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

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, hydrodynamic calculations are used to evaluate and assess hydrodynamic aspects such as wave loading and motion behaviour of offshore structures. When applying calculations based on hydrodynamic theories, it is essentml to be aware of the validity and accuracy of the predicted results. Two aspects need to be considered in assessing the reliability of any prediction. The first concerns accuracy of 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 method 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 hydrodynamic calculations. To assess the Importance of some o f these effects, comparisons with model test measurements may be used Our purpose is to present hydrodynamic calculations as applied to a variety of offshore structures and to show the validity and accuracy of predicted results by comparison with closed form solutions or model test measurements When dealing with offshore operations, one is generally concerned with two types of structures that are fundamentally different from the hydrodynamic analysis point ot view. Firstly, there ale the so-called large volume structures such as barges and full bodied ships: but also semisubmersibles with thick columns and large volume lower hulls may belong to this type of 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 hydrodynamically transparent structures. The Important aspects to consider m the analysis of hydrodynamically compact structures are wave diffraction and radmtion For hydrodynamically transparent structures, hydrodynamic Ibrces are calculated using the two component Morison fommla to account for hydrodynamic inertia and viscosity. Both methods are briefly explained in an Introductory part of Section 2 o f this paper.

Accepted December 1986 Dlscussmn closes December 1987

192

Applied Ocean Research, 1987, Vol. 9, No. 4

For ship-like (slender) structures, the characteristic that the longitudinal length scale IS substantially greater than beam and depth, is used to simplify the hydrodynamic analysis by 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 of hydrodynamic 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 hydrodynamic performance o f structures m seaways that are generally encounteled In offshore opeiations. We shall present a number o f illustrative examples, comparing pledicted hnear hydrodynamic propeltles either with closed form solutions or with measurements. We start with hydrodynamic calculations of a floating sphere, a barge as well as a ship, using the method 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 method incorporating the Morlson formula. Finally. a floating structure will be treated that lends itself to be analysed using both or eIthe~ the diffraction/radiation method and/or the Mollson fornmla. There are rare situations in offshore operataons where nonlinear influences cannot be neglected in hydrodynamic analysis of offshore structmes We shall plesent mtloductory 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 of 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 tleatment 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 hydrodynamically compact structures or ship-like bodies are calculated by a so-called linear potential theory under the condition that the effect of vortex shedding on total hydrodynamic 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 theory) be directly super0141-1187/87/040192-22 $2.00 © 1987 Computational rdechamcs Pubhcatlons

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 ep1 to ep6 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 of the (linear) motion(s) of the structure in waves. Each velocity potential must satisfy the Laplace equation in the fluid domain,

V2%=0

/=o,

1.....

tial at field point (x,y, z) of a source at (~, v, ~') of unit strength which satxsfies the boundary conditions of the fluid domain in the absence of the body. z The source strengths Q] are found by satisfying the body boundary conditions, leading to two-dimensional integral equations of the Fredholm type which are solved by approximating the underwater body surface by a large number o f lateral elements (surface patches), i.e. systems of linear equations replace the integral equations. With ~/ known, we use the linearised Bernoulli equation to obtain added mass and damping coefficients

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 bottom we have ~:/~z

= 0

n/

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 body surface

z = -d

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

0%/an=

(s)

]

=

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 eP1 which are related to the local velocity of the body surface in the ]th degree o f freedom. The second body boundary condition, called radiation condmon, 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 indication o f the direction of the incident waves in order to keep the mathematics as simple as possible.) The unknown velocity potential dp= ~.exp (--twt} is written as

F] = --p(iw).exp (--iwt}" 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 to a vertmal, rigid cylinder in undisturbed surface waves, s 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 of 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, ko =

CD'p'D/2,

k M = CM'p'Tr'D2/4

6

dp= (~b0+~bT)'exp(--i¢ot} + ~

]=1

¢l.~rl/~t

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

¢1(x,Y,Z)=IfQl(~,v,~)'G(x,Y,Z,~,v,~)ds (s) which are integrals over the body 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-

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 dF 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 component of the wave induced water particle acceleration, aBn the normal component 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 of the components o f Urn. The use of 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 CA For a circular cylinder section k.l = CA'p'Tr'D2/4 The Froude-Krylov force is embedded in the waveacceleration term and IS represented by an appropriately chosen value o f CM, the wave-acceleration (inertia) force coefficient Note that k M differs from kA because the fomaer term (containing CM) defines a force acting on a stationary body 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 by CA = 1 and CM = 2 . We shall illustrate the evaluation o f the fluid force toe the general case of 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 UB=V+WXr

where r IS the radius vector from the centre o f rotational motion to the centre of the cyhndrical section The relative velocity between water and body is Ur :

UW--II B

where Uw, the wave-induced water particle velocity at the centrold of 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 of the coordinate system is given in ref. 6. We next define a unit tangent vector, et, directed along the cylinder axis The normal component of the relative velocity IS then given by Urn=e

t X(Ur x e t )

The total velocity-dependent force on the cylindrical member IS obtained by integrating the velocity term 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 body acceleration, aa, is given by

aB = Ov/Ot +

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(Ow/~t)

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where the first term lb the t~anslatlonal accelelatlon ot the centroid of tire cylinder section and the aecond term ll/volves the angular acceleration o f ttu~ section about a centroldal axis of 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 aw

=

[~2(I)0/8X 10t, 02 ePo/OXzOr, 02~o/3X3 3t]

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

ayn = et x (ag 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 member is obtained by integrating the acceleration terms over the immersed length of the cyhnder. A practical simplification of the Morison formula concerns hnearisation of the velocity term. This requires that the quadratic relationship of the drag force component be expressed in a hnearlsed form

CD'p'D/2"uIul +++ CLo'o'D/2"u The equivalent linear drag coefficient CODis 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 of 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 motion as caused by waves o f a certain amplitude. Because motion amplitudes must be known before the system of m o t i o n equations as solved, an iterative 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 hydrodynamic properties m all three translational dvections. Immersed cylinder end plane areas are treated as small parts having hydrodynamic properties in the direction o f the cylinder axis After th~s brief explanation of the two basic methods of linear analysis (potential theory and Monson formula), we proceed to practical examples m order to demonstrate their applicability

2.1. Hydrodynamically compact structures The method used toe the analysis o f hydrodynamically compact structures IS based on linear potential theory, allowing the effects of hydrodynamic interaction between nelghbourlng structural elements to be included m the analysis. The numerical procedure relies on a dlscretlsatIon o f the body surface mto surface elements Both finite as well as infinite water depth can be considered. A computer 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 hydrodynamic 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-

Comparison of 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 surrounding the body. The choice of the number of patches used for computations has a bearing on the quality of the results. Increasing the number of patches generally improves the quality of results. The number of patches necessary for satisfactory predictions can only be determined by repeating 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 dampmg 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

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Applied Ocean Research, 1987, I1ol. 9, No. 4

195

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

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the wetted surtace of the hemisphere). J udglng trom e \ peiience 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 lulfilled with the present ldeahsatlon of the hemispherical surface. From the agreement obtained between numerical and analytical computations, it may be concluded that the use ot this numerical procedure for the analysis of a floating hemisphere results m reliable paedlctions of flast-oldel hydrodynamic forces and motions in waves. Next, we shall employ this method to piedlct the motion behavtour of a rectangular barge, a structure that as very frequently used an offshore operations

2 1.2 Rectangular barge Experimental lesults ot hrstorder oscillatory monons of a rectangular barge floating m regular waves are compared with computed results. The barge selected as replesentatlve of 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 pertbrmed at the Nethellands Ship Model Basra, 14 are used

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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 computed 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 of 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

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Applied Ocean Research, 1987, Vol 9, No. 4

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Heave motion transfer funcnon oJ a sphere

Table i

Partteularsof 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

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

Z

Fig. 9.

Rectangular barge idealisation

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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

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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 of motion amplitudes plotted against full-scale wave period

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Fig. 11. waves

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 2rt

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 of 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 twodimensional 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.

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ship-like structures. In the following we call such structures slender. Slenderness occurs for ocean platforms such as spar buoys, for some bottom supported gravity structures of similar shape, for some articulated towers, and for semisubmersibles 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 denvation of hydrodynamic response forces and wave excitation forces in the equation of motion. Both approaches make use of a velocity potential based on the assumption of 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 body surface

198

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COMPUTED

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Barge sway motion transfer function in beam

Comparison of experimental and theoretical wave actions: C. Ostergaard and T. E. Schellin 1%

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Table 2.

Particulars o f tanker

waves

Program GLSTRIP is used to evaluate five-degree-offreedom 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 of the tanker is subdivided into a total of

Length between perpendiculars (m) Breadth (m) Draft (m) Displaced volume (m 3) C.G abovebase (m) Metacentric height (m) Transverse gyradius (m) Longitudmal gyradius (m) Vertical gyradius (m) Natural heave period (s) Natural roll period (s) Natural pitch period (s)

310 47.17 18.90 234 826 13.32 5.78 14 77 77.47 79.30 11.80 14.20 10 60

?

6 -1C

Fig. 16.

f

5-10

Body plan of tanker

Applied Ocean Research, 1987, Vol. 9, No. 4

199

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

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qiows that consldelable sunphticatlons ,iic made Tile bilge ladnis is neglected completely Results of computations and nleasuiements ol 111st-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 21,22 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 of 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

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waves

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Is31 large number of small elements increases the accuracy of results because small elements better describe the hull surface. Secondly, minimum acceptable ratio of 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)

200

Apphed Ocean Research, 1987, Vol. 9, No. 4

// 04

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Fig. 19. waves

Tanker heave motion transJer function m head

Comparison o f experimental and theoretical wave actions: C. Ostergaard and T. E. Schellin 2r~

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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.)

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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

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• 12 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 dependency 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.

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Fig. 21. waves

Tanker sway motion transfer function in beam

Applied Ocean Research, 1987, Vol. 9, No. 4

201

Om, parison of experimental and theorencal wave actions C Ostergaard and T E. Schelhn 1"[

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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 condltlons 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 4s to 27s lull scale. Results of computations and measurements ot fllstordei oscillatory motions ale gwen m Figs. 25-28 Surge motions, Sl, refer to a point at the upper deck above the

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waves

Is, I assumed equal to one plus added mass coefficients Drag terms are hnearlsed using relative velocity. Hydrodynamic interaction between neighbourlng structural elements is neglected. Finite water depth is accounted for in wave kinematics. Linear theory xs used, and integration o f fluid forces acting on structural members is done up to the still water line. A computer program GLFMTHT of German]scher Lloyd, based on this method, has been developed enabhng the evaluataon of linear vessel motions of and hydrodynamic forces on hydrodynamically transparent structures. 8'2°

2 2.1. Semi-submersible drilling platform RS-35. Experimental results of first-order oscillatory motions o f the semisubmersible drilling platform RS-35 floating m regular waves are compared with computed results. This semlsubmersible is characterlsed by a structural configuration comprxsmg 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 of the structure is dwtded into four slanted cylinders simulating the four columns and 12 small parts slmulatmg the ring hull. Added mass coefficients for cylinders are specified according to the curves given by Lee an his discussion to ref. 19. Added mass coefficients for small parts are set equal to 1.0. Calculations are performed with three different drag coefficients and with two different water depths.

202

Applied Ocean Research, 1987, VoL 9, No. 4

16

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5

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Tanker roll motion transfer function in beam

Table 3. Particularsof semisubmersible RS-35 Outer torus diameter (m) Ring hull diameter (m) Column diameter (m) Draft (operating) (m) Displaced volume (m s) C G. above base (m) Natural heave period (s) Natural pitch period (s)

96 10 12 30 31 220 16.8 21.7 51 0

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

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Fig. 25. waves

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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 identified. 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

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30 Tls]

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

Applied Ocean Research, 1987, Vol. 9, No. 4

203

Compartson o f experimental and theoretwal wave actions COster, eaard and T E Schellm I

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most tavourabl3~ w~th nleasmements m ~ a w s w~th pe~octs nero the natural heave pe~a)d Calculatmn~ with other CD values o f 0 7 and 1 0 ale calued out 111 o~del to demonshate the large influence of tMs value on Dedlcted 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 of 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 of CD = 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 coefficients have an effect on computed results Increasing added mass values generally leads to larger values of heave amphtudes At the same time, however, the maximum 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 conrputatlo 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 of surge and pitch motions, the choice of drag coefficients is of only secondary importance Using cmefully specified hydro-

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

15 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, of course, a function of not only drag coefficient but also wave amplitude 8 Measurements m waves with different amplitudes are, therefore, suitable to demonstrate effects of 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 may not only be due to viscous effects. As the scale o f the model 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 depth has an maportant influence on heave motion 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 of CD = 0 4 compare

204

Apphed Ocean Research, 1987, Vol. 9, No. 4

I

o MEASURED (SCALE 1"100) 150 m WATER DEPTH (~a= 75m 10

Is5[

• MEASURED (SCALE 150) 75m WATER DEPTH ~a 375m

CD=0LI CD = 0 7 CD=I0

05

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10

COMPUTED

15

20

25 30 T[s]

RS-35 ptteh motion transfer functton in head

Comparison o f experimental and theoretical wave actions: C. Ostergaard and T. E. Schellin dynamic coefficients, this numerical method of analysis based on the Morison equation is quite reliable for predictlng first-order motions of hydrodynamically transparent structures in waves. We shall further demonstrate its usefulness by analysing motions and loads o f two articulated towers.

16

191

12

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Total height (m) Diameter (m) Water depth (m) Joint above bottom (m) C.G. above joint (m) Centre of buoyancy above joint (m) Displacement (Mg) Mass (Mg) Moment of mertm (tm 2) Natural pitch period (s)

201 4 20 0 154 5 9.0 51 0 58.5 43 471.8 31 398.4 210 Xl0 ° 45.0

I

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TU-Berlin articulated tower pitch transfer func-

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Particularsof TU-Berhn articulated tower

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2.2.2. TU-Berlin articulated tower. Experimental results of first-order (pitch) motions and corresponding horizontal forces at the umversal joint of the TU-Berlin articulated tower in waves are compared with computed 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 partiulars of this tower are given as full-scale values in Table 4

Table 4

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04

05

06

CO [reid Is] ~20m wave

_

~7

E

E tO

f/f/\\~-'//1/<..'~1(//-,'~7 z//\\~\////

Fig. 29.

Fig. 31. TU-Berlin articulated tower horizontal force transfer function

TU-Berlin articulated tower

A sketch o f the tower model is given in Fig. 29 Experimental measurements are obtained 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 hydrodynamic analysis of 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 hydrodynamically transparent. Computations are done in the time domain, simulating the tower behavlour by direct integration after specified successive Ume steps. Further details are found in refs. 23 and 24. Results of computations and measurements of angular tower deflection (pitch), 0, and horizontal force at the universal joint, Fx, are given in Figs. 30 and 31, respectwely. 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 At = 0.5 s and another with A t = 0.15 s.

Applied Ocean Research, 1987, Vol. 9, No. 4

205

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 of 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 of the transfer functmns are also predicted by computations. The natural pitch period is well predicted, indicating that specified added mass coefficients are well chosen. For 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 of this numerical method for use on articulated towers can be properly assessed, it is desirable to demonstrate its use with a more reahstlc example of 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 of first-order tower motions and corresponding forces at the universal joint o f the HDW articulated loading tower in waves are compared with computed results. This tower is supported near the seabed by a universal joint The column structure, extending vemcally through the water surface, comprises a ballast tank, a steel lattice structure, a buoyancy 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 Instrumented 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 fiom 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 of elements are such as to s~mulate the tower's buoyancy

Table 5

Particularsof HDW articulated loadmg tower

Total height (m) Water depth (m) Joint above bottom (m) C G above joint (m) Centre of buoyancy above joint (m) Displacement (Mg) Mass (Mg)

Natural pitch period (s)

206

Applied Ocean Research, 1987, Vol 9, No. 4

203 5 180.0 12 0 52 74 90.56 11 170 0 11 170 0 42 5

E

~16m

Sectlon A-A

A

II ~

II

A

e~

Sectlon B-B

8

Fig. 32.

B

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 hydrodynamic coefficients for flow normal to the tower One set is based on constant values of added mass and drag coefficients equal to one, a second set is based on coefficients selected from investigations dealing with cylInders in oscillating flow, 26 and a thtrd set IS based on coefficients calculated using potential theory. All values of specified coefficients for sets two and three are given in ref. 24. End plane areas of cylindrical elements need to be treated with care In order to correctly account for hydrodynamic forces acting In the axial direction of the tower. Hydrodynamic coefficients for end plane areas are chosen from tabulated values. 17 Results o f computations and measurements of tower top deflection, Xp, and corresponding horizontal and vertical forces at the joint, F x and F z, are shown in Figs 33-35,

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

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0

5

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20

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30 T Is]

HDW articulated tower transfer function of top

Fig. 33. motion

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400

IF,I co

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30 T Is]

Fig. 34. HDW articulated tower transfer function of horizontal /oint force

300

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CONSTANT COEFF. ICOMPUTED -- • -•

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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 caretully 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.

30 T[s]

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

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

Apphed Ocean Research, 1987, Vol. 9, No. 4

207

Comparison o] expenmentaI and theoretical wave actions. C Ostergaard and T t:: Schelhn computed results SEAGAS comprises a centrally located storage tank sulrounded by eight vertical columns located at the periphery of the central tank. These columns provide the necessary stability of the structure as the central tank IS filled with hquefied gas. Main particulars are given in Table 6 A sketch of the floating LNG tank SEAGAS is shown m Fig 36 In computations as carried out in ref. 9, the central tank is treated as hydrodynamically compact and the eight columns as hydrodynamically transparent. The potential theory is used to determine hydrodynamic masses, damping forces and first-order wave forces for the central tank For numerical calculations, its wetted surface is subdivided Into 132 triangular surface patches The Morlson formula is used to determine the correspndmg hydrodynamlc quantities for the mght columns. Vessel motions are calculated by linearly superposmg these quantities. This means that wave diffraction and radiation effects as far as the columns are concerned are not included an the computations. Conslderlng the tact that hydrodynamic forces acting on the central tank are generally an order o f magnitude greater than corresponding forces on the mght columns, this procedure seems justified for linear hydrodynamic effects of waves on the structure. Expeumental results obtained from model tests gwen an ref 28 are used for comparison. These model tests include runs in legular waves with periods ranging from 7 5 s to 25.2s. Results of computations and measurements of first-order oscillatory motion amplitudes of surge s~, heave s3, and pitch Ss, ale given as transfer functions and plotted against full-scale wave period T in Figs. 37-39, respectively Comparison of measured and computed results shows good agreement in waves with periods less than 18 s for surge, 21 s for heave and 15 s fol pitch motions Differences in surge and pitch motions at higher wave periods may, to some extent, be due to the fact that computations assume the centre ot rotation to be located at the centre o f

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Applied Ocean Research, 1987, Vol. 9, No. 4

-,

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gravity o f the structure. Differences in heave motions occur only m waves with periods close to the natural heave period, suggesting that the calculated heave added mass o f the structure m a y be somewhat small. On the whole, these differences are relatively insignificant as far as motion pre&ctions are concerned because they occur m waves with periods outside the range o f major energy content of the natural seaway. If done with care, this analysis shows that using the potential method in combination with the Morlson formula method results m reliable predictions o f first-order motions for structures comprising large volume as well as small dlammeter parts

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

-----72-----,-

2.4. Hydrodynamic analysis of a semisubmersible drilling platform

I

In general, the purpose of an hydrodynamic analysis IS to obtain reliable predictions of parameters such as motions in waves o f a particular structure. However, at the beginning of the design procedure, it is not always self-evident which computational method to select for the hydrodynamic analysis of the design. Our previous sample computations show that more than one method of analysis can be employed either separately or in combination with each other to yield acceptable results. If uncertainty exists as to which method to choose, it may be advisable to use more than one method. For semlsubmersible drilling platforms comprising twin hulls of relatively large cross section, for example, computational methods that treat the structure as hydrodynamically compact may not necessarily yield more reliable results than methods that treat the structure as hydrodynamically transparent since both effects, diffraction and radiation as well as viscosity, have a significant influence on hydrodynamic calculahons. In order to obtain reliable results, the analysis should be performed using the potential theory as well as the Morlson formula. As an illustrative example, we carry out a hydrodynamic analysis of the ITTC semisubmerslble drilling platform using these two computational methods.

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2.4.1. ITTC semisubmersible drilling platform. In 1984 the 17th International Towing Tank Conference (ITTC) Ocean Engineering Committee conducted a comparative motion study of a semlsubmerslble in waves. 29 This study mcludes model test measurements of first-order vessel motions conducted by Ishikawajlma-Harima Heavy Industries (IHI) Co. m their sea-keeping and manoeuvring tank. These model tests have been carried out at a scale of 1 : 64 and include runs in regular beam waves with periods ranging from 4 s to 29 s full scale. The water depth corresponds to 192 m full scale. We shall compare results of these model tests with computed results The model used for these experiments is a scaled-down replica o f the theoretical full-scale semxsubmerslble drllhng platform shown in Fig. 40. It comprises twin lower hulls and eight circular cylindrical columns supporting a rectangular deck section. The platform is assumed freely floating at survival draft. Mare particulars are given in Table 7. For computations using the potential theory, the wetted surface of the semlsubmersible is subdivided into 268 quadrdateral elements as shown m Fig. 41. For computations using the Monson formula, the underwater part of the platform is divided into eight vertical cylinders simulating the vertical columns and 30 small parts simulating the two horizontal lower hulls and the bracings. In computations using potential theory, hydrodynamic coefficients of added mass and potential damping due to radiation are calculated as outlined m the beginning of Section 2. However, m computations using the Mor~son formula, hydrodynamic coefficients need to be specified as input data. For cylinders, added mass coefficients are specified according to curves given in ref 30, and for small parts, they are set equal to 1.0 Coefficients o f (viscous) damping are set equal 0.5, 1.0 and 1.5. Separate calculations are done with each o f these values. Results of computations of first-order transfer functions of motions in beam waves are compared with measured results m Fig. 42 for sway, s2, in Fig 43 for heave, s3, and in Fig. 44 for roll, s4, respectively. All motions refer to the

Table 7. Partzcularsof lTTC semisubmersibleplatform Length overall (m) Breadth moulded (m) Upper deck elevation (m) Length lower hulls (m) Beam lower hulls (m) Height lower hulls (m) Outer column diameter (m) Inner column diameter (m) Draft (survwal condition) (m) Displacement (Mg) C.G above base (m) Metacentric height (longitudinal) (m) Metacentrlc height (transversal) (m) Natural heave period (s) Natural roll period (s) Natural pitch period (s)

Fig. 41.

115 75 43 115 15 8 10 8 20 35 000 17.5 2.37 2.87 23 9 49 4 57.4

ldeahsation of lTTC semisubmersible

centre of gravity of the platform. Transfer functions are plotted against full-scale wave period T. Phase angles e are given in radlans and are also plotted against full-scale wave period. (Positwe phase angle means that the motion leads the wave elevation.)

Applied Ocean Research, 1987, Vol. 9, No. 4

209

Compartsvn o/ experimentaland theorencal wave acnons" C Ostergaard and T fi: Schellm 2K

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potential theory, v]scous effects are not included at all Consequently, these computations show (unrealistically) zero heave amphtude at the cancellation period T c and infinite heave amplitude at the natural heave period TN. Comparison with measurements shows that calculated heave motions in waves wtth periods near the natural heave period are rehably predicted only when viscous effects are included, that is, when using the MoHson formula. For thls sample analysis, the model tests used for comparison indicate that a relatively small drag coeffloent of 0.5 together with a relatively low wave height of 1.47 m full scale, result m a reasonably good fit with measmed data In waves with periods outside the range of natural heave period, calculations based on the Monson formula result in heave motions that ale very similar to calculations based on potential theory and to measurements. All calculatmns show that water depth has no slgmficant effect on heave Fol roll (Fig. 44), there Ls excellent agreement between calculated results and measurements up to waves with

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For sway (Fig. 42), there is excellent agreement between the calculated results using both methods, and these results also agree well with experimental measurements. Note that measurements show that wave height has virtually no influence on sway and that water depth effects sway only in longer waves with larger periods. Note also that viscosity has practically no influence on sway, demonstrated by the fact that there is no difference between computed results obtained using the Morison formula with &fferent drag coefficients as input. For heave (Fig. 43), there is good agreement between calculated results and measurements m the practically important range of waves with periods less than 21 s. For waves with periods in the vicinity of the cancellation period (T c ~- 22.8 s) and the natural heave period (TN ~ 23.90 s), there is poor mutual agreement between calculated results using the two methods. Computations using the Mormon formula show the strong influence of viscosity, demonstrated by the tact that there are large differences between results obtained with different drag coefficients and fbr different wave amplitudes m the mcmlty of the cancellation period and the natural heave period, respectively. In waves with periods near the natural heave period, measurements performed m different wave heights indicate that also wave height has a strong influence on heave, i e. nonlinear system characterlStlCS m the vicinity of the natural heave period have to be realised. Decreasing the effects of viscosity by decreasing the drag coefficient, or decreasing the wave amplitude, significantly Increases the peak value of heave amplitude These trends are already substantlated by similar results obtained from the analysis o f the semlsubmerslble drdling platform RS-35 (see Figs. 26 and 27). When using the

210

A p p h e d Ocean Research, 1987, Vol 9, No. 4

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Comparison o f experimental and theoretical wave actions: C Ostergaard and T. E. Schellin 0

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In this last section we refer to some nonlinear effects which must not be overlooked in certain (special) design cases o f offshore platforms. This section is meant as an introductory outline of some effects of interest. For our present purpose it Is regarded sufficient if we define linearity of systems in a descriptwe way by the following requirements:

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periods of about 18 s Agreement for longer waves with larger periods is still acceptable for computations based on the Morison formula. However, results based on potential theory tend to be too low at longer waves. Computanons based on the Morison formula using different drag coefficients show no significant Influence of viscosity on roll m this range of wave periods. This is not surprising, because wave periods are far below the natural roll period. Note that wave height has practically no influence on roll in this example. In general, comparison between calculated results based on the Morison formula and those based on the potential theory compare favourably with each other and with measurements. The only exception is heave motion in waves with periods near the natural heave period. In this range of wave periods, viscous effects must be Included, necessitating the use of the method based on the Morison equation. In order to obtain rehable predictions, it is important to select appropriate drag coefficients. Drag coefficients apphcable to model scale predictions can be specified such as to result in a proper fit with measurements. However, drag coefficients apphcable to full-scale predictmns may be difficult to specify realistically. Thus, there is likely to remain some uncertainty m the predictmn of heave for fullscale platform behaviour in waves with periods in the vicinity of the natural heave period Note that the prediction of the natural heave period itself compares favourably with measurements using both analytical methods. Generally, potential theory should be expected to result in better agreement with measurements since added masses are calculated and not estimated. The fact that calculations using the Morison formula agree so well is mainly due a favourable choice of inputted added mass coefficients. The specification of added mass coefficients for the Morison formula is o f prime importance, and a recommended procedure tbr selecting these coefficients is to carry out parallel computations using potential theory.

The fulfilment of these requirements correctly describes linearity though not generally, but a general treatment of the subject :s not needed in the present context. We have already seen that requirement No. 1 is not satisfied by the heave transfer function of semisubmersibles in the vicinity of the natural heave frequency (Figs. 26, 27, 43). Realising that semlsubmersibles and many other floating offshore platforms are designed such that their natural frequencies are outside the range of wave frequencies of high or even medium energy content, the rather hmited violation o f requirement No. 1 generally does not hinder us from applying linear wave response analysis successfully to such structures. Some special nonhnear wave effects, e.g. horizontal drift forces, will be dealt w]th in a separate paper. Not so obvious are violations of the linearity reqmrement No. 2. We exemphfy this with the simplest design case, a vertical pile in linear (i.e. smusoidal) waves. In Fig. 45 the configuration and main characteristics of the pale and wave are presented together w]th the total horizontal wave induced force F x on the pde. Calculations are based on the Morlson formula. We examine the response F x over one half cycle of the smu-

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Applied Ocean R esearch, 1987, Vol. 9, No. 4

211

Comparison o] experimental and theoretical wave acnons. C Ostergaard and T k; Schellm soldal wave amphtude ~'(cot) With increasing wave period T (1 e with increasing wave length), we observe an increasing dewation of this response Fa{oot) away from a harmonic lesponse. Without examining the sources of this phenomenon, which are easily understood from the application of the Morason formula, we recognlse that higher order harmonics affect the time dependence of this response A Fourier analysis reveals the details (see Fig. 46). It is seen that a second harmonic with circular frequency 3co, i.e. with markedly lower period T, occurs with an amphtude of about 15% of the hnear response at co. This second harmonic response at 36o IS nonlinear and could cause serious vibrations If its frequency should coincide with a natural vibration frequency of the pile Our example is theoretical, however, ff a longer pile o f slmdar charactelistics were situated m much deeper water, this nonlinear effect would become a design constraint of practical importance due to the relatively high energy content of the nonlinear component of the horazontal force at its higher frequency It is not always possible to reveal nonlinear effects of the second kind that easily. To exemplify this, we once more

80

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consider the floating LNG tank SEAGAS shown an Fig. 30 Our lineal response analysts yields very accurate ~esults a~ demonstrated in Figs 37 39 However, at certain wave hequencms the tests at the HSVA model basin also ~evealed a so-called parameter excited ~esponse which ~s shown in Fig 47 We observe that large pitch motions. Ss, occm at a period which is twice the wave period, 1.e a nonlinear response must be reahsed since our aequlrement No 2 ~s violated. This phenomenon ~s theorencally described by the so-called Mathleu equation, 31 and cannot be pre&cted using linear m o t i o n analysis

4. CONCLUD~G REMARKS A large amount o f standard or routine problems in morion and related response analyses of offshore structures can be solved rather realistically on the basts of linear hydrodynamics. There are, of course, special problems and design cases where nonlinear hydrodynamics are required. What remains to be mentmned is the fact that design decisions cannot be based on results of h y & odyna mi c analyses alone, however accurate these results may be Natural sea conditions and their analytical description is necessary to decide upon, e g. safe design values for longlasting operations at sea Thus, remaining uncertainties in the presented results o f linear hydrodynamic analyses must be seen also an relation to the uncertainties in their statlsncal and probablhstlcal evaluation. It is the authots' opinion that, in the context of design decisions as a whole, the practical value o f linear hydrodynamics cannot be overesnmated.

REFERENCES

20

1 Newman, J N MarmeHydrodynamzcs, FheMITPress, 1982 2 Wehausen, J V. and Latome, E V Surface Waves Handbuch der Physik, Vol 9, Sprlnger-Verlag, 1969 3 St Denis, M On the monons ofoceamc platforms, Symposium

..... .I

O

0

on the Dynamics of Marine Vehicles and Structures in Waves,

10

30

20

4

T[s] Fig. 46. Fourier analysis o f horizontal wave lorce on c' vertwal pile

5 6 7 8

PqTCH PERIOD

/ ~T~ 1

%[oi 15

9

1o 5 PITCH NOTION

0 -5

,~~

J~~

TIME

10

-10 15

5-AAAAAAAAAAAAAA_ WAVE 0 ELEVATION '- r'E -5_ V V V V V V V V V V V V V V Fig 47. Parameter excited pitch motions o f L N G tank SEAGAS

212

A p p h e d Ocean Research, 1987, Vol. 9, N o 4

11 12 13 14 15

London, 1974 Papamkolaou, A On mtegrat equation methods for the evaluahon of motions and loads of arbitrary bodies in waves, Ingenieur-Archtv, Vol 55, Sprmger-Verlag, 1985 Monson, J. R., O'Bnen, M P, Johnson, J W and Schaaf, S. A The force exerted by surface waves on plies,Petroleum Transactions. AIME, 189. 1950 Paullmg, J R Time domain simulation of semlsubmerslble platform morion with apphcation to tension-leg platform. SNAME STAR Symposium, San Francisco, 1977 Hoerner, S F Fluid Dynamic Drag, 1969 Ostergaard, C and Schellm. T E. On the treatment of vascous effects m the analysis of ocean platforms, wave loads and monons, Schiffund Ha fen, Heft 4, 1977 Ostergaard, C, ScheUln, T E and Sukan, M Hydrodynamlsche Berechnungen fur kompakte Strukturen, Schiff und Hafen, Heft 1, 1977 Sukan, M Berechnung der Wellenwlrkung auf kompakte Oft'shore-Konstruknonen beheblger Form, Schtffstechntk, 29 1984,4 Papamkolaou, A On the evaluation of monons and loads oI arbitrary bodies m waves, Proc Symp Ocean ,Space Util '85. Tokyo, 1985 Kudou, K The drlfnng force acting on a three-dlmensmnal body m waves, JSNA Japan, 141,1977 Maruo, H The drift of a body t'loatmg in waves, Journal o] Ship Research 1960, 4, 3 Pmkster, ] A Low frequency second order wave exciting forces on floating structures, Pubhcatlon No 650, Netherlands Ship Model Basra, Wagenlngen, 1980 Sodmg, H The flow around ship secnons m waves, Scht//s-

Comparison o f experimental and theoretical wave actions: C Ostergaard and T. E. Sehellin technik, 1973, 20, 99 16 17

Grim, O Berechnung der dutch Schwmgungen eines Schlffsk6rpers erzeugten hydrodynamlschen Kr//fte, Jahrbuch der Schiffbautechmschen Gesellschaft, Hamburg, 1950 Grtm, O Die Schwmgungen von schwimmenden zweldmaensionalen Korpern, Reports 1090 and 1171 of the Hamburg-

23 24 25

ische Schiffbau- Versuchsanstalt 18

Van Oortmerssen, G. The motions of a moored ship m waves,

Pubhcatton No. 510, Netherlands Ship Model Basra, Wagen19 20 21 22

mgen, 1976 Hooft, J P. A mathematical method of determining hydrodynamically reduced forces on a semlsubmersible, SNAME Transactions, New York, 1971 Ostergaard, C and Payer, H G Rationale Beurteilung der Festlgkelt von Halbtauchern, Jahrbuch der Schiffbautechnischen Gesellschaft, Hamburg, 1973 Clauss, G. F. Multi-scale model tests with a ring-shaped semisubmersible, Offshore Technology Conference, OTC-3297, Houston, 1978 Bergmann, J Gau#sche WeUenpakte, em Verfahren zur Analyse des Seegangsverhaltens meerestechmscher Konstruktlonen, Dissertation, Technical Umverslty Berlin, 1985

26 27 28 29

Koch, T Benutzerhandbuch fur das Programmsystem Tower, Germanscher Lloyd, Report STB- 745, Hamburg, 1983 ScheUm, T E. and Koch, T. Calculated dynamic response of an articulated tower m waves comparison with model tests, ASME, OMAE-116 , Dallas, 1985 Hattendorff, H. G. and Wyzotski, M. Seegangsversuch mit dem ModeU des Articulated Towers der EG-Studie, Hamburg Ship Model Basin, Report $159/81, Hamburg, 1981 Sarpkaya, T. Forces on cyhnders and spheres m a smusoldally oscillatmg flmd, Journal o f Apphed Mechanics, ASME, 1975, 42,1 Saunders, H E. Hydrodynamics in Ship Design, SN AME, 196 4 Blume, P and Hattendorf, H G. Seegangsversuche mlt dent Modell des SEAGAS-Halbtauchers, Hamburg Ship Model Basin, Report S 126/78, Hamburg, 1978 Takagl, M, Aral, S -I, Takezawa, S., Tanaka, K and Takarada,

N. A Companson of Methods for Calculating the Motion o f a Semt-submerslble, Hitachi Zosen Corp, 17th ITTC OE Com30 31

mittee, 1984 Lee, C M. Dlscussiontoref 19 Grim, O Rollschwmgungen, Stabthta't und Slcherhelt m~ Seegang, Schiffstechnik, Vol 1,1952

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213