Comparison of experimental and theoretical wave actions on floating and compliant offshore structures

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Comparison of experimental and theoretical wave

actions on floating and compliant offshore structures

C. OSTERGAARD and T. E. SCHELLIN

Germanischer Lloyd, Hauptverwaltung, Hamburg, Federal Republic of Germany

1. INTRODUCTION

In general, h y d r o d y n a m i c calculations are used to evaluate and assess h y d r o d y n a m i c aspects such as wave loading and m o t i o n behaviour of offshore structures. When applying calculations based on h y d r o d y n a m i c theories, it is essentml to be aware o f the validity and accuracy o f the predicted results.

Two aspects need to be considered in assessing the reliability o f any prediction. The first concerns accuracy o f numerical predictions with respect to solutions known to be accurate such as exact or closed form solutions of the boundary value problem as posed. Inaccuracies are caused by, for example, a specific numerical m e t h o d used. The second aspect deals with the intended applicability of the stated boundary value problem. Viscous damping, flow separation, or interference between neIghbouring bodies are possably important effects that may not be accounted for in the h y d r o d y n a m i c calculations. To assess the Impor- tance o f some o f these effects, comparisons with model test measurements m a y be used

Our purpose is to present h y d r o d y n a m i c calculations as applied to a variety o f offshore structures and to show the validity and accuracy o f predicted results b y comparison with closed form solutions or model test measurements

When dealing with offshore operations, one is generally concerned with two types o f structures that are funda- mentally different from the h y d r o d y n a m i c analysis point ot view. Firstly, there ale the so-called large volume structures such as barges and full bodied ships: but also semisubmer- sibles with thick columns and large volume lower hulls m a y belong to this t y p e o f structure We shall label them hydrodynamically compact structures. Secondly, there are structures that are made up of one or more cyhndrlcal piles such as some jack-up platforms, or structures that comprise a space flame o f thin cyhndrlcal members such as jacket platforms. Articulated loading columns as well as most semisubmersible drdlmg platforms can generally be designated to this category. We shall label them h y d r o d y n a m i c a l l y transparent structures.

The Important aspects to consider m the analysis o f h y d r o d y n a m i c a l l y compact structures are wave diffraction and radmtion For h y d r o d y n a m i c a l l y transparent structures, h y d r o d y n a m i c Ibrces are calculated using the two c o m p o n e n t Morison f o m m l a to account for h y d r o d y n a m i c inertia and viscosity. Both methods are briefly explained in an Introductory part o f Section 2 o f this paper.

Accepted December 1986 Dlscussmn closes December 1987

For ship-like (slender) structures, the characteristic that the longitudinal length scale IS substantially greater than beam and depth, is used to simplify the h y d r o d y n a m i c analysis b y suitable approxmaatlons based on the slenderness o f the body. The strip theory o f ship motions in waves, for example, is extensively used in naval architecture. Although its application is less in ocean engineering, it has been used to analyse semlsubmersible platforms and spar buoys.

When used properly, all methods o f analysis are generally welt stated to make reliable predictions o f h y d r o d y n a m i c properties that are linear. First-order forces and motions m waves, for example, can be calculated with sufficient accuracy to yield results for rehable decision making, allowmg the assessment o f h y d r o d y n a m i c performance o f structures m seaways that are generally encounteled In offshore opeiations. We shall present a number o f illustra- tive examples, comparing pledicted hnear h y d r o d y n a m i c propeltles either with closed form solutions or with measurements. We start with h y d r o d y n a m i c calculations o f a floating sphere, a barge as well as a ship, using the m e t h o d incorporating effects o f wave dltflactIon and radiation m three and two dimensions Next, a semlsubmersible drilling platform and two altlculated towers will be analysed, using the m e t h o d incorporating the Morlson formula. Finally. a floating structure will be treated that lends itself to be analysed using b o t h or eIthe~ the diffraction/radiation m e t h o d a n d / o r the Mollson fornmla.

There are rare situations in offshore operataons where nonlinear influences cannot be neglected in h y d r o d y n a m i c analysis of offshore structmes We shall plesent mtlo- ductory examples in order to illustrate some of these effects. The first example comprises the action ol high waves on a single vertical pile The other concerns so-called pmameter excited motions o f a floating storage tank There ale more relevant nonlinear phenomena to be observed when analysing mooring or berthing lolces, but this subject deserves a more systematic t l e a t m e n t than possible in this paper. It will be dealt with m a second paper to lolh)w

2. LINEAR HYDRODYNAMICS

Linear wave forces acting on h y d r o d y n a m i c a l l y compact structures or ship-like bodies are calculated by a so-called linear potential t h e o r y under the condition that the effect o f vortex shedding on total h y d r o d y n a m i c pressure is relatwely small and can be neglected The incident waves undergo a certain amount o f scattering (diffraction) at the structure, leadmg to a diffraction wave potential ab7 that may (In linear wave potential t h e o r y ) be directly super-

0141-1187/87/040192-22 $2.00

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Comparison of experimen tal and theoretical wave actions: C Ostergaard and T. E. Schellin

imposed on the incident wave potential qbo. In case the

structure can undergo motions in some or all o f its six degrees o f freedom (surge sl, sway s2, heave s3, roll s4, pitch Ss and yaw s6), waves are created which radiate from the structure. The related radiation wave potentials ep 1 to ep 6 in the otherwise undisturbed water are then superimposed on the incident and diffraction wave potentials in a phase correct manner to yield the total velocity potential ep o f the (linear) motion(s) o f the structure in waves. Each velocity potential must satisfy the Laplace equation in the fluid domain,

V2%=0

/ = o , 1 . . . 7

which is subject to the following boundary conditions. At the water surface z = 0 (small wave height approximation)

a % j / ~ t 2 + z . a % / ~ z = 0

where g is acceleration of gravity and t is time. This condition comprises a linearised free surface condition (the fluid must not penetrate the free surface) and a linearised dynamic condition (the pressure is constant everywhere at the free surface). At the sea b o t t o m we have

~ : / ~ z = 0 z = - d

and at the average position of surface o f the structure we have the body boundary conditions

0%/an

= n/ ] = 1 . . . 6 ~ePT/~n = --~q~o/~n

where d is the water depth and n the outward unit normal vector of the surface of the structure. The first body boundary condition reveals that we use radiation potentials eP 1 which are related to the local velocity o f the b o d y surface in the ]th degree o f freedom. The second body boundary condition, called radiation c o n d m o n , stmply states that the radiation wave particle velocities and the incident wave particle velocities are of equal magnitude but opposite direction at the surface of the body.

Under stationary conditions any of the wave potentials can be written as

% (x, y, z; t) = O] (x, y, z )" exp (--iwt}

For example, the linear incident wave potential as obtained from linear wave theory ~ can be written as

0o = gH/( 2w ).cosh ( k(z +

d)}/cosh

{ kd }

.exp

{ tkx }

where H is wave height, 6~ wave circular frequency, k wave number, i.e. k = 27r/X with X wave length. (We omit indica- tion o f the direction o f the incident waves in order to keep the mathematics as simple as possible.) The unknown velocity potential dp= ~.exp

(--twt}

is written as

6

dp= (~b0 +~bT)'exp(--i¢ot} + ~

¢l.~rl/~t

]=1

with unknown

~1,]

= 1 . . . . ,7, and rl, the response motions. It is possible to show that all solutions of ~j can be written

a s

¢1(x,Y,Z)=IfQl(~,v,~)'G(x,Y,Z,~,v,~)ds

(s)

which are integrals over the b o d y surface S, with

(2]

being unknown source strengths and G being the so-called Green function for the problem, i.e. G Is a known velocity poten-

tial at field point (x,y, z) o f a source at (~, v, ~') of unit strength which satxsfies the boundary conditions o f the fluid domain in the absence o f the body. z

The source strengths

Q]

are found by satisfying the b o d y boundary conditions, leading to two-dimensional integral equations of the Fredholm type which are solved by ap- proximating the underwater b o d y surface by a large number o f lateral elements (surface patches), i.e. systems o f linear equations replace the integral equations. With ~/ known, we use the linearised Bernoulli equation to obtain added mass and damping coefficients

(s)

Bkl = --Pco'Im{~I ¢i'nk dS }

(S)

The wave exciting forces

F],]

= 1, . . . , 6, are obtained from q~0 and q~7, again using the linearlsed Bernoulli equation to obtain the pressure and integrating the pressure over the b o d y surface

F]

= - - p ( i w ) . e x p ( - - i w t } "

I f lj'[~)O +

~b7]

ds

(S)

where

l]

is a generalised direction cosine. 3

Eventually, we use linear motion equations to calculate the response motions

rj,

satisfying the condition that all hydrodynamic forces add to zero (equilibrium). For further details see ref. 4.

In case our first condition, namely that effects of vortex shedding on the total hydrodynamic pressure can be neglected, is no longer valid, e.g. because of comparatively low diffraction wave potentials, the so-called Morlson formula can be used for many practical applications We explain tins briefly as follows.

The most familiar form o f the Morison formula pertains s to a vertmal, rigid cylinder in undisturbed surface waves, and assumes that the total hydrodynamxc force, dF, acting on a unit length dl o f the cylinder comprises two important components' an inertia force associated with the normal component o f the particle acceleration, a, and a drag force proportional to the square o f the normal component of the lnodent particle velocity, u,

dF/dl =

kM'a+ko'ulut

where parameters k D and kMare often given in terms of the dmaenslonless drag and inertia coefficients

Co

and

CM

In the particular case o f a ctrcular cylinder o f dmmeter D,

k o =

CD'p'D/2,

k M = C M ' p ' T r ' D 2 / 4 where p is the mass density o f the water.

In order to apply the Morlson formula to the analysis of hydrodynamically transparent structures, the underwater part of the structure is subdivided into separate elements comprising cylinders and small parts o f winch the added mass and viscous damping coefficient are known. The elements are chosen such that diameters of cylinders and principal dimensions of small parts are small enough for the relative motion principle to be apphcable. The hydrodynamic interactions between nelghbourmg elements is neglected. Total hydrodynamic response forces (added mass and damping) and total wave exciting force on the structure are found by summation over all elements.

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Comparison o f experimental and theorencal wave actions. C Ostergaard and T E Schelhn

The application o f the Morlson formula lequares that it be generalised to include mowng and reclined cylinders and small parts 6 We use the following forin to calculate the force d F on a cyhnder section o f unit length d l

d F / d l = km'awn + k A "aBn + kD'Urn [Urn[

where awn is the normal c o m p o n e n t of the wave induced water particle acceleration, aBn the normal c o m p o n e n t acceleration o f the cylinder section, Urn the normal component o f the relative velocity between the water and the cylinder section and Urn a diagonal matrix o f the components o f Urn. The use o f the normal velocity in the drag force term is due to the so-called cross-flow principle. 7 The paramete~ k A contains the dimensionless added mass coefficient C A F o r a circular cylinder section

k.l = CA'p'Tr'D2/4

The F r o u d e - K r y l o v force is e m b e d d e d in the wave- acceleration term and IS represented by an appropriately chosen value o f CM, the wave-acceleration (inertia) force coefficient Note that k M differs from k A because the fomaer term (containing CM) defines a force acting on a stationary b o d y in an accelerating flow with corresponding pressure gradient, while the latter term (containing CA) defines a force on an accelerating b o d y in a stationary fluid with no pressure gradient The inertia coefficient CM is assumed equal to one plus the added mass coefficient CA. For a deeply submerged circular cylinder, the ideal fluid values are given b y CA = 1 and CM = 2 .

We shall illustrate the evaluation o f the fluid force toe the general case o f a cyhndrical member The absolute velocity u B o f the centroId o f a cylinder section, resulting from rigid b o d y translational and rotational motions v and w, is given by

U B = V + W X r

where r IS the radius vector from the centre o f rotational m o t i o n to the centre o f the cyhndrical section The relative velocity between water and b o d y is

U r : U W - - I I B

where Uw, the wave-induced water particle velocity at the centrold o f the section, is calculated by differentiating the wave potentml ~ o :

T

uw = [~o/~Xi, O'bo/OX2, ~q%/~x3]

where (xi, x2, x3) define a right-handed coordinate system with indices corresponding to motions sj introduced above, e.g sl equals surge. A convenient definition o f the coordinate system is given in ref. 6.

We next define a unit tangent vector, et, directed along the cylinder axis The normal c o m p o n e n t o f the relative velocity IS then given by

U r n = e t X(U r x e t )

The total velocity-dependent force on the cylindrical member IS obtained b y integrating the velocity t e r m o f the force equation over its immersed length. Note that along cylinders piercing the water surface, integration is carried out up to the still water line always.

In order to obtain the acceleration terms of the force equation, the wave- and motion-induced accelerations at the centrold o f the cylinder section are required. The rigid b o d y acceleration, aa, is given b y

a B = Ov/Ot + ( O w / ~ t ) x r

where the first term lb the t~anslatlonal accelelatlon ot the centroid o f tire cylinder section and the aecond term ll/- volves the angular acceleration o f ttu~ section about a centroldal axis o f rotation. Note that the centripetal acceleration is considered comparatively small and neglected in linear analysis. The wave-reduced wate~ particle acceleration a w is evaluated from the wave potential ~0

Z [~2(I)0/8X 10t, 02 ePo/OXz Or, 02~o/3X3 3t] aw =

Smular to normal velocity components, the normal components of acceleration at the centrozd o f the cylinder section are given by

ayn = et x ( a g x et)

wheae Y refers either to the centered o f the cylinder section I Y = B) or to the water particle (Y = W).

Again, the total acceleration-dependent force on the cylindrical m e m b e r is obtained b y integrating the acceleration terms over the immersed length o f the cyhnder.

A practical simplification o f the Morison formula concerns hnearisation o f the velocity term. This requires that the quadratic relationship o f the drag force c o m p o n e n t be expressed in a hnearlsed form

CD'p'D/2"uIul +++ CLo'o'D/2"u

The equivalent linear drag coefficient COD is derived in terms o f the assumed nonlinear coefficient CD by equating energies dissipated per wave cycle for both the linear as well as the nonlinear case, resulting in the expression

6~D = (8/37r) CD'A(u)

where A ( u ) represents amplitude o f the velocity vector of the element For further details we refer to ref. 8.

It IS important to note that damping coefficients result trom the structure m o t i o n as caused by waves o f a certain amplitude. Because m o t i o n amplitudes must be known before the system o f m o t i o n equations as solved, an itera- tive process is necessaly to arrive at acceptable values ol lmearised damping coefficients, s

For small parts, the above details apply in principle as well We assume that each small part can be treated as a cyllnde~ section having distinct h y d r o d y n a m i c properties m all three translational dvections. Immersed cylinder end plane areas are treated as small parts having h y d r o d y n a m i c properties in the direction o f the cylinder axis

After th~s brief explanation o f the two basic methods o f linear analysis (potential theory and Monson formula), we proceed to practical examples m order to demonstrate their applicability

2.1. Hydrodynamically compact structures

The m e t h o d used toe the analysis o f h y d r o d y n a m i c a l l y compact structures IS based on linear potential theory, allowing the effects of h y d r o d y n a m i c interaction between nelghbourlng structural elements to be included m the analysis. The numerical procedure relies on a dlscretlsatIon o f the b o d y surface mto surface elements Both finite as well as infinite water depth can be considered. A c o m p u t e r program G L D R I F T of Germanischer Lloyd has been developed enabling the evaluation o f the linearlsed vessel motions ot and h y d r o d y n a m i c forces on arbitrarily shaped compact structures in regular w a v e s . 9 - 1 1

2.1,1. Floating hemzsphere. For the computations, the wetted surface o f the hemisphere is subdivided into a hum-

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Comparison o f experimental and theoretical wave actions: C Ostergaard and T. E. Schellin

ber of small surface elements. These elements or patches represent a distribution of sink or source singularities, each of which contributes to the potential of the flow surround- ing the body. The choice of the number of patches used for computations has a bearing on the quality of the results. Increasing the number o f patches generally improves the quality of results. The number of patches necessary for satisfactory predictions can only be determined by repeat- ing computations using more patches and comparing results. For this hemisphere, two sets of computations are made, one set with 56 patches and another with 264 patches. In both sets, triangular patches are used as they are well suited for curved surfaces. Figures 1 and 2 show the ldealisations of the hemisphere wtth 56 and 264 triangular patches, respectively.

Computations are carried out with waves approactung the body in the negative x-direction. Computed results of first-order hydrodynamic quantities are compared with closed form analytical results obtained from ref. 12, where an expression given in ref. 13 is used that is based on energy and momentum considerations. The results pertain to a free-floating hemisphere in water of infinite depth.

In Figs. 3 and 4, transfer functions of amplitudes of first-order wave exciting forces in surge and heave, F1 and F3, are compared. Computed values of phase angle e are also given although no analytical values are avmlable for comparison. (Positive phase angle means that the quantity under consideration reaches its maximum positive value before the crest of the undisturbed incident wave passes the centreline of the hemisphere.) Added masses and damp- mg forces in surge, all and b11, and heave, a33 and b 3 3 , a r e

given as coefficients in Figs. 5 and 6, and transfer functions of first-order surge and heave motions, s~ and s3, including

, \

/

t/

Fig. 1. Hemisphere with 56 triangular patches

WOve WOVe

EF 1

IFll

pgV'}4.Ec]

Fig. 3. 0 I I I I | 1 I _ • _ I _ A _ _ A _ _ A _ _ A _ _ • _ _ . . •--

-g~

l _ I - [ 1 I I I I

1.5

1.0

0.5

I I I I

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"i ,,

**" 2/*6 PANE LS (**

~K,,,~ •

56 PANELS

! I I I I

0

0.5

1.0

1.5

2.0 2 5

~ r First-order wave exciting force Ft on a sphere

EF 3

1 L

i

I

t

IF31

Tr. pg r2Eo

1.0

0.5

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! I I I C O M P U T E D I

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21.6 PANELS J

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/

t ~

A

L

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I I I I I /

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1.0

1.5

2.0

2.5

Fig. 4. First-order wave exciting force F3 on a sphere

1.00

, , , ,

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56 PANELS I

0.50

0.25

0 -

,

t

0

0.5

1.0

1.5

2.0

2.5

~ r

Fig. 2. Hemisphere with 264 triangular patches

Fig. 5. Added mass all and damping coefficient bll o f a sphere

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Comparison o1 expenmentat and theorettcal wave actions C Ostergaard and T A &'hellm

1.00 j

,

~ ,

C O M P U T E D

0.75 - ~x

• 21.6 P A N E L S

X ~

t

56 PA NE LS

__

-X~,,a - --. ANALYTICAL

, ~

.

Q50

0 .

2

5 ~

0 ~ I J l

0

0.5

1.0

1.5

2.0

F i g . & sphere 2 . 5

Added mass a33 and damping coefficient b33 o f a

l~s I

I s l l

Fig 7.

1 I__i_i_.i.i_~~__

I _ ----~II-

O !

I

i

I

1.5

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0.5

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l

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0.5

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1.5

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3~r Surge motion transfer Junction o f a sphere

phase angles, are to be found m Figs 7 and 8, respectively. All results m these figures are plotted against the dimensionless frequency parameter k" r, where k = 27r/X (with the wave length X) ts the wave number and r the radius of the sphere Results are always expressed in dimensionless form using quantities such as the wave amplitude, fa, the acceleration of gravity, g, the density o f water, p, and the d~splaced volume, V

From these results at Is seen that c o m p u t e d predictions of first-order quantities agree well with analytical results. Thas is true for both sets o f computations although heave motions near k . r = 1 0 compare shghtly more favourably for computations that are based on the larger number o f 264 patches for the ldealisation o f the wetted surface. In general, predicted results based on a subdlvasaon into only 56 triangular elements seem accurate enough for most purposes (This may not necessarily be so when using quadrilateral patches causing small 'leakages" to occur all over

the wetted surtace o f the hemisphere). J udglng trom e \ peii- ence m the use of this numerical method d" possible, the wetted surface ldeahsation should avoid shaip corne~s between adjacent patches This condition is nicely lul- filled with the present ldeahsatlon of the hemispherical surface.

F r o m the agreement obtained between numerical and analytical computations, it may be concluded that the use ot this numerical procedure for the analysis o f a floating hemisphere results m reliable paedlctions of flast-oldel h y d r o d y n a m i c forces and motions in waves. Next, we shall employ this m e t h o d to piedlct the m o t i o n behavtour of a rectangular barge, a structure that as very frequently used an offshore operations

2 1.2 Rectangular barge Experimental lesults ot hrst- order oscillatory m o n o n s o f a rectangular barge floating m regular waves are compared with c o m p u t e d results. The barge selected as replesentatlve o f a vessel frequently used by the offshore industry as a lay balge or crane vessel Mare particulars are given in Table 1, For computations, the wetted smface of the barge is subdlvaded into 156 quadrilateral elements as shown an Fig. 9.

Experimental measurements obtained from model tests p e r t b r m e d at the Nethellands Ship Model Basra, 14 are used

Es 3

Is31

Fig 8.

2 " '

0 i

- 7

_j-E1 I 1 J I I I I I - - A N A L Y T I C A L 2 0 1 0

I

0

05

1.0

15

20 ~r

Heave motion transfer funcnon oJ a sphere

2 5

Table i Partteulars of rectangular barge Length (m) Breadth (m) Draft (m) Displaced volume (m 3) C G above base (m) Transverse gyradms (m) Longitudinal gyradms (m) Vertical gyradms (m) Natural heave period (s) Natural pitch period (s) Natural roll period (s)

150 50 10 73 750 10 20 39 39 104 121 94

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Comparison o f experimental and theoretical wave actions" C. Ostergaard and T. E. Schellin

Z

Fig. 9. Rectangular barge idealisation

2 ~ Es I 0 12 10 I s l l Fig. 10. w a v e s 0 8 06 0 4 0 2 I I • I I I C O M P U T E D • M E A S U R E D 5 10 15 20 T l s ]

Barge surge motion transfer function in head

for comparison. Carried out at a model scale of 1 : 50, these model tests include runs m regular head and beam waves for a range of wave periods. The water depth corresponds to 50 m full scale.

Results of computations and measurements of first-order oscillatory motion amphtudes in head waves (surge sl, heave s3, and pitch Ss)are given in Figs. 10, 11 and 12, and in beam waves (sway s2, heave s3, and roll s4) in Figs. 13, 14 and 15, respectively.

All motions refer to the centre of gravity of the vessel. Results are presented as non-dimensional transfer functions o f motion amplitudes plotted against full-scale wave period

T. Phase angles e of motions are given m radians and are also plotted against full-scale wave period. (Pomtive phase angle mdtcates that the motion reaches its maximum positive value before the crest of the undisturbed incident wave passes the centre of gravity of the barge )

Comparison of measured and computed results shows that first-order motions are generally well predicted by the computations. Significant differences occur only m roll and sway motion amplitudes near the natural period of roll. These &fferences are mainly attributable to the fact that computations predict larger roll motions due to hnearity and the ommmon of viscous effects of roll damping in the computations. Since sway as coupled with roll, computed sway motions also differ somewhat from measurements.

Phase angles of first-order motions are also generally well pre&cted although at small motion amphtudes, &fferences of phase angles are somewhat larger. These larger &fferences may be due to the greater influence of errors m the measurements when motion amplitudes a r e l o w . 14

In summary, comparison between measurements and computations shows that linear hydrodynamics yield rehable predictions of first-order motions of a typxcal rectangular barge floating in waves

2.1.3. Ship-like structures. Many structures of interest in offshore hydrodynamics have one dxmenslon exceeding the others by about one order of magnitude, e.g. ships or

3T~ Es 3 2r~ "IZ 0 - - ] Z 10 I v I I I I I I Is3[ Fig. 11. waves 0.8 06 0.4 0.2 0 0 C O M P U T E D I 5 10 15 20 T[s]

Barge heave motion transfer function in head

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Cbmpartson o f experimental and theorettcal wave actions C Ostergaard and T k. Schellm E;s 5 2rt T~ --1T 1 2 i 1 8 l I I I 10 O8 0 6 04 02 Fig. 12. waves

COMPUTED

MEASURED

I 5 10 15 20 TIs] Barge pitch motion transfer function in head

ship-like structures. In the following we call such structures slender. Slenderness occurs for ocean platforms such as spar buoys, for some b o t t o m supported gravity structures of similar shape, for some articulated towers, and for semi- submersibles comprising long cylindrical twin-hulls.

In ship hydrodynamics, slender-body approximations based on the so-called strip theory are used routinely to predict ship motions in waves. The basic assumption of this theory is that the time-dependent flow around relatively thm vertical slices (strips) o f the ship's hull is two-dimensional, meaning that the longitudinal components of the flow are considered to be o f second order. Resulting forces and moments on these strips are Integrated over the length o f the hull to obtain total forces and moments.

There is no major difference between strip theory and the three-dunensional (3-D) theory when used for the den- vation o f hydrodynamic response forces and wave excita- tion forces in the equation of motion. Both approaches make use of a velocity potential based on the assumption o f an incompressible, inviscid and irrotatlonal fluid. However, a two-dimensional (2-D) analysis can also make use of conformal mapping techniques, which have been applied extensively in the past. Today, source-sink representations o f the body in the fluid are also used in 2-D theories. In the latter case the evaluation of this velocity potential generally requires an approximate numerical solution based on a spatial discretisation process of the wetted b o d y surface

into a flmte numbm o f h n e (2-D) or surla~e (3-D) elements with the source strength functmn constant over each element Aftra solving an integral equation vm a system or hnear equatmns fo~ the source-sink shengths to obtain tile diffraction and ladlatlon potentials, hneal superposmon with the incident wave potentml ylelda the resultant velocity potential o f the fluid flow (compme mtroductory part of Section 2) for the 2-D snip ot the body Knowing the velomty potential, the hydrodynmmc pressure on the immersed surface is determined using Bernoulh's equation and a hydrostatic component Forces and moments on the surface are then obtained by integrating the pressure distribution.

Strip theory has been tmplemented m the computm program GLSTRIP of Germanischer Lloyd In this program, using a calculation method described in ref 15, the two- dimensional potential flow is generated by a superposmon of two wave-radiating potentials with singularities at the coordinate origin on the water line and a large number of non-radiating, higher-order smgularmes (quadrupoles) located on the reside of the hull section m the vmInlty of the body contour This method is an Improvement of the method presented in refs 16 and 17 and is suitable fol arbitrary frequencies and for a variety of ship sections, that is, it IS not restricted to Lewis sections Boundary conditions imply the assumption of unhinlted water depth.

TE 0 E s 2

-IT

-2r~

12

10 O8 [ s 2 [ O6 0.4 Fig. 13. waves 0 2 I I I I 1 I I I I /

COMPUTED

/

i i I L L 0 5 10 15 20 T [s] Barge sway motion transfer function in beam

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Comparison of experimental and theoretical wave actions: C. Ostergaard and T. E. Schellin 1% I i Es 3 0

f

0 000 • I I

Is31

Fig. 14. w a v e s 1.4 12 10 08 06 0.4 02 I I - - C O M P U T E D • M E A S U R E D 0 5 10 15 20 T Is]

Barge heave motion transfer function in beam

Program GLSTRIP is used to evaluate five-degree-of- freedom first-order motions of a tanker in regular waves using strip theory. Therefore, the sample tanker ]s also analysed using the three-dimensional potential theory, treating at as a large volume, hydrodynamically compact structure in waves.

Experimental results of first-order oscillatory motions of a tanker floating in regular waves are compared with computed results. The tanker selected is representative of a vessel used for permanent storage of crude o11. Main particulars are given m Table 2.

Experimental measurements obtained from model tests described in refs. 14 and 18 are used for comparison. Carried out at a scale of 1:82.5 at the Netherlands Ship Model Basin, these model tests include runs in regular head and beam waves for a range of wave periods. The water depth corresponds to 82.5 m full scale. Rudder, propeller and bilge keels are omitted at the model.

For computations using the strip method, the tanker is subdivided into 20 equally long transverse sections. A body plan showing these sections is shown in Fig. 16.

For computations using the diffraction method, the underwater part o f the tanker is subdivided into a total of

208 surface elements (200 quadrilateral and eight triangular elements) as shown m Fig. 17. According to ref. 18, proper selection of surface elements is based on several consldera- tins. Firstly, element size must be small enough to ade- quately describe the curved surface of the hull. Clearly, a

2 ~ a I I E s4 Tt 0 10

[s~l

x~a 0 Fig. 15. w a v e s i I - - C O M P U T E D M E A S U R E D I 0 5 10 15 20 T t s ]

Barge roll motion transfer function in beam

Table 2. Particulars o f tanker

Length between perpendiculars (m) 310

Breadth (m) 47.17 Draft (m) 18.90 Displaced volume (m 3) 234 826 C.G abovebase (m) 13.32 Metacentric height (m) 5.78 Transverse gyradius (m) 14 77 Longitudmal gyradius (m) 77.47 Vertical gyradius (m) 79.30

Natural heave period (s) 11.80

Natural roll period (s) 14.20

Natural pitch period (s) 10 60

6 -1C

f

Fig. 16. Body plan o f tanker

?

5-10

(9)

Comparison o f expertmental and theoretwal wave actums C Ostergaard and 71 E Schellm

~ : ~" ¢ a 4 t ' ~ ' ~ S L - - ~ " ' ' - - " I I r

Fig. 1 Z Surface element Mealisanon of tanker

2T~ Es I TI~ 0 _.ff 1.2

\

"\

" \

r ' - -

- - -

"

I

I I I I I I Isll 10 0 8 06 Fig. 18. w a v e s 0/+ 02 0 - - - - C O M P U T E D I POTENTIAL) M E A S U R E D

/

%¢ I 5 10 15

/

/.

/

2O T[s]

Tanker surge motion transfer functton m head

large number of small elements increases the accuracy of results because small elements better describe the hull surface. Secondly, minimum acceptable ratio o f wave length to element size should not be less than five. Appreciable errors occur when this ratio is smaller. Since wave periods as low as 6s are of practical interest, selected element lengths should not be larger than 12 m for this tanker. Thirdly, large variations in element size should be avoided, and quadrilateral elements should be close to square-shaped and the shape of triangular elements close to equilateral triangles. Note that the subdivision of the underwater surface of this tanker hull fulfils these requirements quite well. However, a comparison with sectaonal shapes (Fig. 16)

qiows that consldelable sunphticatlons ,iic made Tile bilge ladnis is neglected completely

Results of computations and nleasuiements ol 111 st-Ol dei oaclltatmy motion anaplltudes in head waves (surge sb taeave s3 and pitch ss) me given m Flga. 18, 19 and 20 and in beam waves (sway s2, heave s3 and loll $4)111 blgs 2 1 , 2 2 and 23, respectively All motions refel to the centre ol gravity of the vessel. Results are presented as nero-dimensional transfer functions of motion amplitudes plotted against lull-scale wave period. (Positive phase angle means the motion leads the wave elevation )

Comparison of measured and computed results shows that flrst-order motions are generally welt predicted by the computations Significant differences occui only in roll and sway motion amplitudes Near the natural period of roll, these differences are mainly caused by the lact that con> putatlons piedlct laiger ~oll motions due to the omission of viscous effects of loll damping in the computations The computed sway motion curves in beam waves show humps occurring at the natulal roll period These humps ale due to coupling telms

Phase angles o f first-order motions are also generally well predicted although at small motion amplitudes, differences o f phase angles ale sonlewhat larger. These largei differences of phase angles may be due to the greater influence of measurement errors when motion amphtudes a~e l o w . 14

Comparison of numerical results obtained from strip theory and diffraction theory shows that both theoiles generally predict motmns of this tanker quite well. At large wave periods, however, differences occur They are due to

E s 3 2 ~ "lZ 10 I I I

,TG',

i 11 i I I I 08 06

Is31

0 4 0 2 Fig. 19. w a v e s 0 i 0 - - STRIP T H E O R Y I c o M P U T E D POTENTIAL M E A S U R E D / /

/,

A.._ I 5 I0 15 20 T [s]

Tanker heave motion transJer function m head

(10)

Comparison o f experimental and theoretical wave actions: C. Ostergaard and T. E. Schellin Es 5 2r~ 11: 0 -11; I I | I I I I I

1.0

08

0.6 I .1

0,4

0.2

I I I Fig. 20. w a v e s

- -

STRIP THEORY 1

POTENTIAL

COMPUTED

M E A S U R E D

0 L ~ I I I

0

5

10

15

20 T[s]

Tanker pitch motion transfer function in head

the fact that the influence of fimte water depth is not included in strip theory.

Both methods predict almost the same natural heave and pitch periods (see Figs. 19, 20 and 22). Predicted natural roll periods differ only 8% from each other (14.8 s using strip theory vs. 13.6 s using general 3-D theory), and both values are within 4% of the value given in the hst of particulars.

In summary, comparison between measurements and computations shows that both of the numerical methods used here yield reliable predictions of first-order motions of a typical tanker floating in waves. The 3-D potential theory is perhaps to be preferred, particularly at shallower water depths or in longer waves where the effect of finite water depths must be considered. In ref. 18 is shown that the influence of water depth on hydrodynamic response can be extremely important and that the frequency de- pendency of this response is obvious, especially in very shallow waters. In addition, due to the three-dimensional description of the velocity potential, end effects are accounted for. However, computations show that even for a full bodied ship such as a tanker, end effects seem small enough to not significantly influence the results. This may not be the case, of course, when analysing other bodies or ships with appreciable forward speed.

An important disadvantage when using 3-D diffraction theory is that a considerable effort is required to prepare input data and that relatively large computer storage capacity must be available for computations. Strip theory as used here, even with its relatwely complex description of the two-dimensional velocity potential, requires relatively little effort. (Note that strip theory cannot predict surge amplitudes, and these motions are, therefore, computed using the 3-D diffraction method only.)

2.2. Hydrodynamically transparent structures

The method used for the analysis of hydrodynamically transparent structures is based on the so-called Hooft method. I9 This method employs the Morison equation modified for relative flow past slender, cylindrical members (see introductory part of Section 2) The structure is divided Into cylinder sections and small parts, and the fluid forces are integrated to obtain total forces and moments on the structure.

Drag and added mass coefficients must be specified for the considered structural parts. Inema coefficients are

E

0

~s 2 _ - - T [ ~ - -211;

12

1 0 - 08

Is21

0.6

0.4

0.2

Fig. 21. waves ' J I l

I

• •

t -

I

• I

I

• I

I

I t I I -I

STRIP THEORYJ

POTENTIAL

f COMPUTED

• M E A S U R E D

I I J

/

/ ° 1 I 5 10

L

_i

_/t/

I

15

20 T[s]

Tanker sway motion transfer function in beam

(11)

Om, parison of experimental and theorencal wave actions C Ostergaard and T E. Schelhn Es 2 1"[ _ --T'~ - 2/* i i I I I I , , , | - - STRI P THEORY) POTENTIAL f COMPUTED MEASURED [ ~ 5 10 15 20 T[s] 2 0 16 12 08 0/* Fig. 22. w a v e s

Tanker heave motion transfer function m beam

assumed equal to one plus added mass coefficients Drag terms are hnearlsed using relative velocity. H y d r o d y n a m i c interaction between neighbourlng structural elements is neglected. Finite water d e p t h is accounted for in wave kine- matics. Linear t h e o r y xs used, and integration o f fluid forces acting on structural members is done up to the still water line. A computer program G L F M T H T o f German]scher Lloyd, based on this m e t h o d , has been developed enabhng the evaluataon of linear vessel motions o f and h y d r o d y n a m i c forces on h y d r o d y n a m i c a l l y transparent structures. 8'2°

2 2.1. Semi-submersible drilling platform RS-35. Experi- mental results of first-order oscillatory motions o f the semisubmersible drilling platform RS-35 floating m regular waves are compared with c o m p u t e d results. This semlsubmersible is characterlsed b y a structural configuration com- prxsmg an underwater ring hull and four shghtly slanted columns carrying the deck structure. Main particulars are given in Table 3.

A sketch o f the RS-35 is shown m Fig. 24.

For computataons, the underwater part o f the structure is dwtded into four slanted cylinders simulating the four columns and 12 small parts slmulatmg the ring hull. A d d e d mass coefficients for cylinders are specified according to the curves given b y Lee an his discussion to ref. 19. Added mass coefficients for small parts are set equal to 1.0. Cal- culations are performed with three different drag coefficients and with two different water depths.

Experimei]tal measulements obtamed I tom model test~ performed at the Techmcal Unwersity Berhn 2x are used f i , compallson These model tests have been caTHed out at the two model scales ot I 50 and 1 100 undel tile same con- dltlons m the wave tank, thus simulating the two full-scale water depths of 75 m and 150 m, lespectwely. Tests include ~uns in regulai head waves with periods trom 4 s to 27s lull scale.

Results o f computations and measurements ot fllst- ordei oscillatory motions ale gwen m Figs. 25-28 Surge motions, Sl, refer to a point at the upper deck above the

2 r t I I I E ~ I • •

['F

l J

I I I 2/. 20 1 6

Is, I

12 0 8 0/* Fig. 23. waves - - S T R I P T H E O R Y C O M P U T E D P O T E N T I A L M E A S U R E D I I

I

..fj

5 10

\

0 15 20 Tts]

Tanker roll motion transfer function in beam

Table 3. Particulars of semisubmersible RS-35

Outer torus diameter (m) 96

Ring hull diameter (m) 10

Column diameter (m) 12

Draft (operating) (m) 30

Displaced volume (m s) 31 220

C G. above base (m) 16.8

Natural heave period (s) 21.7

(12)

-

""°

1 ,,,I

Comparison o f experimental and theoretical wave actions: C Ostergaard and T. E. Schellin

D i m e n s i o n s in m

y

Fig. 24. Semisubmersible drilling platform RS-35

C.G., heave motions, s3, refer to the C.G. (pxtch mot:ons are designated Ss). Results are presented as transfer functions of motion amphtudes plotted against full-scale wave period, T. Phase angles of motions, e, are given in radlans and are also plotted against full-scale wave period. (Positive phase angle means that the motion leads to wave elevation.) Comparison of measured and computed results shows that surge and pitch motions (Figs. 25 and 28) are generally well predicted by computations. Experimental data at both model scales correspond well with theoretical predictions. No significant influence of water depth can be identi- fied. However, considerable scatter of measured values is noticeable. Note that changing the drag coefficients in the computations does not signifcantly affect calculated surge and pitch motions. The scatter of measured results is, therefore, not likely to be due to viscous effects.

Comparison of measured and computed heave motions shows good agreement in waves with periods less than 14 s. Experimental data at both model scales correspond well with theoretical predictions, no sigmflcant influence of water depth can be noticed, and computations with different drag coefficients and wave amplitudes result in heave motions that are practically the same, indicating that viscosity has no effect on heave motions within th:s range of wave periods.

However, this is not the case in longer waves with periods greater than 14 s. Computations with different drag coefficients (Fig. 26) and wave amplitudes (Fig. 27) result

T~ 2 E s 1 0 2 15 10 - I s l J 0 5 - Fig. 25. w a v e s I I I I 0 0 I I I o

MEASURED

(SCALE 1 100) 150 m WATER DEPTH Ea= 7 5 m

•-

.7//~"

• MEASURED (SCALE 1 50) o / . ~ 75m WATER DEPTH • o ~ - [~a = 375m • oO~/~¢" o

• ~ f

COMPUTED

/

- co:o

/ - CD--07 i l l A I I N -Y I I I 5 10 15 20 25 30 T [ s ]

RS-35 surge motion transfer function in head

0 E s 3 TC 2 -11 15

is3[

10 05 I I I I 1 I I I I I I 1 I I I I

I

• MEASURED(SCALE 1100} p.

COMPUTED

./'

CD= 071~=7.5m

C

D

= 1.0 / / I I 0 20 25 30 T l s ]

_J

5 10 15

Fig. 26. RS-35 heave motion transfer function in head waves (150 m depth)

(13)

Compartson o f experimental and theoretwal wave actions COster, eaard and T E Schellm Es 3 IT. I I I I I I I I I I 2 5 I I I

I] /

I • M E A S U R E D _i

(SCALE 1 50}

[

\

2 0

is31

C O M P U T E D

[CD=I O)

\

~a

~a = 1 m

_{/ " N _ _}

\ •

15 -

~a

=

5m

I /

\ \ k

-

~a = 10m

/

I,

~.o / I

05

1

0 - I I I I

0

5

10

15

20

25

30 T[s]

Fig. 27. RS-35 heave motion transfer funcUon in head waves (75 m depth)

m heave amplitudes that are markedly different from each other, indicating that viscous effects are, indeed, slgmficant m longer period waves. Although hnearlsed, viscous damping is, o f course, a function o f not only drag coefficient but also wave amplitude 8 Measurements m waves with different amplitudes are, therefore, suitable to demonstrate effects o f wscosity experimentally. Figures 26 and 27 show that measured heave amphtudes are not the same m waves with different amphtudes o f 7 5 m and 5.0 m full scale, respectively. However, this difference m a y not only be due to viscous effects. As the scale o f the m o d e l tests is not equal m these experimental runs, the corresponding (full scale) water depth is not the same, and differences in heave amplitudes are, therefore, also a result of different water depths.

The water d e p t h has an maportant influence on heave m o t i o n characteristics. The cancellation period is reduced and heave amphtudes are generally less in the vicinity of the cancellation period, whereas heave amplitudes are increased in the vicinity o f the natural heave period, s

Results in 7.5 m waves (Fig. 26) show that computations with the smallest drag coefficient o f CD = 0 4 compare

most tavourabl3~ w~th nleasmements m ~ a w s w~th pe~octs nero the natural heave pe~a)d Calculatmn~ with other C D values o f 0 7 and 1 0 ale c a l u e d out 111 o~del to demon- s h a t e the large influence o f tMs value on D e d l c t e d heave amphtudes m waves ol large period

Results m 5 0 m waves (Fig 27) show that computed pledlCtlons with 5.0 m amphtude waves do not compalc well with measurements m the wcimty o f the natural heave period. Predictions with the smaller 1.0 m amphtude waves compare more favourably. This is due to the choice of the drag coefficient o f C D = 1.0, which is too high As seen m Fig 26, a lowel &ag coefficient would lesult m bettel a g~ eement.

Not only drag coefficients but also added mass coeffi- cients have an effect on c o m p u t e d results Increasing added mass values generally leads to larger values o f heave amphtudes At the same time, however, the m a x i m u m ol the t~ansfer function for heave would occur at lalgel wave periods, 1 e. the natural heave period would increase. In om example, peak values o f calculated heave amphtudes occm at wave periods very close to the natural heave period, that is, the natural heave period is well predicted by conrputa- tlo ns.

In summary, comparison between measurements and computations show, firstly, that added mass coefficients are well chosen for this example and, secondly, that drag coefficients need to be specified with great care for the prediction o f heave motions, whiners for the prediction o f surge and pitch motions, the choice o f drag coefficients is o f only secondary importance Using cmefully specified hydro-

__K 2 Es 5 o I£ 2 15 10

Is5[

Fig. 28. waves I I I I I I l I I I 1

o MEASURED (SCALE 1"100)

150 m WATER DEPTH

(~a= 75m

• MEASURED (SCALE 150)

75m WATER DEPTH

~a

375m

05 CD=0L I

CD = 0 7 C O M P U T E D C D = I 0

5

10

15

20

25

30 T[s]

RS-35 ptteh motion transfer functton in head

(14)

Comparison o f experimental and theoretical wave actions: C. Ostergaard and T. E. Schellin

dynamic coefficients, this numerical m e t h o d o f analysis based on the Morison equation is quite reliable for predict- lng first-order m o t i o n s o f h y d r o d y n a m i c a l l y transparent structures in waves. We shall further d e m o n s t r a t e its useful- ness by analysing motions and loads o f two articulated towers.

2.2.2. TU-Berlin articulated tower. Experimental results of first-order (pitch) m o t i o n s and corresponding horizontal forces at the umversal j o i n t o f the TU-Berlin articulated tower in waves are compared with c o m p u t e d results. This tower consists o f a single ctrcular cylindrical pile connected to a b o t t o m foundation with a universal joint. Main par- tiulars o f this tower are given as full-scale values in Table 4

Table 4 Particulars of TU-Berhn articulated tower

16 12

191 ~o

8 4 l~g. 30. tion Total height (m) 201 4 Diameter (m) 20 0 20 Water depth (m) 154 5

Joint above bottom (m) 9.0

C.G. above joint (m) 51 0

Centre of buoyancy above joint (m) 58.5 16

Displacement (Mg) 43 471.8

I~1

Mass (Mg) 31 398.4

Moment of mertm (tm 2) 210 Xl0 ° ~o 12

Natural pitch period (s) 45.0

E w a v e _ i E tO f/f/\\~-'//1/<..'~1 (//-,'~7 z //\\~\////

Fig. 29. TU-Berlin articulated tower

~ 2 0 m ~7 I I I I I COMPUTED - - - A t = 0 5 0 s • - - A t = 0 1 5 s • MODEL TESTS

~ _ _ _ _ T _ _ --

, I

01 0 2 0 3 04 0 5 6 co [rcld Is] TU-Berlin articulated tower pitch transfer func-

#

I 01 i i i 1

COMPUTED

1 ---At=O50s

- - A t = 0 1 5 s • MODEL TESTS \ • 0 2 03 0 4 0 5 0 6 CO [reid Is] Fig. 31. TU-Berlin articulated tower horizontal force

transfer function

A sketch o f the tower m o d e l is given in Fig. 29 Experi- mental measurements are o b t a i n e d from model tests performed at the Technical University Berlin. 22 Carried out at a scale of 1 : 100, these tests include runs in regular waves with full-scale circular frequencies ranging from 0.12 s -1 to 0.57 s -1 (corresponding full-scale wave periods range from 11 s t o 54s).

Computations are carried out using the computer program GLTOWER o f Germanischer Lloyd that has been developed especially for the h y d r o d y n a m i c analysis o f articulated towers. Using the Morlson equation modified for relative flow, this program is based on the assumption that the tower structure comprises only thin cylindrical structural elements, 1.e that it is h y d r o d y n a m i c a l l y transparent. Computations are done in the time domain, simulating the tower behavlour by direct integration after specified suc- cessive Ume steps. Further details are found in refs. 23 and 24.

Results o f c o m p u t a t i o n s and measurements of angular tower deflection (pitch), 0, and horizontal force at the universal joint, Fx, are given in Figs. 30 and 31, respec- twely. Results are gwen as transfer functions plotted against wave frequency co. All results are gwen as full-scale values.

Two sets o f computaUons are done, one set with a specified time step A t = 0.5 s and another with A t = 0.15 s.

(15)

Compartson oJ experimental and theoretical wave acnons" C Ostergaard and T E. Schelhn

Comparison with measured results shows generally good agreement in waves with frequencies not in the vacmlty o f the natural patch period. Note that computed results using the smaller time step are always somewhat closer to measured values, indicating that a smaller time step leads to more rehable predictaons. In the vicinity ot the pitch natural period, agreement between measurements and computations is less favourable, however, agreement as good enough to be of paactical value because large measured values o f the transfer functmns are also predicted by computations. The natural pitch period is well predicted, indicating that specified added mass coefficients are well chosen.

F o r the computational analysis, the tower is adealised as a series o f five circular cyhndrical elements (strips). For each element, added mass and drag coefficients in the normal direction are assumed equal to 1 0 and 0.6, respectively.

Before the rehabdlty o f this numerical m e t h o d for use on articulated towers can be properly assessed, it is desir- able to demonstrate its use with a more reahstlc example o f an articulated tower with a more comphcated structural configuration. Such an example as represented by the articulated tower designed by Howaldtswerke-Deutsche Werft AG (HDW), Hamburg. This tower, intended to be used as a loading platform for LNG transfer to a tied-up tanker, will be treated as our next sample case

2 2 3. HDW articulated loading tower. Experimental

results o f first-order tower m o t i o n s and corresponding forces at the universal joint o f the HDW articulated loading tower in waves are compared with c o m p u t e d results. This tower is supported near the seabed by a universal joint The column structure, extending v e m c a l l y through the water surface, comprises a ballast tank, a steel lattice structure, a b u o y a n c y unit (also known as mare float), and a chtmney supporting a platform above the water surface Main particulars are given m Table 5. Mare dimensions are shown in the sketch o f the tower (Fig. 32). All particulars and dimensions refer to the full-scale structure.

Experimental measurements are obtained from model tests performed at the Hamburg Ship Model Basin (HSVA) The model has been built to a scale of 1 32.75 and Instru- mented to measure tower top motions and horizontal and vertical forces at the umversal joint Model tests include runs in regular waves at periods ranging from 6 s to 15 s (lull scale) and wave heights ranging f i o m 1.9 m at low peiiods to 5.6 m (full scale) at high periods Model tests include runs in regular waves o f two different heights. They Indicate that measured responses are reasonably linear with wave height. A more detailed description o f model tests is given in ref. 25.

For computations, the loading tower as ldealised as a series of 27 circular cylindrical elements (strips). Diameters o f elements are such as to s~mulate the tower's buoyancy

Table 5 Particulars of HDW articulated loadmg tower

Total height (m) 203 5

Water depth (m) 180.0

Joint above bottom (m) 12 0

C G above joint (m) 52 74

Centre of buoyancy above joint (m) 90.56

Displacement (Mg) 11 170 0

Mass (Mg) 11 170 0

E e ~ A II ~ II A ~16m Sectlon A-A 8 Sectlon B-B B

Fig. 32. HDW articulated loading tower

distribution, meaning that equivalent diameters are specified for those structural components that are not circular cylinders such as the lattice structure and ballast tank Three sets of computations are performed, each set with different h y d r o d y n a m i c coefficients for flow normal to the tower One set is based on constant values o f added mass and drag coefficients equal to one, a second set is based on coefficients selected from investigations dealing with cyl- Inders in oscillating flow, 26 and a thtrd set IS based on coefficients calculated using potential theory. All values o f specified coefficients for sets two and three are given in ref. 24. End plane areas o f cylindrical elements need to be treated with care In order to correctly account for hydrodynamic forces acting In the axial direction of the tower. H y d r o d y n a m i c coefficients for end plane areas are chosen from tabulated values. 17

Results o f computations and measurements o f tower top deflection, Xp, and corresponding horizontal and vertical

forces at the joint, F x and F z, are shown in Figs 33-35,

(16)

Comparison of experimental and theoretical wave actions: C Ostergaard and T. E. Schellin

Ix.l

Co

1 0 Fig. 33. motion I I I I I V A R I A B L E COEFFIcoMPUTE o / - - - - - - CONSTANT COEFF~ / • MEASURED / /

/

• ,"1" I I I I 5 10 15 20 25 30 T Is]

HDW articulated tower transfer function of top

IF, I

co 500 4 0 0 300 200 100 I I I I I .k

J

/

/ • MEASURED

VARIABLE COEFF f COMPUTED

- - ~ - - CONSTANT C O E F F 1 I I I ] I 5 10 15 20 25

/

30 T Is]

Fig. 34. HDW articulated tower transfer function of hori- zontal /oint force

300 I I I I I

VARIABLE COEFF )

250 CONSTANT COEFF. ICOMPUTED

| - - • - - REDUCED AXIAL C~4 j • MEASURED / 200 150 / • / loo so

i III

0 " ' % ~ l " • I • OI I I 0 5 10 15 20 25 30 T[s]

Fig. 35. HDW articulated tower transfer function of vertical ]oint force

respectively. For all three sets of calculations, agreement between measured and calculated results of tower top deflection m quite good over the entire range of wave periods compared. Agreement of horizontal force is also good except for waves with relatively small periods where differences are in the order of 8-14%. However, for all three sets of calculations correlation of vertical force is poor. Except for the shortest wave with 6 s period, calculated vertical forces are much higher than measured, and comparison with model test results show that they are unrealistic.

In ref. 24 it is shown that the poor correlation between measured and calculated results of vertical force may be due to a wrong choice of hydrodynamic coefficients for flow in the axial direction. Therefore, as an example, simulating the action of much reduced axml inertia forces, in ref. 24 are carried out computations with smaller axial inertia coefficients equal to 10% of previously selected values. In Fig. 35 results show that, except for the smallest wave period, measured and calculated vertical forces now agree quite well. Both magnitudes as well as trends of calculated vertical force follow measurements. Of course, not only reduced inertia forces but also reduced added masses, or a combination of the two, could have been simulated by selecting appropriately reduced hydrodynamic coefficients for axial flow, thereby possibly obtaining a still better correlation of vertical force. In any case, it appears that detailed knowledge of actual flow conditions is necessary to attempt an accurate specification of axial hydrodynamic coefficients.

Computations are done for a range of wave periods extending beyond the range of measured values. For the set with constant coefficients, periods range from 3 to 28 s. It can be seen that the cancellation period of vertical forces is reduced from 21 s to 15 s when axial inertia coefficients are decreased.

In summary, comparison between measurements and computations shows that this analytic method based on the Morlson equation can be used to make reliable predictions of tower top deflection and forces at the universal joint provided hydrodynamic force coefficients are properly chosen. For the example given here, coefficients for flow normal to tower centre line need not be specified as care- tully as coefficients for flow in the axial direction. In general, this phenomenon is likely to be so for articulated towers with a structural arrangement comprising a transparent lattice structure connected to a main float.

2.3. Composite structures

Offshore structures cannot always be categorised either hydrodynamically compact or transparent. Sometimes their structural configuration IS such that certain parts can only be analysed as large volume compact structures whereas other parts comprise small diameter cylindrical piles and are thus transparent. For such structures, a hydrodynamic analysis may necessitate using the potential theory for the large volume parts, whereas the Morxson formula can be used for small diameter piles of the structure As an example of such a composite structure, we analyse the motion behavlour of the floating storage tank SEAGAS in waves.

2.3.1. Floating storage tank SEAGAS. Experimental results of first-order oscillatory motions of the floating storage tank SEAGAS in regular waves are compared with