A SIMULATION MODEL FOR A SPM TANKER.PDF

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


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)\

L-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 theorien 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.


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'quipement 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


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


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


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


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.\"

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 [111 ], [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.

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/ /

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_mtion_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*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) = 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) = - 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 ^.(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) = x1}(t) + x2)(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) (2.2.9) 00

r {(M +m )x(1)+ 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) = xX)(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 ) = x2)+XL(t) (2.2.14)

;4. r,,^lK, JV\ = vW, S ((H +a ( ) ) x i ; + b M i 1 J + c X 1 J | = X ^ ' ( t j = l , 3 , 5 J J J J J J J

(2.2.15)

and

_ 2 {(Mkj+akj(oo))x^bkj(Oi..+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=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

a^co.O) j lj(-.il(2)) = a1;]((,0) + l i , xj

x^

, . C ) \ ^ b (),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) x

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)

x^

in which a^co.O), bj.((,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) + c n.x(^ = x { ^ V .0,0) + (Ij xl +

34

ax^feW.o.o) m -jj- iJz;+X(t) (2.3.5) x^

and

n * a (u,0) Z { M x 1 ) + (a .O) + ^ .x)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 x)

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.cos)t

Relative to the slowly oscillating vessel this regular wave can be written as:

C(t) = Ca cos(wet + K Xl) (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^>((e) 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.,

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

* m tfs2/m m m m m

m

m m

"2 ID'

tfs2/m ts2/m tfs2 ts2 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 bH^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

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(5Ci),o,o)

+ 77(2) *!

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 "

^

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 - B 1 1 2 *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 U 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

-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.

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(lHi

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

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()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 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 (Kh)

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 t 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 )Q the frequencies 002 will shift to smaller values by the term

^ 3 . yS^ ^

V c

+z3

0 3 S

r^ ' ' \^/

! x i 'aO,

"0= U J3

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 = ^ + inffeh)

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

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_25EH2i22_2_lS_i2_Yi2i.Z_SE2nt waY_Ei_E5

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:

((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 (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 = 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) = -Pt(x(j),t;) or P(x(j),t) = p0(x(j),t) + xp^xCj)^) (2.6.2.3)

in which:

P0((J).t) = " PaT * 0 (- ( j ) , t )

Pl(x(J),t) --pgl^K^.t) -pi._rTy0((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 = - 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

=

- 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

" iwehki = 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.2Wd_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 are taken at the mean position of the point. The material derivative, D/Dt, results in a /t, and a con-vective term -U./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 = - / 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'' 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)) +

-^p|v$|2 n dS - -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

O\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

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_ comonents_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 + ) (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((>.-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-u

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