary propellers
The hydrodynamic forces X arise from changes in the relative motions of the ship and the surrounding fluid. In unrestricted water the forces are independent of the co-ordinates x(l) and x(2).
According to classical hydrodynamic theory the hydrodynamic forces will not be dependent on higher derivatives of the displacement than the second [3.lJ and are usually expressed as:
XJJ = f(u,v,x6(u,y),x6>x,x) (3.2.8)
in which: u,v = components of the steady stPte drift velocity x^(u,y) = steady state drift angle x = variations about the steady state condition
98
Expansion of the Taylor series about the steady state condition results in the following expression for the hydrodynamic forces and moment:
v If 9 _,. 9 , • 9 j. • ° j_ •• ° ' ; X„ = E -j- LXA + xi + XT + XA + xi + n=0 ox, ox. ox„ ox, ox. 6 1 2 6 1
+ x 2 —^- + x6 -^-] * f(u>v,x6(u,v),x6>x)x) ox» 9x, x =0,x=O,x=0
° (3.2.9)
If the expansion is carried out for the first or higher order terms a number, of terms are generated. The coefficients, in general, are assumed to be constant while the magnitudes have to be determined by means of model tests or calculations.
Contrary to normal manoeuvring applications (high speed, high rate of turn and relatively small drift angles) for a tanker moored to an SPM specific requirements have to be fulfilled:
- large drift angles (0 - 360 degrees); - small values of the drift velocity (current speed); - small values of oscillating rate of turn; - relatively large transverse motions; - for zero rate of turn the hydrodynamic forces and moment correspond to the steady current loads.
Specifically for these conditions the equations of motion and the hydro-dynamic viscous forces have been derived for still water and current, see Sections 3.3 and 3.4.
99
3^31_H^drod^namic_viscous damping forces in still_water
3.3.1 Equations of motion in still water
In deriving the low frequency fluid reactive forces in calm water, the external force X^ in equation (3.2.7) will be considered only:
M(x+Dx) = X (3.3.1.1)
Because of the low frequency motions it can be assumed that the disturbances of the free surface of the fluid are negligible. Assuming an ideal and Irrotational fluid, Norrbin [3-15] derived for the forces exerted on the vessel:
.2 X1H ~ allxl + a22x2x6 + a26X6
X2H = _a22X2 " allXlX6 " a26X6
X6H = _a66X6 " (a22-all)xlX2 " a62 ( x2 + Xl X6 ) (3.3-1.2)
where a, . = added mass coefficient at low frequency.
The above equations lead to the well-known d'Alembert paradox since the right-hand sides are equal to zero for:
*1 = x2 = x6 = °
The term ~(a22-aii)^.i_ is the only term arising in an ideal and irrotational fluid and is often referred to as the Munk-moment [3-16]. The force distribution of the Munk-moment caused by the linear velocity is shown in Figure 3.5. In real fluid, however, viscosity is involved. The viscosity introduces additional damping forces. Further it may be assumed that the acceleration dependent terms are hardly affected by viscosity and may be determined by means of the 3-D potential diffraction theory.
100
Xfi = -(a22"air)'r^2
Figure 3.5 The force distribution of the Munk-moment caused by linear velocity
For the equations of motions the following formulations are assumed:
.2 = (M+a22)x2x6 + a 2 6x 6 + X ^ (M+au)x1
(M+a22)x2 + a26x6 = - ( H f a ^ x ^ + X ^
<I+a66)X6 + a62X2 " _(a22-all>iii2 " a62ili6 + X6SW (3.3.1.3)
in which X-,gw> X2SW an<* X6SW a r e the '*'ow f r e < 3 u e n c v viscous fluid resistance force/moment components in still water. The low frequency viscous fluid resistance terms are determined by means of physical experiments.
3.3.2. Test set-up and measurements
In order to determine the low frequency viscous resistance force/moment components in still water caused by the sway and yaw modes of motion
101
planar motion mechanism (PMM) tests were performed. For the surge mode of motion extinction tests in still water were carried out. The tests were done with the 200 kTDW tanker at scale of 1 to 82.5, see Section 2.4.1. The PMM tests were carried out in the Shallow Water Laboratory of MARIN. The water depth amounted to 1 m. The basin measures 15.75 m x 210 m and is provided with a carriage.
For the PMM tests a hydraulic oscillator was used. The oscillator was fixed to the carriage and was driven by means of two hydraulic pistons. When separately adjusted the pistons can perform a prescribed displacement, frequency and phase angle with a high degree of accuracy. The test set-up of the oscillator is shown in Figure 3.6- By means of two ship-bound two-component force transducers the vessel was connected to the oscillator allowing pitch, roll and heave motions. By means of the transverse forces measured with the force transducers located at the fore and aft part of the vessel the total transverse force and the moment can be determined.
Figure 3.6 Test set-up with the hydraulic oscillator
102
Both sway and yaw oscillation tests were performed. For the oscillator tests the stroke was kept constant, but the frequency of the motions was changed for each test. The amplitude of the yaw angle amounted to approximately 16.2 degrees, while for the sway motion the amplitude was approximately 30.2 m. The frequencies applied were 0.0111, 0.0179 and 0.0248 rad.s . The values given are for full scale.
To evaluate the viscous damping forces for the surge mode of motion extinction tests were carried out.
All presented results were obtained by scaling the measured model results to prototype according to Froude's law of similitude.
3.3.3. Viscous damping in the surge mode of motion
Computations by means of 3-D potential diffraction theory have shown that the radiated damping can be neglected for to/L/g < 0.5 , see Figure 2.1. This means that for the low frequencies the viscosity will dominate the damping. Since in the low frequency range the amplitudes of oscillation will be large, the steady current force formulation is normally used. The equation of the low frequency motion for a motion decay test can be described as:
(M+au(u1))x1 - IjpLTC^C^pxJ + c ^ = 0 (3.3.3.1)
in which: Cif.C'lv,-) = resistance coefficient in current cb = relative current direction Ycr
Based on equation (3.3.3.1) thé decaying surge motion of a linearly moored and loaded 200 kTDW tanker in 82.5 m water depth was computed. The computations were performed for two spring constants viz. c,. = 53.72 tf.m ' and 251.15 tf.rn" . The resistance coefficients were derived from Figure 2.9 and amount to C1(,(180°) = -0.032 and C^ (0°) = 0.038.
103
The results of the computations are presented in Figure 3.7. For the same conditions physical extinction tests were carried out. The results of the model tests are also given in Figure 3.7. Comparing the measured and calculated results it can be concluded that for the determination of the surge damping in calm water the relative current concept is not applicable.
Test No. 8053 Test No. 8069 200 kTDW - c „ = 53.72 tf.nf1- »,, = 0.046 rad.s"1 200 kTDW - c ,, = 251.15 tf.m"'- „,, = 0.099 rad.s"
^ k L V v 1
L l
\ 1
^v,
ft \
1
^ }
COMPUTED B n = 13.71 tf .m" .5
Sb
k ^
|"V<
^ N
N ***-s
.o
MEASURED B n = 41 .2 t f .m" ' . s
^ h 0
N
—
o,
—
V ^ ■ * >
t
l
[ \ \
x, l
\ ;
■^
I N
i
\
-o,
1
^
COMPUTED B n = 8 .43t f . ro" , . s
^ c, ■ 2J
M d l oi \
MEASURED B ( 1 = 23 .4 t f .m" ' . s
L t; V
D
i
e ï
~2
20 19 18 17 16 15 14
13 12
11
10
9
8
7
6
5 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
N Number of oscillations '"*
Figure 3.7 Results of decaying surge motions in still water
A possible explanation for the discrepancies between the measured and calculated motion decay is a different boundary layer behaviour. The longitudinal frictional resistance coefficient C l c was determined for a steady current. In this steady current the boundary layer is assumed to be turbulent. For the oscillating ship in still water it can be assumed that the boundary layer is laminar. In order to study the laminar case the theory of an oscillating plate with a laminar boundary layer can be used, see Article 328 of reference [3-17]. For the laminar boundary layer theory the general Navier-Stokes equation can be written as:
104
ö 2 P?rv. + pV.fv.v.) = -V .p + T)V v. Kot j y iv i y y J
in j-direction for i = 1,2,3 (3.3.3.2)
in which: T) = dynamic viscosity = v.p
v = kinematic viscosity p = mass density
and the equation of continuity:
V i = ° (3.3.3.3)
For a flat plate oscillating in its plane the sign convention for the displacement, velocities and forces is given in Figure 3.8.
+x 3
x3
T
T
+ i i_ 3X3
dx
d x i
3
dx3
+x,
'1 Figure 3.8 Sign convention
For the plate the equation can be reduced as follows: öv 1 dt p Vi ( vi vl } nvV o (3.3.3.4)
Local Convective term Viscous term term (destabilizing term) (stabilizing term)
105
The viscous term can be written as:
ox dx- (3.3.3.5) 3
öv in which the laminar shear force: x = v.p -*—
ox,
Assuming that the non-linear convective term can be neglected the equation will be linear:
öv 5 v. P-OF" V ^ = °
ÖX3 öv. ö v. ^ - ^ = 0 (3.3.3.6)
5x3
For: x3 = 0 , v^ = vla-sin ut x3 = », V l = 0
the solution will be:
-px, / = vla-e J.sin((0t-Px3) (3.3.3.7)
in which:
P = /u/2v and u = 2n/T
The distribution of the velocity in the laminar boundary layer alongside an oscillating surface is shown in Figure 3.9. Following equation (3.3.3.5) and equation (3.3.3.7) the laminar shear tension along the plate can can be derived as follows:
T = v.p.p.Vla/ï.cos(-| - ut) = p/onv. vlacos(^ - ut) (3.3.3.8)
106
-1. O / / / / / / / / X / / / / / / / 0 Oscillating +1 . 0 surface
'1a
Figure 3.9 Distribution of the laminar flow
Applying equation (3.3.3.8) to the resistance force on a tanker oscillating in surge direction we obtain:
/ ft C = p/cü.v .S. vla.cos(x - ut) (3.3.3.9)
11
in which: p = 0.1045 tf.s2.m~ v = 1.18831*10~6 m 2.s - 1
S = wetted surface = (3.4 * V 1 / 3 + 0.515*L)V1/3 3 V = displacement in m
L = length between perpendiculars
107
The theoretical resistance damping coefficients were determined for the loaded 200 kTDW tanker and for 4 different spring constants. The theoretical results were compared with the damping values as obtained from physical extinction tests. The results are presented in Table 3.2.
Tanker kTDW
200
S (m2)
22,804
Model
Scale
\
82.5
S
(m2)
3.35
Ü)
(rad.s-1)
0.2162 0.4178 0.9027 1.5531
Prototype
0)
(rad.s-1)
0.024 0.046 0.099 0.171
Calculated En (tH.nf1)
10.97 15.6 23.0 30.1
Measured B,.
, H -1\ (t.s.m x) 18.2 23.4 41.2 92.6
Table 3.2 Results of calculated and measured surge damping coefficients in calm water
In Table 3.2 the damping values are derived for model scale and scaled to prototype values according to Froude's law of similitude. The results show that the calculated values based on the oscillating linearized laminar boundary layer theory, for which no correction is made for form resistance, are lower than the measured values. Because of the relative large differences between the calculated and the measured results, for the applications use will be made of empirical surge damping coefficients. To this end extinction tests for various vessel types and linear spring constants were performed. The results, scaled according to Froude's law of similitude, are presented in Figure 3.10.
108
50
I/)
E
^-4-> c
40
30
20
10
Ta ffkpy"* C ") 7 P '
e>' 200 kDWT (100% loaded) • 250 kDWT (100% loaded) + 55 kDWT ( 80% loaded) 3 o Symmetrical v = 213,717 m
LNG car r ie r : ., X 1250,000 nT Ref. [2-2]
■
o
• o
+
5*10-w * S 3 / 2 in m3 . s " 1
Figure 3.10 Measured viscous damping coefficient for the surge mode of motion as function of wetted hull area and surge frequency
3.3.4. Viscous damping due to sway and yaw motions
By means of PMM tests in still water the loaded and ballasted 200 kTDW tanker was subjected to sinusoidal oscillations in both the sway and the yaw direction. Based on the known displacement, frequency and the phase lag of the oscillator pistons and the measured oscillating force/moment components the low frequency resistance force/moment components were derived. The low frequency resistance force/moment components in sway and yaw direction due to the motion in sway and yaw direction, X22>X62 and xtg>X26 respectively, are given in Figure 3.11.
109
o x62 X X „
SWAY MODE OF MOTIONS - moment about CG - ver t ica l scale for
posit ive sway veloci ty
i l l a t i o n tests (19B*)
-300
xzz n t f
-ZOO
-100
0
-
'Jrr
tZ2/
/yf\i
i i i
■ («,,.«)' in m'.s"'
VAW NODE OF MOTION - moment about CG - ver t ica l scale for
posit ive yaw veloc i ty
* X66 rotat ion tes t ! (1982)
■ . v osc i l l a t ion tests ( * 1984, • 1985)
—Z0,000
*66 in t f .m --10,000
r-
-
■
^
LOADED TANKER (100ÏT)
*— * ' * *>
1 i
hb^.
Hi .
•
' 1.0 1.0
( » , , . . ) ' In r .o ' .s
t-0 2.0 3.0 4.0 S.C .,2 . . . . „ ! . " ! (< j , . . ) ' In r.d'.r
Figure 3.11 Hydrodynamic viscous resistance force/moment due to motions in sway and yaw direction
Instead of employing the coefficients as can be derived from the Taylor expansion, see equation (3.2.9), an approximation has been made to express the coefficients in terms of damping resistance coefficients averaged over the length of the vessel. For the sway direction the damping coefficient B 2 2
an^ B6? w e r e obtained as follows:
FP X22 = ~^pTB22 J ^ l ^ l 0 * = _^PTB22 CFP-AP)x2 I i2 I
AP i ' '
(3.3.4.1)
FP „2 .„2, X62 = "^pTB62 ' 2 * 2 * d A = - i ;P T B62 ( F P -AP ^ 2 * 2 (3.3.4.2)
AP • '
while in yaw direction the coefficients B 2 6 and Bg6 can be determined from:
FP X26 = -^TB26 i V | V | d A " ' £ PTB26(FP3
+AP3)x6 AP ' '
(3.3.4.3)
110
FP X66 " "*PTB66 1 V l V l * dJl " - l< ,TB66(Fp4+Ap4> i6| i6| AP
(3.3.4.4)
in which:
X99, X A ? > X?A> X,, = measured forces/moment, in-quadrature with the ap-^22' ^62' 26' A66
AP FP
plied displacements = ordinate of aft perpendicular = ordinate of fore perpendicular
In terms of damping resistance coefficients the results of the oscillation tests for the sway and yaw modes of motion are presented in Figure 3.12.
— • FULLY LOADED TANKER ._# BALLASTED TANKER
SWAY MODE OF MOTION
? n
22
1.0
n
KC = 4
-1
(1.333)
(1.01)
KC = 4
-
( 4 . 0 9 ) '
— ■ m -
(2 .28)
0 0.025 0.050 0.075
■fê 0 0.025 0.050 0.075
7?\
YAW MODE OF MOTION
<d.l)
66
1.0
n
•— •—
—•-■ - > "
(1.333) •
— — • 1
(1.01)
4
26
2
n
r •
• f-
(4.09) •
«._ (2.28)
0 0.025 0.050 0.075
V? 0 0.025 0.050 0.075
\S 9 V g Figure 3.12 Low frequency viscous damping coefficients for the sway and
yaw mode of motion in still water
111
Although the stroke was kept constant and the frequencies of the motion were changed during the tests, the results show that the coefficients B22' B62' B26 a n d B66 a r e frequency-independent. The fluid flow along an oscillating body can be characterized by the Keulegan-Carpenter number. Because the stroke was kept constant the results for the sway mode correspond to a constant Keulegan-Carpenter number:
KC x_ T 2a 2nx, 2a = 4 (3.3.4.5)
in which:
2a = amplitude of oscillating velocity T = period of oscillating velocity B = breadth of tanker
Faltinsen et al. [3-18] show that for other KC numbers in the same low range the transverse resistance coefficients for tanker-shape cross sections are approximately constant. The results of their measurements of the transverse coefficients are shown in Figure 3.13.
B22. 1
LOADED 66 kDWT TANKER - DEEP WATER
In spite of the fact that B22~B66 a n d B62~B26 b u t
because B22~B66 ^ B62~B26 a
non-homogenous distribution of the transverse coefficient over the length of the vessel must exist. This implies that the resistance force and moment due to a
Figure 3.13 Transverse resistance coeffi- combined sway and yaw motion cients as function of the cannot be determined direct-Keulegan-Carpenter number ly on the basis of the re-[3-I8] sistance coefficients.
• • 1 • •
• 1
• 1
112
In order to determine the resistance due to a combined sway and yaw motion the distribution of the resistance coefficient along the length of the vessel has to be known. For the approximation a simplified strip theory approach has been used'.
For the approximation the length of the vessel is divided in sections assuming constant local resistance coefficients for each section. Since according to equation (3.3.4.1) through equation (3.3.4.3) four equations are available, four unknown local resistance coefficients can be solved. To this end the length of the vessel can be divided in four sections. Taking into account the body plan as is shown in Figure 2.4 four sections with each a typical cross section were chosen: - section 0- 2; - section 2- 4; - section 4-18; - section 18-20.
The resistance coefficients as function of the longitudinal position of the considered sections along the tanker centreline, as is indicated in Figure 3.14, can be found as follows:
for n=l, B k j = B 2 2
for n-2, B k. = ^
-c0_2(^n)-c2_4(^)+<:4_18(^)+c18_20(^-^) = B k.(^)
for n=3, B k j = B 2 6
for n=4, Bk:j = B & 6 (3.3.4.6)
113
*i
'2
1
1
o FULLY LOADED • BALLASTED
l 5
14
1 1 ! ! ' I i i |
1 1 1 ^ W !
"■■■— 0 ''
G ♦ * t*= («P> (FP)
Figure 3.14 Transverse resistance coefficient as function
By means of equation (3.3.4.6) the four resistance coefficients can be solved. The results are given in Figure 3-14.
Applying the assumed distribution of the transverse viscous fluid resistance coefficients along the length of the vessel and assuming a decoupling in the surge direction the low frequency resistance of the longitudinal po
sition along the tanker forces/moment can be described as: centre line
X1SW ~ ~B11X1
FP 2SW -5jPT ƒ C(A)(x,+x,A)|x?+x,A|dA
AP FP
2 A 6 ^ r 2 ' " 6 '
X6SW = "*pT J' CW(x2+x6A)|x2+x6*|A dSL AP
(3.3.4.7)
From Figure 3.14 it can be seen that the mean transverse resistance coefficient (B-,) is much higher than found from steady current load* measurements at a current angle of 90 degrees (see Section 3.5). This is analogous to the results found for the surge mode of motion.
Based on the derived equations of motion, time domain simulations may be carried out for the non-current condition. To evaluate the proposed description of the equations of motion the results of the time domain computations have to be compared with the results of model tests. This validation is presented in Chapter 5.
114
3.4. Hy_drody_namic_yiscous damping forces in current
3.4.1. Equations of motion in current
In deriving the low frequency fluid reactive forces in current, only the external force Xg in equation (3.2.7) will be considered:
M(x + Dx) = JC (3.4.1.1)
The relative velocity of the vessel with respect to the fluid is:
Vcr = (ur + vr ) % (3.4.1.2)
in which the relative velocity components are:
u = x, - V cos(<|> - x , ) r 1 c c 6'
v = i . - V sin(<|> - x . ) ( 3 . 4 . 1 . 3 ) r 2 c c b
and the relative acceleration components:
u = x, - v x. sin(4< -x.) r 1 c 6 c b
v = x, + V i. cos(c|* -x£) (3.4.1.4) r ^ cb c o
while: V = current velocity 4>c = current direction Xf. = yaw angle in global co-ordinates
Because of the low frequency motions it can be assumed that the disturbances of the free surface of the fluid are negligible. Assuming an ideal fluid, Norrbin [3-15] derived for the forces exerted on a vessel:
X1H " ll^r + a22V6 + "26*6
115
X2H = _ a 2 2 \ " allurX6 " a26X6
6H = ~a66x6 " <a22-all)urvr " a 6 2 ( V u r i 6 ) (3.4.1.5)
where a. . = added mass coefficient at low frequency
Following the equations (3.4.1.3), (3.4.1.4) and (3.4.1.5) we will obtain:
X1H = "allxl " (a22-aU)Vc s i n < W X 6 + a22x2x6 + a26X6
X2H = _a22X2 " a26X6 " (a22_all)Vc C08< W ^ ~ allxlx6
X6H = _a66X6 " a62X2 " (a22-all)urVr " a62XlX6 (3.4.1.6)
Equation (3.4.1.6)) leads to the well-known d'Alembert paradox since the right hand sides are equal to zero for x\ = x^ = x, = 0. The term -(a22-all^urvr *s t*le on^y t e r m arising in an ideal fluid and often referred to as the Munk-moment [3-16].
In a real current, however, viscosity is involved. The viscosity leads to modifications of the velocity dependent terms and/or introduces additional damping terms. The degree of modification to the Munk-moment as a result of the viscosity is shown in Figure 3.15. Further it may be assumed that the acceleration dependent terms in the relative low current speed will hardly be affected by the viscosity. It is assumed that these terms may be determined by 3-D potential diffraction theory.
Replacing the destabilizing Munk-moment by the steady current moment .2 • • formulation and neglecting the small contributions of a2(-x, and a.„x x,
in respectively the x,- and x6-direction we rewrite equation (3.4.1.1) and equation (3.4.1.6) by the following formulation thereby combining expressions for a real and ideal fluid:
116
POTENTIAL THEORY + V + X 1 IDEAL FLUID
Figure 3.15 Flow stream and force distribution along a body in an ideal and a real fluid
(M+a11)x1 = (M+a )i x + X. + X, , 11 2 6 lstat ldyn
(M+a22)x2 + a26x6 = " ( M + a ^ x ^ + X 2 s t a t + X ^ ^
(I +a )'x. + a,,x, = X, _ k + X, , 6 66 6 62 2 6stat 6dyn (3.4.1.7)
in which:
X, H t = hp LTC, (4< )V^ lstat v lc cr' cr
X,ot. . = ip LTC„ (4- )V2 2stat 2c cr' cr
\ ,. . = ^P L2TC. (<|> )V2 6stat 6c cr cr (3.4.1.8)
being the quasi-steady current forces/moment components
117
where: V = (u 2 + v 2 ) ^ vcr v r vr '
= relative current velocity 4>cr = arctan(-vr/-ur)
= relative current angle of incidence
and the dynamic current load contribution is assumed' to be:
Xldyn " -(a22_all)Vc sint+c-Jt6)i6
X2dyn = -(a22_all)Vc cos< W X 6
X = 6dyn L
+ X1D
+ X2D
+ X6D i
viscous
i
part
( 3 . 4 . 1 . 9 )
potential part
which consists of a potential part and a viscous part.
For the viscous parts the following three formulations based on the local cross flow principles have been used in the past. The viscous part is assumed to take into account the additional force/moment components caused by the yaw motion in the relative current velocity field. The parts are assumed to be described as follows:
1. The integration over the length of the vessel of the relative transverse current force minus the undisturbed transverse current force (1979), [3-12]:
X1D = °
X„„ = ^pTC. (90°) ƒ [(v -x,X)lv -i,A I - v |v I ldA 2D w 2cv ' LV cr 6 ' cr 6 cr cr J A P ' < I '
FP X 6 D = ƒ A X 2 D a (3.4.1.10)
AP
118
2. The integration over the length of the vessel of the current forces based on the local transverse velocity and the local relative velocity, the undisturbed transverse and total velocity and the constant transverse current coefficient (1980), [3-13]:
X1D = ° FP
X,n = ijpTC, (90°) ƒ [(v - * , * ) ( ( v -x,A)2+u2 )^-v V ]dl 2D 2c .„ L cr 6 *■ cr 6 c r ' c r c r J AP
FP \ D = / AX2Da (3 .4 .1 .11 )
AP
3. The integration over the length of the vessel of the relative current force minus the undisturbed current force (1983), [3-14]:
X1D = °
X = 0 (was not taken into account in [3-14]) 2D FP
,2. 2 -, . „ , , N„2 x,. = hpr ƒ [c. (4 W)((v -x,x) +u ) - c„ (cp )v ]A dA 6D ' L 2cv crx 'JK cr 6 ' cr' 2c cr crJ
AP (3.4.1.12)
where: u c r = -ur
v = —v cr r v -x.,.1 <\> (£) = arctan cr 6 rcrx ' u
cr In order to judge the reliability of the assumed approximations of the low frequency hydrodynamic viscous reactive forces in a current field model tests were carried out. The model tests are dealt with in the next section.
3.4.2. Test set-up and measurements
The mathematical approximations as treated in the previous section are based on relative current formulations. To evaluate the relative current concep-ts the following model tests have been carried out:
119
- steady current force/moment measurements; - extinction tests in the surge mode of motion; - planar motion mechanism (PMM) tests in the sway mode of motion; - PMM tests in the yaw mode of motion to measure the dynamic current
contribution.
The tests were carried out with a model of the 200 k.TDW tanker as discussed in Section 2.4.1.
The tests, except the extinction tests, were carried out in the Shallow Water Laboratory of MARIN. The water depth was 1 m. The basin measures 15.75 m * 210 m and is provided with a towing carriage.
For the PMM tests both an electrically driven oscillating rotator and an hydraulically driven oscillator were used. The test set-up with the electrically driven oscillating rotator is shown in Figure 3.16.
Figure 3.16 Test set-up of the oscillating rotator.
120
In order to apply the yaw mode of motion to the tanker in the PMM tests, use was made of the oscillating rotator. A description of the rotator set-up is given below. The vertical shaft of the rotator was connected to the carriage. A horizontal yoke was mounted at the lower end of the shaft. The vessel was connected to the yoke by means of three rods, two in transverse direction and one in longitudinal direction of the vessel. In the three rods the force. transducers were incorporated. On each end the rods were fitted in ball joints, allowing displacements in heave, roll and pitch direction. In still water the position of the rods was located in the horizontal plane through the centre of gravity of the vessel. The center of the vertical axle was located above the centre of gravity. By means of the transducers in the transverse rods the force in sway direction and the moment in yaw direction were obtained, while the transducer in the longitudinal rod gives the force in surge direction.
Current was simulated by running the carriage. Current speeds, corresponding to 2 and 4 knots were applied.
The yaw oscillation tests were performed for sector steps of 45 degrees. For 5 current angles, 4 yaw frequencies and 2 steady current velocities the force/moment components of the dynamic current contribution may be derived as a function of the relative current velocity/angle and the yaw velocity.
Besides the dynamic contributions the steady current force and moment components were also determined. For that purpose the rotator was set in a fixed position and the tanker was towed through the basin at a constant carriage speed.
To evaluate the relative current concept for the sway mode of motion oscillation tests were carried out. These tests, however, were performed for a restricted number of current angles. The sway oscillation tests were carried out with the hydraulically driven oscillator, as described in Section 3.3.2.
121
For both the rotator and oscillator tests the stroke was kept constant, but the frequencies of the motions were changed. The amplitude of the yaw angle amounted to 16.2 degrees, while for the sway motion the amplitude was 30.2 m. The frequencies applied to both the rotator and the oscillator were 0.0071, 0.0111, 0.0179 and 0.0248 rad.s-1.
To evaluate the relative current concept for the surge mode of motion extinction tests were carried out in current. The results and the evaluation are presented in the next sections.
3.4.3. Current force/moment coefficients
Since a part of the low frequency viscous damping is related to the steady current forces/moment, the results of these latter quantities will be presented first. In Figure 3.17 the sign convention of the current forces and moment, ship's heading and current direction is given.
Figure 3.17 Sign convention for The current data presented as the the current forces non-dimensional resistance coeffi-
ficients are shown in Figure 3.18. The data concern the loaded and ballasted condition for a water depth of 82.5 m.
The non-dimensional coefficients are as follows:
Xlc (W C. = ~— (longitudinal direction)
X 2 c ( W '2c pLTV
■y— ( t r ansve r se d i r e c t i o n )
122
'6c DC C O
2 2 ^ PL TVc (yaw direction) (3.4.3.1)
in which: 4< -xft = undisturbed current angle of incidence V = undisturbed current velocity c
L = length between perpendiculars of the tanker T = draft of the tanker p = specific density of sea water
0.05
'1c '• 0
Loaded : » 1.03 m.s • 2.06 m.s o 2.57 m.s
Ballasted: • 1.03 m.s"
0.1
-1 -1 h/T = 4.37
h/T = 10.9
-0.05
1.0
L2c 0.5 y«^*W,
180
"6c
-0.1
<* X •
*k 1) \ \
v. \
Based on CG
\ \
V.
p /4
' i i
1 i i
é
180 Uc-x6) in deg
90 (*C-Xg) in deg
Figure 3.18 The current force/moment coefficients
3.4.4. Relative current velocity concept for the surge mode of motion
In order to quantify the oscillatory surge damping of a tanker in current the motion decay will be studied. It is assumed that during the decay motions the following relation exists:
123
X1Nf V IN « V (3.4.4.1)
in which: current velocity frequency of the slowly oscillating vessel amplitude of the Nth oscillation in surge direction
»*1 X1N x.. = low frequency surge velocity during the Nth oscillation
Under this condition it is plausible that the laminar boundary layer will be disturbed by the current and will be dominated by turbulence. In this case the relative current concept may be used. The relative current concept will be applied to the equations of the low frequency surge motion of a linearly moored tanker in steady current with velocity V and direction (|>c. For the degree in freedom in surge direction and following the sign convention as given in Figure 3.3, the equation of motion can be written as follows:
( M f a ^ ^ + c ^ - *PLTClc«Pcr)Vc (3.4.4.2)
in which:
v2 cr u cr
V c cr
= = = =
u 2 + cr
V cos c V sin c
2 V c <v < * c arctan v
V " -v /u c cr
x l
Expanded in a Taylor series the current force becomes:
" i c ' V W V " Xlc<Vc'*-Vil) vx6(0)'xr°
+ X . ÖXlc(Vc>'t'-VXl)
Si, x6=x6(0),Xl=0 + higher order terms (3.4.4.3)
Taking the appropriate terms the equation of motion will read:
124
(Mfa11(u1))x1 + B n i 1 + c u X l = X l c ( V c , V x 6 ) ( 3 . 4 . 4 . 4 )
in which:
B-Q = current damping coefficient Xg = Xg(0) = constant yaw angle of tanker
The current damping coefficient can be derived as follows:
ÖX lc 11 ox,
- ^PLT Xl=°
3(0, (<|» ) - V lc cr cr ox, Xl=o
- *PLT(V 2 ac
1c<*c> ox,
av xi=°
+ . c l c ( V - 6 ) • ÖX, xl=°
in which: (3.4.4.5)
ÖC. (<!< ) lc cr ox,
dC. (<P ) 3* ö(v In ) 3u lc cr cr c c c
V° Scp Ö(v /u ) c c 3u 3x, V° oC (<|* -x,) sin(<(, -x,) lc c 6 c 6
3(|> V (3.4.4.6)
and
dV cr ax,
av cr av 2 2 a(u +v ) au cr c cr xr° av 2 2 3(u +v ) v cr c' au ax, xr°
= - 2 V cos (<|< -x,) c v c 6' (3.4.4.7)
Using the equations (3.4.4.6) and (3.4.4.7), equation (3.4.4.3) can be rewritten as follows:
125
( M + a u ( u 1 ) ) x 1 + {2Xlc(cPc- x 6 ) cos(*c-x6) a(xlc(»c-x6)
a<i< sin(<l» -x )
^ } ix + C n . X l = X l c(Vc ,<Pc-x6) ( 3 . 4 . 4 . 8 )
In case the tanker performs low frequency motions in head current (<k,-x,-= 180°) and|x I « V then the equation of motion can be represented as follows:
(M+an(u1))x1 2Xlc((VX6)=18°°) . . xx + c11x1 - X1(. (3.4.4.9)
It must be noted that for the current damping the frequency dependency as found for the still water damping does not exist.
l e s t No. 6360 - c . , - 19 .3 tf.m - T-233 s - i
>, ■a,
N>L " n ,
■v k k r ■ ^
'S > L
> o - X
NUMBER OF OSCILLATIONS It
Figure 3.19 Results of measured decay motions in surge direction for head current
To evaluate the relative current concept physical model tests in current were carried out with a loaded 200 kTDW tanker moored in 82.5 m deep water, see Section 2.4.1. The tanker was moored by a linear spring with c 11 19.3 tf.m and exposed to 2 kn head current. The measured resistance coefficient amounted to C l c = -0.040 (Xlc = -13.1 tf). In terms of logarithmic decrements the results of the tests are presented in Figure 3.19.
126
Using the formulation of the viscous damping coefficient according to equation (3.4.4.9) the damping coefficient can be computed and amounts to:
2*X B n = y±£- = 25.4 tf.s.in
c
From Figure 3.19 it can be seen that the logarithmic decrement 6 for the large low frequency amplitudes may be considered as constant.
Using the constant logarithmic decrement the linear damping coefficient according to equation (2.4.2.6) can be derived from Figure 3.19. The linear damping coefficient derived from the decay test amounts to Bii = 27 tf.m.s-1.
Comparing the computed and measured result it may be concluded that the relative current concept for the surge mode of motion is applicable for head current.
Evaluating equation (3.4.4.8) reveals that for current angles approaching beam current the current damping coefficient becomes negative. In [3-13] model test results of extinction tests in surge direction under different current angles are presented. The results of the measurements show, however, that the damping decreases to some extent for current angles approaching beam current but- remain clearly positive.
The results in [3-13] indicate that contrary to the theory of the relative current concept the experiment shows that the current damping coefficient stays positive for the cross current condition. A possible explanation is that due to the cross current velocity, the current velocity 'along the hull side at the leeward-side is almost absent. Due to these phenomena a part of the larger stiil water damping is involved and may dominate the damping force. It is therefore recommended that the still watef damping be used as the lower bound for the damping in current.
127
3.4.5. Relative current velocity concept for the sway mode of motion
To evaluate the relative current concept for the sway mode of motion model tests were carried out in a current field. For the model tests use was made of the hydraulically driven oscillator.
COMPUTED loaded 200 kDWT MEASURED 82.5 m water depth
Test No. 19452
Istat ( t f )
I x f c l 0 * 30.27 nï ; «I = 0.0111 rad.s
40 h
v 800 *2stat ( t f )
20,000h X6stat 0 c (tf.m) 0
.03 "-S
1000 I
2000 (s)
Test No. 19421
I x j j l - 30.1 m ; uc = 0.025 rad.s"
1S" -KAAA/NA/VW (tf)
100,000 6stat (tf.m)
1000 Time in s
Figure 3.20 Computed and. measured relative current force and moment components for the sway mode of.motion
Based on the known oscillatory sway displacement and frequency, the known added mass coefficients and the inertia properties of the vessel, the relative current force and moment components Xlstaf X2stat a n d X6stat can be derived from the measurements using equation (3.4.1.7). The dynamic current load contributions are assumed to be dependent on the yaw velocity.
For some specific conditions the results of the derived measurements are presented in Figure 3.20. In Figure 3.20 the results of the computed steady relative current force and moment components based on equation (3.4.1.8) have also been plotted. The coefficients used for the computations were obtained from Figure 3.18.
128
From the results it can be concluded that the application of the steady relative current force concept for the sway mode of motion gives satisfactory results.
3.4.6. The dynamic current contribution
The part of each of the equations of motion in equation (3.4.1.7) in-phase with the yaw velocity (x/-=0) represents the components of the quasi-steady current load and the dynamic current contribution. By subtracting the components from the steady current load, the components of the dynamic current contribution remain.
By means of the oscillating rotator a total of 70 yaw oscillating tests in current were carried out to determine the dynamic current contribution, see Section 3.5.2. By means of a post-processing computer program applied to the stored digitized data of the measured oscillating forces and moment the amplitudes of the dynamic contributions in-phase with the negative and positive yaw velocities have been derived.
The force and moment components in-phase with the positive yaw velocity were transformed into negative yaw velocities taking into account the appropriate sign and current angle:
X l d y n ( V W x6a = P° S° = ^dyn^c' 3 6 0'"^"^) .*6a = n e S °
X 2 d y n < W x 6 > *6a = P° S° = "X2dyn( V 3 6 0 ' - < V x 6 > >x6a = neS->
X6dyn<Vc'V-x6> x6a = P ° S ° = - X 6 d y n ( V c > 3 6 0 ° - < W >x6a = ne8->
(3.4.6.1)
The yaw oscillation tests were carried out for 5 current angle sectors U*c" x6 ) = °*' 4 5 ° ' 9 0 ° ' 1 3 5° and 1 8 0 ° ) - Applying equation (3.4.6.1) to the results a presentation of the dynamic contribution is obtained for current angles covering 360 degrees. This presentation corresponds to a rotation of the tanker with a quasi-steady yaw velocity in a current
129
field and is analogous to the presentation of the results of the rotation with steady yaw velocity as is shown in Figure 3.2. The difference is, however, that the present method takes into account the oscillatory hydrodynamic effects In the current field such as vortex shedding etc.
For the loaded and ballasted tanker both in 2 kn and 4 kn current and based on the negative yaw velocity of the tanker the dynamic current contributions in surge, sway and yaw direction respectively XiJ n, ^?dvn and X/-j derived as function of the undisturbed current angle of incidence or the relative current angle of incidence are given in Figure 3.21 and Figure 3.22.
Based on the results of the model tests a description has to be established in order to formulate the components of the dynamic contribution. Following the relative current velocity concept and the quantities varied during the tests, being - the frequency of the yaw oscillation; - the current angle of incidence; - the current velocity, it is assumed that the tanker motions may be considered as a combined surge, sway and yaw motion. Therefore the proposed formulation will be expressed in terms of relative current speed and angle.
The force and moment components of the dynamic current contribution as shown in Figure 3.21 and Figure 3.22 are described in the following form:
Xldyn = Cldyn<*cr>r,>^LTVcr
X2dyn - ^ d y n ^ c r - r ' ^ P ^ c r
*6dyn = W ^ ' ^ X r (3"4-6-2)
130
200 kTDW tanker - 82.5 m water depth - V =1 .03 m.s"
-250 40,000
30,000
5 20,000
"6a1
*6al
0.002 rad.s" 0.003 rad.s"
10,000
x |*6 | = 0.005 rad.s" o — - |* 6 a | = 0.007 rad.s" '
Fourier approximation
10,000
5,000 -
Model tests
BALLASTED
360
Figure 3.21 Dynamic current contribution in surge, sway and yaw direction due to motion in yaw direct ion in 2 knot current, based on negative yaw ve loc i ty
131
200 kTWD tanker - 82.5 m water depth - Vc = 2.06 m.s -1
-250
> 6a
= 0.002 rad.s = 0.003 rad.s"
-1
• x — u , | = 0.005 rad.s ' 6a' - i
o — l x 1 = 0.007 rad.s , 1 6a' Fourier approximation
Model tests
- 25
A\
BALLASTED
5,000]
180 360
*cr in deg
Figure 3.22 Dynamic current contribution in surge, sway and yaw direction due to motion in yaw direction in 4 knot current, based on negative yaw velocity
132
in which: V = (u 2 + v , 2 ) ^ vcr v r r ' *cr = arctan(-vr/-ur) C . j = dynamic current coefficient for j=l,2,6 r' = ift-L/V = dimensionless yaw velocity
By means of Fourier theory applied to each of the appropriate curves the dynamic current coefficient can be approximated in terms of Fourier coefficients:
N Cjdyn = C 0 j ( r , ) + J^nj^ 1') «>°("-*cr> + Snj(r') «»i°(n.<|>cr) )
for j=l,2,6 (3.4.6.3)
Each of the Fourier coefficients will represent a function of r' and is shown in Figures 3.23, 3.24 and 3.25 for the components in surge, sway and yaw direction.
These functions can be described by polynomial terms. As a consequence of equation (3.4.6.1) for the description of the components in sway and yaw direction as function of the yaw oscillating velocity at a certain relative current angle a proper approximation of the Fourier coefficients must be applied, being:
Cnj = Vj + Vj r' + cnjr* lr* I V j " dnj + enj| r'| + f n j r ' 2 forj-1,2,6 (3.4.6.4)
By .substituting the Fourier coefficients in equation (3.4.6.3), equation (3.4.6.2) and equation (3.4.1.9) and after some re-arrangement of the terms the following expressions were found for the components of the dynamic current contribution:
Xldyn " "°-4 * <a22-all> * s i n < W * Vc * *6
133
X2dyn " - < a 2 2 - a l l > * c °*( W * Vc * *6 + X2D
X, . = X,_ ( 3 . 4 . 6 . 5 ) 6dyn 6D
in which:
X2D = [X2Vr*Vcr**6 + X2V |r) *Vcr* j ^ | + X'2/h*\ + X2 | r | r * L * V N +
+ X 2 r 3 / V ^ 6 * L 2 / V c r + X2 | r 3 | / v * | i 6 | 3 * L 2 / V c r ] ^
and
X6D = K v r ^ c r S + X6V | r | *Vcr* | ^ 6 1 + ^ / ^ + *6 | r | r * L * V | *6 | +
+ X' *£3*L2/V + X' , * | i , | 3 * L 2 / V lijp.L3.T , 3 ,„ 6 er r 3 ,„ 6 c r J 6r /V 6 | r | /V '
where for the l a t e r a l viscous par t we w i l l have the following
c o e f f i c i e n t s :
X' , = 0.06435 - 0.03996 cos2<|* '+ 0.02654 cos3c|* + 2Vr cr cr
+ (0.00683 cos2(p + 0.06634 cos3<); )Q v ^cr c r "
X 2 V | r | = ( - 0 - 2 2 0 7 + 0.1309Q)sincPcr
X' = (0.3285 - 0.1527Q)sin<J> 2r 2 C r
X' = 0.02157 + 0.01484cos2(J- - 0.03886COS3I|J + 2r r | cr cr
+ (-0.00838 cos2<|> - 0.10804 cos3<|; )Q
X' = (0.01168 + 0.03286Q) cos3<l< 2r /V C
134
2 rJ|/V = (-0.0664 + 0.05801Q) sin* er
while for the moment component the coefficients of the viscous part will be:
6Vr
X' 6V|r|
X ^ 2 6r
6r |r
= -0.05659 - 0.00656 cos* + 0.0225 cos2* + er er
+ (-0.02981 + 0.03072 cos* + 0.01994 cos2* )Q er er
= (0.00722 + 0.01847Q).sin*
= (-0.03753
er
0.01069Q).sin* er
6r3/V
= -0.01706 - 0.00512Q + (0.019122 - 0.0350O2Q).cos*
(-0.007587 + 0.009023Q).cos*
6 r /V (0.00982 + 0.00391Q)'.sin*
0.4U, >,,)
\ Ï O O S T 0.2
40%T \
-2 -1
-O.t
-0.2
■
\ \ — r
Figure 3.23 The derivation of the coefficients of the dynamic current contribution in surge direction due to motion in yaw direction in a current field
The denominator in the viscous parts of equation (3.4.6.5) contains the current velocity. For this reason it is clear that the derived formulations are only valid for sufficiently high values of the current velocities. Because of the restricted amount of data the validity will hold for current speeds of 2 knots or higher.
From Figures 3.23 through 3.25 it is assumed that the Fourier coefficients are dependent on the draft of
135
the vessel. The factor Q used for the interpolation can be formulated as follows:
T-T Q = 40
T -T 100 40 (3.4.6.6)
in which: l100 r40
loaded draft = draft at 40% of loaded draft
The comparison between the Fourier approximation and the test results is given in the Figures 3.21 and 3.22.
-2
0.4
0.2
-1
-0.2
-0.4
- «— 100%T — o — 40ÏT
y ^ ^ *?
— +r'
-
-
CQ2 = 0.06435r' * 0.02157r' | r ' |
\ 0.4
° X » s \ . 0 . 2
-0.2
-0.4
—— +r'
V *1 +2 \ \
' \ \
(a22 - a l l '
A °-4 A. \ \
V\ 0.2 \\ . -2 -1 - V
-0.2
1 1
// ft ~—= A +2
—— +r'
S.-(IOOXT) = - 0.08981r' | + 0.1758r'? - 0.00839 | r ' | S,2< 40ST) = - 0.2207|r'| + 0.3285r'd - 0.0664 | r ' |
0.2.
-2 -1
-0.2
+r' ♦ 1 +2
C22(100«T) = - 0.03313r' * 0.00646r'|r ' | Cj£( 40J5T) = - 0.03996r' + 0.01484r
-0.2
r ' |
C„(100ST) = ♦ 0.09288r' - 0.1469 r' | r ' | + 0.044541-':; C||( 40JST) = ♦ 0.02654r' - 0.03886r' | r ' j * 0.01168r,':'
Figure 3.24 The derivation of the coeficients of the dynamic current distribution in sway direction due to motion in yaw direction in a current field
136
C06(100%T) =
CQ6( 40%T)
"06
0.08640r' + - 0 .02218r ' | r ' |
- 0.05659r ' + - 0.01706r' I r '
16
S 6(100ÏT) 0.02569 I r ' - 0.04822r ,2
+ 0.01373|r ■ i3
S,6( 40%T) = + 0.007221r'1+ 0.03753r ,2
0.00982 I r '
-.2 -1
-0.1
-0.2 -
0.2
0.1
'16 " 2 \
0.02
0.01
-1
yi -0.01
-0.02
-
I s , \ \ 1 \2
— - + r ' \
C16(100%T) = + 0.02416r' - 0.01588r ' | r ' | + + 0.001436r'3
C16( 40CT) = - 0.00656r' + + 0.019122r' |r ' | +
'26
0.2
0.1
-2 -1
^ -0.1
-0.2
1 2 — - +r'
C26(100ÏT) = + 0 .04244r ' C26( 40%T) = + 0 .02250r '
- 0.007587r ,3
Figure 3.25 The derivation of the coefficients of the dynamic current contribution in yaw direction due to motion in yaw direction in a current field
137
3.4.7. Evaluation of the semi-empirical mathematical models in current
By means of the present formulation for the dynamic current contribution according to equation (3.4.6.5) the semi-empirical mathematical models based on relative current velocity concepts [3-12j, [3-13j and [3-14] can be evaluated.
In accordance with theory the components of the dynamic current contribution in surge and sway direction consist of a potential and a viscous part, see equation (3.4.1.9). The experimentally derived component in surge direction shows that the component mainly consists of a potential part. The magnitude of the potential part, however, was found to be lower than predicted by theory. The component in sway direction consists of a potential and a viscous part. The magnitude of the potential part was in good agreement with theory.
To evaluate the dynamic current load contribution, computations on the theoretical models according to the equations (3.4.1.9), (3.4.1.10), (3.4.1.11) and (3.4.1.12) and the present formulation were carried out; the results were then compared. The computations concerned the loaded and ballasted tanker in a 2 knot current and a yaw velocity x, = -0.007 rad.s- . The results are presented in Figure 3.26.
The results show that the yaw moment of the dynamic current contribution will be underestimated by applying the theoretical models. The sway cont-ponent shows the importance of the potential part, while the viscous part seems to be to some extent too small by applying the theoretical models.
In accordance with equation (3.4.1.7) the total low frequency hydrodyna-mic viscous resistance components in current consist of a relative current load part and a dynamic current contribution. The magnitudes of each part will depend on the values of the relative current speed, the relative current angle and the oscillating yaw velocity.
138
82.5 m water depth 200 kDWT loaded tanker V = 1.03 m.s Xg = -0.007 rad.s
160
80 -
-80
-160
r\ - \ v i ' /
Method Obokata [3-14] — Method Mol in [3-13] i Method Wichers [3-12]
Present formulation • Oscil lation tests
40
82.5 m water depth 200 kDWT ballasted tanker
-1 1.03 m.s -0.007 rad.s ■1
-40 90 180 270 360
90 180 360
180 270 360
50
n'
50
* * " • < .
^
V '1
/ / /
90 .180 270 360
2.5E4
* c r in deg
3.26 Comparison of the results of the dynamic current contribu
tion components due to yaw mode motion following the
existing methods and the present formulation
139
To evaluate the relative differences in the results of the theoretical models computations have been carried out on the total low frequency viscous resistance components in both the sway and yaw direction (XJ t a t + Xjdyn f o r J=2.6>-
The computations were carried out for the loaded and ballasted 200 kTDW tanker in 2 and 4 knot current and yaw velocities x, = -0.002 and
_ i 6
-0.007 rad.s . The results are shown in the Figures 3.27, 3.28, 3.29 and 3.30.
It can be concluded that for low values of the oscillating yaw velocities the results of the theoretical models of Molin [3-13] and Obokata [3-14] are close to the results of the present formulation. For higher values, however, the differences with the present formulation increase considerably.
By means of the present formulation for the low frequency hydrodynamic viscous force/moment components time domain computations of the low frequency tanker motions moored in current may be carried out.
To evaluate the proposed description of the equations of motion the results of the time domain computations have to be compared with the results of model tests. This validation is presented in Chapter 5.
140
Draft 100* = 18.9 m Current = 1.03 m.s" Yaw velocity = FP = 148.4 m AP = -161.6 m
-0.002 rad.s
Obokata 1983 Mol in 1980 Wieners 1979 Present formulation
-1 Draft 40% = 7.56 m Current = 1.03 rad.s ' Yaw velocity = -0.002 rad.s FP = 144.54 m AP = -165.46 m
200
100
0
-100
-200
-300
r —
ƒ
60
30
-30
-60
-90
If // Jl
It 1 1
N \
\ \ \\ \\ \\ \\
/l ^r.y
20,000
10,000
-10,000
5000
2500
-2500
\ 1 V / \>
Ij
/
f\ \
360
* c r in deg
90 180 270 . 360
Figure 3.27 Comparison between r e s u l t s of t o t a l low frequency viscous r e s i s t a n c e components in sway and yaw d i r ec t i on due to yaw mode of motion following the ex i s t i ng methods and the p r e sent formulation
141
Obokata 1983 Mol in 1980 Wieners 1979 Present formulation
Draft 100% = 18.9,m Current = 2.06 m.s Yaw velocity = -0.002 rad.s FP = 148.4 m AP = -161.6 m
.1000
500
-500
-1000
60,000
30,000
\
\^,
-30,000
L
W / A
/ \
\ / A. /
A \
Draft 40% = 7.56 m , Current = 2.06 m.s" Yaw velocity = -0.002 rad.s" FP = 144.54 m AP = -165.46 m
300
150
-150
-300
15,000
7,500
\
\ -
180 270 360 ' 7 ' 5 0 0 t
♦ c r in deg
m 90 180 270 360
Figure 3.28 Comparison between results of total low frequency viscous resistance components in sway and yaw direction due to yaw mode of motion following the existing methods and 'the present formulation
142
Obokata 1983 Hol in 1980
— Wieners 1979 Present formulation
Draft 100% = 18.9 m. Current = 1.03 m.s Vaw velocity = -0.007 rad.s" FP = 148.4 m AP = -161.6 m
Draft 40% = 7.56 m . Current = 1.03 m.s Yaw veloci ty = -0.007 rad.s" FP = 144.54 m AP = -165.46 m
100
-100
-150 V ^ -
50,000
25,000
-25,000
/ \._y /
/'
\
/ / / / \
V \
15,000
10,000
5,000
0
\ A S' '..,
\
/ 1
/ /
'"N \
- \
V V \\
90 180. 270 360 0 . 90 * in deg
180 270 360
Figure 3.29 Comparison between results of total low frequency viscous resistance components in sway and- yaw direction due to yaw mode of motion following the existing methods and the present formulation
143
Obokata 1983 Mol in 1980 Wichers 1979 Present formulation
Draft 100% = 18.9 ra. Current = 2.06 m.s Yaw velocity = -0.007 rad.s" FP = 148.4 m AP = -161.6 m
1000
-1000
-1500
.100,000
_50,000
-50,000
Draft 40% = 7.56 m Current = 2.06 m.s Yaw velocity = -0.007 rad.s" FP = 144.54 m AP = -165.46 m
300
-450
30,000
15,000
-15,000
, in deg
\y \
/ . / /
s
^_\
\\
Figure 3.30 Comparison between r e s u l t s of t o t a l low frequency viscous r e s i s t a n c e components in sway and yaw d i r e c t i o n due to yaw mode of motion following the ex i s t i ng methods and the p re sent formulation
144
REFERENCES (CHAPTER 3)
3-1 Abkowitz, M.A.: "Lectures on ship hydrodynamics, steering and manoeuvrability", Hy A Report Hy S, 1964.
3-2 Inoue, S., Hirano, M., Kijima, K. : "Hydrodynamic derivatives on ship manoeuvring", International Shipbuilding Progress, Vol. 28, 1981.
3-3 Hirano, M. and Takashina, J.: "A calculation of ship turning motion taking coupling effect due to heel into consideration", Transaction of the West-Japan Society of Naval Architects, No. 59, March 1980.
3-4 Gerritsma, J., Beukelman, W. and Glansdorp, C.C.: "The effect of beam on the hydrodynamic characteristics of ship hulls", Proc. 10th Symposium on Naval Hydrodynamics, Boston, 1974.
3-5 Glansdorp, C.C.: "Ship type modelling for training simulator", Proc. 4th Ship Control Systems Symposium, The Hague, October 27-31, 1975, Volume 4.
3-6 Sharma, S.D. and Zimmermann, B.: "Schragschlepp- und Drehversuche in Vier Quadranten- Teil 1", Schiff und Hafen/Kommandobrlicke, Heft 10-33 Jahrgang ,1981.
3-7 Sharma, S.D.: "Schragschlepp- und Drehversuche in Vier Quadranten-Teil 2", Schiff und Hafen/Kommandobrlicke, Heft 9-34 Jahrgang, 1982.
3-8 Faltinsen, O.M., Kjaerland, 0., Liapis, N. and Walderhaug, H.: "Hydrodynamic analysis, of tankers at single point-mooring systems", Proc. Symposium on Behaviour of Offshore Structures, London, August 1979.
145
3-9 Ractliffe, A.T. and Clarke, D. : "Development of a comprehensive simulation model of single point mooring systems", Royal Institute of Naval Architects, Paper 9, London, 1980.
3-10 Remery, G.F.M. and van Oortmerssen, G.: "The mean wave, wind and current forces on offshore structures and their role in the design of mooring systems", OTC Paper No. 1741, Houston, 1973.
3-11 OCIMF: "Prediction of wind.and current loads on VLCCs", OCIMF, 6th floor, Portland House, Stag Place, London, 1977.
3-12 Wichers, J.E.W.: "Slowly oscillating mooring forces in single point mooring systems", Proc. Symposium on Behaviour of Offshore Structures, London, August 1979.
3-13 Molin, B. and Bureau, G.: "A simulation model for the dynamic behaviour of tankers moored to SPM", International Symposium on Ocean Engineering and Ship Handling, Gothenburg, 1980.
3-14 Obokata, J. : "Mathematical approximation of the slow oscillation of a ship moored to single point moorings", Marintec Offshore China Conference, Shanghai, October 1983.
3-15 Norbinn, N.H.: "Theory and observation on the use of a mathematical model for ship manoeuvring in deep and confined water", Proc. 8th Symposium on Naval Hydrodynamics, 1970.
3-16 Munk, M.: "The aerodynamics of airship hulls", NACA Report No. 184, 1924.
3-17 Lamb, H.: "Hydrodynamics", 6th edition 1932, Cambridge University Press, London
3-18 Faltinsen, O.M., Dahle, L.A. and Sortland, B.: "Slow drift damping and response of a moored ship in irregular waves", Proc. OMAE, Tokyo, 1986.
146
CHAPTER 4 EVALUATION OF THE LOW FREQUENCY SURGE MOTIONS
IN IRREGULAR HEAD WAVES
^.l^Introduction
In Chapter 2 the speed dependency of the potential theory regarding the second order wave drift forces in head waves was discussed. In Chapter 3 the low frequency viscous damping caused by the low frequency motions in the horizontal plane, including surge direction, have been dealt with. In this chapter the results as derived in the previous chapters are applied to the computations of the low frequency surge motions of a moored tanker in irregular head waves with and without current. The current is co-linear with the waves. Computations were carried out for both the frequency and time domain.
To illustrate the effect of the wave drift damping on the low frequency motions, frequency domain computations were performed for sea states with increasing significant wave height [4-1]. The results were compared with the results of model tests. The frequency domain computations are based on the wave spectra as were adjusted for the model tests. As a consequence of the frequency domain approach the results of the computations can only be presented in terms of statistical quantities. For irregular waves combined with current a similar computation procedure can be applied [4-2].
To show the deterministic procedure for simulation of the low frequency surge motions in irregular waves with and without current time domain computations were carried out. The time domain computations are based on the wave train registrations as were adjusted for the model tests. Prior to solving the equation of motion the mean wave drift damping coefficient and the registration of the wave drift force with and without current were computed. The results of the computed wave drift force registration with and without current in terms of spectral densities were compared with the results of model tests. Finally the results of
147
the computed low frequency motions were compared with the results of model tests.
4.2. Frequency domain computations in irregular head waves without current
4.2.1. Theory
The equation of the low frequency surge motion of a l i n e a r l y moored tan
ker exposed to i r r e g u l a r head waves can be wr i t t en a s :
( M f a u ( u 1 ) ) x 1 + B n ( u 1 ) x 1 + B ^ + c ^ = X ^ t ) ( 4 . 2 . 1 . 1 )
added mass coefficient at the natural frequency |j.. still water damping coefficient at the natural frequency n, mean wave drift damping coefficient linear spring coefficient wave drift force registration
In the frequency domain the quantities considered are expressed in terms of spectral densities. Since the equation of motion is in a linear form, the spectral density of the low frequency surge motion can be written as:
S (u) = S x ( ^ . ( ^ u ) ) 2
1 1 la
while the variance of the low frequency surge motion will be:
a2 = / S Y (a) ( T T ^ U ) ) 2 da (4.2.1.2) xl 0 Xl Xla
where: S (n) = spectral density of the longitudinal wave drift force Xl
in which: all^l> =
Bn(u1) = Sl cll X,(t) =
148
Xla (|i) = surge amplitude per unit longitudinal wave drift force
la 1
\/ T~2 - TT-^(cu-muH ) +(B11+B1) u
\i = frequency of low frequency part of the second order forces m u = M + a11(u1)
For systems with a small damping, the response of the surge motion at the natural frequency dominates (has a peak). Therefore the spectral density can be kept constant over the frequency range. Following [4-3] the variance reduces to the following form:
ax = SX < M / h r ^ ) ) 2 ^ (4.2.1.3) Xl Xl l 0 Xla
which yields:
°l = r 2 S (Ü,) (4.2.1.4) 1 2<B11+VC11 l
where:
\ Fïi \i. = \ / = natural frequency of the system 1 V n S ((O = spectral density of the wave drift force at
frequency u.-, To solve equation (4.2.1.4) the input data have to be known. The still water damping coefficient can be read from Figure 3.10. The mean wave drift damping coefficient B. in an irregular sea with N wave components can be determined as follows:
in series notation or in spectral notation N o _
B = Z CT.D(o) ) B = 2 ƒ S (oo).D(u)) du (4.2.1.5) 1 i=l X l 0 V
149
in which: S-(w) = spectral density of the irregular sea
B (co)
The quadratic transfer function of the wave drift damping coefficient D(io) can be obtained by means of computations or model tests as is described in Chapter 2.
The spectral density of the wave drift forces in an irregular sea state with spectral density S(co) can be computed following [4-3]:
oo
Sv CM-) = 8 ƒ Sr(co).Sr(u+u) (TCio.uH-^))2 dio (4.2.1.6) Xl 0 Q C
in which: T(w,itH-|i) = amplitudes of the quadratic transfer function of the wave
drift force dependent on u and uH-(i
Because the natural frequency in surge direction for a moored tanker is small the spectral density S„ (^) will approximately be equal to Sx <[i=0), or:
S (u=0) = 8 ƒ S2(a>) (T(to,a>))2 dco (4.2.1.7) Xl 0 C v
in which:
4 2 ) ( W ) T(io,io) = 5
It should be noted that in equation (4.2.1.6) the spectral density of the wave drift forces is directly related to the theoretically derived spectral density of the wave group in a sea state. To show this relation the derivation of the wave group spectrum will be given. Because the drift force is related to the square of the wave amplitude, the square of the wave envelope for a regular wave group as given in equation
150
(2.7.2.4) will be considered. The regular wave group consists of two regular waves with wave amplitudes C. and C„ and with frequencies u, and to, respectively. The square of the wave envelope can be written as
follows:
A2 = C^ + Q\ + 2C1C2.cosnt (4.2.1.8)
in which: \i = co-, - co
in accordance with the definition of a spectrum the spectral density of
the low frequency part of the square of the wave envelope will be:
S (u) A<o = \{2C, C ? ) 2 (4.2.1.9) A
while for wave component n the spectral density is:
Sc(un) Aco = JgC2 (4.2.1.10)
which w i l l y i e l d :
S 2 ( u ) Aw = 8 S C ( O J 1 ) A U S C (Ü) 2 )AU) ( 4 . 2 . 1 . 1 1 )
or for all the wave groups with the frequency-difference |i in a wave spectrum:
CO
S 9(t0 = 8 / S (u) S (urt-u) dw (4.2.1.12) A 1 0 ^ <*
Actually equation (4.2.1.12) represents a spectral density of the wave groups for a random process of wave components in the spectrum S-(a> )
151
for n = infinity. To obtain this result the sea state will have to last infinitely. The consequence is that the variance in equation (4.2.1.4) represents the steady state value. Because the sea state or the design storm with a prescribed wave spectrum lasts only for a short duration the wave train has to be considered as one realisation. As a consequence the slowly oscillating wave drift force present in the wave train has to be considered as one realisation also. The result is that the slowly oscillating motion is a representation of that realisation. In terms of the variance the result represents one unique value.
If we consider a number of independent wave trains, statistical theory can be applied to the wave drift force excitation, which results in sta-tictics of the motions. For this alternative approach, which necessitates the replacement of frequency domain by time domain, reference is made to [4-4J.
Restricted to the theory on the frequency domain procedure the computations can be carried out.
4.2.2. Computations
The computations were carried out for the loaded 200 kTDW tanker. The particulars and the body plan are given in Table 2.1 and Figure 2.4 respectively. The tanker was moored by means of a linear spring in surge direction with a spring constant of 13.6 tf.m . The water depth corresponded to 206 m full scale. The tanker was exposed to wave spectra as presented in Section 4.2.3. These spectra were applied to the computations.
For the spectral density of the wave drift forces Sx (n=0) use is made of the computed transfer function of the wave drift forces as given in Figure 2.12 and Table 2.3. The mean wave drift damping has been computed by means of the data given in Figure 2.11.
152
Wave spectra Nos. 1, 2, 3 (f, - K s) Wave spectra Nos. 4, 5, 6 (ƒ,= 12.0 s)
For the still water damping use is made of measured data, see Section 4.2.3, which resulted in B,, = 24.4 tf.s.nf1. The results of the computations of the spectral densities of the wave drift forces and the mean wave drift damping are given in Table 4.1. The results of the computations with and without wave drift damping
''ui/31 '"m in terms of the root-mean-Figure 4.1 Root-mean-square values of the square values of the low
low frequency surge motion ver- frequency surge motion are sus significant wave height given in Table 4.1 and in squared Figure 4.1.
• without wave d r i f t damping J - o with wave d r i f t damping —J—
A measured /
i 7
y fl V
k
/
ja
MARIN T e s t
No.
3601
3591
3582
3621
Wave s p e c t r u m
No.
1
2
3
4
5
6
' - V 3
m
1 2 . 5 0
9 . 4 0
7 . 5 0
1 2 . 5 0
9 . 8 0
6 . 1 0
? 1
s
1 4 . 1
1 3 . 7
1 3 . 6
1 2 . 0
1 2 . 0
1 1 . 4
C 1 1
t f . m " 1
1 3 . 6
1 3 . 6
1 3 . 6
1 3 . 6
1 3 . 6
1 3 . 6
M e a s u r e d
0 *1 m
13.50
10.36
12.80 6.74
Ca lcu la ted data according to frequency domain
m with wave
d r i f t damping
14.5
10.5
7.2
18.4 13.4
5.5
without wave d r i f t
damping
25.1 15.4 9.0
32.8
20.9
6.60
S p e c t r a l dens i ty d r i f t
f o r c e s S (u=0>
*1 t f 2 s
132,625
49,835
17,161
227,741
92,087
9,321
mean wave d r i f t damping
= 1 t f . s . m " 1
48.75 27.60 14.14
53.7 34.4 10.6
Table 4.1 Results of computation and model tests
4.2.3. Model tests
To validate the results of the computations model tests were carried out. The model tests were carried out in the seakeeping laboratory of
153
MARIN, see Section 2.4.1.
Prior to testing the wave spectra were adjusted with the wave probe at the projected location of the centre of gravity of the tanker model. The wave spectra are presented in Figure 4.2 (spectra No. 1, 2, 3, 5 and 6). Each sea state was adjusted for a test duration of 2.5 hours prototype t ime.
2
3
4
0.5 .1 -0 0.5 tu i n rad.s u in rad .s
Figure 4.2 The spectral densities of the wave spectra
The tests were performed in the wave spectra No. 1, 2, 5 and 6. The statistical properties belonging to the measured wave spectra are given in Figure 4.3 and 4.4.
The results of the tests in terms of the root-mean-square values of the low frequency motions are given in Table 4.1 and in Figure 4.1.
Prior' to the tests in waves a surge extinction test was performed in still water, to measure the natural period and the still water damping coefficient. The results in terms of the logarithmic decrement are shown in Figure 4.5.
154
DISTRIBUTION FUNCTION OF «AVE ELEVATION
it 10
.A h^ o =3.16m Camax. + = 14.2m
-10 m Ü m 10
WAVE SPECTRUM No. 1 \0
Theoret ical ly From time record
DISTRIBUTION FUNCTION OF SQUARE OF THE ENVELOPE
50
E
.
0 /
r .
)
— V
-N '*v
\\ V \
\
=====
:
' r \ ■
N ^
v> 0.5 ,1.0
to in rad.s
Figure 4.3 The theoretical and measured distribution function of the wave elevation and square of the wave envelope belonging to spectrum No. 1 (Spectrum 16346)
50
40
30
20
10
^ ^
4 crests " I t "*■ " ' " o troughs D fi 11
B = 24.4 t f .s .m
^ 4 ^ c ^ ^ *---c > ^
. -1
n^— (
2 3 4 5 Number of oscillations xu
Figure 4.5 The measured still water damping
155
SPECTRUM 16346
4000
3000
2000
1000
0
T , ■
\
\
4.1 s
* ^
From time record
SPECTRUM 16386
Theoretical W 3 ' 9 - 4 ™ fl • 13.7 s
Theoretical From time record
0.25 0.50
u in rad.s"
\ V
0.25 0.50 u in rad.s
SPECTRUM 16368
T, . 12.0 s
SPECTRUM 16415
- Theoretical - From time record
— Theoretical — From time record
•'~\ \ \
\ \ \ \ \ \ \ \
\ \ *__
200
150
'Z. 100
<
so
0
\ \ \
tv \
ÏZ — 0.25 0.50
v in rad.s" 0.25 , 0.50
Li in rad.s
Figure 4.4 The spectra of the wave groups
156
4.2.4. Evaluation of results
The measured still water damping coefficient as derived in Figure 4.5 is B u = 24.4 tf.s.m , which a be derived from Figure 3.10.
Bll = 24.4 tf.s.m , which approximately corresponds to the data as can
The results of the statistical quantities of the surge motion obtained by computations and model tests will be correlated. To be able to relate the results the statistical properties of the undisturbed wave specCra, adjusted for a duration of 2.5 hours full scale, have to be verified with the theoretical ones.
The theoretical properties of the wave spectra will be reviewed. The theory is based on the random wave model. Given the spectral density of the wave record the random model assumes a normal distribuCion of Che wave elevations (Gaussian model). The random model predicts then the spectral density of the low frequency part of the square of the wave envelope as presented in equation (4.2.1.12).
The theoretical distribution of the low frequency part of the square of the wave envelope for a random model is given by Davenport and Root [4-
P(A2) = ~ ^ - e 2 m0 (4.2.4.1) L 0
in which:
00
m = ƒ S (oo) dio U 0 ^
The low frequency part of the square of the wave envelope is exponentially distributed.
For the measured spectrum No. 1 (C . ,- = 12.5 m and f^ = 14.1 s) the statistical properties will be compared with those derived from theory. By means of low pass filtering of the square of the wave record the low
157
frequency part of the square of the wave envelope can be determined as follows:
A2(t) = 2cJ(t) (4.2.4.2)
in which:
2 2 C„(t) = low pass filtering of the square of the wave elevation C (t)
The distribution function of the elevation of the wave and the square of the wave envelope as derived from the experiment and theory are given in Figure 4.3. Furthermore the wave trains of all applied wave spectra were subjected to the analysis on the spectra of the wave groups. The results are presented in Figure 4.4. A good agreement can be found between the actual and the theoretical statistical properties of the wave groups. Knowing the good correlation of the input the statistical quantities of the motions (the output) obtained from computations and model tests can be compared.
The results of the computations and the model tests on the characteristic data of the low frequency surge motion are given in Table 4.1 and Figure 4.1. The agreement is good.
It can be concluded that the mean wave drift damping coefficient has to. be taken into account to predict the correct motion characteristics of a moored vessel exposed to irregular head waves.
The present wave spectra with approximately the same mean wave period were adjusted in the basin by increasing the stroke of the wave generator. By increasing the stroke the energy of the spectrum increases but the sequence of the wave trains will be approximately the same. The result is that the statistical properties will be the same.
From a point of view of statistics the present realizations of the wave train were more than unique: although the duration was restricted to 2.5
158
hours full scale the statistical properties of the wave groups correspond well with the theoretical ones.
As was mentioned the theoretical spectrum of the low frequency part of the square of the wave heights will be obtained by an infinite long test duration. In this wave train the variance of the low frequency motions will reach the steady state value as given in equation (4.2.1.4). If the sea state is stationary during an assumed period of time the variance of the variance of the steady state value of the low frequency motion can be determined. This is demonstrated in [4-4]. The theory is given by Tucker [4-6], who gives the expression to determine the expected variance of the variance of a normally distributed process as function of the test duration.
LOW-FREQUENCY SURGE MOTIONS
NORMAL DISTRIBUTIONS
- THEORETICAL Test No. 3601 SURGE
r NO
.0
'
.0
0
Ï
\ \
\ fv \
o
\ J V V A \
Figure 4.6 The normal distribution of the low frequency surge motion
It is assumed that the low frequency surge motion is normally distributed. The distribution of the low frequency surge motion as occur in wave spectrum No.1 is shown in Figure 4.6. In terms of standard deviations the distribution shows a normal distribution. For more support of the assumed distribution reference is made to [4-4J.
According to Tucker the variance of the variance can be written as follows:
<A 2-re ƒ S2 (u) du (4.2.4.3)
159
where: T = duration of the record S (u) = spectral density of the motion record Xl
Making use of equation (4.2.1.2) for Sx ((J.) and applying the low damping assumption we obtain the following result for the root-mean-square value of the variance:
°l \JTèh 2<B11«1)c11 Xl
in which: 6' = non-dimensional damping
6 Bll+51 = 2^ = — (4.2.4.5) cri
where in accordance with equation (2.4.2.6) 6 is the logarithmic decrement and
B i = critical damping
= 2/c u(M+a ur (4.2.4.6)
The part in square brackets in equation (4.2.4.4) is recognized as the steady state value of the variance as given in equation (4.2.1.4).
Dividing by the steady state variance and assuming that 6' « 1 equation (4.2.4.4) results in the following assessment for the non-dimensional root-mean-square value of the motion variance:
a' = l (4.2.4.7) a1 / T O 7 ] ^ x 1
160
From this expression we may see which factors are of importance with respect to the root-mean-square value of the variance. We see that, for a given vessel/mooring system, which determines the natural frequency n-, and the non-dimensional damping 6', only the test/computation duration T will influence this quantity. It is seen that given a particular requirement with respect to the root-mean-square value of the motion variance a system with low damping and low natural frequency will require a longer test/simulation duration. For examples reference is made to [4-4].
4^3^_Time_domain comgutations_in_irregular h£aQ,_waves_with and without current
4.3.1. Theory
For the time domain computation of the low frequency motions of a moored tanker in irregular head waves with and without current the following equation of motion has to be solved in the time domain:
(M+au(u1))x1 + B 1 1(n 1)i 1 + B ^ + c ^ = X ^ t ) (4.3-1.1)
in which: a, ,((i,) = low frequency added mass coefficient Bll^l) = s t l H water or current damping coefficient B. = mean wave drift damping coefficient c,, = linear spring coefficient in surge direction X1(t) = wave drift force in the time domain ji. = natural frequency of the system
The damping consists of the viscous and the mean wave drift damping. The viscous damping is either the still water or the current damping. The viscous damping coefficients B-^ can be determined as is indicated in Chapter 3.
161
Further for the time domain computation the wave drift force registration and the mean wave drift damping coefficient are required. To compute the registration and the damping coefficient respectively the wave train and the spectrum have to be known. It is assumed that for zero speed the transfer function of the wave drift forces P(tOj,u.) and the wave drift damping coefficient D(U)J) are known. The transfer functions can be derived from either computations or model tests.
Applied to the wave spectrum the mean wave drift damping coefficient can be calculated:
B (V ) = 2 ƒ S (oo) D(o)*) do> (4.3.1.2) 1 c 0 c,
in which: Sr(oj) = wave spectrum as measured in situ W* = 00 + K.VC K = wave number V = current velocity D(u)) = B ^ ^ / C g 2
The transfer function of the wave drift force in current can be approximated by the gradient method:
P(OJ*,OJ*) = P(u. ,u.) - D(w.).V v i' i i l i c
The time history of the wave drift force can be obtained by means of the quadratic impulse response function technique applied to the record of the measured wave train as proposed by Dalzell [4-7 ]. By means of the Fourier transform the quadratic impulse response can be written as:
+oo +00 (id) t -iu T ) g(x1,t2) = l ^ ) 2 ƒ ƒ G(2)(oJl,co2)e 1 1 2 2 d ^ doo2 (4.3.1.3)
~00 —00
in which:
162
(2) G (w. ,(!)„) = complex quadratic transfer function = P("1>w2
) + iQ(ui»u)2)
t.,T2 = time shifts
Because of the very low natural frequency of the system the matrix (2) G (oo1 ,o)9) may be composed by means of the in-phase components P((D. ,U>2)
approximated on base of P(oo. ,co.) and P(w2,a>2), see equation (2.7.3.3). The in-quadrature component Q(io, ,(*)«) will be neglected.
On the base of linear interpolation of u^ and u>. in the matrix the Fourier transforms have been applied to obtain the quadratic impulse
(2) response functions g (t.,t.)-
Using the record of the adjusted waves C(t) the time domain simulation of the wave drift force can be written as:
X,(t) = ƒ ƒ g(i T ) C(t-t ) C(t-T ) dt dx (4.3.1.4) 0 0
Using the theory, the wave drift force registration can be determined. After the determination of the viscous damping coefficients by means of model tests or the data given in Chapter 3 the equation of motion can be solved.
4.3.2. Computed wave drift forces and mean wave drift damping coefficient
The quadratic transfer function for the wave drift force P(tü^,u^) for zero speed was computed for the loaded 200 kTDW tanker in 82.5 m deep water. The quadratic transfer function of the wave drift damping coefficient D(oo.) was derived from the experiments as shown in Figure 2.11. Based on the gradient method the transfer functions of the wave drift force P(io* a)*) and the wave drift damping coefficient D(co*) for 1.03 m.s current velocity were determined. The results are shown in Figure 4.7.
163
^
Table 4.2 Matrices of the quadratic transfer functions of the wave drift forces with and without current
164
With the transfer functions P(wi,u^) and P(u)*,u*) the matrices of the quadratic transfer functions P(io. ,<o.) and ~P(u*,u*)t respectively without
■J i J and with current, have been composed and are presented in Table 4.2.
x < 2 > J1
''a in t f .nT2
Fully loaded 200 kDWT tanker Water depth 82.5 m * t r = 180 deg.
%z \ YCurrent \ \ 2 kn.
Current 0 kn.
Current 2 kn.
JCurrent 0 kn.
fV'a
'a in t f .s .m""
Figure 4.7 The quadratic transfer functions of the wave drift force and wave drift damping coefficients with and without 2 kn current
The impulse response function g^ ^ ( ^ . T . ) has been determined for each of the matrices. For the deterministic approach the convolution was applied to the calibrated wave trains as adjusted in the model basin. For the spectra of the wave trains reference is made to Section 4.3.4. The results of the computed wave drift force registration with and without current are presented in Figure 4.8 in the form of spectral densities over the frequency range of interest.
Using the transfer functions of the wave drift damping as given in Figure 4.7 and applied to the spectra with and without current the following results were computed:
B1(Vc = 0) = 48.2 tf.s.m
B.(V = 1.03 m.s 1) = 42.8 tf.s.m l
1 c
After obtaining the input data for the viscous damping coefficient the equation of motion can be solved.
165
V =1.03 m.s~' O COMPUTED
MEASURED
Vc=0.0 m.s~' • COMPUTED
MEASURED
ü<2> in t f COMPUTED Vc=0 Vc=1.03 m.s
-135.4 -166.6
MEASURED Vc-0 Vc=1.03 m.s'
-145.1 -166.3
0 0.05 0.10
u in rad.s
Figure 4.8 The spectral densities of the measured and computed wave drift forces
4.3.3. Computed motions
The viscous damping coefficients were obtained from the model tests, see Section 4.3.5. For the still water damping and the current damping the following values were obtained:
Bu(still water) = 25.3 tf.s.m-1
B u ( V c = 1.03 m.s"1) = 27.0 tf.s.m-1
The total damping coefficient for the condition of waves without current is equal to 73.5 tf.s.m , while for the condition of waves with current the total damping amounts to 69.8 tf.s.m • For the spring constant c-,,
-1 = 19.3 tf.m was chosen. The computed results are presented as time domain plots in Figure 4.9. While the test duration was 2.5 hours for full scale, only the last 1.5 hour have been presented. The first hour was
< 0 >
/ 1
1 0 /
/ /
I / / / /%
1
o °
\ \ \ \ \ 1
> 0
0 0
\ \ \ » \ *
166
necessary to account for the transient phenomena in the computations.
LOW FREQUENCY SURGE MOTION OF TANKER IN IRREGULAR «AVES WITH AND WITHOUT CURRENT
4 . 3 . 4 . Model t e s t s
0 m -15 m
0 m 15 m
Test No
CURRENT
63444-63621
= 0 m .s " '
^
Test No CURRENT
63514-63592 = 1.03 m.s"'
CALCULATED MEASURED
V^
3600 4000 5000 6000 7000 8000 91
Time in s
Figure 4.9 The computed and measured low frequency surge motion of the tanker in irregular waves with and without current
In order to correlate the computed registration of the wave drift force and the low frequency surge motion a series of model tests were carried out with the afore-mentioned tanker.
The tests were carried out in the Wave and Current Laboratory of MARIN at a water depth of 1 m.
Prior to the wave calibration a homogeneously distributed current field was adjusted with a velocity corresponding to 2 kn for the full scale. Combined with current or without current the same wave spectra were adjusted at the projected location of the centre of gravity of the moored tanker. Each sea state was prepared for a test duration of 2.5 hours full scale time. The spectra and the distribution functions of the waves and the wave groups are shown in the Figures 4.10 and 4.11 respectively.
Both wave drift force and motion measurements were carried out. For the wave drift force measurements a vertically positioned cylinder hinged with air lubricated bearings was used to keep the tanker model on station. The bearings were earth fixed. As is illustrated in Figure 4.12 the test set-up allows for heave and pitch motions. As a consequence of the set-up the tanker was not able to perform high frequency surge motions. This approach is allowed, because it can be shown that the contribution of the high frequency surge motions to the wave drift forces is negligibly small.
167
(5 CYCLES) WITHOUT CURRENT (5 CYCLES) WITH CURRENT 1.03 n.s"
WAVE ELEVATION
Jd ^3
rrK -,
WAVE 8749
° t = 2.88 m Camax+ = 14 .6 m
^amax* = 1 0 . 3 m
Smax = 2 3 . 0 m C w l / 3 = 11 .6 m
E l e v a t i o n i n m
WAVE SPECTRUM
MEASURED : 4 , « ~ . 11 .5 niiT, .
THE0RETICAL(P.M.):4^iJÖ= 11 .5 m;T, ,
J
ƒ f //
r\
\\
s \ \ ^ - ^
0.3 s
10.8 s
0.5 1.0
u in rad.s"
WAVE ELEVATION
J F" \
W WAVE 8751
° t = 2 . 9 0 m camax* = 14 .2 m
^amax = 9 . 8 m ;wmax = 2 4 . 0 m ; w 1 / 3 = 11 .6 m
-10 0 10 E l e v a t i o n in m
WAVE SPECTRUM
n -MEASURED :>^\ö" " - S n i T , ■ 10 .3 s THE0RET1£A,L,(P.M.):4.C^Ö"- 11.5 iti.T, . 1 0 . 8 s
J
"
^ \\
\\ \\ \ \ \
0 0 .5 1.0
u in r a d . s
Figure 4.10 The adjusted wave spectra with and without current.
Since a force transducer was mounted to the lower side of the cylinder the force was measured in the horizontal direction. The measured horizontal force consisted of the first order and the second order wave forces. By means of low pass filtering techniques the low frequency second order wave drift forces were obtained. The measured drift forces are presented in Figure 4.8 in terms of the spectral density.
For the surge motion measurements the tanker was kept on station by a spring system, see Figure 2.5, having a linear spring constant in surge direction of c,, = 19.3 tf.m- . Prior to the tests in waves with and without current, extinction tests were carried out; if applicable the current load was measured. The current force amounts to 13.1 tf. The results of the extinction tests in terms of the logarithmic decrement are shown in Figure 4.13.
168
■ DERIVED FROH LOU FREOUENfY PART OF SOUARED HAVE RECORD ■ DERIVED THEORETICALLY BASED Ott SPECTRUM OF MEASURED WAVE
1500.0
1000.0
500.0
0.0
Test «0.
/ \ f \
8749
\ \
\ \ N \
"•-
\\ \ \ »,
V \\
0.25 0.50 u in rad.s
0.25 0.50 in rad.s"
■ DERIVED FROM LOW FREQUENCY PART OF SQUARED WAVE RECORD ■ DERIVED THEORETICALLY BASED ON SPECTRUM OF MEASURED WAVE
i Test No. 8749
\5 :
=E§§|EEËË
50.0 100.0 fl' i n m'
150.0 0.0
0
,
2
1
4
\ \ \ *-\ \
\\
\\ \A * \ A
M \
Test No. 8751
_,
50.0 100.0
A2 in „2 '
Figure 4.11 The spectra and the distribution functions of the wave groups
169
CYLINDER WITH AIR LUBRICATED BEARING
WEIGHT OF CYLINDER INCORPORATED IN WEIGHT DISTRIBUTION TANKER
TRIM-DEVICE
t HINGE AND FORCE TRANSDUCER
19.3m
HEAVE MEASUREMENT POTENTIOMETER
GYROSCOPE
Figure 4.12 Test set-up for the wave drift force measurements
c,,=19.3 tf .m- ; T=233 s ; v. = .02697 rad.s
--
" 1
-SS^ h r* Ics
Tost No. 6360 {with current)
„ „ , . . - -1 B l
Test No. 6358 (without current)
B ( ) » 25.3 t f . s .m" 1
VsCS?"-
aY^T 0 | ■• 0 >
During the tests the surge motions were measured in the centre of gravity (CG) by means of a high accuracy optical tracking device. The result in the form of time traces is shown in Figure 4.9.
10 20 Number of osci l lat ions N'
Figure 4.13 Surge extinction tests with and without current
4.3-5. Evaluation of results
According to equation (2.4.2.6) for the still water damping and the current damping the following values were found:
170
B11(still water) 25.3 t f . s . m
BU(VC = 1.03 m.s L) = 27 t f . s . m - 1
These values correspond approximately to the values as derived in
Chapter 3, being:
BU(VC = 0, ux = 0.02697 r ad . s x ) = 19 tf.s.m"
Bn(V = 1.03 m.s i, X, = -13.1 tf) = 25.4 tf.s.m" 1c
In Figure 4.8 the spectral densities of the computed and measured wave drift forces without current and with 2 kn current are shown. The agree-pent is good for the frequency range of interest. At higher frequencies the computed spectral densities increase due to the approximations of the off-diagonal terms. The effect of the current on the wave drift excitation is clearly demonstrated. By means of the theoretically derived hydrodynamic input data the equations of motion with and without current were solved separately. The weather conditions that were used are presented in Figure 4.10 and reviewed in the table below:
Pierson-Moskowitz wave spectra
Without current
j-wl/3 = U - 5 m
T = 10.8 s
V = 0 m.
With cu r ren t
'wl/3 11.5 m
10.8 s
1.03 m . s - 1
/ / waves
171
The computed and measured low frequency surge motions are presented as time domain plots in Figure 4.9. While the test duration was 2.5 hours for full scale, only the results of the last 1.5 hours are presented, see Section 4.3.3. From the results it can be concluded that a good agreement is achieved between theory and experiment.
172
REFERENCES (CHAPTER 4)
4-1 Wieners, J.E.W.: "On the low frequency surge motions of vessels moored in high seas", OTC Paper No. 4437, Houston, 1982.
4-2 Wichers, J.E.W.: "Progress in computer simulations of SPM moored vessels", OTC Paper No. 5175, Houston, 1986-
4-3 Pinkster, A.J.: "Low frequency phenomena associated with vessels moored at sea", SPE Paper No. 4857, European Spring Meeting of SPE-AIME, Amsterdam, 1974.
4-4 Pinkster, A.J. and Wichers, J.E.W.: "The statistical properties of low frequency motions of non-linearly moored tankers", OTC Paper No. 5457, Houston, 1987.
4-5 Davenport, H.B. and Root, W.L.: "An introduction to the theory of random signals and noise", Mc Graw-Hill, NY, 1958.
4-6 Tucker, M.J.: "The analysis of finite length records of fluctuating signals", British Journal of Applied Physics, Vol. 8, April 1957.
4-7 Dalzell, J.F.: "Application of the fundamental polynomial model to the ship added resistance problem", 11th Symposium on Naval Hydrodynamics, University College, London, 1976.
173
CHAPTER 5 EVALUATION OF THE LOW FREQUENCY HYDRODYNAMIC VISCOUS DAMPING FORCES AND LOW FREQUENCY MOTIONS IN THE HORIZONTAL PLANE
5.1. Introduction
In Chapter 4 the large amplitude surge motions of a tanker exposed to survival conditions were discussed. The tanker was assumed to be moored directly by the bow by means of a mooring system as is indicated in the lower part of Figure 1.1. For this kind of mooring system it may be assumed that if the system is sufficiently stiff the tanker stays in line with the co-linearly directed weather components.
In this chapter a tanker will be considered moored by means of a bow hawser to a fixed pile. This system is shown in the upper part of Figure 1.1. In order to absorb sufficient kinetic energy at acceptable force levels the hawser lines normally consist of synthetic material, so that the load-elongation characteristic of the mooring line will be non-linear. Because of the limited strength of a hawser these kinds of mooring systems are often used for areas with mild or moderate weather conditions.
One of the features of hawser moored tankers is that the system can be dynamically unstable. The result of a dynamically unstable system *is that a tanker exposed to certain weather conditions can perform large amplitude low frequency motions in the horizontal plane even in the absence of low frequency excitation. The large amplitude motions may induce considerable loads in the hawser.
In weather conditions without low frequency excitation the tanker can perform large amplitude unstable motions. For these large amplitude motions the description of the equations of motion including the nonlinear hydrodynamic viscous damping forces as were derived in Chapter 3 will be evaluated by comparing the results of the low frequency motions obtained from computations and physical model tests.
For still water the large amplitude low frequency motions for dynamically unstable conditions of a tanker exposed to long-crested regular waves will be considered first.
For current a dynamically unstable system which performs the large amplitude low frequency motions will be selected. In order -to select the conditions the stability criterion will be studied. For steady current and wind velocities under different angles of incidence and for different loading conditions the unstable conditions were determined as a function of the length of the bow hawser. Derived from the stability criterion, both the stable and unstable conditions are considered in this validation study.
5.2. Tanker_moored_by__a_bow hawser exgos'ed_to_regular waves
5.2.1. Introduction
500 ƒ Load 1 in tf /
250 "/
0 10 20 30 Elongation in m
Figure 5.1 Bow hawser load-elongation characteristic
To evaluate the viscous damping values in still water the low frequency motions of a tanker moored by means of a bow hawser and exposed to regular long crested waves have been computed. The results are compared with the results of model tests. The tanker concerns the loaded 200 kTDW tanker moored in 82.5 m water depth, see Section 2.4.1.
The bow hawser was connected to a fixed pile. The length of the unloaded hawser amounted to 75 m. The non-linear load deflection curve is shown in Figure 5.1.
176
For the regular waves the following characteristics were used:
Test No.
7219-7215 7224-7218 -
7220-7217
T in s
6.45 7.14 8.00* 9.10
2Ca in m
4.74-2.75 4.72-3.23 5.00-3.00 4.98-3.07
* theory only
5.2.2. Computations
Following equations (3.2.7), (3.3.1.3) and (3.3.4.7) the equations of motion can be written as follows:
M(X ri 2i 6) = xlp + x l s w + xl(d,Cr) - 51i1 + xlm
M C x ^ i g ) = X 2 p + X2SW.+ X2(*Cr) - B2i2 + X 2 m
Ix, = X, + X,„„ + X,(<lv ) - B.x, + X. 6 6p 6SW 6V Cr' 6 6 6m (5.2.1.1)
in which: X. = potential inertia parts of the reaction forces/moment kp XkSW viscous parts of the reaction forces/moment
V*. Cr > = mean wave drift forces/moment in a regular wave as function
"Cr o f <\>r Cr <|>r - x, = relative wave angle mean wave drift damping coefficient in a regular wave moorir
1,2,6
X, = mooring force components caused by the bow hawser
The wave d i r e c t i o n as defined in Figure 3.3 was <|>j- = 180 degrees .
177
The potential inertia parts of the reaction forces/moment are given in equation (3.3.1.3), while the values of the coefficients are presented in Table 2.1. The coefficients of the viscous part of the reaction forces/moment in surge direction can be found in Figure 3.10, while the values of the transverse resistance coefficients are C(Jl) = 2.402, 0.49, 1.439, 0.361 along the length between sections 0-2, 2-4, 4-18, 18-20 respectively as can be read from Figure 3.14.
The mean wave drift forces/moment in the regular waves as function of the relative wave angle were computed by means of the direct pressure integration method [2-3]. The results in terms of quadratic transfer functions are presented in Figure 5.2.
V 135 deg 150 deg 160 deg 170 deg 180 deg.
-30
-20
-10 / J
1 V*
^ '%
/ i
\ / i-M*
/
150
100
50
)>
/ i / 1 ' 1/
\ /
_--
/
4000
2000
0.5 1.0 0.5 1.0 -2000
^t % W 0.5 1.0
Wave frequency in rad.s
Figure 5.2 The quadratic transfer function of the wave drift force components as function of relative wave direction
For the mean wave drift damping coefficient in the regular waves a constant coefficient Bi is taken into account only. The value is derived from the quadratic transfer function as is given in Figure 2.11. With regard to the value of the still water damping B u the value of B^ contributes significantly. As is shown in [5-1] it may be assumed that for the sway and yaw modes of motion the mean wave drift damping is small. Moreover, due to the appropriate large viscous damping, the wave drift
178
damping in the sway and yaw modes of motion is deleted. The values for the still water and wave drift damping coefficients for the surge direction as applied to the computations are presented in Table 5.1.
10
r a d . s
0.974
0.88 0.785 0.69 0.924 0.88 0.785 0.69
T
in s
6.45 7.14 8.0 9.1 6.45 7.14 8.0 9.1
2Ca
m
2.75 3.23 3.00 3.07 4.74 4.72 5.00 4.98
B l l
t f . s . m - 1
10.1 10.1 10.9 10.9 11.63 12.40 12.00 11.63
■ 5 i t f . s . m - 1
3.9 4.7 4 .1 4.7
11.8 10.0 11.4 12.4
Table 5.1 The damping coefficients for the surge mode of motion
5.2.3. Model tests
The model tests have been carried out in the Wave and Current Laboratory of MARIN at a scale of 1:82.5. During the model tests the horizontal motions at point A (= fairlead, located 4.5 m in front of the fore perpendicular) were measured in an earth-fixed system of co-ordinates as indicated in Figure 5.3. The yaw motion and the hawser force were also recorded. Both the linear and the rotational motions were measured by means of optical tracking devices.
Under influence of the environment the tanker was kept under the following start condition:
x6 = +7.5' xA(2) = 0 m
179
+x
FIXED SPM
BOW HAWSER
x6 (0) = 7-5°
«2 —"
—
(1)
A
—1 Gl
\\
\ \
( ^ ^ = 180°
ROD .
RELEASE JOINT
+X1
*x6
ROD .
UNIVERSAL JOINT
and in the xA(Indirection restrained by the bow hawser force. At t=0 the tanker was released and the measurements were started.
5.2.4. Evaluation of re-sults
UNIVERSAL JOINT
RELEASE JOINT t = 0 : PULLING UP OF RODS
—c5= =
In Figure 5.4 the comparison of the results of the simulation and the model test is presented for 2C = 4.72 m and T = 7.14 s. As a result of the unstable behaviour of the system the dynamic loads in the hawser increase to 160 tf instead of 49 tf, which would have been the result of the static mean wave drift force in that particular regular
wave. A plot of the behaviour of the tanker and the bow hawser force is given in Figure 5.5. Of the steady parts of the computations and measurements the amplitudes of the motions xA(l)> xA(2) and Xg and the maximum bow hawser force are plotted in Figure 5.6. The results of the yaw periods are presented in the same figure. A good agreement is found between the computed and measured values.
^VTT-^T-CTT-TV^C^V: \*«.».vvvvvv\vv
STAND
Figure 5.3 Set-up of the model test
5.3^_Tanker moored_b2_a_bow_hawser_ex22sed_to_current
5.3.1. Introduction
To evaluate the resistance force and moment components in current, low
180
-10S 100
2; = 4.72 m - T ■ 7.14 s
COMPUTED MEASURED
-'V \f, M u <] f V 1/ A V 1
AfnVA ^-vv fnV -A" /-\A ^M^^\ A i i\ f
^Kj irfVT^ wnöiü iör/zi
D -
^^niDIET ^S\/IW Ji\J V* IFVrYl \JV\1 *MmM 4000 0
Time in s 2000 4000
Figure 5.4 Time domain results from simulation and model test (bow hawser length 75 m)
frequency motion decay simulations of a tanker moored by means of a bow hawser were carried out. The simulations are compared with the results of model tests. The tanker concerns the fully and intermediately loaded 200 kTDW tanker moored in 82.5 m water depth. The bow hawser was connected to a fixed pile. For both the computations and the model tests an unloaded hawser lengths of 45 and 75 m were applied. The non-linear load deflection curve as used for both hawsers is presented in Figure 5.1. The tanker was exposed to current with speeds of 2 and 3 knots.
5.3.2. Computations
Following the equations (3.2.7), (3.4.1.7) and (3.4.6.5) the equations of motion can be written as follows:
(M+a.-Vx. = (M+a„)x„x, + X, „ ,. + X, , + X. 11 1 22 2 6 lstat ldyn lm
(M+a0,)x + a„.x, = -(Mfa,.)i.x, + X + X,. + X. 22 2 26 6 11 1 6 2stat 2dyn 2m
(l,+a.,,)'i, + a.„x„ = X, + X, . + X, 6 66 6 62 2 6stat 6dyn 6m (5.3.2.1)
181
x(l)
FF(I80°)=49 t:
The current direction as defined in Figure 3.3 was 4>c = 180 degrees. The values of the coefficients used for the potential inertia parts are given in Table 2.1. The coefficients for the steady relative current forces/moment components are shown in Figure 3.18. For the viscous part of the dynamic current contribution the coefficients are used as derived in Section 3.4.6.
5.3.3. Model tests
Figure 5.5 Relation between the tanker mo- The SPM model tests were tion and the bow hawser force carried out in the Shallow
Water Laboratory of MARIN. The description of the basin is given in Section 3.3.2. Current speeds of 2 and 3 knots were simulated by towing the set-up, as is shown in Figure 5.3, through the basin. During the tests only the yaw angle was measured.
5.3.4. Evaluation of results
For the initial conditions for both the computations and the model tests a similar procedure is followed as for the still water case with regular waves. In Figure 5.7 the results of the motion decay tests as obtained from the simulations and the model tests are presented.
Sensitivity computations using the present formulation and the formulations given by Molin [3-13] and Obokata [3-14] showed that the value of
182
&
-^
\ \ £
o ^ - 1
\ > V
■
o 400
" S " ' ^
ü
0
1
a
A
X o
, - ■
tt
2.^
■,
■ "
o
—"' '
5 7 9 II WAVE PERIOD in s
5 7 9 11 «AVE PERIOD in s
BOW HAWSER LENGTH 75 m WAVE HEIGHT 3.0 ID WAVE HEIGHT 5.0 m
COMPUTED MEASURED
5 7 9 11 WAVE PERIOD in s
Figure 5.6 Results of computations and model tests
the steady relative current force in longitudinal direction will influence the decay of the motion in the horizontal plane. In spite of the very small value in relation to the other force/moment components the influence on the yaw motion was considerable. To fit the results of the model tests the longitudinal steady current force has to be increased to some extent. Tne reason may be found in some scatter in the experimentally determined resistance coefficient in head current C, (180°) but also may be caused by the dynamic behaviour of the tanker. The dynamic force component in surge direction due to the coupling of the low frequency surge and the low frequency yaw motion was not investigated. The simulation model was modified by multiplying the longitudinal steady current force component by a factor 2. This factor was applied to all time domain computations for the 3-DOF model.
In the mentioned cases the tanker performs stable decaying motions after the release from the initial start position.
183
LOADING CONDITION: 100% T
xA (1) 2 5 r
i n 11 0
Hawser length = 45 m
Vc = 1.54 m.s"1 ; t c = 180°
Computed
xA(2) 2 5 r in m 0
H 10 in deg n
Measured: Test No. 13241
3600
I n i t i a l cond i t ions:
x ( l ) = -205.97 m ; x(2) = -27.11 m
xc = +6.33°
Hawser length = 75 m Vc = 1.03 m.s"1 ; * = 18
25 r Computed 0 _ = = _ _ = = = = =
3600
Measured: Test No. 13211
x(1) = -231.12 m ; x(2) = -30.31 m
xc = +5.6°
LOADING CONDITION: 70* T
awser length = 45 m Hawser length = 75 1.54 m.s"1 ; if = 180° u = 1 ""■ m c ' * =
xA(1) 2 5 r
in m 0 Computed
xA(2) 25 in m 0
x6 iu in deg
FF 50 in tf „
\ _
[A A. — 7 V v^^—
1800 3600
10
in deg
Time in s
Measured: Test No. 13231
Computed
1800 3600
Time in s
Measured: Test No. 13221
I n i t a i l condi t ions: x(1) = -198.2 m ; x (2) = -21 .99 m x(1) = -227 .5 m ; j t (2) = -29.95 m x . = +6.33° Xg = +7.5°
Figure 5.7 Results of the motion decay in current
184
To evaluate the low frequency viscous terms in the equations of motion, conditions have to be found in which the tanker performs large amplitude unstable motions in the horizontal plane. By means of the Routh criterion the dynamic instability can be established.
5.4^_Tanker moored by a bow hawser exposed to wind and current
5.4.1. Dynamic stability of a tanker moored by a bow hawser
xA(1) xA(2)
xA(1) xA(2)
DYNAMICALLY STABLE Monotonically Oscillatory
DYNAMICALLY UNSTABLE Monotonically Oscillatory
The procedure of determining the dynamic stability of a tanker moored by a bow hawser has been extensively presented and evaluated by Wichers in [5-2 ]. In the following the method will be briefly described. For stable or unstable behaviour of the moored tanker a steady current and wind field will be considered. For the sign convention reference is made to Figure 3.3. Definitions of the dynamic stability in general are graphically shown in Figure 5.8.
Figure 5.8 Graphical representation For the determination of the sta-of dynamic stability bility of the system the characte
ristic equations have to be solved. The solution of the characteristic equations gives the motion characteristics due to small deviations around the static equilibrium position of the tanker in the steady current and wind field. To establish the equations of motion equation (3.2.7) has to linearized. Linear equations of motion were obtained by means of a Taylor expansion to the first order yielding the following equations:
185
£ ((M, .+a, . ) x . + b, .x . ) = X, + X, for j = 1,2,6 and k = 1,2,6 , j = 1 kj kj 3 kj y ke Tem
(5 .4 .1 .1 )
in which: a. . = added mass coefficients at low frequencies in k direction due to
j mode of motion bk . = potential damping coefficients at low frequencies
and
, ÖXke(V0)) ÖXke^6(0)1 x .
ox. J öx. J J J
ax (x (o)) ax (x.(0)) \m = *|JV0» + ^ Xi + ^ Xi C5.4.1.3) km km o öx_ j ^ j
J J in which: X, (x,(0)) = mean external force component in k direction due to wind
and current X, (x,(0)) = mean bow hawser force component in k direction x (0) = tanker heading in the equilibrium position o x^ = tanker yaw with regard to the equilibrium heading x,(0)
Considering the linearized equations of motion about the equilibrium position and taking into account the appropriate coefficients we obtain:
6 E m x. + B x. + c .x. = 0 for k = 1,2,6 and j = 1,2,6
J_I KJ J fcJ J KJ J (5.4.1.4)
in which: X j - V e -A- = constant, dependent on the initial disturbance a = complex coefficient mkj " Mkj+akj
186
ÖXke(x6(0)) 3kj
'kj
ai. j + b, . kj
Ö X J X 6 ( 0 ) ) Ö\e^6 ( 0 )) öx. ox.
The complex coefficient defines the characteristics of the motion. When the real part of a is minus, the motion converges. The motion diverges in the plus case. Therefore, the condition of the motion can be determined by the sign of the real part of a. Equation (5.4.1.4) can be written as:
A M A.a + B. .A.ö + c. .A. = 0 kj J kj j
or in matrix form:
[ma + Ba + C]{A} = 0
for k = 1,2,6 and j =1,2,6
(5.4.1.5)
(5.4.1.6)
Since it is assumed that {A} é 0, the following determinant results 2 ma + Ba + c I = 0
On expansion this determinant may be arranged as
S, a + Sca + S, a + S,a + S.a + S, a + S. = 0 6 5 4 3 2 1 0
(5.4.1.7)
(5.4.1.8)
in which the coefficients for j = 1,2,6 are symbolically expressed as determinants:
lj D2j D3j
m l j m2j B3j
+ "lj B2i m3j
+ V m2j m3i
187
s, = n l j m2j C3j
+ 'ij B2j B3j
+ ml.i C2j m3j
+ B U m2j B3j
+ BU B2j m3j
+ CU m2j m3j
s„ = m l j B2j C3j
+ "IJ C2j B3j
+ "ij m2j C3j
+ B U B2j B3j
+ BU C2j m3j
+ CU m2j B3j
+ Cl.i B2j m3j
"ij C2j C3j
+ Blj B2j C3j
+ BU C2j B3J
+ Clj m2j C3j
+ Cl.i B23 B3J
+ Cl.i C2j m3j
B U C2j C3j
+ CU B2j C3j
+ CU C2j B3j
so = c l j C2j C3j
(5.4.1.9)
Following the definition In Figure 3.3, the linearized damping and spring coefficients respectively Bj. and Cj. can be written as follows (neglecting the damping due to the wind):
cosU-x (0)) ax U - x (0)) sinU-x (0)) Bll = 2Xlc^c-X6(°)) V •- ^ + b
a l i
R ,v f, „ ^ S l n ( V X 6 ( 0 ) ) . 8 X i A - V ° » c o s ( V x 6 ( 0 ) ) B12 " 2X l c ( ( P c -x 6 (0)) + ^ ^ + b 12
B 1 6 = 0 . 4 ( a 2 2 - a u ) V c s i n ^ - x ^ O ) ) + b 16
B21 = 2 X 2 c < V x 6 ( 0 > >
c o s U - x , ( 0 ) ) OX, (* - x , ( 0 ) ) s i n ( 6 - x , ( 0 ) ) ' c "6 2cVMc 6V
a<i> + b cr
21
188
sin(cp -x (0)) ÖX U -x (0)) cos(<P -x (0))
*22 = 2 X 2 c ^ c - X 6 < ° ^ S7— + *C £ V 7 — + b22
B26 = ( a 22- a U ) V c " - ( V X 6 ( 0 ) ) + b 26
cos((l> - x , ( 0 ) ) ÖX, O -x (0 ) ) sln(cb -x ( 0 ) . B, , - 2XA (* - x , ( 0 ) ) V 5 6 C ° 6
v 6 . + b
61 6cv c 6 -1 V_ Z><\> V _ ö<|> V ' " 6 1 c er c
sln(c(,c-x6(0)) ÖX6c(<|,c-x6(0)) cos(<|.c-x6(0).; «62-2x6c(Vx6(o)) - y '+ DC'V. " —^r— + b6 "62 c er e
_ _ (L/2-FB)3 + (L/2+FB)3
B66 ~ 3L B22 + D66
2 ^ FF . 2 c x l = CE cos y + j-g- s in Y
FF C12 = ( C E ~ LË^ C 0 S Y S l n Y
Ö X l eK ( ° l l c = e12AG + FF slny O
C21 °21
2 FF 2 c 2 2 = CE sin Y + ^Ë c o s r
_ ax2e(x6(0): C - , £ = C T ) A G " F F C 0 S ^ " "26 " "22"" " " " ' öx£
C61 = C 21 A G
189
C62 " C 22 A G
cfifi = ( c o 9 A G ~ F F cosy)AG "66 Vl"22 öx, (5.4.1.10)
- vo ) in which: Y LE = length of hawser with mean load in the equilibrium position FF = mean hawser load in equilibrium position CE = derivative of the static load deflection curve at position LE and
FF AG =longitudinal distance between centre of gravity and position of
fairlead
By varying the parameters of the system and solving equation (5.A.1.8) the convergence or divergence of the motion can be determined by the sign of the real part of a. Of the 6 complex solutions, all real parts have to be negative for the motions to be convergent (stable).
5.4.2. Determination of the stability criterion
. y / /
y
■'—■"*■». 0.5
2»
^ ^ ~ ' ^
>S Based on
Station 10
(V*6* i n d e g 'VK6' i n d e 9
Figure 5.9 The wind forces/moment coefficients [5-3]
For the 200 kTDW tanker in loaded, intermediate and ballasted condition (respectively 100%, 70% and 40% of the loaded draft) the stability criterion has been determined. Therefore the tanker was exposed to 2 kn current and 60 knot wind speed. The stability criterion was determined as function of the hawser length and angle between current and wind. The
190
static load-deflection curve of the hawser was assumed to be independent of the hawser length; it is shown in Figure 5.1.
For the computation of the coefficients use has been made of the mass coefficients as given in Table 2.1 (the potential damping coefficients are assumed to be zero), while for the current loads and their derivatives the results as presented in Figure 3.18 were applied. For the wind loads on the tanker the data were used as presented by Remery and van Oortmerssen [5-3]. The wind coefficients are presented in Figure 5.9.
The results of the computed stability criterion are given in Figure 5.10. For the fully loaded tanker unstability will occur for a bow hawser longer than 120 m. From the result it can be concluded that lengthening of the hawser may result in unstable behaviour of the tanker. This agrees with the conclusion of Strandhagen et al. [5-4J, who indicate that a towed vessel will become unstable in the horizontal plane with increasing length of the towing line.
In order to evaluate the equations of motion for the unstable behaviour of the tanker in 2kn current, 60 kn wind an unloaded hawser length of 90 m has been chosen. From Figure 5.10 it can be read that the 70%T and 40%T loaded tanker exposed to the specified parallel directed wind and current will be in the unstable region.
Time domain simulations were carried out of which the results were compared with the results of model test. Finally time domain simulations were carried out to check the stability of the system for the 70%T
= 1.03 m.s = 60 kn computed
Figure 5.10 Stability criterion based on the bow hawser length and angle between wind and current
191
loading condition with an angle a of 45 and 90 degrees between the wind and the current, see Figure 5.10.
5.4.3. Computations
The tanker with the specified loading conditions was moored in 82.5 m water depth and connected by means of the hawser to a fixed pile. The load-deflection curve of the 90 m long hawser is presented in Figure 5.1. Following the equations (3.2.7), (3.4.1.7) and (3.4.6.5) the equations of motion can be written as:
(M+a,.)x, = (M+a.„)x,x. + X. ,. „ + X, , + X. + X, 11 1 22 2 6 lstat ldyn lw lm i') *
(M+a )x, + a.,x. = - (Mfa.Jx.x, + X, _ _ + X. + X0 + X. 22 2 26 6 11 1 6 2stat 2dyn 2w 2m
C^+a,-,-)*, + a£ o X - = X, „ + X, . + X, + X, (5.4.3.1)
6 66 6 62 2 6stat 6dyn 6w 6m '
in which: X, , X- , X. = the steady wind force/moment components lw 2w 6w
For the definition of the system of co-ordinates and the definition of the weather directions, see Figure 3.3. The values of the coefficients used for the potential inertia parts of the reaction forces/moment are given in Table 2.1, while the potential damping coefficient b, . = 0. The coefficients for the steady relative current are shown in Figure 3.18. For the viscous part of the dynamic current contribution the coefficients are used as derived in Section 3.4.6.
The wind forces are defined as:
,2 Xlw = AW^TS + (H"T)B\
X„ = V . C 0 ((|/ )(ATO + (H-T)L)V2 2w A 2w wr v LS ' wr
192
X6w " ^AC6w^„r^ALS + ( H" T ) L) L Vwr " X2w F B (5'4-3-2>
in which: p. = specific density of air = 0.00013 tfs2m .
For the further nomenclature, see Table 2.1. The wind coefficients are shown in Figure 5.10. Following [5-3] the wind force coefficients are defined on base of linear interpolation of the loading condition.
5.4.4. Model tests
The SPM model tests were carried out in the Wave and Current Laboratory at a model scale of 1:82.5. The wind field in the basin was generated by means of a battery of portable electrically driven wind fans. The fans were placed some distance from the testing area. The width of the battery was as large as was necesssary for the adjustment of a homogeneous wind flow over the testing area. At the projected location of the tanker the wind field was adjusted by means of an anemometer. The test set-up is shown in Figure 5.11.
Figure 5.11 Test set-up with the tanker in the wind and current field.
193
During the model tests the horizontal motions at point A (= fairlead) were measured in an earth-fixed system of co-ordinates as indicated in Figure 3.3. The yaw motion and the hawser force were also recorded. Both the linear and the rotational motions were measured by means of optical tracking devices.
5.4.5. Evaluation of results
In Figure 5.12 and Figure 5.13 the comparison of the results of the simulation and the model test for the unstable tanker conditions are presented. It can be concluded that the results of the computations are in good agreement with the results of the model tests. Contrary to the 60% loaded condition the tanker in ballast condition performs considerably large motion amplitudes. As a result of the unstable behaviour of the
1.03 m.s~] ; i|> = 180° 30.9 m.s" ; c = 180°
Computed
0 1800 3600 1800 Time in s
Initial conditions: x(1) = -253.20 m ; x(2) = -19.64 m ; xg = +7.5°
Figure 5.12 Computed and measured behaviour of the unstable tanker in wind and current (60% loaded)
v4- = Measured:
x (1) - 4 0 r T e s t No. 23883
in m -90
xA (2) 50
in m
50r in deg g
FF 100 in t f
n
194
ballasted tanker the dynamic loads in the hawser increase to values of up to 255 tf instead of 87 tf as results from the static calculation.
-1 V =1.03 m.s"' ; *c = 180° V = 30.9 m.s"1 ; * , = 180° w w
Measured: -40 r Test No. 2393
xA(D in m
xft(2)
x6 in deg
FF in tf
-90 50
0
50 0
100 0
-40
-90
rComputed
.^^AA/VVVA/WWV
3600 1800
Initial conditions:
Time in s
x(1) = -250.493 m ; x(2) = -19.64 m ; xg = +7.5°
Figure 5.13 Computed and measured behaviour of the unstable tanker in wind and current (25% loaded)
The computed results of the stable conditions with the 70%T loaded tanker are shown in Figure 5.14. In these conditions the model tests showed a stable behaviour of the system also. The measured and computed equilibrium positions are shown in Figure 5.15. A good agreement exists between the computed and measured results.
For the integration procedures of the computations of the low frequency oscillations reference is made to Section 6.5.
195
LOADING CONDITION: 70% T Hawser length = 90 m
180° V = 1 . 0 3 m.s , ; $ V,c = 30.9 m.s"1 ; * c = 225°
x A (1 ) i n m x A (2 ) in m
x 6 in deg
0
-200 50
0
50
n
Vr = 1.03 m.s" V„ = 30.9 m.s" w
FF l O O f v / W ^ ^ -in t f
0 1800
I n i t i a l condit ions: x(1) = -218.97 m x(2) = -127.42 m x, = +36.49°
200L 50"
50
0
100
0 3500
' Time i n s
xA (2)
1800
x(1) = -120.88 m x(2) = -22-1.513 m x , = +66.9°
if = 180° * c = 270° w
3600
Figure 5.14 Computed behaviour of a s t a b l e SPM system
Tesc No. 2390
X(2) 100 m
1 —
Loading condition: 70% T Hawser length = 90 m Vr = 1.03 m.s V,, = 30.9 m.s"1
100. m
Test No. 2389
Computed Measured
Figure.5.15 Computed and measured stable equilibrium position of the stable SPM system
196
REFERENCES (CHAPTER 5)
5-1 Huijsmans, R.H.M. and Wichers, J.E.W.: "Considerations on wave drift damping of a moored tanker for zero and non-zero drift angle", Prads, Trondheim, June 1987.
5-2 Wichers, J.E.W.: "On the slow motions of tankers moored to single point mooring systems", OTC Paper No. 2548, Houston, 1976/Journal of Petroleum Technology of SPE-AIME, SPE Paper No. 6242, June, 1978, pp. 947-958.
5-3 Remery, G.F.M. and van Oortmerssen, G.: "The mean wave, wind and current forces on offshore structures and their role in the design of mooring systems", OTC Paper No. 1741, Houston, 1973.
5-4 Strandhagen, A.G., Schoenherr, K.E. and Kobayashi, F.M.: "The dynamic stability on course of towed ships", SNAME Spring Meeting, Cleveland, Ohio, 1950.
197
CHAPTER 6 SIMULATION OF THE LOW FREQUENCY MOTIONS OF A TANKER MOORED BY A BOW
HAWSER IN IRREGULAR WAVES, WIND AND CURRENT
6.1. Introduction
In Chapter 5 the theory of the equations of motion in a current field and also of the stability criterion in wind and current including time domain simulations was evaluated. In this chapter simulations will be discussed when the tanker is exposed to current, wind and irregular waves.
For the simulations in wind, waves and current the same mooring system will be used as described in Chapter 5. The tanker will be the 60% loaded 200 kTDW tanker. The weather conditions are assumed to correspond to operational conditions. The weather components consist of a 2 kn current, a 60 kn wind and a wave spectrum with a significant height of 3.9 m and a mean period of 10.2 s. The weather components were kept constant; three different combinations of directions will be applied for the computations. A review of the environmental conditions is presented in Figure 6-1.
Because the significant wave height for the operational condition is relatively small the effect of the wave drift damping coefficients on
■ current: Vc = 2.o kn ' the tanker motions will be » Wind : Vw = 60.0 kn ^\^~ «ave .- j w l / 3 . 3.9 m T, = io.2 s neglected.
In Section 5.4 it was found that under the influence of 2 kn current and 60 kn wind a hawser length of 90 m causes dynamic unstability for environment 1, while stability was obtained for environment 2 and 3. To
©
<>
A
Figure 6.1 Review of the environmental conditions on the SPM system
199
simulate the behaviour of the tanker in the mentioned environments a hawser with a length of 90 m was used. The results of the computations are compared with the results of the model tests.
In the following sections the equations of motions, the computations, the model tests and the evaluation of the results will be dealt with.
6^2^_Equations_of_motion
For the simulation a computation scheme was followed as is given in Figure 6.2. This scheme tdkes into account the computation procedure for the high frequency motions (six degrees of freedom) and for the low frequency motions (three degrees of freedom in the horizontal plane).
ENVIRONMENTAL FLOW FIELDS
CURRENT
UINO
WAVES
CURRE
—
n
WIND
WAVE FIELD
CUMMINS DESCRIPTION
INERTIA MODEL
RELATIVE CURRENT CONCEPT
RELATIVE WIND MODEL
IMPULSE RESPONSE FUNCTION
MOORING CHARACTERISTICS
HIGH FREQUENCY FLUID REACTION FORCES
-| LOW FREQUENCY FLUID REACTIVE FORCES
FIRST AND SECOND WAVE FORCES
MOORING LOADS
TRANSFORMATION
SOLUTION IN LOCAL SYSTEM
TRANSFORMATION
TRANSFORMATION
TIME STEP INTEGRATION IN GLOBAL SYSTEM
(EARTH-BOUND) POSITION ORIENTATION VELOCITIES
Figure 6.2 Computation scheme for SPM simulations
200
To solve the motions of the tanker moored by a bow hawser, the equations of motion can be split up into a high frequency and a low frequency part, viz.:
The low frequency part as a function of the low frequency motion in the horizontal plane can be written as follows:
in which: (2) (2) x = XJ V ' = low frequency motion components for j = 1,2,6 (2) Xl = hydrodynamic loads caused by the added masses and induced by
accelerations and centrifugal effects, see equation (3.4.1.7) (2) X = relative current loads, see equation (3.4.1.8) -stat > T \ / (2) X: = dynamic current load contributions, see equation (3.4.1.9)
X ( 2 ) = wind loads -w (2) X = loads induced by the mooring system (2) XJ = time varying second order wave drift force components in the
horizontal plane as function of the low frequency position of
the tanker in the wave field
The matrices M and D are defined according to equation (3.2.7).
The high frequency part is expressed in linear hydrodynamic terms. The impulse response technique according to Cummins has been applied, see equation (2.2.5):
6 °°
+ c -x^1? + X ^ (x(2), x(1),t) = x£1}(x(2).t) for k=l(l)6 (6.2.2)
201
where: Xj^ ' = high frequency motion in j-direction Xk (x >c) = h l S h frequency time varying wave forces in the In
direction as function of the low frequency position of tanker in the wave field
Mj. = matrix of inertia of the vessel m^j = matrix of added inertia KJ^J . = matrix of retardation function c^j = matrix of hydrostatic restoring forces XJJJ J*- •'(x*- ',x^ ') = mooring force in k-direction due to high frequency
motion in j-direction as function of the low frequency position of the tanker
The wave exciting loads are functions of the vessel's position and of time. For a long-crested irregular wave train CgCt) defined in a fixed position, the wave load on the floating structure moving near this position can be computed as has been shown by Wichers and van den Boom [6-1 ]. The computation of the wave loads using convolution integrals based on the wave height at the actual (instantaneous) position of the vessel in the wave field can be formulated as follows:
CD
X ^ C x ^ . t ) = ƒ g^Cx^,-^) C(s,t-T)dT for k=l(l)6 (6.2.3)
x[2)(x(2),t) = ƒ / g k 2 ) ( x 6 2 > ' V V £( s> t _V C(8,t-i2)dT1,di2
for k=l,2,6 (6.2.4)
in which C(s,t) stands for the wave elevation at the actual location of of the vessel (CG) and can be obtained from the wave elevation at the space fixed point 0, being CQ(t).
The distance s is defined as the length between the space-fixed point 0 and the instantaneous low frequency position of the CG of the vessel projected in the direction of the wave propagation. The transformation
202
of the reference wave CnCt) t 0 ttie required position will be:
GO
C(s,t) = ƒ w(s,t) C (t-x)dT (6.2.5) 0 u
-H» 2% where W(S,T) = — ^ / W(s,u) e i W t du
and W(s,w) = — C(s,t) .2it
e 1 ^ (6.2.6) C(0,t) in which: F { } denotes the Fourier transform.
It should be noted that this transformation of the wave elevations is only valid for neighbouring locations.
The linear and quadratic kernels g ^ ' and g^ ' in equation (6.2.3) and equation (6.2.4) are found from the Fourier transform of the corresponding frequency domain transfer functions:
i1^'^ -2Ï /^(«fUe^-d» (6-2.7)
(2), (2) , 1 f" f~„(2), (2) , i(ü)lTr W 2 V , A
4 (x6 W =7TT2 ' / Gk (x6 ' W e dVu: (2it) -<= -°°
1 2
(6.2.8)
in which:
a ( ^ ^(x^ 2^,^) = P(x6(2>,u)lfu2) + iQ(x6(2> .^.Uj)
By applying this computation procedure the instantenèous phase relation between the motions and the waves will be taken into account properly.
203
The high frequency hydrodynamic reaction coefficients, the low frequency added mass coefficients, the transfer functions of the first order wave forces G^ ' and the quadratic transfer functions G ^ ' can be computed with potential theory.
6.3. Computations
In the previous section, for sake of completeness, the description of the equations for both the high and low frequency motions were presented. Since the present computations concern the low frequency motions only the high frequency part will be deleted from now on.
For the computations use will be made of the formulation given in equation (6.2.1). The equations of the low frequency motions, except for the wave drift force components, correspond to equation (5.4.3.1). For the present computation the same input data will be used as presented in Section 5.4.3.
In current and wind Figure 5.11 shows that the motions are dynamically unstable for environment 1, while in accordance with Figure 5.12 dynamically stable motions were obtained for environments 2 and 3.
In order to account properly for the relatively large amplitude motions for environment 1 the computations of the wave drift force components have to be carried out in accordance with equation (6.2.4), being the large amplitude model. For the large amplitude model the wave loads are supposed to be a function of X]^2'(t), x2' '(t) and x6^2^(t).
Due to the dynamic stability, the small amplitude model is applied to environments 2 and 3. For the small amplitude, model the wave loads are a function of the mean position of the tanker in the wind, current and wave field, being x(l), x(2) and x, . The mean position can be computed by applying the wind, mean wave drift, and current load components to the tanker.
204
For the large amplitude model, however, the correction of the wave drift force components to the low frequency surge motion with regard to the length of the appropriate wave group components will be small. Therefore in equation (6.2.4) the correction can be reduced to a constant value, being the distance between the position of the reference wave height and the mean position of the tanker.
For the computations the wave drift force components can be simplified as:
CO 00
x[2)(x(2),t) = / ƒ 4 2 ) ( x 6 2 ) , W « ^ - V ■C(ï,t-t2)dT1dx. O O 2
for k=l,2,6 (6.3.1)
in which: (2) xfiv ' = instantaneous yaw angle in environment 1 (2) Xgv ' = mean heading of the tanker for environments 2 and 3
s = projected distance (CG tanker-wave) for the wave transformation
The low frequency added masses ak1 and the quadratic transfer functions of the drift forces P^^ in regular waves were computed for the tanker considered. The results of the quadratic transfer function P u ^ a s function of the wave direction are given in Table 6.1. For the computations the tanker hull was schematized, see Figure 6.3.
Because of the very low natural frequencies of the system the matrices Gi. were composed by approximating the in-phase components P^-M by taking P, ,., see equation (2.7.3.3). The quadrature components QT.II were neglected.
By means of the cubic spline interpolation method the main diagonals for different angles of wave attack were determined from.the results given
(2) -1 in Table- 6.1. The matrices G^ with frequency differences 0.02 rad.s for the relative wave directions <p = 160°, 170°, 180°, 190°, 200° for
205
ON
direction/ frequency
ui
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
0 ' 20" 45" 70" 90" 110" 135" 160" 180"
xf'fc.2
0.0
0.0
0.9
5.3
7.2
7.2
10.2
15.6
U . 5
5.6
9.1
3.9
7.1
2.4
6.6
2.6
6;5
-0.1
-0.4
-0.8
4.7
7.4
8.4
9.7
14.2
13.1
4.6
9.5
5.5
7.7
3.3
7.3
6.0
■7.1
0.0
0.1
0.5
2.2
6.1
10.0
13.2
9.4
10.9
12.5
11.1
6.7
8.7
7.1
8.0
9.5
8.9
0.0
0.1
0.2
0.7
3.5
9.0
11.3
20.7
23.2
18.9
14.7
12.4
8.4
6.0
8.3
9.5
8.9
0.0 0.0
0.0 -0.1
0.0 -0 .3
0.0 -1.0
-0.1 -4 .1
-0.7 -9.5
-2.4 -10.5
-4.5 -16.7
-3.9 -18.9
-2.4 -16.5
-1 .1 -13.1
0.0 -9.9
0.5 -0.7
0.6 -5.2
0.7 -4.9
0.8 -5.9
0.7 -7.4
-0 .1 -0 .3
-0.2 -0 .3
-1 .0 -1.9
-3.8 -6.0
-7.9 -9.4
-12.0 -11.0
-15.7 -14.3
-13.0 -17.4
-12.4 -16.2
-15.3 -10.9
-12.6 -11.3
-10.5 -11.9
-10.4 -10.4
-12.5 -9.7
-11 .3-10 .6
-11.9 -9.9
-12.1 -10.6
-0.3
-0.4
-2.3
-6.4 .
-9.3
-9.6
-14.5
-18.5
-14.9
-11.7
-11.4
-9.5
-10.0
-7.4
-9 .9
-6.4
-10.0
0 ' 20" 45 ' 70" 90' 110" 135" 160" 180"
■ xfW
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.8
4.7
7.4
15.9
21.6
24.1
18.6
23.9
24.2
27.7
27.9
33.1
35.0
27.1
45.4
0.0 0.1 0.2 0.2
0.2 0.1 0.3 0.4
0.5 0.6 1.3 1.3
2.3 2.3 4.3 4.0
9.8 12.4 24.3 18.0
25.5 53.5 118.0 57.0
51.8 60.9 120.3 66.3
73.9 97.4 158.2 102.8
81.9 120.1 160.3 114.7
84.2 126.9 152.1 120.7
83.7 131.1 145.8 127.4
85.2 136.5 144.2 128.9
0.2
0.4
1.4
5.0
14.5
31.7
62.4
86.2
95.0
97.8
96.4
99.0
87.2 135.2 146.2 131.8 100.1
91.4 122.2 145.8 122.4 103.7
90.6 U5.5 148.1 119.5 104.4
89.3 133.3 158.3 135.5
89.6 153.3 168.8 147.2
95.0
98.8
0.1
0.3
0.9
3.4
8.5
19.4
26.1
28.7
24.8
29.3
31.9
35.2
37.3
42.4
45.4
38.4
55.7
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0" 20' 45" 70" 90" 110" 135" 160" 180"
* > 2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
-110.9 -68.3
-290.7 -256.9
-625.0 -659.0
-932.1 -1206.5
-907.0 -1723.0
-548.3 -1581.9
-191.0 -1451.8
200.0 -167.5
-1107.1 129.3
-178.0 -264.0
-493.0 -1241.8
-135.5 -1212.5
-571.0 -1581.8
-229.1 -1486.9
-512.8 -2043.6
-2750.0 -2828.6
-H2.2 -2384.2
-24.8
-100.7
-260.5
-640.1
-1662.0
-142.6
-546.7
-1849.2
-3527.7
-3598.7
-3242.6
-3085.2
-2224.9
-1219.6
-1273.7
-2024.9
-2300.8
0.3
0.1
2.4
13-2
69-6
601.5
1070.9
914.9
-276.0
-879.0
-1009-5
-1025.8
-998.7
-1019.5
-943.7
-626.5
-793.1
25.4
97.0
265.2
668.3
1767.7
741.3
1785.9
2869.6
2493.6
1780.6
1595.2
1149.0
522.3
-401.3
-913.5
-195.6
419.7
100.7
319.1
770.1
1414.1
1977.1
1662.5
1602.3
-35.0
-773.5
-352.5
396.0
562.8
745.3
952.1
1230.4
1390.1
1040.0
136.7
340.0
695.1
983.5
928.9
464.2
83.1
-203.2
845.1
138.8
237.5
68.5
272.2
167.5
271.0
2025.4
235.9
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
a» 200 knw - 60% loaded - 82.5 m t a t e r depth frequency in rad.s" . - i . *<2> /Ca In tf .up* or tf .m"1
environment 1 and C|J (X,) = 218.55° and 205.8° for the environments 2 and 3 respectively were composed. The last mentioned wave angles were results of the computed mean headings of the tanker for the appropriate weather conditions.
Figure 6.3 Facet distribution of the tanker hull, (60% loaded) (symmetrical starboard side)
On base of linear interpolation of w. and u. in the matrices the Fourier transforms have been applied to obtain the quadratic impulse response
(2) functions g^ ' •
For the computation of the registration of the wave drift force components use was made of the wave registration as simulated in the model basin. For the computation of the mean heading of tanker the spectra of the adjusted waves were applied. The spectra are shown in Figure 6.4. The wave registrations are presented in the Figures 6.6, 6.7 and.6.8.
For the mentioned relative wave angles the registrations of the wave drift force components were computed and stored for the simulations. During the simulation for environment 1 the drift force components were obtained by linear interpolation. of the mentioned pre-processed registrations on base of the instantaneous value of the relative wave angle.
The results of the time domain computations of the low frequency motions and the hawser force are presented in Figures 6.6, 6.7 and 6.8. Although
207
the wave train duration was 2 hours for the full scale, only the computed registration of the last 1 hour is shown. Due to zero speed in the start condition the first hour is required to account for the transient phenomena.
6^4^_Model tests
For the description of the tests, the basin and the measuring set-up, reference is made to Section 4.3.4. and Section 5.3.3.
After the adjustment of the homogeneously distributed current speed of 2 kn the waves were calibrated. The waves were adjusted at one location in the basin. For the waves running parallel with and perpendicular to the current approximately the same spectrum type was adjusted. Each sea state was prepared for a test duration of 2 hours. The spectra are shown in Figure 6.4.
Current velocity 2.0 kn. wave / / current (?
w wave 1 current (
3=3.97 m;Tt=10.1 s;Test No. 2369) =3.91 n; =10.3 s; 2371)
I
/
7\.A
; \ 1.0 • 1 . 5
Prior to the actual tests, trial tests in wind, waves and current were carried out to shift the mooring pile in order to adjust the mean position of the CG of the tanker above the location were the waves were calibrated. During the tests the bow hawser force and the horizontal motions at the location of the fairlead and the yaw angle of the tanker were measured.
Figure 6.4 Measured wave spectra
208
The measured results presented as the mean positions of the tanker are shown in Figure 6.5, while the time domain plots are presented in Figures 6.6, 6.7 and 6.8. The presented time domain results were not filtered. All signals were recorded with 25 samples per second. The results are given relative to the starting position, see Figure 6.5.
' 2\i 5Y2i"a£i2ÏL2Ë results
In Figure 6-5 the mean positions of the tanker exposed to environments 2 and 3 are presented. With regard to the model test results the computed headings were found to be 9.4 and 4.0 degrees smaller for environment 2 and 3 respectively. Due to the combined action of wind, waves and current an explanation is hard to give. The effect of doubling the longitudinal current component as used for all time domain computations will result in a surge and sway displacement only. Furthermore the deviation can not be explained by the increase of the drift force in current. The wave potential will not be affected by current, which is directed perpendicular to the direction of the wave propagation.
In Figures 6.6, 6.7 and 6.8 the results of the model tests are presented as time domain plots. The results are given relative to the initial start condition as indicated in Figure 6.5. As mentioned earlier the measured signals were not filtered. For the considered conditions it can be concluded that the high frequency, components hardly contribute to the motions or the hawser force. The system is dominated by the low frequency motions of the tanker. As discussed before the mean positions as derived from computations and model tests did not completely correspond to each other. In order to compare the low frequency results the mean values of the computed results were shifted to coincide with the means of the measured signals. From the results in Figures 6.6, 6.7 and 6.8 it can be concluded that the computed low frequency components are in reasonably- good agreement with the measured results.
209
MEAN POSITION TANKER
START POSITION
x(1)
x(Z) Environment 3
100 m
START POSITION 100 m
Figure 6.5 The measured and computed mean position of the bow hawser moored tanker in wind, waves and current
210
2.50
FF tf
50.00
Test No. 23884 (wave test No. 2369) Environment 1
I|I =180° ; ii =180° ; I|I =180°
MEASURED COMPUTED
+ 10.00-
*A<2> J A
3600 4000 5000 6000 Time in s
7000
Figure 6.6 The measured and computed results of the tanker exposed to
weather environment 1.
211
2.50
Test No. 2389 (wave test No. 2371) Environment 2 - tanker 60X loaded
* =270° ; + =225° ; if =180°
MEASURED COMPUTED
+10.00
♦10.00 xA(2)
i ■ ■ i ■ ■ ■
3600 4000 5000 6000 Time in s
7000
Figure 6.7 The measured and computed results of the tanker exposed to
weather environment 2.
212
2.50
Test No. 2390 (wave t e s t No. 2371)
Environment 3 - tanker 60% loaded
50.00 FF t f
M^MJifrnA^A
+10.00 A6 deg.
MEASURED COMPUTED
-10.00 x A ( 1 )
x A ( 2 ) +10.00
I ■■ I 3600 4000 5000 6000
Time in s
7000
Figure 6.8 The measured and computed results of the tanker exposed to weather environment 3.
213
Finally some remarks will be given on the computational procedure with respect to the integration procedure and the integration technique. For the computation of the input data for the simulation program pre-processing programs were used. For the simulation the Advanced Continuous Simulation Language packet (ACSL) was used [6-2]. All programs were run on a Cyber 175 or 170-855.
For the time domain simulation of the low frequency motions the computational scheme was followed as presented in Figure 6-2. It is assumed that the position and the velocities of the ship in the global system of co-ordinates x(j) and x(j) for j=l,2,6 are known.
To compute the values of the separate terms in the model the position of the vessel in the global system of co-ordinates x(j) and the velocities in the vessel-bound or local system of co-ordinates x. have to be known. The transformation of the velocities from the global to the local system of co-ordinates can be computed with:
x = T x (j)
in which:
cosx6 -sinx6 0
sinxg cosxg 0
Knowing the values of the terms in the equations of motion the accelerations x. in the local system of co-ordinates can be solved. For the computation of the large amplitude motions the following integration procedure has been applied. According to equation (3.2.6) the absolute ship's accelerations along the instantaneous directions of the local system of co-ordinates are:
214
X1E xl X2X6
X2E x2 + xlx6
X6E = X6 ( 6 - 5 a )
The components of the acceleration along the earth-fixed system of coordinates can be written as follows:
x (j) = T x -Jfc
which is in accordance with equation (3.2.4) or:
x(j) = Tx + T x for j = 1,2,6 (6.5.2)
By means of numerical integration the global velocities and displacements x(j) and x(j) respectively can be determined for the prediction of the next time step. By means of these data the computations of the new acceleration can be carried out.
Some attention has to be paid to the integration technique. Although the natural periods of the system are large, difficulties can occur with the applied integration method. Examples of results of different integration methods are shown in Figure 6.9. The examples concern the 60% loaded tanker exposed to the co-linear directed wind and current (see Figure 5.12). The current velocity was 1.03 m.s , while the wind velocity amounted to 30.9 m.s . The initial conditions in the global system of co-ordinates were:
x(l) = -254.87 m i(l) = 0 m.s"1
x(2) = - 19.64 m x(2) = 0 m.s"1
x(6) = + 7.5° x(6) = 0 rad.s-1
215
In the first example the first order Runga-Kutta or Euler method was applied. The time step was chosen to be 1/50 of the smallest natural period being in the order of 5-10 seconds. The computation was carried out with a time step of 10 seconds. A higher order integration method was used in the second example. Both higher order Runga-Kutta methods and methods with variable step and variable order, for instance according to Gear[6-3], gave consistent results.
EULER OR RUNGA-KUTTA FIRST ORDER
\ A / \ / \ A / \ / \ A / I A A A A A \M\\pJ\\i v j v v
GEAR
A r ?r A A A nWW / 111/ v 1/ \ 1 .u
A A A A A i ' \ / \ / \ / \ / " W V v V M M
0
\ A A . " \ / \ / \
vy v \ P> A f
/ \ / \ ' J \J \J
\
Figure 6.9 Euler and Gear integration method
Because of the occurrence of peak loads in the hawser, as can be seen in Figure 5.12, the integration method of Gear has been used for all the computations. This integration method has proven to be reliable [6-4J. Besides the reliability of the method, also the execution time may not be more than for the Euler method with small time steps. The efficiency is caused by the variable time steps for a relative smooth signal.
216
REFERENCES (CHAPTER 6)
6-1 Wichers, J.E.W. and van den Boom, H.J.J.: "Simulation of the behaviuor of SPM-moored vessels in irregular waves, wind and current", Proc. 2nd International Conference and Exhibition on Deep Offshore Technology, Valletta/Malta, 1983.
6-2 Mitchell, E.E.L. and Gauthier, J.S.: "Advanced continuous simulation language (ACSL)", Simulation, March 1979, pp. 72-76.
6-3 Gear, C.W.: "Numerical initial value problems in ordinary differential equations", Prentice-Hall Inc., Englewood Cliffs, New Yersey, 1971.
6-4 Wichers, J.E.W. and Drimmelen, N.J.: "On the forces on the cutter head and spud of a cutter suction dredger operating in waves", 10th World Dredging Congress, Singapore, April 1983.
217
CONCLUSIONS
As a result of the investigations the following conclusions may be drawn up:
1. For a tanker moored in waves the velocity dependent second order wave drift forces have to be taken into account.
2. For regular head waves and relatively deep water the quadratic transfer function of the velocity dependent wave drift forces for storm wave frequencies can be computed by means of 3-dimensional potential theory at small values of forward speed. The computed results show a satisfactory agreement with values obtained from measurements on small scale models.
3. By means of the quadratic transfer function of the velocity dependent wave drift force in regular head waves the transfer function of the total wave drift force is obtained. In case of still water and by means of the gradient method the total transfer function of the wave drift force can be split up in the transfer function of wave drift force for zero speed and the transfer function of the wave drift damping coefficient. In combination with co-linearly directed current and by means of the gradient method the total transfer function of the wave drift force can be split up in the transfer function of the current speed dependent wave drift force and the transfer function of the wave drift damping coefficient.
4. For a moored tanker with a low natural frequency and exposed to irregular head waves combined with or without co-linearly directed current the complete matrix of the quadratic transfer function can be composed on base of the main diagonal. The determination of the main diagonal of the transfer function of the wave drift damping coefficient will be sufficient. The contribution of the low frequency oscillating wave drift damping force to the motions is negligibly small.
219
5. For the simulation of the low frequency surge motion of the tanker besides the total wave drift forces, also the hydrodynamic low frequency reactive forces have to be known. The added mass coefficient may be computed by means of 3-dimensional potential theory, while for still water the viscous resistance coefficient and for current the current resistance coefficient must be derived from experimental data. Computed results of the surge motions show a good agreement with the results of model tests. The results clearly demonstrate the importance of the application of the velocity dependency of the wave drift forces.
6. To describe the low frequency behaviour of an SPM moored tanker in the horizontal plane sets of equations of motion can be drawn up. Distinction has to be made between the still water and the current condition. The low frequency viscous resistance coefficients for the 3 modes of motion in the horizontal plane can be determined by means of PMM tests.
7. Based on the results of the PMM tests in still water the distribution of the transverse resistance coefficient along the length of the tanker can be approximated.
8. Time domain simulations on the 3 degrees of motion with the bow hawser moored tanker exposed to regular waves were performed. The computed results show a good agreement with values obtained from model tests.
9. For the current condition the viscous resistance force and moment components can be formulated as a steady relative current force/moment contribution and a dynamic current force/moment contribution. The experimentally determined results have been compared with results based on the relative current concept. The recently formulated semi-theoretical models based on the relative current concept, can only be applied to the very low frequencies of the yaw velocities.
220
10. Time domain computations of the motions for 3 degrees of freedom of a bow hawser moored tanker in a current field were carried out. The results show that a satisfactory agreement is obtained with the results of model tests when the relative current force in surge direction is doubled.
11. Using the linearized criterium of Routh the stability of a bow hawser moored tanker exposed to current and wind was judged. The predictions were in good agreement with the model test results.
12. Computations for the b.ow hawser moored tanker exposed to various operational conditions were carried out. To compute the wave drift force registration a large and a small amplitude motion model was used. The large amplitude model was used for the unstable conditions, while the small amplitude model was used for the stable conditions. The result from the motion simulations show a reason-greement with the results of the model tests.
221
APPENDIX
THREE-DIMENSIONAL DIFFRACTION THEORY WITH LOW FORWARD SPEED
Introduction
This appendix gives a short account of the underlying theory and method of computation of the velocity potential for an arbitrary tanker-shaped body sailing at low forward speed in regular, long-crested waves and in deep water as given by Hermans and Huijsmans [l-ll]. In keeping with fill] the nomenclature x(l), x(2) and x(3) as used until now will become x, y and z, see Figure A.l. Furthermore the underlying theory is general for drift angles, defined as p. The frequency used must be read as the frequency of wave encounter.
Descri2tion_of_theory
Boundary condition and expansion of the source strength and potential function to small values of forward speed
The total potential function will be split up in a steady and a non-steady part in a well-known way:
$(x,t) = -Ux + 5(x;U) + Kx,t;U) (A.l)
In this formulation U is the incoming unperturbed velocity field, obtained by considering a coordinate system fixed to a ship moving under a drift angle p. In this approach the angle does not need to be small. The time dependent part of the potential consists of an incoming wave at frequency co, a diffracted and/or a radiated wave contribution. In the computations all these components will be taken into account. A restriction is made to a general theory concerning the wave components.
223
Figure A.l System of axes
The equations for the total potential <3? can be written as:
A$ = 0 in the fluid domain D (A.2)
At the free surface we have the dynamic and kinematic boundary condition:
gC + $ t + h VS.V4 = const.
z x^x y y t at z = C(x,y,t) (A.3)
When it is assumed that the waves are high compared with the Kelvin wave pattern and that they both are small, the free surface condition can be" expanded at z = 0. Elimination of C leads to the following non-linear condition:
a2* a$ . a ,v®.v®^ + % 'JT+ ■£ (V$.V*) +-^.V[~^) = 0 at z at az at (A.4)
To compute the wave resistance at low speed thé free surface must be treated more carefully, because " the wave height is of asymptotically smaller order. Ttiis problem has been studied extensively by Eggers [A-l], Baba [ A - 2 ] , Hermans [A-3] and Brandsma [ A - 4 ] . The velocity field is well described by the double body potential with a small wave pattern.
224
Therefore the double body potential is taken into account and the stationary wave pattern is neglected..
For the wave potential $(_x,t;U) the free surface condition now becomes:
4> + g4> + 2U* + 2VÏ.V$ + (U2 + 2UÏ + ï 2)^ + 2(U + 5 )ï Ï + tt e z yxt t x x'Txx v x' y xy
+ $2<)) + (3U$ + 5 Ï + $ * )5 + + (2U$ + $ <£> + $ $ ) $ + y yy xx x xx y xy x xy x xy y yy y
+ L(2){*} = 0 at z = 0 (A.5)
The boundary condition on the hull can be written in a similar way for all radiating and diffracted modes. Generally the condition exists:
(n.?*) = (a + Vx (axW)).n (A.6)
at the mean position of the hull, in which a is the oscillation motion _ T
and W = (u,v,o) . The speed components will be:
u = W cos8 v = W sinB,
with 8 the drift angle of the ship with respect to the velocity field as defined in Figure A.l.
For the six modes of motion the body boundary condition now reads:
T~ <|> = i ioa + Wmfc k = 1 ( 1 ) 6 (A .7 )
225
in which:
Wmk = -(n.V)w = 0 k = 1,2,3
Wmk = -(n.V)(r x w) k = 4,5,6
which leads to:
m^ = -sinfi ^
n6 = sinp n. - cosp n~
where n^ is the direction cosinus.
The non-linear operator on $ will be neglected as well. The first line in equation (A.5) contains linear terms in U. The Ansatz is that in order to obtain the first order appoximation with respect to U the second
t order terms with respect to U may be neglected in the free surface. In the next section it is shown that in general this is true, but first the construction of the regular part of the perturbation problem with the complete linear free surface condition will be discussed.
It is assumed that <t>(x,t;U) is an oscillatory function:
~ -iwt <t>(x,t;U) = <|>(x,U)e (A.8)
The free surface condit ion i s wr i t t en as :
o2<(> - 2iwU<t> + U2<(> + g* = D(U;i){<(>} a t z = 0 (A.9)
226
where D(U;$) is a linear differential operator acting on <)> as defined in equation (A.5). We apply Green's theorem to a problem in Di inside S and to the problem in Dg outside S where S is the ship's hull. The potential function inside S obeys equation (A.8) with D = 0, while the Green's function fulfills the homogeneous adjoint free surface condition:
-u>2G + 2iu>UGr + U2Grr + gG,. = 0 at z = 0 (A. 10)
This Green's function has the form:
G(x,5;U) = - i + i - - (Kx,5;U) (A.11) 1
where r - | S - £ | rx = I x - 5'| 5' = the image of £ with respect to the free surface.
Combining the formulation inside and outside the ship we obtain a description of the potential function defined outside S by means of a source and vortex distribution of the following form:
" J/Y(5) |n^(5»S)dS5 - //a(5)G(x,£)dS£ - llSS. ƒ Y(£)G(x,§)dTi S S WL
+ J- I l>(§)§gG<2.5> " {\yt(D + aTYT(§)}G(x,5)]dTi 2
+ j - ƒ a a(5)G(x,5)dTi + -^ ƒƒ G(x,5)D{(|>}dS = 4n<|)(x) (A.12) 8 WL g FS ^
a = cos(Ox.t) aT = cos(Ox.T) a = cos(0x,n) n -
where n is the normal and t the tangent to the waterline and T = txn the bi-normal.
227
It is clear that with the choice y(5) = 0 the integral along the water-line gives no contribution up to order U. The source distribution we obtain this way is not a proper source distribution, because it expresses the function <t> in a source distribution along the free surface with strength proportional to the derivatives of the same function $. However, this formulation is linear in U and moreover the integrand tends to zero rapidly for increasing distance R. So finally we arrive at the formulation:
- 2*0(5) - JS o(£) -X- G(x,?) dS + ~- ƒ a a(g)|-G(x,§)dT, S x ^ s WL
+ T^ / / izr G(ï>^) °{*}dsF = ^v(x), x e s (A.i3) g FS x **
and
4n<t>(x) = - ƒƒ o(|)G(x,£)dS. + 7- / a 0(g)G(x,§)dn S ^ g WL
+ J 7 G(x,§)D{<t>} dS , x SD (A.H) g FS c, e
Now small values of U will be considered, keeping in mind that there are two dimensionless parameters that play a role in the limit. It is considered that 1 = — « 1 and v = 772- » 1- It turns out that the source
8 u
strength and the potential function can be expanded as follows:
o(x;U) = aQ(x) + ta^x) + a(x;U)
<>(x;U) = $0(x) + t()i1(x) + $(x;U) (A.15)
where a and $ are 0(t2) as T+0, while the expansion of the Green's function is less trivial.
The Green's function
In this section an asymptotic expansion of the Green's function will be provided. The Green's function follows from the source function pre-
228
sented in Wehausen and Laitone [ A - 5 ] .
In the case T<1/4 the function (Kx,^;U) is written as:
■n/2 n <Kx,|;U) = I2" / d0 ƒ dk F(0,k) + -^ ƒ d0 ƒ dk F(0,k) (A.16)
0 LL n/2 L2
where:
v,a M - kexp(k[z + C + i(x-5)cos0])cos[k(y-n)sin0] ^ ö ' ^ " gk - (ui + kUcos0)2 CA.1/;
The contours L, and hn are given as follows:
k1 k2
o V_V ^ — l' k3 k4
\y o* ~L* Figure A.2.
These contours are chosen such that the 'radiation' conditions are satisfied. The radiated waves are outgoing and the Kelvin pattern is behind the ship. The values k are the poles of F(0,k). For small values of t these poles behave as:
/gk^, /gk^" ~ id + 0(T) as T+0 (A. 18)
/^2* - / iS~^i0 + °(1) aS T*° (A-19)
A careful analysis of the asymptotic behaviour of <Kx,5;U) for small values of U leads to a regular part and an irregular part:
<Kx,£;U) = <l>0(x,£) + -t(PL(x,<i;) +.. .+ ïQ(x,5) + v ' ^ x . g ) +. . . (A.20)
229
where
, k(z+C) V * > P " 2§ / gk - V k R ) d k (A-21>
2 2 k(z+C)
+!<ï.5) = 4ig2cos0' / (gk - M ^ 2 J^lcRjdk (A.22) L2
where
R2 = (x-5)2 + (y-n)2
0' = arctg ( Ejr) and
■a/2 <|J0(X,§) = -4v ƒ exp[v(z+Qsec26]sin[v(x-£)sec Q] *
* cos[v(y-T))sin0sec2e]sec20 d0 ' (A.23)
Hermans and Huijsmans [A-6] have shown that, due to the highly oscillatory nature, the influence of (A.23) may be neglected In the first order correction for small values of t.
Expansion of the source strength
In this section an approximation solution of equation (A.14) will ba derived. Inserting the equations (A.16) and (A.22) in equation (A.12) one obtains for like powers of x the following set of equations:
2u aQ(x) - ƒ ƒ aQ(|) (x.^dS^ = 4nV0(x), x € S (A.24)
and
230
8G - 2 ^ ( 5 ) -Ha^l) ^ - (x , | )dS 5 = - / / ö 0 (5 )a ip ^ ( x . ^ d S ^ +
ü X o X
o + T~ H Ö^T ( ^ } VÏ-V<"odsF + 47 tVi(5> (A.25)
where
G„(x,£) = - — + - 'Iv.Cx.g) is the zero speed pulsating wave source, .and V(x) = VQ(x) + T V ^ X ) + 0(-c2)
This perturbation leads to a fast algorithm that takes into account speed effects once a fast method is available for the zero speed diffraction problem. Therefore the diffraction program has been extended with the Newgreen subroutines of Newman [ A - 7 ] .
The potential functions (equation (A.15)) now become:
V*> ■ - h IJ ffo<§> e0<*»£>ds
£
and
♦i(ï) = r // v ^ v ^ s ) ^ - h a o1(g)G0(x,5)dsc + b b
+ 2 g~ M G0(x,5)VÏ.V$0dS& (A.26) FS
231
REFERENCES (APPENDIX)
A-l Eggers, K.: "Non-Kelvin dispersive waves around non-slender ships", Schiffstechnik, Band. 8, 1981.
A-2 Baba, E.: "Wave resistance of ships in low speed", Mitsubishi Technical Bulletin, No. 109, 1976.
A-3 Hermans, A.J. : "The wave pattern of a ship sailing at low speed", Report 84A, University of Delaware, 1980.
A-4 Brandsma, F.J. and Hermans, A.J.: "A quasi-linear free surface condition in slow ship theory", Schiffstechnik, April, 1985.
A-5 Wehausen, J.V. and Laitone, E.V.: "Surface waves", Handbook of Physics, Vol. 9, I960.
A-6 Hermans, A.J. and Huijsmans, R.H.M.: "The effect of moderate speed on the motion of floating bodies", Schiffstechnik, Band 34, Heft 3, sept. 1987.
A-7 Newman, J.N.: "Three dimensional wave interactions with ships and platforms", International Workshop on Ship and Platform Motions, Berkeley, 1983.
232
NOMENCLATURE
Symbols not included in the list below are only used at a specific place and are explained where they occur
affix ( '•■' )>' ) ' ' ' denotes whether a quantity is of zero, first, second, third order, etc.
A envelope of the wave train ALg lateral wind area of superstructure Arpg transverse wind area of superstructure AP ordinate from CG to section 0 (negative) B breadth of the ship B. wave drift damping coefficient in regular wave in
k-direction B mean wave drift damping coefficient in k-
direction . Bi, • damping coefficient in the k- mode due to a
motion in the j-mode
C wave velocity C. coefficient in k-direction C , n-th Fourier cosine coefficient in k-direction D. . components of the quadratic transfer function of
the wave drift damping coefficient dependent on (o. and u. i J
E wave energy FF force in bow hawser FB centre of buoyancy forward of section 10 FP ordinate from CG to section 20 (positive) G Green's function G^ ' linear transfer function
(2) Gv complex quadratic transfer function GMt metacentre height transverse GMi metacentre height longitudinal
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body axes with origin in CG depth of the vessel moment of inertia in the k-th mode Bessel function retardation function in the k-th mode due to motion in the j-th mode centre of gravity above keel length of the ship mass of the vessel inertia matrix number of oscillations number of wave components earth-bound system of co-ordinates atmospheric pressure components of the quadratic transfer function dependent on u. and to. added resistance spectrum of quantity u total wetted surface of the hull mean wetted surface of the hull surface element of S or S n-th Fourier sinus component in k- direction wave period draft of the vessel test or computation duration transformation matrix amplitude of the quadratic transfer function mean wave period towing speed transport of wave energy velocity static or mean- waterline on the hull of the body force or moment component in the k-th mode
added mass coefficient in the k-th mode due to motion in the j-th mode damping coefficient in the k-th mode due to motion in the j-th mode spring coefficient in the k-th mode due to motion in the j-th mode line element of the waterline 2.718 (constant of natural logarithm) acceleration of gravity first order impulse response function second order impulse response function water depth imaginary unit frequency-independent added mass coefficient in the k-th mode due to motion in the j-th mode area of spectrum first moment of spectrum ordinate along base line with origin underneath CG (forward positive) normal vector, pointing outside the body generalized direction cosine hydrodynamic pressure dimensionless yaw velocity projected length between location of wave reference and CG time velocity component in x^-direction amplitude of an oscillatory quantity velocity component in X2-direction laminar flow velocity displacement in the j-th direction
first order angular motion vector angle between wind and current direction of bow hawser drift angle logarithmic decrement phase angle between wave and some oscillatory quantity u random phase angle of i-th frequency component free surface elevation significant wave height dynamic viscosity ratio between two time scales wave number wave length low frequency natural frequency in surge direction kinematic viscosity specific density sea water complex coefficient source strength root-mean-square value laminar shear force time shift total velocity potential steady velocity potential oscillatory velocity potential angle of direction wave frequency vector operator volume of the mean submerged part of the body Laplace operator
subscripts
a amplitude c current D wave drift force
viscous part dyn dynamic current load contribution e encounter
external force components due to wind and current G centre of gravity h point on the hull H hydrodynamic reaction i integer number k,j direction or degree of freedom m restoring due to mooring system n integer
normal directed N number of oscillations p potential origin stat steady current load contribution SW still water t tangential directed T thrusters r relative w wind C wave
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SUMMARY
It is the intention of this thesis to formulate a simulation model, which can be used to compute the behaviour of a tanker moored to a single point.
Exposed to current, wind and long crested irregular waves the motions of the tanker and the forces in the mooring sytem consist of both high (= wave) frequency and low frequency components.
When computing the low frequency motions, difficulties arise in the description of the mean and second order wave drift forces and the low frequency hydrodynamic reactive forces. For survival and operational weather conditions the wave drift excitation and the hydrodynamic reactive forces are discussed in this thesis.
For the survival condition the computer model is restricted to co-linearly directed current, wind and long crested irregular waves. The water is. assumed relatively deep. For most of the mooring systems this condition will determine the design. Experiments on model scale showed that the magnitude of the amplitudes of the low frequency surge motions will be influenced by the low frequency velocity dependency of the wave drift force excitation. The velocity dependency is caused by the Doppler effect on the vessel in a wave field. To account for this effect use is made of the velocity potential for small values of forward speed of the tanker. For small values of forward speed the first order motions were solved. By means of the direct integration method the low velocity dependent second order wave drift forces in regular waves were computed. For the simulation the velocity dependent wave drift force is split up into a current velocity dependent wave drift force and a wave drift damping coefficient. The complete matrix of the wave drift force is approximated employing the main diagonal only.
In the low frequency range the wave drift damping and the wave radiated damping can be derived from potential theory. The wave radiated damping
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is negligibly small. Except for the damping forces of potential origin also damping forces of viscous nature are present. The viscous damping terms cannot be computed and have to be derived from physical experiments.
By means of computations and verified by model tests the importance of the velocity dependency of the wave drift excitation has been confirmed.
In operational conditions the combination of current, wind and irregular long crested waves can be arbitrary in terms of occurrence and directions. In order to formulate the simulation model for the low frequency motions of the tanker in the horizontal plane two problems must be solved. First, a set of equations of motion must be drawn up, which describe the large amplitude low frequency motions. Secondly, the components in the equations of motion, which adequately describe the low frequency hydrodynamic resistance forces, must be derived while the low frequency viscous resistance coefficients must be known.
The flow pattern around the tanker, which performs low frequency oscillations, will be different in still water or in a current field. For both conditions semi-theoretical mathematical models were derived. The viscous resistance coefficients were experimentally determined for a 200 kTDW tanker in a water depth of 82.5 m. The derived results were compared with existing formulations of the viscous resistance.
The equations of motion are evaluated by means of results of physical model tests for a 200 kTDW tanker moored by means of a bow hawser to a fixed mooring point. To determine the stability of this kind of mooring system the stability criterium of Routh has been applied to the tanker exposed to wind and current.
In order to demonstrate the validity of the derived formulations with respect to arbitrary weather direclons in operational conditions computer simulations were carried out. It is shown that the model tests confirm the results of the computations.
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SAMENVATTING
De bedoeling van dit proefschrift is een simulatiemodel te formuleren dat gebruikt kan worden voor het berekenen van het gedrag van een tanker afgemeerd aan een een-punts afmeersysteem. Blootgesteld aan stroom, wind en onregelmatige golven bestaan de bewegingen van de tanker en de krachten in het afmeersysteem zowel uit hoog-frequente (in de frequenties van de golven) als laag-frequente componenten. De grootte van de laag-frequente bewegingen zijn in vele gevallen bepalend voor het ontwerp van het afmeersysteem.
Bij het berekenen van de laag-frequente bewegingen ontstaan problemen bij het beschrijven van de gemiddelde en laag-frequente tweede-orde golfdriftkrachten en de hydrodynamische reaktiekrachten. De hydrodyna-mische excitatie- en reaktiekrachten worden behandeld voor de tanker in weerscondities tijdens het overleven en het operationeel zijn.
Voor de overlevingsconditie beperkt het model zich tot storm omstandigheden, waarbij de richtingen van stroom, wind en onregelmatige lang-kammige golven samenvallen. De waterdiepte is relatief groot. Voor de meeste afmeersystemen bepaalt deze weersconditie het ontwerp. Experimenten op modelschaal toonden aan dat de grootte van de amplitudes van de laag-frequente schrikbeweglngen van het afgemeerde schip in belangrijke mate beïnvloed wordt door snelheidsafhankelijke golfdriftkrachten. De snelheidsafhankelijkheid wordt veroorzaakt door het Doppler-effeet van het schip in het golfveld. Om dit effect mee te nemen is gebruik gemaakt van de snelheidspotentiaal voor lage voorwaartse snelheden. Voor kleine waarden van voorwaartse snelheid zijn de eerste orde scheepsbewegingen opgelost. Door toepassing van de direkte-druk integratiemethode over het natte oppervlak kunnen de snelheidsafhankelijke tweede orde golfdrift-krachten worden bepaald. Voor het simulatiemodel wordt de snelheidsafhankelijkheid verdeeld in een stroomsnelheids-afhankelijke golfdrift-kracht en in een golfdriftdempingscoefficient. De waarden van de complete matrix van de golfdriftkrachten worden benaderd met behulp van de waarden, die berekend zijn voor de hoofddiagonaal.
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In het laag-frequente gebied kunnen golfdriftdemping en golfdemplng uit de potentiaaltheorie afgeleid worden. De golfdemplng, die onstaat door de uitgestraalde golven als gevolg van het laag-frequente bewegen van het schip, blijkt verwaarloosbaar klein te zijn. Behalve dempingskrachten van potentiaaloorsprong zijn ook weerstandskrachten van visceuze oorsprong aanwezig. De visceuze krachten kunnen niet berekend worden en zijn experimenteel bepaald.
Met behulp van berekeningen, die geverifieerd worden met modelproeven, wordt de belangrijkheid van de snelheidsafhankelijkheid van de golfdrif tkrachten in hoge golven bevestigd.
Tijdens operationele weeromstandigheden kunnen stroom, wind en onregelmatige langkanunige golven zowel in voorkomen als in richting willekeurig zijn. Bij het formuleren van de berekeningswijze van de laag-frequente bewegingen in het horizontale vlak van een afgemeerde tanker treden twee problemen op. Allereerst moet een stelsel bewegingsvergelijkingen opgesteld worden voor de grote bewegingen in het horizontale vlak. Ten tweede moeten de componenten van de visceuze weerstandskrachten in de bewegingsvergelijkingen voldoende beschreven worden en moeten de waarden van de viskeuze weerstandscoefficienten bekend zijn.
Het stromingsbeeld rond een laag-frequent bewegende tanker zal verschillend zijn in stil water of stroom. Voor de componenten van de viskeuze weerstandskrachten in de bewegingsvergelijkingen zijn voor beide omstandigheden formuleringen afgeleid. De formuleringen zijn gebaseerd op theorie en empirie. Voor een 200 kTDW tanker afgemeerd in 82.5 m diep water zijn de visceuze weerstandscoefficienten experimenteel bepaald. De resultaten zijn vergeleken met bestaande formuleringen.
De bewegingsvergelijkingen zijn geëvalueerd met behulp van modelproeven. De validatie betrof een 200 kTDW tanker afgemeerd met een meerlijn, die bevestigd is aan een in de zeebodem geplaatse paal. Onder invloed van de weerscondities kunnen dergelijke systemen instabiel zijn. Instabiele systemen kunnen grote krachten in de meerlijn induceren. Met behulp van
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het criterium van Routh is de stabiliteit van het systeem bepaald onder invloed van wind en stroom.
Met het doel de geldigheid aan te tonen van de afgeleide formuleringen zijn tenslotte computersimulaties uitgevoerd met de afgemeerde tanker. Voor de operationele weerscondities zijn verschillende combinaties van stroom, wind en onregelmatige golven toegepast. Er is aangetoond dat de resultaten van de modelproeven in overeenstemming zijn met de theoretische berekeningen.
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