The simulation of slow-drift motions of offshore structures

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0141-1 187/96/$! 5.Ü0

The simulation of slow-drift motions of offshore

structures

**O. J . Emmerhoflf* & P. D. Sclavounos**

Deparmant of Ocean Engineermg, Massaciiusetts Institute of Teclinology, Cambridge, MA, USA (Received 19 March 1996)

The large amplitude surge-sway-yaw 'slow-drift' motions of a floating body con-strained by weak restoring forces in random waves are considered. A multiple time scales approximation is employed to separate the fast time scale associated with the linear motions f r o m the slowly varying motions. The ideal fluid free surface flow is approximated by a perturbation series expansion f o r small slow-drift veloc-ities and wave-steepness, and is solved around the instantaneous position o f the body. The linear zero-speed and forward-speed velocity potentials are solved for arrays of vertical cylinders, using exact interaction theory. The horizontal mean d r i f t forces and the wave-drift damping are obtained, and results for realistic configurations are compared with well-established methods. The surge-sway-yaw equations o f the slow-drift motions are solved numerically in the time domain under the influence o f short-crested, random waves, including viscous forces. The random wave-signal is generated by the filtering of white Gaussian noise. The slowly-varying forces are obtained using the Newman approximation and effi-cient summations o f time series. The results are compared with f u l l QTF-matrix (Quadratic Transfer Function-matrix) computations of the exciting force. The use of a robust random number generator and the Fast Fourier Transform allows for efficient simulations o f long records of the slow-drift motions, and the study o f its statistical parameters. The sensitivity upon the simulation length, transients, drag-coefficient and directional spreading are demonstrated. Copyright © 1 9 9 6 Elsevier Science Limited

E L S E V I E R P I I : 8 0 1 4 1 - 1 1 8 7 ( 9 6 ) 0 0 0 1 5 - 6

1 I N T R O D U C T I O N

Floating production systems have become a very impor-tant part of the exploration and extraction of hydrocar-bons f r o m the sea-bottom, and are considered the only feasible solution for safe and cost efficient oil and gas production at large depth. Unlike gravity platforms, such systems undergo large excursions f r o m their mean posi-tion, due to the influence of wind, waves and current. I t is therefore vital for the design of moorings and riser systems that the magnitude of the motions are predicted well and that they are kept within preset bounds.

The motions of moored floating platforms in deep wa-ter are typically characwa-terized by two distinct time-scales; the fast and the slow time-scales. The former is of the or-der of 10 seconds, which is the dominant period in the wave-spectrum of interest. The slow time-scale is defined by the natural period of the mooring system, which is an

1 *Present address: Hydro A l t j m i n i u m a.s., R & D Centre, N-4265 Havik, Norway

order of magnitude larger than the wave-frequency due to the relatively large mass of the system and the weak restoring forces. The slow motions, which are excited by the slowly varying forces f r o m the environment, are typi-cally much larger than the fast motions, due to the weak restoring forces of the mooring system and the highly tuned low-frequency resonance of the system. The reso-nant slow-drift motions are only bounded by the relevant hydrodynamic damping. The damping is due to both ideal fluid effects and the damping associated with the viscos¬ ity o f the fluid. Ideal fluid effects are dominated by the wadrift damping, which is linear i n the slow-drift ve-locity and second-order i n the wave-amplitude. The vis-cous forces mainly arise f r o m the flow-separation around the structure, the riser and the mooring lines.

The motions of floating structures in real sea-states are considered random processes, due to the random excita-tion. It is therefore necessary to obtain sufficient infor-mation about the statistical properties of the motions in order to predict extreme motions. There exist statistical

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models, based on the statistical properties of the driving forces of the system.' However, the applicability of the models is limited, due to simplifications of the physical model and the complexity of solving the Fokker-Planck-Kobnogorov equation for multi degree-of-freedom sys-tems. Slow-drift motions in the frequency domain are also used in the design of floating production systems. A major limitation of this method is that i t accepts only linear terms in the motions, which excludes, for example, responses to Morison-type drag forces and nonlinearities due to the large excursion of the structure. The use of time-simulations combined with statistical models have proven valuable in the predictions of extreme motions, due to their flexibility with respect to nonlinearities i n the physical model. However, the slow time-scale of the slow-drift motions requires that the length of the simula-tions are very long, in order to obtain useful information about the statistical properties. The hydrodynamic forces are obtained f r o m time-series o f the frequency-domain forces, since the direct solution of the linear and second-order problem in the time-domain would lead to a pro-hibitive computational task.

I n the current study, efficient routines have been devel-oped for the simulation of the slow-drift motions. Real-izations of the random wave-elevation is achieved by fil-tering of white Gaussian noise using the 'random phase-random amplitude' approach. The slowly varying forces are expressed in terms of products of single Fourier inte-grals, which are evaluated by Fast Fourier Transforms. A simple Morison-type formula is employed to account for viscous effects i n the fluid. The method is flexible w i t h re-spect to including other nonlinearities i n the model. The method has been implemented i n the computer program S W I M , which is sponsored by the industry, and results are presented in order to establish convergence of statis-tical quantities and to investigate the sensitivity to simu-lation parameters.

2 M A T H E M A T I C A L F O R M U L A T I O N

The differential equations which governs the slow-drift \surge-sway-yaw motions and the simulations of the

rel-evant forces are presented next.

Slow-drift equations of motion

The equations of the slow-drift motions of a floating body can be derived f r o m Newton's second law and a multiple time-scales analysis to separate the fast time-scale mo-tions f r o m the slow time-scale momo-tions. We will foflow the approach i n Sclavounos,^ which derives all potential flow forces f r o m first principles. A simple model to ac-count for the effects of viscous drag in the fluid will also be included.

The slow-drift surge-sway-yaw motion of the struc-ture is defined by the horizontal position (Xoit), Y^it), 0)

and the yaw angle 6(;) of the slow-drift coordinate sys-tem xyz, relative to the mertial coordinate syssys-tem XYZ, as illustrated in Fig. 1. The magnitude of the slow-drift velocities XoiO, Yo(t) and 0 ( 0 are here assumed smaU, whereas no assumptions are made with respect to the mo-tions Xoit), Yoit) and 0 ( 0 . I t is here convenient to define the hydrodynamic forces i n terms of the slow-drift frame xyz, where the slow-drift motions and velocities appear as parameters. However, the slow-drift equations will be given w i t h respect to the inertial frame, by the transfor-mation matrix [S], defined below. Based on the deriva-tion in Sclavounos,^ and neglecting second-order terms in the slow-drift velocities, the surge-sway-yaw slow-drift equation matrix is thus written

[S][M+A][Sf Yo [S][Bit)][Sf

(Di{t)\

+fAXo, Yo, 0,0 + [C] Yo = [S] D2(t) [p6(t)J

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where [S] is the transformation matrix, [M] is the inertia matrix, [A] is the added-mass matrix, [B{t)] is the wave-d r i f t wave-damping matrix anwave-d [C] is the restoring matrix. The matrices are defined as follows:

COS0 - s i n 0 0 ' / l l 0 0 "

[S] = s i n 0 COS0 0 , [ M ] = 0 hi 0

0 0 1 0 0 _

[A] =

An A\2 A\e All All A26

A(,\ A(,2 A(,6

[C] =

CxX CxY CxQ

CYX CYY CYQ

C@x CQY CS@

[B{t)] =

Buit) Bnit) Buit) Bidt) Biiit) Bjdt) Bexit) B(,iit) B(,dt)

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The subscripts 1, 2 and 6 refer to surge, sway and yaw in the slow-drift system, while the subscripts X, Y and 0 refer to the corresponding directions in the inertial system. lu = hi is the mass of the structure and he the moment of inertia about the z-axis. The center of gravity and the origin of the slow-drift frame are assumed to be along the same vertical line, such that hj = 0 for 1 = 1,2, j = 6 and / = 6, 7 = 1,2. The Aij coefficients are the double-body flow added-mass coefficients. The viscous drag force J^y iXo, %, 0, 0 wiU be discussed in the following section for a single degree of freedom system.

Simulation of second-order forces

The simulation o f t h e second-order wave forces 5,7(0 and A ( 0 , which for simplicity wül be denoted Fit), is dis-cussed next. We will only consider long-crested irregular waves, but the extension to include short-crested waves follows along the same lines, and is given in Emmerhoff.-'

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Slow-drift motions of offshore structures 57

Y A

W [x.((),r.(0] ^

O

Fig. 1. The inertial coordinate system XYZ, and the slow-drift coordinate system xyz following the slow-drift motions o f the structure, here illustrated by four vertical

cylinders.

The time-history of F{t) i n an irregular sea-state is ob-tained by the following expression:

Fit) = 2 ^ Z ( a ) i , Xo, Yo)Z{(V2, Xo, Yo)

J V — CO

xF{coi,co2,@)e~'^'"''""'^'dwid(j02 (3) where F{coi, W2, @) is the force due to waves with unit

amplitudes and frequencies coi, CÜ2 and Z{(Vi,Xo, Yo), Z{iü2, Xo, Yo) are the complex wave amphtudes. As written i n eqn (3), the force F{t) includes both high¬ and low-frequency components, and the double inte-gral is not easily computed f o r very long simulations. The high frequency components are removed by i n -troducing the low-frequency force F~ (coi, 002. @) = 2U - sgn(coi)sgn(a)2)]F((oi, C Ü 2 , 0 ) , where sgn(a)) = ± 1 for CO < 0. Further, the Newman approximation is employed i n order to obtain the following expression for the low-frequency force F'(t):

F-{t)

= ( 2 T T ) - 2 | I Z ( a ) i , Z o , 7o)i^~ ( 0 ) 1 , - 0 ) 1 , 0 ) e-*'""daj,

X Z ( a ) 2 , Xo, Yo) e - ' " ' ^ ' d c ü 2

sgn(a)i)Z(a)i, Xo, Yo)F-{wi, - w i , 0 ) e-'^-'do).

-Foo

s g n ( a ) 2 ) Z ( a ) 2 , Z o , Yo) doo2 (4)

The double Fourier integrals in eqn (3) have here been re-placed by single Fourier integrals, which are evaluated by efficient FFT-routines. Realizations of the random wave-amphtudes are obtained as described later i n this section.

The expression (3) can alternatively be evaluated us-ing the exact values of F~(.(joi, ~cü2, 0 ) , when the

off-diagonal terms are poorly approximated by the off-diagonal terms due to low resonant periods of the excited modes or wide-banded wave-spectra. A n efficient method has been developed by Emmerhoff et al. which makes use of a spline representation of F " ( c o i , -0)2, 0 ) with respect to the variables coi, 0 ) 2 in order to write the double integral (3) i n terms of products of single integrals. The compu-tational effort of obtaining the second-order force F{t) without using the Newman approximation is thus of the same order as i n expression (4).

The random wave-amplitudes Z(ooi,Xo,Yo) and Z ( a ) 2 , Xo, Yo) are functions o f the slow-drift motions Xo and Yo, and the force F~ {coi, 002, 0 ) is a function of the yaw rotation of the structure, when given with respect to the slow-drift system. The second-order forces BijU) and D , ( 0 therefore depend on the slow-drift motions, which complicates the solution of eqn (1). We will show, however, that the forces can be computed at a mean slow-drift position, without introducing much error.

Simulation of the random wave-amplitudes

The simulations o f the random wave-amplitudes Z ( a ) i , Xo. Yo) i n eqns (3) and (4) are described next. Con-sider the complex variable W{co) = WR{IÜ) + iWi((v), where i is the complex number such that i ^ = - 1 , and WR{W) and Wjico) are independent white Gaussian noise (WGN) processes, with zero mean and variance equal to T T . The wave-amplitudes Z(co, Xo, Yo) are then obtained by the ffitering of l^(co) as follows:

Z(60, Xo, Yo) = H(uj)W(co) CO > 0

Z*{-(jo,Xo, Yo), o x 0 (5)

where H(co) = 5'^(a))i/2ei'<(ôcos;io+rosin^„)^ ^ê asterisk ( * ) denotes complex conjugate and SI^{(JO) is the two-sided wave-spectrum. Here, /3o defines the angle between the direction of propagation and the X-axis. Assuming deep water, the wave number i< and the frequency cu are related by u)^ = gK. The wave-elevation ît) follows f r o m the Fourier identity and is given by

2TT Z(a), Xo, Yo) e-''"'da) (6)

By the theorem of superposition of Gaussian variables it follows that ^(t) is a Gaussian process, and i t can be shown that the power spectrum of expression (6) is iden-dcal to S^icjo). The representation of ^ ( 0 is referred to as 'random phase—random amplitude', since the com-plex wave-ampKtudes Zi(v,Xo, Yo) can be rewritten as Z{w,Xo, Yo) = A{(jo)é'^"'\ where A(.cu) and e(a)) are real random processes w i t h Rayleigh distributions and uniform distributions, respectively.

Realizarions of the W G N processes Wnito) andWiico) for discrete values of co are obtained by efficient routines which employ a u n i f o r m random number generator and

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20 25 P-[DEG1

Fig. 2. Relative error of slow-drift motion rms using fixed wave-heading /3 relative to structure.

an analytic filter to obtain the Gaussian distribution. The random number generator is capable of producing se-quences of 2^^ independent numbers, which corresponds to a non-repetitive signal with length T = 136 years, us-ing 1 second time-steps.

Evaluation of forces at mean position

As mentioned earlier, the second-order forces are func-tions of the slow-drift surge, sway and yaw response, due to the phase change of the incident wave and the change of wave-heading, cf eqn (3). The amplitude o f t h e surge and sway slow-drift motions are typically of the order o f the dimension of the structure, while the slow-drift yaw amphtude is of the order of 10°. The major contribution to the slow-drift response is attributed to waves that are much longer than the dimensions of the structure, and the forces are therefore not very sensitive to the change of slow-drift posirion. The forces are however more sensitive to the change of yaw-rotation, due to the rapid change of the wave field between the cylinders f o r different wave-headings.

The difference between using the exact slow-drift pa-rameters Xo{t), Yoit), @{t) and the mean values Xo,

Yo, 0 , in the calculations of the second-order forces are shown in Fig. 2 for long-crested seas. I t is seen that an er-ror o f approximately 10% is introduced by assuming zero yaw-rotation in the computation o f the forces at wave-heading /? = 5 M n short-crested seas the error is expected to be proportional to a weighted average of the values i n Fig. 2, depending on the dominant wave-direction. The error of calculating the forces at the mean position was also investigated and was found to be less than 1%. The computation time o f t h e second-order forces is greatly re-duced by calculating the forces at the mean surge, sway and yaw positions, which allows the expression (4) to be computed by F F T prior to solving for the slow-drift mo-tions.

Based on the above discussion, we conclude that the second-order forces can be computed at the mean

posi-tion of the structure, except when the yaw response is large or is considered critical for the design of the structure.

3 H Y D R O D Y N A M I C F O R C E S

I n the following section the evaluation of the hydrody-namic forces i n eqn (1) are discussed. The effects due to potential flows and viscous flows are accounted for sep-arately

Potential flow forces

Due to the small forward speed of the structure, the iner-tial forces are modelled by the double-body flow added-mass, assuming that wave-effects are negligible at low speeds.

The computation o f the second-order forces Bij{t) and Di{t), represented by eqn (4), requires the knowledge of the mean force F~{oj, -UJ, 0 ) in regular waves with fre-quency CO. The mean force on a floating structure w i t h a small forward speed is obtained by solving for the linear hydrodynamic flow, expanded i n the slow-drift velocities UJ, and calculating the mean forces correctly io O {Uj), as derived i n Emmerhoflf & Sclavounos^ and Emmerhoff.-' The index j denotes surge, sway and yaw i n the body flxed frame of reference. The linear flow around vertical cyhnders is calculated by employing the theory of L i n t o n & Evans,^ with extensions to include the forward-speed effects. The pontoons are accounted for by a long-wave theory. Neglecting terms of 0{Uj), the mean force on the structure in direction ; is then written as

Fi = Di BijUj (7)

where a summation over the index j is understood. The term D; is the zero-speed mean force on the structure. The coefficients Bij denote the wave-drift damping coef-ficients, and represent the correction of the mean force due to the forward speed.

The computarions of the wave-drift damping coeffi-cients B\ 1, B(,(, and the drift forces D\, D(, for typical plat-f o r m are shown in Figs 3-5. Comparisons are made with resuhs f r o m Nossen et alJ and the radiation/diffraction code W A M I T , developed at M I T .

The off-diagonal terms F " { t ü i , - C ü 2 , 0 ) of the zero-speed exciting force were obtained in order to study their influence on the slow-drift motions. The QTF-elements were computed efficiently by the indirect method using the technique o f integration i n the complex plane for the free-surface integrals. The surge exciting force o n a platform in bi-chromatic long-crested waves is shown in Fig. 6, compared with the results f r o m the second-order module of W A M I T .

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Ka

Fig. 3. Wave-drift damping Bi\ for a four-leg plat-f o r m with cylinder radii=10 [m] centered on a square w i t h sides=70 [m] and draft=30 [m]. The cylinders are standing on four pontoons w i t h breadth=20 [m] and

height=14 [ m ] .

(0),+tD2)/2

Fig. 6. Surge second-order exciting force (zero-speed) on a concrete floater for the difference frequencies Acu =

W l - 0 ) 2 = 0, 0.1 and 0.2.

Fig. 4. Wave-drift damping yaw moment due to slow-d r i f t yaw motion o f the platform i n the previous figures.

Fig. 5. Surge d r i f t force and yaw d r i f t moment on the platform used in Fig. 4.

Viscous forces

A significant part o f the damping force on floating struc-tures is due to viscous effects. They arise f r o m the sep-aration of the flow around the structure and its moor-ing or tether system caused by large amphtude slow-drift responses. Significant viscous effects are also known to arise f r o m the wind flow around the superstructure. A n ambient current may also amplify the viscous effects and alter the solution o f the free surface problems. A l l these effects may be accounted for by the multiple time-scale framework employed in the present study, assuming ap-propriate physical models are available.

Comprehensive surveys on the physics and modeUing of viscous drag due to the steady and unsteady flow around bluff bodies are given by Sarpkaya & Isaacson^ and Faltinsen.^ I n connection with the slow-drift response of offshore structures, viscous separation must be at-tributed to the large amplitude of the slow-drift responses, rather than the small amplitude of the fast-scale oscflla-tions. Viscous forces are thus expected to vary apprecia-bly over the slow time scale and their effect must be ac-counted for i n the slow-drift equations of motion. While extensive research is underway on the modelhng of un-steady flow separation around bluff bodies, models based on Morison's equadon enjoy widespread use i n practice. It expresses the viscous force as a nonlinear function of the relative velocity between components of the structure and the ambient flow.

Two models are presented for the viscous force on a single degree of freedom system. I n one model the effects due to the slow-drift velocity U and the hnear velocity u are added separately, eqn (8), thus allowing f o r separate coefficients f o r the slow and the fast-scale responses. The other model is a funcrion o f t h e total relative velocity U + u, thus combining the slow and the fast-scale velocities as given by eqn (9). The force d F on a horizontal strip o f a vertical cylinder and the integrated force i ; are given by

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or

dF=^pDdz{CDsU\U\ + CDLu\ii\), (8)

dF = ipDdzCs{U + u)\U + u\. (9)

dF (10)

-T

KN=11 KN=12

-10 O 10 •10 O 10

The force is integrated along the vertical axis, includ-ing effects f r o m the wave-zone, which contributes signifi-cantly to the mean force on the structure.

A steady current is often present and must be ac-counted for i n the modeUing of the slow-drift responses of floating structures. The presence of current alters significantly the nature of the viscous flow around the structure by introducing an appreciable ambient flow with steady velocity. Therefore, when modelled after Morison's equation, the drag coefficients must be se-lected to reflect the presence of a current. The effects of a current may be easily accounted for in eqns (8) and (9) by introducing an apparent slow-drift velocity which is the sum of the structure slow-drift velocity and the neg-ative current velocity. This apparent slow-drift velocity replaces the slow-drift velocity in the definition o f the slow-drift damping coefficients Bjj.

4 R E S U L T S

Results f r o m simulations o f the slow-drift motions of floating platforms in random waves are presented. The computer program S W I M , which is the implementation of the method presented in this article, was used i n the simulations.

Results f r o m the simulations of the wave-elevation, which is a major underlying stochastic process o f the slow-drift motions, are presented flrst in terms o f its probability density function (pdf) and power spectrum. The following are resuhs f r o m simulations o f the slow-drift motions. The convergence with respect to simula-tion length and transients are studied. The sensitivity to short-crested seas are investigated. Two models for the viscous drag were employed and the sensitivity to the drag-coefficients are shown, as well as the effects o f a steady current.

A l l results are presented in terms of the mean and rms values of the slow-drift motions. The values are obtained f r o m time-averaging of the simulated records. Due to the nature of the slow-drift motions, very long records are required in order to get rehable estimates of the statistical parameters. The mean and rms are therefore computed f r o m a large number of relatively short runs (3-6 hours), where transients due to the start up of the system are removed.

KN=13 KN=14

Fig. 7. Computed probability density functions f o r the wave-elevation (bars) compared with the Gaussian

dis-tribution (solid line), for simulation lengths T = 2 * ^ .

Wave-elevation

Simulations of the wave-elevation in long-crested random seas, expressed by eqn (6), are presented. The conver-gence of the p d f with respect to the simulation length is compared with the Gaussian distribution, and is shown in Fig. 7. The results for the length T = 2''*[ sec] = 4.55[ hours] seems very close to the expected distribu-tion. The energy contents of the different spectral com-ponents i n the signal are compared with the input JON-SWAP spectrum, illustrated i n Fig. 8. The spectrum was computed using a maximum entropy method with some smoothing. The spectrum of a single reahzation is not ex-pected to compare well with the input spectrum for short simulations, since the wave-amplitude and thus the com-puted spectrum are stochastic processes. The variance of the wave-elevation, which is an integrated quantity o f the spectrum, does however converge for increasing simula-tion length.

Slow-drift simulations

The slow-diift motions are obtained by solving the gov-erning eqn (1). Realizations of the slowly varying forces B,jit) and Z), (f) are calculated at the mean position prior to solving for the motions. The equations are solved nu-merically by a Runge-Kutta scheme. A sample o f the slow-drift motions of a typical platform is shown in Fig. 9. A time-step of 1 second was found to be an appropriate resolution of the time axis, and was used i n obtaining the results that follow.

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KN=14 KN=15

2 0.4 0.6 0.8 1 2 0.4 0.6 0.8 1

l<N=16 KN=17

2 0.4 0.6 0.8 1

Fig. 8. Computed spectral density o f the wave-elevation (dashed-line) compared w i t h the input spectrum (solid

line), for simulation lengths T = 2*'^.

+ R m s of mean X R m s of rms X )( + + X ₊ 15 16 17 m

Fig. 10. Top and center plots: boxplots which display mean and rms o f the surge slow-drift motions, based on 1000 samples, each o f duration t = 2'^^ [sec]. 50% o f the data lie within the upper and lower edges o f the box and all data lie within the whiskers. The horizontal line inside the box marks the average o f the data. Bottom:

rms o f quantities i n the plots above.

Tims (min]

Fig. 9. The slow-drift response o f a typical p l a t f o r m i n long-crested waves w i t h ^ = 3 0 ° . The surge and sway motions are given in [meters] and the yaw rotation i n

degrees.

Length of simulations

The measured mean and rms values o f t h e simulated slow-d r i f t motions slow-depenslow-d strongly on the length of the sim-ulations. Figure 10 shows boxplots of the mean and rms values, which are based on 1000 samples, each with length 2 ^ ^ [sec]. The natural period of the platform was approxi-mately 200 [sec], and a simulation length of 2' ^ [sec]

there-fore corresponds to approximately 40 slow-drift cycles. The rms of the mean and of the rms becomes smaller for longer simulations. The large scatter for relatively short simulations is due to the small number of cycles of the slow-drift motions.

Transients

The simulations of the slow-drift motions are started up with a prescribed position and velocity of the body at t = 0. These specified initial conditions are most likely differ-ent f r o m what would have been the case i f the simulations had started at some time before t = 0. The start-up of the simulations are therefore usually associated with a tran-sient period, before the motions have reached a 'steady-state'.

Figure 11 displays the change of the calculated rms of the slow-drift motions as a function of time. The rms was calculated for consecutive segments, each of 10 slow-drift cycles. Some 25 000 simulations were performed and the mean value of the rms is shown in the upper plot. The lower plot shows the rms of the rms. The results clearly show different statistical parameters for the first 10 slow-drift cycles compared to the converged values. The initial position o f the body in the simulations was selected at the mean position and the initial velocity an average velocity (based on the average of the slow-drift amplitudes). A longer transient period was found when the initial position and velocity simply were set equal to

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• O 10 20 30 40 Number of slow-drift cycles

Fig. 11. The rms o f the surge slow-drift motion (top) and rms o f the rms (bottom), based o n the data o f 25000 simulations. Each bar represents the simulation

of approximately 10 slow-drift cycles.

7.0 F

0.0 1.0 2.0 3.0 4.0

Fig. 12. Mean deflection and rms o f t h e surge slow-drift motion as a function o f the drag-coefBcient CDL, using

Cos = 0.7.

zero (not shown in the figure).

Viscous forces and current

The sensitivity to the drag-coefficients CDS, CDL and Co, which appear i n eqns (8) and (9), are shown in Figs 12¬

14 i n terms of the mean and rms of the surge slow-drift response. The values o f the drag-coefficients f o r design purposes must be determined f r o m model tests or ex-perimental data, in order to properly model the viscous damping of the system.

The effects of a steady current are demonstrated i n Fig. 15. The current enters into the equations of motion as an apparent slow-drift velocity, which alters the

wave-7.0 F

.1

6.0 1.0 -0.0 t 1 1 1 1 1 1 : 1 1 1 1 , , , I . . , _ 0.0 1,0 2.0 3.0 *0 Co

Fig. 14. Mean deflection and rms o f the surge slow-drift motion as a function o f the drag-coefficient Qi.

drift and viscous damping. Since the current is indepen-dent o f the slow-drift velocity, the forces due to the cur-rent act as i f they were exciting forces, which can be seen f r o m the results. The effects of the current should be re-flected in selecting the drag-coeflRcients.

Short-crested seas

The simulations of the slow-drift motions i n short-crested seas have been studied. I n the present method the simu-lation of the slow-drift motions i n short-crested seas are performed by adding a number o f 'long-crested' inde-pendent waves f r o m different directions, accounting for the interactions between waves. The directional wave-spectrum is

^ „ ( a ; , ^ ) = . S „ , ( a j ) [ c o s / 3 ] N C ° ^ (11) where a cosine distribution was used i n the simulations.

The JONS WAP spectrum was used for S^iuo) and ^ de-notes the angle between the wave-direction and the X-axis. NCOS is an even integer between 2 and oo, where

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Slow-drift motions of offshore structures 63 7 5

Current-velocity [m/s]

Fig. 15. Mean deflection and rms value o f slow-drift motions in surge as a function o f the current-velocity U.

Table 1. Tlie rms of the slow-drift surge, sway and yaw re-sponse. A/? denotes the spacing between the wave-headings

and A'^ is the number of wave-headings. A ^ Surge Sway Yaw 2.8 63 2.912 0.9293 1.410 5.6 31 2.926 0.9344 1.409 11.2 15 2.905 0.9388 1.419 22.5 7 2.916 0.9273 1.407 45.0 3 2.985 0.9489 1.147

the upper limit corresponds to long-crested seas. The sim-ulations are sensitive to the number of wave-directions Np and the spacing between the directions, as shown m Table 1. The incident waves were distributed between

f^min = -90° and p,„ax = 90°. I t is seen that the re-sults using Np = 1 directions differ by less than 0.25% f r o m the results using Np = 63. Since the simulations in short-crested seas increases the computational time by the square of Np, the statistical parameters can be obtained accurately with as few as Np = 1 wave-directions. The sensitivity to the spreading parameter NCOS is shown in Fig. 16. The surge rms is reduced by approximately 35% f r o m NCOS = oo to NCOS = 2.

Exact off-diagonal elements

The simulations of the slowly varying forces using the ex-act off-diagonal elements of the exciting force, presented in the previous section, were studied. The time history of the forces were efficiently simulated using the method described in Emmerhoff al.'^ Preliminary resuhs f r o m simulations with the exact exciting force QTF-matrix for the surge slow-drift motions i n long-crested waves show differences of less than 5% i n the rms values compared with the results usmg Newman's approximation. More research remains, however, in order to determine the

dif-1.0 h

' I . 1 . 1

0.0 0.1 02 03 0.4 0.5

1 / N C O S

Fig. 16. Mean and rms of the surge slow-drift motions as a function o f the inverse o f the directional spreading

coefficient NCOS.

ferences under more general conditions and for vertical modes of motions.

5 C O N C L U S I O N S

A method for simulations of the surge-sway-yaw slow-d r i f t motions of offshore structures in short-cresteslow-d ran-dom waves is presented. The slow-drift motions are de-coupled f r o m the fast (linear) motions by a multiple time scale analysis, which allows the equations of motion to be written in terms of the slowly varying forces only. The wave-induced forces are computed by the summation of quadratic time series, which are here reduced to products of single time series whether the Newman approximation or the exact Q T F elements were used. The viscous forces are modelled by a Morison-type formula. Realizations of the random wave amphtudes, which enter the expressions for the slowly varying forces, are obtained by the filtering of white Gaussian noise produced by a random number generator. The method allows for very long simulations of the random forces and motions, which are obtained numerically by a Runge-Kutta scheme.

Results are presented for the second-order frequency domain exciting forces and wave-drift damping coef-ficients where good comparison with other numerical methods are shown. Simulations of the slow-drift mo-tions and sensitivity studies of the different parameters involved are presented. The convergence of statistical pa-rameters for increasing simulation length are established and the length of transients associated with the start-up of simulations were detemiined to approximately 10 slow-drift cycles. The slow-drift motions are sensitive to the selection of drag-coefficents, which must be deter-mined f r o m model tests or experimental data. The effects

i of a steady current affect the mean and rms values of the

slow-drift motions significantly. The difference between simulations i n long-crested and short-crested waves are shown to be moderate, and relatively few discrete

(10)

wave-directions are required in order to obtain accurate results. Ttie simulations of the surge-sway-yaw slow-drift re-sponse of offshore structures are presented. The vertical modes heave-roll-pitch, however, may be o f significance for a certain type of structures. The present method will therefore be extended to include these vertical modes of motion, which will apply f o r a wide range of offshore platforms and ships.

A C K N O W L E D G E M E N T S

Financial support for this study was provided by the M I T Sea Grant College Program and by a consortium of industrial sponsors of the S W I M (Slow Wave Induced M o -tion) Project, consisting of Aker, Amoco, Conoco, Norsk Hydro, Norwegian Contractors, Saga, Statoil and Veritas Research.

R E F E R E N C E S

1. Johnsen, J. M . & Naess, A . , Time variant wave drift

damp-ing and its effect on the response statistics of moored off-shore structures. Proc 1st Int. Offoff-shore & Polar Engng Conf. (ISOPE-92), Edinburgh, U K (1991).

2. Sclavounos, P. D., Slow-drift oscillations o f comphant float-ing platforms. Proa 7th BOSS Conference, Boston, M A , U S A (1994).

3. Emmerhoff, O. J., The slow-drift motions o f offshore struc-tures. P h D thesis, M I T (1994) pp. 525-68.

4. Emmerhoff, O. J., K i m , S. & Sclavounos, R D . , Time-simulations of second-order forces. Proa 10th Int. Workshop on Water Waves & Floating Bodies. Oxford, U K (1995). 5. Emmerhoff, O. J. & Sclavounos, P. D., The slow-drift motion

o f arrays o f vertical cylinders. J. Fluid Meek, 242 (1992) 31-50.

6. Linton, C M . & Evans, D . V , The interaction o f waves with arrays o f vertical circular cylinders. J. Fluid Mech. 215 (1990) 549-69.

7. Nossen, J., Grue, J. & Palm, E., Wave forces on three-dimensional floating bodies with small forward speed. J. Fluid Mech. Ill (1991) 135-60.

8. Sarpkaya, T. & Isaacson, M . , Mechanics of Wave Forces on Offshore Structures. Van Norstrand Reinhold, N Y , 1981. 9. Faltinsen, O. M . , Sea Loads on Ships and OJfshore