Simplified methods applied to nonlinear motion of spar platforms

NTNU Trondheim

Norgesiteknisk-naiurvilmiskapel ige

universitet

Doktor ingenioravhandling 2000:101

hist iUiI I for marl hydrodynam ikk

MTA-rappori 2000-141

cn

at-Simplified methods applied to

nonlinear motion of spar platforms

A thesis submitted in partial fulfillment of the

requirements for the degree of

Doktor Ingenior

by

Herbjorn Alf Haslum

Trondheim, 2000

DEPARTMENT OF MARINE HYDRODYNAMICS FACULTY OF MARINE TECHNOLOGY

Abstract

Simplified methods for prediction of motion response of spar platforms are presented. The

methods are based on first and second order potential theory. Nonlinear drag loads and

the effect of the pumping motion in a moon-pool are also considered.

Large amplitude pitch motions coupled to extreme amplitude heave motions may arise when spar platforms are exposed to long period swell. The phenomenon is investigated theoretically and explained as a Mathieu instability. It is caused by nonlinear coupling

effects between heave, surge, and pitch.

It is shown that for a critical wave period, the envelope of the heave motion makes the pitch motion unstable. For the same wave period, a higher order pitch/heave coupling excites resonant heave response. This mutual interaction largely amplifies both the pitch and the heave response. As a result, the pitch/heave instability revealed in this work is

more critical than the previously well known Mathieu's instability in pitch which occurs if

the wave period (or the natural heave period) is half the natural pitch period.

The Mathieu instability is demonstrated both by numerical simulations with a newly

de-veloped calculation tool and in model experiments.

In order to learn more about the conditions for this instability to occur and also how it

may be controlled, different damping configurations (heave damping disks and pitch/surge

damping fins) are evaluated both in model experiments and by numerical simulations.

With increased drag damping, larger wave amplitudes and more time are needed to trigger the instability. The pitch/heave instability is a low probability of occurrence phenomenon.

Extreme wave periods are needed for the instability to be triggered, about 20 seconds for a typical 200m draft spar. However, it may be important to consider the phenomenon in

design since the pitch/heave instability is very critical.

It is also seen that when classical spas platforms (constant cylindrical cross section and about 200m draft) are exposed to irregular seastates with long wave periods, linearly

iv

excited large amplitude heave resonance may occur. This is relevant for design seastates West of Shetland arid in the Northern North Sea.

Acknowledgments

This work has been carried out under supervision of Professor Odd M. Fabtinsen at the

Department of Marine Hydrodynamics, the Norwegian University of Science and

Technol-ogy. I'm grateful for his important contributions arid time for discussions inspite of his tight schedule. With his wide experience in hydrodynamics he has saved me a lot of time

by guiding me away from dead end roads.

I am also grateful for interesting discussions with my colleagues arid friends at Unme and

at the Departments of Marine Hydrodynamics and Marine Structures, NTNU.

This work was made possible by the financial support of my employer lime Oil and Gas. Thanks to I.E. Namork for making that possible. Funding was also provided by the

Re-search Council of Norway through the DEEPER programme.

During my year in France, I was at Technip Geoproduction assigned to making a paramet-ric motion study of a deep draft platform. To do that by 'conventional numeparamet-rical' programs

would have been a painstaking data job. That forced me to develop simplified programs.

I am thankful to P.A. Thomas because without that background, I might not have been on

vi

Abstract

Acknowledgments

Nomenclature xi

1

Introduction

1.1 Background and motivation 1

1.2 Previous work 2

1.3 Important effects 3

1.4 Outline of the thesis 3

2

Linear Frequency Domain Method

2.1 The hydrodynamic problem 5

2.2 Hydrodynamic forces based on long wavelength theory 6

2.2.1 Heave excitation force

2.2.2 Heave added mass 7

2.2.3 Horizontal excitation forces 8

2.2.4 Horizontal added mass 9

2.3 Horizontal excitation forces based on McCamy & Fuchs theory 10

2.4 Restoring forces 12

2.4.1 Hydrostatic restoring forces 12

2.4.2 Restoring forces due to a mooring system 12

2.5 Damping effects 13

2.6 Linear Transfer Functions of Motion 14

2.6.1 Motion Behaviour 16

2.6.2 Influence of mooring forces on RAOs 17

2.6.3 Eigenvalue problem and influence from mooring 19

2.7 Spectral analysis 20

2.8 Alternative Hull shapes 22

vii

Contents

. . .

viii CONTENTS

2.8.1 Increased heave damping . .

2.8.2 Increased natural heave period . ,

2.8.3 Reduced heave excitation forces .. .

23 24 24

2,9

Chapter summary ... .

Improved Wave Frequency Response Model 29

3.1 Coupled Heave/Moonpool response ...

3.1.1 Equations of motion

30

30

3.1.2 Linearization of quadratic drag forces 35

3.1.3 Verifications of linearized results . . . , 39

3.2 Horizontal drag forces - crossflow principle ....

,,.

3.3 Generation of time series of irregular sea . . . 43

3.3.1 Straight forward summation . . . y .1 amV le U. a 1. a Imm 43

3.3.2 Using the Fast Fourier Transform . 44

3.4 Time stepping 45

3:5 Time domain results . . . . 45

3.5.1 Verification of results . . . um. .. ma *. imm - 45

3.5.2, Time domain difficulties 46

3.5.3 Parameter sensitivity analysis of coupled heave/moon-pool response 50

3)6 Discussion on element discretization and heave motion 55

3.6.1 Stochastic linearization and response amplitude dependent drag

co-efficient 56

3.7 Chapter summary ., 58

4 Second order slow drift response

4.1 Perturbation theory 60

4.2 Second order difference frequency loads

...

4.2.1

Newman's approximation ...

, ,

4.2.2 WAMIT study - isolation of different effects .

61

62

4.3 4.4

Simplified Second order Heave QTF . . .

Simplified Second order Surge QTF ..

63,

68

4.5 Simplified Second order Pitch QTF . 72.

4.6 Slow Drift Response Calculations 73

4.6.1 Generation of time series of difference frequency forces 73

4.6.2 Wave drift damping 74

4.6.3 Simplified second order frequency domain response; 76

4.7 Chapter summary 76

5 Mathieu unstable heave/pitch response

5.1 Mathieu instability in heave, simplest case 77

5.2 Coupled heave/pitch instability - theory 79

5.2.1 Pitch equation of motion . 79

5.2.2. Heave motion envelope . ,82

MN

CONTENTS ix

5.2.3 Non-linear heave excitation, due to surge and pitch 83

5.2.4 Mutual amplifying pitch/heave interaction 84

5.2.5 Other Mathieu unstable wave periods 85

5.3 Model test results vs Numerical results 86

5.4 Horizontal quadratic drag forces 92

5.4.1 Choosing drag coefficients (CD) 92

5.5 Parametric sensitivity analysis of instability 97

5.5.1 Variation of wave periods and wave amplitudes 97

5.5.2 Effect of irregular sea 100

5.5.3 Variation of drag coefficients CDs and CD: 103

5.5.4 Model test results of drag variation 105

5.6 Comments on the drag damping 107

5.7 Controlling the instability 108

5.8 Period doubling 109

5.9 Chapter summary 109

6

Conclusions and recommendations for further work

A Drag calculations

A.1 Empirical estimates of the drag coefficients KC dependence 121

A.2 Vertical drag force on strakes 124

A.3 Horizontal drag on rnodel-test-strakes 126

B Model experiments

WI Experimental setup 127

B.1.1 Damping devices 129

B.2 Free decay tests 131

B.2.1 Heave drag (CD,) estimated from decay tests 131

B.2.2 Pitch drag (CD,) estimated from decay tests 132

B.3 Numerical decay tests using KC-dependent drag model 138

.. . . . . . . . . . . . . . . .

....

General

Symbols are generally defined where they appear in the text for the first time. Symbols for vectors and matrices are written in boldface.

Overdots signify differentiation with respect to time.

Subscripts a denotes the amplitude value of an oscillating function.

Subscripts i generally denotes direction i.e. motion (or velocity, acceleration, force)

in generalized force direction i. Here i = 1, 2,3 denotes x-, y- and z-direction re-spectively and i = 4,5,6 denotes the moment components about the same axis. For a ship, these i directions (or degrees of freedom) are denoted (1=Surge, 2=Sway, 3=Heave, 4=Roll, 5=Pitch, 6=Yaw).

Roman symbols

Added mass matrix element ij

Linear damping matrix element ii (i.e damping coefficient in DOF i). Restoring force matrix element

CD Drag coefficient.

CT? Drag coefficient at KC > oo.

Diameter of platform. Radius of platform. Draft of platform. Excitation force in DOF Acceleration of gravity.

/55 The platform's moment of inertia in Pitch (about y-axis).

Rgyr The platform's gyration radius of inertia.

xi

Nomenclature

xi i CONTENTS

KC Keulegan-Carpenter number (= 27r4ii, see 5.4.1 P. 92).

Re Reynolds number (= see 5.4.1 p. 92.

Time variable.

1' Oscillation period.

TN,i Eigenperiod in DOF

Tp Spectral peak period.

Hs Significant wave height.

Wave number. (=

Imaginary unit.

Normal vector (Positive direction defined into the fluid). ni Component of normal vector in direction i.

Greek symbols

Wave amplitude.

A Wave length.

Linear damping ratio in DOF i (= B..

7r The constant 3.14159...

Mass density of fluid.

Standard deviation of response in DOF

Velocity potential.

Frequency of oscillation (=

WN,ti Undamped eigenfrequency in DOF

77i Motion response of platform in DOF

77MP Heave response of water in moon-pool.

Kinematic viscosity.

V Submerged volume.

Abbreviations

LWL Long Wave Length approximation.

VVF Wave Frequency.

LF Low Frequency (associated with difference frequencies).

2D Two dimensional.

3D Three dimensional.

DOF Degree Of Freedom (1=-Surge, 2=Sway, 3=Heave, 4= Roll, 5=Pitch, 6=Yaw). RAO Response Amplitude Operator (linear transfer-function).

QTF Quadratic Transfer Function (second order force). VCG Vertical center of Gravity.

9),,

Li

Background and motivation

The deep draft spar platform has been considered as a competitive alternative for deep-water field developments. The spar platform concept in general has been thoroughly

de-scribed in the literature, see for instance Sante!' arid deWerk (1976), Glanville et al. (1991),

and Halkyard and Horton (1996). So far, spar production platforms have only been

in-stalled in the Gulf of Mexico, but the concept has several times been proposed for fields in

more hostile environment such as the Northern North Sea and West of Shetland, see for

instance Converse and Bridges (1996).

By a 'classical spar' production platform is here meant a large vertical circular cylinder with constant cross section and with a draft of approximately 200 meter, see Figure 1.1. The idea behind this concept, or rather what is justifying the use of this enormous hull is that due to the large draft, the motion response of the platform should be adequately low to permit installation of rigid risers with dry wellheads. Therefore, the motion response (in particular heave and pitch) is crucial for the concept.

Motion response optimization of 'dry wellhead' platforms is an important issue. Extensive

computational tools are often used in the design process as 'black box' analysis tools. Since physical understanding is important in design, it is desirable with supplementary simplified computational methods. Therefore several new, smaller arid simpler computer programs have been developed in this work. Sensitivity of different hull configurations

is easily verified without extensive input files. In addition, simplified calculation tools

may be tailor-made to solve special problems such as the Mathieu instability (occurring at difference frequencies) which is revealed in this work.

CHAPTER 1

Introduction

=Fa

_Rt.

I.

mop1,C1

ION

Chapter I. Introduction

Figure 1.1: Principle sketch of a classical spar platform with strokes, outboard profile left and inboard profile right. (Illustration from Del Norske Veritas).

1.2

Previous work

Many authors have presented methods for numerical prediction of motion response of spar platforms. The majority of these methods are based on time domain calculations and they

predict uncoupled motion response (i.e. dynamic effects of mooring lines and risers are neglected). Some of the authors solve the scattered first- and second- order diffraction potential while others use a slender body approximation i.e. they assume that the wave

field is virtually undisturbed by the structure and that a combination of the incoming wave

potential and Morisons equation (Morison et al. 1950) can be used to calculate the wave

loads on the spar:

Emrnerhoff and Sclavounos (1996) presented analytical first and second order diffraction solutions for vertical floating cylinders. The diffraction problem was solved in the frequency

domain and the motion response of the platform was solved in the time domain.

Ran et al. (1995) used a higher order boundary element method to solve thesecond order

1.3. Important effects 3

compared to experiments.

Mekha et al. (1995) presented a time domain calculation method using Morisons equation

to estimate the loads on the spar. They also included different nonlinear modifications to

Morisons equation in order to account for diffraction effects.

Weggel (1997) used a nonlinear frequency domain diffraction analysis to identify important nonlinear effects, and he presented empirical (curve fitting) results.

Cao and Zhang (1996) used a Morison model combined with a hybrid wave model consid-ering second order wave kinematics.

Chitrapu et al. (1998) have studied motion response of a spar platform using a Morison model and time domain calculations. They also give a review of different studies on spar

platform response.

Dern (1972) studied the stability of the motions of a spar buoy both theoretically and

experimentally. It was shown that when the incoming wave period is half the pitch (or

roll) period, unstable pitch (or roll) motions could occur.

Huse (1992) analyzed the spar motion response due to current fluctuations both numerically and by experiments.

Lately, some authors have presented so-called coupled analysis methods where slender

elements (such as mooring lines and risers) are also modelled, see Colby et al. (2000) and

Ma et al. (2000). It is stated that dynamic effects of risers and mooring lines tend to be

important for large water-depths.

1.3

Important effects

The design philosophy behind deep draft floaters in general implies that the draft is ade-quately large to reduce first order heave excitation sufficiently. As a result, second order

difference frequency excitation can be an important contributor to the total heave response.

In this work, first order and second order difference frequency excitation in surge, heave

and pitch are considered.

In a survival condition, wind is an important contributor to the total surge and pitch response. However, such aspects are believed to be more important when designing mooring system. In this study only wave arid current effects are considered.

1.4

Outline of the thesis

Basic theory is in general not described in this thesis, but reference is made to textbooks. However, in cases where basic theory is needed for the physical understanding of a

4 Chapter 1. Introduction

In this work, the main focus has been on identifying important physical effects for the mo-tion response of spar platforms and trying to describe these effects by simplified theoretical

models. It therefore felt natural to divide the thesis into chapters containing the different

effects. Each chapter contains theory, description of model, and discussion of results.

Chapter 2 presents a simplified linear frequency domain method for prediction of wave

frequency surge, heave, and pitch response of spar platforms.

In Chapter 3, the simplified calculation method is improved to include nonlinear drag clamping and the effect of a moon-pool. The heave response of the platform and the pumping heave motion of the fluid in the moon-pool are coupled. The coupled dynamic response is solved in the frequency domain by a linearization technique and it is also simulated in the time domain.

Chapter 4 deals with second order difference frequency excitation and slow drift response. Simplified methods to estimate difference frequency excitation in heave, surge, and pitch are presented.

Chapter 5 describes the newly discovered unstable heave/pitch response of spar platforms

theoretically. A numerical calculation method is also presented, and results from this

nu-merical method are compared to results from model experiments. A parameter sensitivity

analysis (variation of drag damping parameters and wave description) is also carried out in order to better understand when the instability occurs and how it may be controlled.

Chapter 5 is considered as the main part of this thesis. However, physical effects described

in Chapters 2, 3, and 4 are also important for the instability phenomenon. Parts of this

CHAPTER 2

Linear Frequency Domain Method

In order to learn more about the motion behaviour of spar platforms and also to see which effects are important for the motion response, a simplified calculation method is developed. This simplified calculation method is based on linear potential theory and the superposition

principle, i.e behaviour in irregular sea is modeled by linearly superposing results from

regular waves. Hydrodynamically, it is therefore sufficient to analyse a spar platform exposed to regular sinusoidal waves. This simplified method is described in Faltinsen

(1990).

2.1

The hydrodynamic problem

Assuming linear damping, the linear equations of motion for surge, heave, and pitch can be

solved in the frequency domain. The damping represents non-potential flow effects. Due

to symmetry, the waves can be assumed to propagate along the positive x-axis with no roll,

sway, and yaw -response of the spar. The heave equation of motion is un-coupled while pitch and surge are coupled. The wave elevation and the velocity potential of incoming

waves respectively may be written:

(ag

k-=

It is assumed that the wavelength is much longer than the diameter of the spar (A > 5D). A consequence of this long wavelength assumption is that no waves are generated by the hull. Then the diffraction problem may be solved in a simplified manner. The excitation

forces are obtained from the incoming wave potential and using analytical expressions for

5

6 Chapter 21 Linear Frequency Domain Method

the added mass. The goodness of this long-wave-length approximation is evaluated by

comparing results with a numerical panel method program. No internal flow effects in the

rnoonpool are considered (i.e. the spar bottom is closed). The equations to solve are the coupled surge/pitch equations of motion;

and the heave equation of motion;

[ 7,, [ Fj(t) 1 F5(t)

(2.2)

(Al A33)i)3 -I- B3377/3 + (1337/3 = F3.(t) (2.3)

but before the equations can be solved, all the coefficients (A0,B15,Cij, and Fi) have to

be determined. These coefficients are representing hydrodynamic forces and determining these coefficients, "the hydrodynamic problem", can be divided into two sub-problems:

"The diffraction problem": The forces and moments on the body when the body is fixed and there are incoming regular waves. These hydrodynamic forces are again divided into the Froude-Kryloff forces (pressure forces and moments due to

undis-turbed fluid flow) and the diffraction forces (pressure forces occurring since the body changes the pressure field by its presence in the water). Ft = FFK 4-FDIF i = 1,3, 5.

"The radiation problem": The forces and moments on the body when, the body is forced to oscillate and there are no incident waves. These hydrodynamic toads are

identified as added mass, damping, and restoring terms. (i113, B13, C1 j = 1, 3, 5). Note that due to the long wavelength assumption, there is no radiation damping, since

it is assumed that no waves are generated by the hulk Consequently Bo consist of non potential flow effects only.

These are basic principles in hydrodynamics and are described in text books,. see Faltinsen (1990).

2.2

Hydrodynamic forces' based on long wavelength

theory

'The heave excitation force is obtained by integrating the dynamic pressure over the wetted

hull surface. The pressure is found by using Bernoulli's equation. Formally theexcitation,

force can be written:

=

f Ptornids =pt°

s

n =< n1, n2, n3 > is the vector normal to the body surface defined to be positive into

the fluid. But as previously mentioned, a simplified method based on a long wavelength assumption will be applied.

[M

2.2. Hydrodynamic forces based on long wavelength theory

2.2.1

Heave excitation force

The Froude-Kryloff heave force is obtained by integrating the undisturbed fluid pressure from the incoming wave potential over the bottom of the spar. The diffraction force is

obtained in a simplified manner as previously described.

Due to the long wavelength assumption, the diffraction term may be simplified and the integral over the wetted surface can be replaced by the quantities at the center of the spar

(x = 0). Due to the normal vector of the body surface, only the bottom surface of the spar

contributes to the heave force (see Figure 2.1).

F3

f

S,..,.../

_...-'

FK

= (a(pg Au, w2A33)e-k1 sin(cot) (2.5)

The first term on the left side is the Froude-Kryloff force, while the second term is an

approximation for the diffraction force. For a spar platform, the Froude-Kryloff term is an order of magnitude larger than the diffraction term, due to the low added mass. Therefore, a spar platform does not take advantage of the heave cancellation effect which is important for sernisubmersibles and for tension leg platforms.

The heave cancellation occurs when the two counteracting terms in Equation (2.5) are equal. It will later be seen that the heave added mass for a spar can be estimated A33

:171- pr3. Substituting this A33 value and Au, = 7rr2 into Equation (2.5), the cancellation

period is found:

(Pg At, w2A33) <=> Tcanceltation =

3g 8r

(2.6)

For a typical spar platform with diameter around 40m, this heave excitation cancellation

occurs for wave periods around 7 sec. For such low wave periods, the heave response is very limited. Hence, the cancellation effect is not taken benefit of. However, it should be noted that such low wave periods are outside the range of validity for the long wavelength theory. Heave excitation forces and cancellation effects are discussed in more detail in Section 2.8.

2.2.2

Heave added mass

The heave added mass A33 appears both in the expression for the excitation force. Equa-tion (2.5), and as a mass term in the equaEqua-tion of moEqua-tion, EquaEqua-tion (2.3). In order to solve the equation of heave motion, it is necessary to estimate the added mass A. Es-timating the heave added mass is complicated without help of numerical methods. New-man (1985) calculated numerically the axial added mass for a semi infinite cylinder to be

A33 = 2.064pr3. Numerical calculations by WAMIT for a 228m draft spar with radius=21 II gave A33 = 2.01/Yr3 . This indicates that for a typical spar the free surface effects have small

8 Chapter 2. Linear Frequency Domain Method Zstrip Z d fstrzp n,= 0 n3=-1

Figure 2.1: Horizontal and vertical excitation forces.

influence on the heave added mass i.e. the added mass is basically an end effect. It may be noted that 2.064pr3 is fairly close to the displaced mass of the semisphere 7rpr.3 = 2.09pr3. The added mass is low compared to the total mass of the spar, and has therefore a relatively small effect. The differences in the added mass estimates above have a negligible effect on the heave response. A33 = :17pT3 is used in most of the calculations presented here.

When the moon-pool is taken into account, it is assumed that the added mass is reduced further. However, the response of the water in the moon-pool may affect the added mass

estimate (see Section 3.1.1). In the linear analysis here, it is assumed that the spar bottom

is closed (no moon-pool). The effect of the moon-pool is investigated in more detail in

Section 3.1.

2.2.3

Horizontal excitation forces

The excitation forces and total added mass for lateral motions are estimated using strip

theory and the two-dimensional added mass for a cylinder in infinite fluid.

For a two dimensional cylinder section in infinite fluid, the excitation force can be written

(2D)

f strip= (A inrr2)a. The first term is the diffraction force and the second term is the

Froude-Kryloff force. RID) is the two dimensional added mass of the section, r =radius of the cylinder, and a is the undisturbed water particle acceleration. This is awell known

result which is discussed in many textbooks, see Faltinsen (1990, p. 59). Note that this

expression for the force corresponds to the inertia term in Morison's equationwith inertia coefficient Cm = 2.

2.2. Hydrodynamic forces based on long wavelength theory 9

The total surge and pitch excitation forces are obtained by integrating the unit-length force on each horizontal strip along the wetted hull surface. The pitch excitation moment

is taken about VCG (i.e. Z strip = [z ± + BO])', see Figure 2.1:

ro F1 = j_d i str tpdz = 2pg7rr2(1a eke')cos(wt) (2.7) fo

F5 =

2pg7r2([(d

BC+

Here, BC is the distance from the center of buoyancy (B) to the vertical center of gravity

(VCG).

2.2.4

Horizontal added mass

The added mass coefficients i,j = 1,5 are determined by considering forced surge

and pitch oscillations of the spar, see Figure 2.2. Under combined surge/pitch oscillations,

every strip along the hull has the acceleration astrip = 1i -I-Zstrip1i5. Z ,trip is again the

vertical distance from the strip to the vertical center of gravity, VCG. Water can not penetrate the spar hull, so when the strip is accelerated by astrip, a pressure field is set up on the hull's surface to displace the water. The strip will "feel" a counteracting inertia

force, a5tripA(121D).

The global reaction forces due to the forced oscillations (FI,RAD arid F5,RAD) are obtained

by integrating the reaction force on each strip. The added mass coefficientsAk are then

found based on the definition of added mass:

Fk = 11kAi (2.9)

Fl RAD =, AD(2)alocalaz = Th- ( fnrr2 d) 715 (girr2 (IBC)

Ail 0 4(2D) F5,RAD '11 alocalZstrapd.Z

ia

( pir r d + 1nu2dBG2) (int-r2 dBG)

AG1

A55

(2.10)

Au,

10 Chapter 2. Linear Frequency Domain Method

Figure 2.2: Radiation problem, horizontal added mass.

2.3

Horizontal excitation forces based on McCamy &

Fuchs theory

The simplified "long-wavelength-theory" assumes no waves are generated by the spar. This

is not a good approximation for short waves, that is for wave length to spar diameter less than 5. For typical spars, this corresponds to a wave period of 11 sec.

Here, the simplified method will be modified to include diffraction effects in a simplified

manner. McCamy and Fuchs (1954) presented an analytical solution for the diffracted

wave potential from a bottom mounted cylinder in regular waves. The solutionis exact to first order. Here McCarny and Fuchs theory for infinite water depth is used to calculate horizontal excitation loads on the spar. The wave potential corresponds to the one for a bottom mounted cylinder with infinite length. However, the forces are only applied on the truncated cylinder (spar) i.e. the cross flow at the keel is neglected. Added mass and

damping coefficients are handled as in the simplified long-wave-length theory.

In this way, the scattered wave potential is partly taken into account. Partly, since in a consistent analysis not only the excitation force but also the hydrodynamic coefficients should depend on the scattered wave potential.

According to Mc Carny & Fuchs, the linear horizontal force per unit length of the cylinder in regular waves (wave profile as defined in Equation 2.1) may be written:

fstrip,McCF(Z) = 4pg(a eosh(k[z + hp

k cosh(kh) A(kr)cos(wt a) (2.11)

where

A(kr) = ([,11(kr)]2 + [n(kr)1)-1 (2.12)

and

1111

2.3. Horizontal excitation forces based on McCanty & Fuchs theory

I LWL [ - - - McC&F 2 z 1.5 0 u_ 2.5x 109 LWL - McC&F

Figure 2.3: Transfer functions for excitation forces from McCamy and Fuchs theory ('McCE4F) and from the simplified long wavelength theory (IWO, for a spar platform

with draft=202.5m, D=37.5m.

It is seen that the results based on the two methods coincide for long wavelengths while

they disagree for low wave periods. Asymptotically, for long wave periods, the horizontal force based on McCamy and Fuchs theory corresponds to the 'LWL' solution (which again

corresponds to the inertia force in Morison's equation with mass coefficient Cm = 2). It

is also seen from Figure 2.3 that the LWL surge force seems to "flatten out" for low wave periods while the pitch force is still increasing for low wave periods. The reason for this is

For infinite water depth, Equation (2.11) may be written:

f stripati ce F(Z) = 4P kg( a A(kr) ekz COS(Wt

-

Here, J1 is the Bessel function of the first kind and first order, and Y1 is the Bessel function

of the second kind and first order. J and means the derivative of these Bessel functions.

Surge and pitch excitation forces are obtained by integrating this force per unit length along the hull. The pitch excitation moment is taken about the VCG:

= fstrip,McCFdZ 4Pg("a A(kr)(1

-

k2

F5 =

_f

d

kd kd

4P9(a A(kr)-1[e-kd

-

BCk+

) - (1 - BCk - )]cos(wt)

k2 2

In Figure 2.3, the horizontal McCamy and Fuchs excitation forces from these expressions are compared to the excitation forces from the long-wavelength theory.

0.5 0.5

00 10 20 30 40 oo 10 20 30 40

T [sec] T [sec]

Fr

12 Chapter 2. Linear frequency Domain Method

that the pitch force's moment arm (about VCG) is increasing for low wave periods. Short waves are not penetrating to a large depth. Hence, for short waves, the horizontal force

is "attacking" close to the free surface i.e. far away from the VCG.) Since both the surge

force and the pitch force are obtained by integration of the same unit force along the hull, the pitch force will increase relatively to the surge force for short waves.

2.4

Restoring forces

2.4.1

Hydrostatic restoring forces

Since the spar is free floating, only hydrostatic terms are contributing to the restoring

matrices:

C11 = 0, C33 = pg.A, and C55 = pgVGM (2.17)

Here, Au, is the waterplane area, GM is the metacentric height, and V is the displaced

volume of the spar.

2.4.2

Restoring forces due to a mooring system

For traditional floating platforms and ships, the forces from the mooring system are

usu-ally more than an order of magnitude smaller than the linear excitation forces. A mooring

system is in such cases only used to keep the platform in position by counteracting

envi-ronmental mean forces (and not the linear wave frequency forces). The mooring system

has therefore usually a negligible effect on the linear wave frequency motion.

However, due to the large draft of a spar, the mooring forces could have a very large moment arm. This large moment arm could result in important mooringforce-moments. In order to investigate whether this effect has any influence on the linear WF response,the effect of a mooring system is included in the simplified frequency domain analysis.

Mooring systems may have many different configurations. Important parameters are the

geometrical shape, number of lines, material (chain, polyester, steel rope or combinations of these with buoys). pretension etc. However, in a linear global analysis all these different mooring system configurations may be described by 2 important parameters: initial vertical stiffness (kv), and initial horizontal stiffness (kH), see Figure 2.4. The pretension (Tv) may

affect the stiffness matrix by changing the metacentric height (GM). However, thiseffect

is by designers often considered as part of a load condition and it will therefore not be

included here. The effect of the mooring system is added to the hydrostatic restoring

terms. From Figure 2.4 it is seen that the mooring system may result in coupling terms between pitch and surge. The reaction forces due to the mooring system can be found by

considering a surge and a pitch displacement and by requiring moment equilibrium. The

restoring coefficients (Ckj) are then found from the definition of Cki:

.?. 5. Damping effects 13

2.5

Damping effects

In general, both generation of waves (radiation damping) and viscous forces (non potential flow effects) are contributing to the total damping of a floating body.

In the simplified analysis it is assumed that wave generation by the body is negligible, i.e. there is no radiation damping. This approximation is relevant for survival conditions (long wave periods). For shorter wave periods on the other hand, where radiation effects are more important, damping effects have a small influence on the linear wave frequency

response.

In the region around resonance, which is important in this study, the radiation damping is small. It is therefore assumed that the important damping effects are caused by viscous

forces on the platform hull, on mooring lines, on risers, and other appendices. It is believed

that these viscous drag forces have a quadratic behaviour. However, only linear damping forces will be included in this simplified linear frequency domain analysis. Non linear

damping effects will be considered later in this thesis. For simplicity, the linear damping

coefficients are here calculated as ratios of the critical damping = B I Bcriticat): kv

Figure 2.4: Principle sketch of the mooring system,.

&floor

F5mOOT

(qi +7/5zin)kii

B55 = OA55 155)C55 and B33 3211(A33 M)C33 (2.21)

=

Cllnaoor Cl5moor

= kHZ7n. R1 4,2 or, (2.20)

14 Chapter 2. Linear Frequency Domain Method

2.6

Linear Transfer Functions of Motion

When all the coefficients (A,j,Bij,Cij, and Fi) are established, the equations of motions are solved by assuming steady state solutions oscillating with the same frequency as the

excitation. The assumed solutions (ni = Thee') are substituted into the equations of

motion (2.2) and (2.3). The motion response amplitude is complex.

Motion transfer function or response amplitude operators (RAO) are defined as the fre-quency dependent (T = steady state motion response amplitude divided by the wave

elevation amplitude:

Phase angles describing the phase shift between the wave elevation, at x = 0, and the

motion response (see Figure 2.8) are defined as:

= arctan2(Imag {rh} , Real{:1jic/j}) (2.23)

LWL McC&F WAMIT

10 20 30 40

T [sec]

Figure 2.5: Surge and pitch motion transfer functions (RA Os) based on the simplified methods CLWL', and 'McC&F") compared with results from a panel method ('WA MIT').

There is no additional damping in surge and pitch.

In Figure 2.5 and 2.6 the motion transfer functions in surge, pitch, and heave and the respective phases are plotted for a typical free floating spar with draft=202.5m. The main particulars of the spar geometry used in this example is given in Table 2.1. The mo-tion transfer funcmo-tions based on the two simplified calculamo-tion methods (Long wavelength

approximation and McCamy and Fuchs theory) are compared to the results from the com-mercial panel method program WAMIT (1995). The agreement between the three methods

is good. 0.8 0.7 0.6 TO.5 70.4 cr) c.f.)0.3 02 0.1 oo 3.5 3 2.5

-E-t

RA0i(T) =

RA03(T) =

2.6. Linear Transfer Functions of Motion 15

Table 2.1: Main particulars of the two platform geometries used in most examples in this

work. 7 6 5 ff

I.

Figure 2.6: Left: Heave motion transfer function from the simplified method (1WL') com-pared with results from a panel method ('WA MIT'), for a 202.5m draft spar. In the

simpli-fied analysis ('LWL') the linear heave damping ratio (3 = 0.03. In the panel method solu-tion(' WAMIT'), the total damping is the sum of the linear heave damping ratio = 0.03

and potential (radiation) damping. The latter has a very small effect. In the simplified method A33 = :i7i-pr3 is used. The main particulars of the spar geometry used in this

example is given in Table 2.1. Right: Phase angles for 'LWL' RA Os.

draft (d) 202.5m 228m

diameter OD) 37.5m 42m

radius of gyration (R,,n,r) 80m 77.5m

vertical center of gravity (VCG) ref WL -97.25m -119m

metacentric height (GM) 4.4m 5.4m

oo 10 20

T (sec] ILWL

16 Chapter 2. Linear Frequency Domain Method

2.6.1

Motion Behaviour

It can be seen from the phases and the RAO's in Figure 2.5 and 2.6 that the coupled surge

and pitch motion are in phase, which means that both surge and pitch are contributing to displacements of the deck simultaneously. By combining the pitch and surge response at different z-levels, it is seen that for a typical spar platform, the center of pitch rotation (a point where the combined pitch/surge motion is zero) can be far below the center of

gravity. Actually, as will be seen later (in Figure 2.10) the center of wave frequency rotation

is a function of the wave period. For low wave periods, the platform rotates about a point between the center of gravity and the keel. For larger wave periods this center of rotation

is below the keel.

Motion transfer functions in pitch and roll are often represented as an angle divided by the wave height (i.e [rad/m] or [deg/m]). This way of representation angular motion has

practical reasons. A more physical representation is to divide the angular motion by wave slope. The wave slope is obtained by differentiating the wave elevation:

= k(a cos(wt kx) (2.24)

dx

The non-dimensional pitch RAO in such a dimension-less form is obtained by dividing the

RA05 from Equation (2.22) by the wave number (k), This non-dimensional pitch RAO is plotted in Figure 2.7. The pitch response is smaller than the wave slope for wave periods

up to around 40 sec.

In Figure 2.6, it is seen from the phase plot that the pitch response is 90 degrees out of

phase with the wave elevation for wave periods below the natural pitch period (i.e for all

wave periods containing significant energy). The wave elevation is again 90 degrees out of

phase with the wave slope. Hence, the max pitch amplitude is 180 degrees out of phase with the wave slope, see the phase diagram in Figure 2.7. This may at first glance seem

surprisingly. However, one should remember that the pitch motion is caused by 'horizontal

loads'. From the phase angles in Figure 2.6, it is also seen that for wave periodsbetween heave force cancellation (T 8sec) and heave resonance (T 30sec), the heave motion is 180 degrees out of phase with the heave elevation. The platform position at different wave positions is illustrated in Figure 2.9.

2.6. Linear Transfer Functions of Motion 17

Figure 2.7: Nondimensional pitch. 715,M AX

115

Figure 2.8: Phase diagram.

Figure 2.9: Motion behaviour for a typical spar platform. The pitch motion is 180 degrees

out of phase with the wave slope. For all relevant wave periods, the pitch motion is 180

degrees out of phase with the wave slope. For wave periods in a range from approximately 8 to 30 sec, the heave motion is 180 degrees out of phase with the wave elevation.

2.6.2

Influence of mooring forces on RAOs

The mooring system's effect on linear wave frequency motions is illustrated in Figure 2.10.

A very stiff mooring system with a large pretension is chosen in order to see the effect clearly. The mooring system used is a 16-point taut system with polyester lines and it has

the following physical parameters: k = 270[kNim], kh = 350[kN/m], T = 12000[k/V]. Two different vertical locations for the mooring system are evaluated (the Zn, from equation

2.19 is varied). For Zm= 0 the mooring force attacks at the level of the center of gravity

('mooring VCG') and for Zrn, = KG the mooring force attacks at the level of the keel

('mooring keel'). It is seen from Figure 2.10 that even this very stiff mooring system has a quite small effect on the WF motions. It tends to move the center of WF rotation

downwards for large wave periods.

40

10 20 30

T [sec]

18 Chapter 2. Linear Frequency Domain Method 0.8 0.7 0.8 f0.5 7;0.4 (7)0.3 0.2 0.1 20 15 00 5 00 no moor - moor 0 VCG moor 0 keel 10 20 30 40 T isecj no moor - moor VCG moor keel X10-3

3.5

7

Figure 2.10: Mooring system's influence on linear motions. Transfer functions (surge,

heave, and pitch) and pitch center of rotation: without mooring ('no moor), with fairleads

at vertical center of gravity ('moor t VCG') and with fairleads at the keel ('moor

The main particulars of the 202.5m draft spar is given in Table 2.1.

JOE 20 30 40 T (sec] moor no moor - moor 0 VCG 0 keel

2.6. Linear Transfer Functions of Motion 19

2.6.3

Eigenvalue problem and influence from mooring

When the equations of motion are established, the eigenvalue problem for free undamped

oscillations may be solved. The eigenvalues and eigenmodes are relevant for resonant response. Both the second order low frequency response and the Mathieu unstable pitch

response, which will be studied later, are resonance dominated response.

The heave motion is uncoupled from the other degrees of freedom. The natural heave

frequency is found directly:

C33

C33 - Lil2(M ± A33)= 0 <#.

W'3 = (M + .433) (2.25)

The eigenvalues for the coupled surge/pitch motion may be solved in a quite similar way, starting out with the general eigenvalue problem:

[C w2(M Anti; = 0 (2.26)

Here, C and (M+ A) are the stiffness and inertia matrices defined in Equation (2.2). This general problem may be transformed to the so-called special eigenvalue problem, see for

instance Thomson and Dahleh (1998):

[B u.)2I]x = 0

Here, B = L-1C(LT)-' where L is the Cholesky factorization of (M + A). Standard methods for solving the special eigenvalue problem (2.27) are described in many textbooks

(see for instance Kreyszig (1988)).

The solution of an eigenvalue problem consists of both eigenrnodes x and eigenvalues w2.

The eigenvalues gives the natural frequencies and the eigenrnodes may be used to obtain the center of rotation of the coupled pitch/surge motion.

(2.27)

Table 2.2: Mooring system's influence on the eigenvalue problem. Rotation center is where the combined pitch/surge is zero, and it is defined in [ral above VCG. The main particulars of the 202.5m, draft spar geometry used in this example is given in Table 2.1.

The mooring system's influence on the eigenvalues (natural periods) and eigenmodes

(ro-tation center) are evaluated using the same mooring system variations as were used in Section 2.6.2. The oscillations with the natural pitch frequency have a rotation center in the vicinity of the VCG, even for the case with a stiff mooring system at the keel. When

t he platform is free floating, there are no natural oscillations in surge.

no mooring 94.7 0 co - 29.4

mooring k_k VCG 94.7 0.006 227.4 -167.1 29.0

mooring Tit keel 80.1 0.2 268.8 -4.98 29.0

Mooring description TN,5 [sec] Rot. ctr. [m] TNJ [sec] Rot. ctr. [m] TN,3 [sec]

20, Chapter 2. Linear Frequency Domain Method'

2.7

Spectral analysis

When the linear motion transfer functions are established, the motion behaviour in long-crested irregular waves can be estimated by introducing a wave spectrum and by linearly superimposing the different frequency components. This is basic theory in offshore engi-neering and it is explained in textbooks and design guidelines, see for instance Faltinsen

(1990) and DriV (1991)i. The motion response spectrum is written:

Sresponse,i(w)' = IHi(w)12.9wave(w) (128)

Here, 11,(w) is a frequency representation of the motion transfer function i(RA0z). Many different wave spectra can be found in the literature. A recent review of various

spectrum formulations is given by Michel (1999).

in this work, the spectral response analysis is carried out using 3 different wave spectra:

Pierson Moskowitz, JONSWAP and a 2-peaked spectrum by Torsethaugen (Torsethaugen

1996). These spectra are combined with environmental data for the Northern North Sea, here taken from Haver, Gran, and Sagli (1998). The wave data for North Atlantic fields

such as West of Shetland are not much different.

If the wave elevation is a steady state Gaussian distributed process, the process has a narrow band spectrum, the maxima are statisticallyindependent, and the dynamic system is linear, then the motion response amplitudes may be predicted by the Rayleigh probability

distribution. The most probable largest response in degree of freedom i (Zz) in a seastate may then be estimated as:

Zz= az\l2 I n (i) 1(2.29)

Here, o is the standard deviation of the response (al =

seA,

Tz = 27rVrno/rn2 and ?Tin = focc Sresponsc,iw"dw N denotes the number of oscillations,

'is the zero up-crossing period of the response, and r is the duration of the seastate. For simplicity, a contour line approach is here followed instead of carrying out a long term statistical analysis. The idea behind the contour line approach is (Winterstein and Engebretsen 1998) that the contour line specifies all pairs of (Hs,Tp) that are candidates by virtue of their relative rareness to produce the 100-year level' of any load or response 'quantity. However, the short term statistics based on the contour line approach does not always realize the long term response statistics. In Haver, Gran, and Sagli (1998), the response statistics from the contour approach is compared to results from a full long-term statistical analysis. They studied different transfer functions and used wave data for the northern North Sea. It was shown that in many cases, the most probable maximum response (Z; in Equation 2.29) based on the 100-year contour curve underestimated the 100 year response from the long-term statistical analysis. For the contour lineapproach to reproduce the 100-year long-term response, higher fractiles (from the 80% fractile up to

about the 90% fractile -dependent on the shape of the transfer function) should be used.

"response or load with 100 year =turn period

2.7. Spectral analysis 21

The Heave RAO from Figure 2.6 with 1% of critical damping was used as input in a

long-term statistical program (Fames and Passano 1989) and combined with wave data for

the northern North Sea and the Pierson-Moskowitz wave spectrum. In accordance with Haver et al. (1998), the long term statistical results from this analysis indicated that the

contour approach underestimates the long term statistics with about 30 %. However, such

statistical aspects are not further studied in this work.

fin

0

05

10 15 20

Tp [sec]

Figure 2.11: Contour curve for 100-year

de-sign storms in the Northern North Sea. The

sea-state with largest significant wave height

is marked Hs10°max. Sea-states critical for

linearly excited heave resonance response of the classical spars are hatched.

100 7, BO z, 60 (T) 40 20 Torsethaugen ' JONSWAP 25 30 5 10 15 20 25 30 35 40 T [sec]

Figure 2.12: JONSWAP wave spectrum (7=1.05) and 2 peaked spectrum. Tp=20sec and Hs=14.3m for both spectra. The JON-SWAP spectrum contains considerably more energy at the heave resonance peak ("--40 sec) than the 2-peaked spectrum.

In order to illustrate the response, results for a classical spar (draft=228m, diameter=42m)

is presented in Figure 2.13. The spectral analysis is carried out for several seastates along the contour curve (Figure 2.11). For each seastate, the most probable maximum response amplitudes (Z) is estimated. This procedure is repeated for different damping levels and the results are represented as function of the peak period, see Figure 2.13.

It is seen that the damping has no influence on the linear surge and pitch response. The

reason for this is that damping effects are most important in the close vicinity of resonance. For both surge and pitch, wave frequencies are considerably larger than the resonance fre-quencies. The heave response in seastates with large peak periods (Tp) is much influenced by the damping level.

Estimates of a linearized heave damping using stochastic linearization gives values lower than 1% of critical damping for spar platforms without helical strakes and up to

approxi-mately 2% for platforms with strakes, see Section 3.6.1.

The spectral analysis results in large resonant heave motions for sea states with long peak periods. In this context, "long peak periods" means peak periods (Tp) greater than the one

2.8

Alternative Hull shapes

Classical spar platforms with cylindrical shape and constant cross section area may expe-rience excessive heave motions in sea states with long peak periods due to its low damping,

4 r - E

i

e-

Tp (sec) Tp [sec) Tp [sec]

Figure 2.13: Statistical results: Extreme values for a 230m draft spar,, assuming Rayleigh

distributed response amplitudes and 3 hours duration of the sea-state. Most probable max.

response amplitudes in surge, heave, and pitch (Z1, Zs, Z5) for different damping levels ,R) plotted versus peak period (Tv). The contour curve in Figure 2.11 combined with a

JONS WA P spectrum is used as input in the analysis. The seastate-duration is r=-3 hours. with the largest significant wave height, denoted 'Hs100max' in Figure 2.11. Unfortunately,

only this sea-state with 'Hs100max' has been used in many design cases. The peak period corresponding to this sea-state is not very high compared to the resonance period and consequently it does not result in critical linearly excited resonant heave response for a spar. This problem emphasizes the importance in design of using more than one sea-state

from the contour curve or even the complete frequency table of Hs and Tp.

Especially when the JONSWAP or Pierson-Moskowitz spectra are applied,, the resonant

heave response due to sea states with moderate wave heights and large peak periods become

more important. When using a 2-peaked wave spectrum, this heave resonanceproblem is

somewhat reduced since this 2-peaked spectrum contains less energy at periods close to

the heave resonance peak, see Figure 2.12.

A classical spar with 230m draft 'has larger wave frequency' heave response in sea-states

along the hatched line in Figure 2.11 than in the sea-state HslOOntax. This is the case both

for 1% and 2% of critical damping in heave and using both JONSWAP and the 2-peaked

;spectrum by Torsethaugen.

Whether these sea-states with peak periods up to 25sec are realistic or not is a subject for

discussion in other fora. More environmental data observations may be needed. However,

both governmental regulations and good design practice require use of the most, critical

&sea-state along the contour line.

22 Chapter 2. Linear Frequency Domain Method

10 7

2

2

2

2.8. Alternative Hull shapes 23

and relatively low natural heave period. The wave frequency heave resonant response may

be reduced by: (1) increasing the damping of the system, (2) increasing the natural heave

period out of the range of the wave energy, (3) further reducing the linear heave excitation forces.

A combination of all three possibilities is actually what is tried here. a

Figure 2.14: Alternative hull shapes with improved heave motion characteristics. The alter-native hull shapes b), c), d), e) and g) all have a counteracting effect in the heave excitation force (illustrated by arrows).

2.8.1

Increased heave damping

Increasing the damping of the system may be achieved by increasing the length of the helical strakes or by a disc at the keel, see Figure 2.14 hullshape b). Increasing the length of the strakes will result in larger environmental forces due to viscous excitation from current and waves. These forces have to be taken up by the mooring system. The disk at the keel has the advantage that the viscous excitation from waves is limited due to the exponential decay with depth. Using such disks to increase the heave damping is evaluated by model tests in scale (1:300). Two different disks were evaluated, one with diameter 1.32D and another with 1.1617 (here D is the cylinder diameter of the spar). Decay tests indicated that the linear heave damping was more than doubled for a spar with a damping disk with diameter 1.16D compared to a bare cylinder. Using the disk with diameter 1.32D, the linear heave damping was more than four times larger than in

the case of a bare cylinder. These model experiments are further described in Appendix B. Use of such discs to increase the heave damping of a spar is also investigated in Fischer and

Gopalkrishnan (1998), in Lake et al. (1999) and in Thiagarajan arid Troesch (1998). An important damping effect of the disk is reported by all these authors. The latter reported

that a thin disk will have more damping than a thick disk and also that separating the disk

and the cylinder (relevant for a truss-spar platform) considerably increases the damping and the added mass.

24 Chapter 2. Linear Frequency Domain Method

2.8.2

Increased natural heave period

The heave natural period increases by increasing the draft, the added mass and/or

decreas-ing the waterplane area. Disregarddecreas-ing the heave added mass, the natural heave period is proportional to the square root of the draft of the spar platform (see Equation 3.4). In-creasing the draft to increase the natural period in heave does not seem an efficient and economical way of increasing the natural period further. The added mass may be in-creased by applying a pontoon or a disc at the keel or by making the hull structure with

non constant cross section, see Figure 2.14.

2.8.3

Reduced heave excitation forces

The exciting force can be reduced by increasing the draft further or by changing the hullshape. When increasing the draft, the heave force is reduced due to the exponential

decay with depth effect alone, (ekz). When applying the alternative hullshapes (hullshape cg), a combination of two effects are reducing the heave excitation force.

Counteracting the Froude-Kryloff force with the diffraction force. Reducing the Froude-Kryloff force.

In Section 2.2, it was seen that in the long wavelength theory, the heave excitation force

consists of two counteracting components, namely the Froude-Kryloff force (FFK) and the

diffraction terms (FD,F). Semisubmersibles, TLPs and deep draft floaters with pontoons

have improved heave motion characteristics due to the column/pontoon interaction which has a cancellation effect on the heave excitation force. Briefly one may say that the

Fronde-Kryloff force counteract the diffraction force. Optimization of heave force cancellation

effects are further described in Haslum (1995). Classical production spar platforms, on the

other hand, for which the FFK-term is an order of magnitude larger than the FD/F-term,

do not benefit from this cancellation principle, see Figure 2.15.

A spar hullshape with heave force cancellation is not a new idea. For instance the Brent

Spar from the 1970s had a hullshape similar to 'hullshape d' in Figure 2.14.

An increase in the added mass (A33) increases the diffraction term and consequently reduces

the heave excitation force. Increasing the added mass by applying a thin disk at thekeel

('hullshape b' in Figure 2.14) would increase the diffraction term without affecting the

Froude-Kryloff term. (The Froude-Kryloff pressure is equal on both sides of the part

of the disk outside the cylinder). In Figure 2.15, the heave excitation forces for a bare cylinder and for a cylinder with a disk (1.3D) at the keel are compared. It is seen that the Fronde Kryloff forces are still dominating, arid that the disk has to be very large to have an important effect on the heave excitation force. From a practical construction point of

view, such large disks may be troublesome.

.1.

2.8. Alternative Hull shapes 25 a) 6 - FK 1 DI F 10 15 20 25 30 35 5 4 '6) o 3 u_ CZ2 x 10' FK DIF 00 5 10 15 20 25 30 35 T [sec] T [sec]

(a) Bare cylinder (hullshape a) (b) Cylinder with disk (hullshape b)

Figure 2.15: The effect on the heave excitation force of applying a disk at the keel on a

spar with draft=228m and radius=21m. Note that the Froude Kryloff (FK) and Diffraction (DIF) components are 180 degrees out of phase. Consequently, the resulting potential theory heave force is represented by the difference between the two components. In a) is used the

half sphere added mass estimate A33 = 1rpr3Pe 1.99 107[kg]. In b)433 2.7 107[14. Using WA MIT, it is seen that this corresponds to a disk with diameter Ddi,k R".: 1.3D Defining the direction of the normal vector to the hull surface to be positive into the fluid,

then the alternative hull shapes presented have a submerged surface above the bottom with normal vector pointing upwards. Briefly one may say that the dynamic Froude-Kryloff pressure force acting on this upward pointing surface to some extent counteracts the exciting force at the bottom of the spar and that this effect reduces the total heave

ex-citation force. See arrows in Figure 2.14. Similar to semisubmersibles, the heave exex-citation

may have a cancellation if properly tuned (as for 'hullshape e' in Figure 2.16).

In Figure 2.16, the effect of increasing the draft is compared to the effect of changing the hullshape. It is seen that counteracting the heave force by an alternative hullshape seems

much more efficient than just increasing the draft. Even an increase in draft from 200m to

260m on the classical spar has a smaller effect than tapering of the diameter in the upper

part close to the water surface. From Figure 2.16, it is also seen that for low wave periods,

the heave excitation force for 'hullshape d' is considerably larger than for 'hullshape a'. This may result in more motion response in an operational condition. However, since the

dynamical transfer function is very limited for these low wave periods, this increased force has a small effect.

Due to the exponential decay with depth of the wave effects (wave particle acceleration

and dynamic pressure), the same counteracting effect may be achieved by a small surface

24-0

U.

26 Chapter 2. Linear Frequency Domain Method

o

0

Figure 2.16: The effect of increasing the draft vs counteracting the heave excitation. The draft of the ordinary spar hullshape 'a' is increased from 200m to 230m and to 260m and the heave excitation forces on these are compared to the one for an alternative hullshape

'e' with 200m draft. The platform dimensions in the figure are given in meters.

close to the water surface or by a larger surface at a larger depth. This means 'hullshape d)' and 'hullshape e)' in Figure 2.14 are equivalent regarding linear heave excitation. For

spar production platform with a large topside weight and a large airgap, the metacentric

height becomes a limiting factor. In this respect, the 'hullshape e)' is more advantageous

than 'hullshape d)' since the center of buoyancy of the former is more elevated. However,

this up-pointing surface should by all means be placed adequately deep to avoid it from penetrating the free surface in waves. When the non constant cross section penetrates the water surface, the heave restoring force changes. This may, as will be shown later, cause

Mathieu instability in heave.

2.9

Chapter summary

A simplified calculation of linear motion response of spar platforms is presented. The

method is based on long-wavelength theory and the problem is solved in the time domain. For long wavelengths, the motion response of a typical spar platform based on this simpli-fied method is in good agreement with results from a commercial panel method program. The simplified calculation method has been improved by taking into accountthe scattering

of the wave potential when estimating the horizontal excitation forces i.e. by using a Mc

Gamy and Fuchs formulation. Using this improved force formulation, good agreement with

50 60 40

10 20 30

T [sec]

2.9. Chapter summary 27

the panel method was also obtained for the lower wave periods.

The effect of a mooring system on the linear motion response is investigated. It is seen

that even a very stiff mooring system has a small influence on the linear wave frequency

surge/pitch response.

A spectral analysis, using different spectrum formulations combined with environmental data for the Northern North Sea is carried out. It was seen that in seastates with long

wave periods, typical spar production platforms may be exposed to linearly excited heave

resonance with large amplitudes. Alternative hull shapes with improved heave motion

29

CHAPTER 3

Improved Wave Frequency Response

Model

In the linear frequency domain analysis, large resonant heave motions were observed when

a classical spar platform was exposed to design storms with large peak periods. In such

cases, the heave response is dominated by linearly excited resonance. Resonance response is controlled by damping effects. Physically, the linear damping assumption is not a good

approximation when large response amplitudes occur. Viscous drag effects dominate the damping and consequently the damping is quadratic.

Before we conclude that spar platforms may suffer linearly excited large amplitude resonant

heave when exposed to seastates with large periods, the effect of nonlinear drag forces should be investigated. We therefore need a more physical description of the damping

effects

The linear frequency domain method could be improved by introducing a linearization' technique (i.e. the linear damping coefficient is dependent on the response amplitudes), or the problem may be solved in the time domain where the nonlinear drag forces can be

included without modification.

In this chapter, both linearization techniques and nonlinear time domain simulations are used to estimate the motion response of the platform.

There are several reasons motivating the use of different methods to solve the same problem: In a sensitivity analysis where different physical phenomena and their relative contribution

to the total response are studied, it may be easier to see physical trends using a frequency domain method (frequency domain results do not vary from realization to realization).

30 Chapter 3. Improved Wave Frequency Response Model

However, frequency domain methods have limitations and they become complicated when several effects are included.

From a method developers point of view, the use of the time domain methods an almost

unlimited number of effects may be included (provided that one knows how to

mathemat-ically model these effects). A drawback, however, is that interpreting time series may be troublesome. In particular when dealing with extreme value statistics and a system with

low damping.

Using different methods to solve the same problem also serves as a verification of the

different methods. As will be seen, a harmonic linearization technique is verified by both

nonlinear time domain simulations with regular waves and by a linear frequency domain method. Time domain methods with irregular sea is verified by a stochastic linearization

technique and by the linear frequency domain method.

3.1

Coupled Heave/Moonpool response

Spar production platforms have a flooded centerwell, moon-pool, serving to locate the

risers. In motion response calculation methods for spar platforms, presented by other

authors, see for instance Mekha, Johnson, and Roesset (1995) arid Weggel (1997), the internal flow in the moon-pool is neglected and the spar is considered a closed cylinder. However, resonance oscillations of the water column in this moon-pool may occur. If this centerwell is modeled by a panel method computer program, assuming potential theory, the

vertical fluid motion in this centerwell is practically undamped and very large resonance amplitudes may occur. For a classical spar platform, the natural period of this water column is close to the platform's natural period in heave and the response of this water column in some cases destroys the panel program solution. In reality there are internal structures in the moon-pool; risers with buoyancy cans, riser guides etc, that will cause obstruction and flow separation. This implies nonlinear damping of the flow and also

damping of the heave motion of the platform.

It may be argued that the moon-pool is more or less closed at the keel and consequently the water in the moon-pool is forced to follow the platform response. However, if the centerwell is configured to let a blowout preventer or other large equipment to be lowered through it,

there must be a large passage in the full length of the ceriterwell. In such cases there are

reasons to believe that the resonance response of this water column could be important.

3.1.1

Equations of motion

Assuming one-dimensional flow and no flow separation effects, the Bernoulli's equation and

RI

3.1. Coupled Heave/Moonpool response 31

motion' of the fluid in the pool, denoted rimy. Considering an element of the moon-pool, see Figure 3.2, the continuity equation requires:

A(z)imp(z) = const (3. 1 )

It is assumed that the surface area of the rnoonpool is constant A(z) = Amp. The moonpool

velocity (imp) is then equal for all elements. Bernoulli's equation is differentiated with

regard to z:

Op

0'0

p + pgz

pat =

0z pg + pOzot = 0 (3.2)

The kinematic boundary condition of the free surface; limp = (2 is substituted into

equa-tion (3.2). Equaequa-tion (3.2) is then integrated from z = d to the instantaneous posiequa-tion = .rimp This results in the equation of motion for the moon-pool response (lbw p):

, 1 00,

P _{71/11Phazlz=d =} 0 (3.3)

It is important to integrate the Bernoulli's equation to the instantaneous free surface

elevation in the rnoonpool since this causes the restoring force on the water column. The

dynamic pressure from the wave at the bottom of the moonpool (keel) is the driving force.

The natural period of the water column's response is found from the equation of motion.

(3.3):

TN,mp =

similarly, the natural period of the heave motion ()lithe platform is:

\e/M +A33 TN,3 27t

C33

(3.4)

(3.5)

Assuming that the platform is free floating (C33 = pgrr2 Amp), and neglecting the added

mass (A33 = 0) then the natural period of the heave motion coincides with the natural period of the moonpool response (TN,3 = TN,Mp). One can therefore say that the heave added mass is keeping the two natural periods apart. In Section 2.2.4, the heave added mass for a vertical closed cylinder was estimated as A33 = irpr3. Using this estimate for the added mass, the ratio of the natural periods for a closed bottom spar and the natural

period of the pumping moon-pool response becomes:

Tiv,3 2r

= 1 + (3.6)

TN,MP

For a typical spar platform, there is only about 3% difference between these two natural periods. Introducing a mooring system would increase the stiffnessC33 and consequently

move these two natural periods even closer. There are therefore reasons to believe that a

resonance motion of the platform will affect the moon-pool response and vice versa. z

9

3d