E L S E V I E R

Applied Ocean Research 15 (1994) 351-370 1994 Elsevier Science Limited

Printed in Great Britain. All rights reserved 0141-1187/94/$07.00

Wave and current induced motions of floating production systems

Odd F a l t i n s e n

Division o f Marine Hydrodynamics, Norwegian Institute o f Technology, 7034 Trondheim - NTH, Norway

Mathematical models for slow drift motions of floating production systems are discussed and weaknesses are pointed out. Results from a comparative study of numerical prediction programmes of slow drift motions are presented and discussed. A large scatter in the results is reported. The main error source is due to damping. Model tests of floating production systems in waves and current are discussed and the necessity to use an ocean basin is documented. The effect of the platform on the hydrodynamic loads on risers is discussed.

INTRODUCTION

Environmental loads due to wind, waves and current cause mean and oscillatory motions of floating pro- duction systems that are critical for mooring lines and risers. The wave loads can be classified as wave frequency loads, low frequency loads and high frequency loads. By high and low frequency loads we mean relative to the wave frequencies of practical interest. There are mean loads included in the low frequency loads. The high frequency loads can cause ringing and springing of Tension Leg Platforms (TLPs). The low frequency loads cause important slow drift oscillations and for a large-volume structure with small waterplane area it causes also slowly varying heave, pitch and roll oscillations. By a large-volume structure we mean that the incident wavelength is sufficiently small relative to characteristic dimensions of the structure, so that the incident waves are modified by the presence of the structure. A rough estimate for a vertical circular cylinder is that the cylinder is a large- volume structure when A/D < c.10(D = diameter).

The wave field around a large-volume structure can be significantly influenced by the presence of a current. 1 This implies that there is an interaction between wave and current loads. The changes in fluid accelerations and velocities due to the presence of a large-volume structure will obviously influence the wave and current loads on the risers.

In the main text we will concentrate on second-order mean and slowly oscillating wave loads and response. The interaction with the current loads will be discussed. The linear wave loads and responses are, in general, possible to determine accurately by model tests or numerical codes like three-dimensional (3-D) diffraction

351

programmes for large-volume structures. One exception is rolling close to the roll resonance of a ship or a barge, where viscous damping is important. Viscous effects cannot be totally neglected for an accurate determina- tion of heave and pitch motions of ships with sharp corners like bilge keels. 2-4 Application of a 3-D diffraction programme requires that the hull is physi- cally modelled correctly. This means for instance that the dimensions of the surface elements used in the numerical approximation are sufficiently small relative to a wavelength and that a higher density of elements is used in areas of high surface curvature.

We will give results for 'academic' bodies like a half sphere and truncated vertical cylinders floating in the free surface. Results for real structures like a TLP, a turret moored production ship (TPS) and a moored deep draft floater (DDF) will also be presented. The D D F and the TPS are presented in Figs 1 and 2 and were used in the Norwegian research programme FPS 2000. Twenty-three institutions from all over the world participated in a comparative study of numerical predictions of wave frequency and low frequency oscillatory loads and responses of the TPS and the DDF. Both an operational condition and a design condition were selected. The operational condition corresponded to significant wave height HI~ 3 - - 6 m and mean wave period T 2 = 8.5s, while the design condition corresponded to H1/3 = 15.5m and T2 = 1 3.5 s. Longcrested waves were assumed. The waves had a 10 heading for the TPS and head sea was used for the DDF. Current was accounted for. The mooring system was represented by linear springs. Some results from this study have been reported by Hertford and Nielsen 5'6 and Nielsen and Herfjord. 7 They concluded that first-order quantities can be computed with a high level of

352

I

I

Odd Faltinsen

cc=70m

c.g . , , z = 9 8 . 0 n

I I

)

z= -150m

~ Anchor li connecti(

Do = 119.0m

: I I I

y

Geometry of Deep Draft Floater.

Fig. 1. Main dimensions of the DDF.

accuracy, while second-order loads and motions show a computed results are also due to differences in physical considerable amount of scatter. Some of the scatter in models. HerOord and Nielsen 6 concluded that c. 80% of the results can be explained by human errors due to the scatter in the low frequency motion response is due input errors and how the structures were modelled, to the variation in damping, The remaining 20% is However, some of the reasons for the differences in the mostly due to variations in drift force estimates. We will,

Wave and current induced motions of floating production systems 353

AFT SHIP

z=3.0m~7

v

8,9, 10

765 4 3 2 1 0

z FORE SHIP

/ z =9.0 m

z=-15.0m !0 19 18 17 16

(a)

J c.o. 0 1 2 3 4 5 (~ 7 B 9 10 1'1 1'2 13 1'4 1'5 16 17 1'8 1'9 20

~t1~'1 no.

x= 115.m x= -115.m AP (b) FP

Fig. 2. Profile and sections of the TPS; (a) Bodyplan distance between sections 11.5m. Half breadth= 20.5m. Draft= 15.0m. (b) Profile of ship. Definition of coordinate system, y-axis positive in starboard direction.

in the main text, discuss the damping formulation in detail. Main contributions come from wave drift damping, anchorline damping and viscous damping from the hull. The discussion of the viscous damping term has relevance for the hydrodynamic analysis of risers. The formulation of the viscous damping term is similar to the drag term in Morison's equation. A difference is that the Keulegan-Carpenter number is small when studying viscous damping from the hull, while it can be large in the riser analyses. Hydrodynamic analyses of risers are discussed more in the main text.

Table 1 gives numerical results for extreme low frequency motion amplitudes and extreme combined low frequency and wave frequency motion amplitudes for the DDF and the TPS in the design and the operating condition. The translatory motions are for a point on the structure in the still water plane. The horizontal coordinates for the motion reference point are the same as the horizontal coordinates for the centre of gravity of the structure. The results in Table 1 are obtained by averaging all computed results except results which were obviously due to human errors. For the low frequency motions the standard deviations of the different computed results for each motion

amplitude were typically of the order of magnitude of 50% of the mean values presented in Table 1. The heave and pitch motions of the TPS are mainly due to linear wave effects and are not presented in Table 1. Table 1 shows that low frequency motions, in general, dominate the motion amplitudes in surge, heave and pitch for the DDF and for surge, sway and yaw for the TPS. In the design condition the yaw amplitude for the TPS is 13.7 i.e. larger than the mean wave heading of 10 . The surge amplitude of the TPS is more than twice as large as the surge amplitude of the DDF. The extreme value of the surge amplitude of the DDF in the design condition is of the order of magnitude of the column radius.

In the main text we will start with formulating the equations of slow drift motions and concentrate on the discussion of mean wave loads and slow drift damping. For additional details and background on linear wave frequency and slow drift motions the reader is referred to Faltinsen. 8 The work presented in this paper does not represent a state of art survey of all the contributions made by researchers in this field. It concentrates on earlier work in which the author has been involved. We will only discuss results for long- crested waves. This does not mean that shortcrested seas

354 Odd Faltinsen

Table 1. Estimates of extreme values o f motions of a D D F and a T P S (see Figs 1 and 2) in operating condition Hi~3 = 6.0 m, T 2 = 8.5 s and design condition H1/3 = 15.5m, T 2 = 13-5s. Head sea for DDF; 10 heading for T P S

Motion Platform Condition Extreme Extreme combined low frequency wave frequency amplitude and low frequency

amplitude

Surge DDF

TPS

Heave DDF

Pitch

Sway TPS

Yaw

Operating Design

Operating Design

Operating Design

Operating Design

Operating Design

Operating Design

3.05 m 3.09 m 5.05m 9.17m

7.76m 8.64m 17.52 22.15m

0.084m 0.12m 0.74m 0.95m

1.0 1.01" 2.1 o 2.89

3.55m 3.68m 7.03 m 7.79 m

5.63 5.71 13.57 13.72

and widely spread crossing waves are not important. It is an area where research is going on. Krokstad 9 has given a state of the art survey and presented analytical and numerical work on second-order loads in multi- directional seas.

EQUATIONS OF MOTIONS

In this section we will describe the equations of motions for slow drift oscillations of a moored structure in irregular seas. Longcrested seas will be assumed. By slow drift we also include a mean displacement. Wind loads will not be discussed. This does not mean that its influence can be neglected. For instance, wind gusts can cause horizontal slow drift oscillations of a moored platform.

We will define a Cartesian coordinate system that is fixed in space. The origin of the coordinate system is in the average mean water plane. The z-axis is vertical and positive upwards. When the structure is in the average position, the z-axis goes through the centre of gravity of the structure. At the same position the x-z plane is a symmetry plane for the submerged part of the structure. For a ship the x-axis is in the aft direction. Let the translatory displacements in the x, y and z directions with respect to the origin be rh, ~72 and rl3, respectively, so that ~71 is the surge, rl2 is the sway, and r/3 is the heave displacement. Furthermore, let the angular displacement of the rotational motion about the x, y and z axes be r/4, r/5 and r/s, respectively, so that r/4 is the roll, r/5 is the pitch and r/6 is the yaw angle. The linear wave fre- quency motions and the slow drift motions will be denoted 77(1) and r/(2), respectively.

The six coupled differential equations of slowdrift motions can be written, using subscript notations, in the

following abbreviated form:

6 r _(2/+ & + RA Z [ ( M j k + Ajk)/'J~ 2) + BjkC?~ 2) + "~jk'lk

k = l

= Fj(t), j = 1"6 (1)

where Mjk are the components of the generalized mass matrix of the structure; Ajk, Bjk and Cjk are the linear added-mass, damping and restoring coefficients; Dj and Rj are damping and restoring terms that have a nonlinear dependence on the slow drift velocities and motions; Fj(t) are the slow drift excitation force and moment components; F1, F2 and F3 refer to the surge, sway and heave exciting forces, while F4, Fs and/76 are the roll, pitch and yaw exciting moments. Dots stand for time derivatives; Ajk can be calculated by the 'rigid,wall' condition on the free-surface, and Cjk are due to the mooring system as well as hydrostatic and mass considerations. Nonlinear restoring effects due to the mooring system can matter, but will not be discussed in detail. This means Rj = 0 are set equal to zero in the presented results. We will discuss the excitation loads and damping terms later in more detail. We will then see that Dj are interpreted as viscous damping effects. In practice, many of the coupling terms in eqn (1) are zero. For instance, the slow drift surge motion of a ship is normally analysed as an uncoupled motion. Further, the slow drift sway and yaw motions are coupled, while slow drift heave, pitch and roll motions of a ship are generally disregarded.

Limitations of mathematical models

Bjk and Fj(t) in eqn (1) depend strongly on the linear wave frequency motions of the structure, while it is

Wave and current induced motions of floating production systems 355

common to assume that the slow drift motions do not influence the first-order motions. This may not always be true. Consider for instance a sea state with a moored ship performing large amplitude roll reso- nance oscillations. The slow drift sway motions cause a frequency of encounter effect, that changes the energy spectrum of the linear wave frequency excita- tion loads. This may influence a lightly damped roll response. This can be illustrated by Fig. 3, which shows results from free decay tests o f slow drift sway motions o f a moored ship model in regular beam sea waves. The ship is described by Faltinsen et al. lO The wave period was 1.92s, and the natural roll period derived from roll decay tests was 2.3s. There is obviously a frequency of encounter effect on the measured roll motion, but we cannot explain com- pletely the results by assuming a steady-state response of the roll motions as a function of the frequency of encounter between the waves and the slow drift motion. There seems to be a memory effect in the roll motion, due to small roll damping.

Resonance roll motion also causes other difficulties in the analysis. Nonlinear viscous effects are significant in the estimation of roll at resonance. This implies that a linear first-order roll motion does not exist at resonance. This causes inaccuracies in the prediction of second- order hydrodynamic forces.

The linear wave frequency motions are calculated for the mean wave heading. It was pointed out in the Introduction that the slow drift yaw motions can be large. The linear wave frequency motions for the instantaneous wave heading due to the slow-drift yaw motion will differ from the linear motions in the mean wave heading. This may influence the results.

Instabilities of the lateral motions may occur in wind, waves and current of a turret-moored ship. This depends on the longitudinal position of the turret and can be analysed in a similar way as described by Faltinsen et al.ll for a single point mooring system. The instabilities cause slowly varying motions that cannot be modelled by the equation system described above.

Slowly varying motions of a D D F in current can occur due to 'lock-in'. 'Lock-in ' may also occur in small sea states when a current is present. The ' lock-in' oscillations are connected with vortex shedding around the columns. Presently there exists no theoretical method that can analyse the phenomena. The 'lock-in' oscillations of DDFs have been observed in model tests and can be critical for the mooring lines and risers. The possibility of ' lock-in' for the D D F (see Fig. 1) can be examined by relating the vortex shedding period Tv to the natural period of surge, sway, roll, pitch and yaw motions of the DDF. Tv can be determined from the Strouhal number. We will exemplify this by studying a case where the current is in the x-direction. The uncoupled natural periods in

1.0

0.5

0.0

-0.5

-1.0

-1.5

SWAY (t~

i\ 4 / V

"Iv

" 1 r - q l l r , - q F - r ' , I w , . -

220. 240. 80. 100. 120. 140. 180. 180. 200.

30?.

20..

10.,

0.'

-10.

-20.

-30. 80. 100. 120. 140. 160. 180. 200. 220. 240.

Fig. 3. Slowly varying motion and first-order roll motions measured in free decay tests of a moored ship in sway. Ship is described in Faltinsen et a1.1 Incident waves are regular and in beam sea direction. Natural roll period is in vicinity of wave

period.

sway and roll are, respectively, 176 and 46.5 s. To our knowledge there does not exist any published results for the Strouhal number for a cluster of four columns as in Fig. 1. However, we can use published data for two cylinders in tandem and side-by-side arrange- ments. We will denote the distance between the cylinder axes by s. An important parameter is s/D, where D is the cylinder diameter. We will first study a situation with only the two front cylinders in Fig. 1 present. According to data presented by Zdravko- vich, 12 the interaction between the cylinders will not be strong when s/D = 3-5. His data are for subcritical flow, but there is no reason to believe that the interaction should be strong for transcritical flow. As a first-order approximation we can, therefore, neglect the interaction between the two rows of cylinders in Fig. 1. Let us then study the interaction between two cylinders in tandem configuration corresponding to the flow around the two cylinders in one of the rows of cylinders in Fig. 1. Also, in this case s/D = 3.5. Okajima 13 has presented results for two circular cylinders of equal diameter in tandem in transcritical flow. The front cylinder was rough. However, this is not believed to have a significant effect on the measured Strouhal number in the transcritical flow regime. Okajima presented data for s/D = 3.0, which we can use. These data show a Strouhal number of 0.13, i.e. a quite low value relative to the Strouhal number for one single cylinder. Due to the negligible interaction between the flow around the two rows, we will use 0.13 as the Strouhal number for the whole platform. The Strouhal number is defined as

356 Odd Faltinsen

St =D/(TvU), where U is the current velocity. For instance when U = l m/s, Tv - -54s . This illustrates that 'lock-in' oscillation in sway are possible.

The amplitudes of sway oscillations can be of the order of magnitude of the column diameter. In reality, it is important to study the influence of sheared current profiles on 'lock-in'. In current and large waves, where the waves cause motions over substantial part of the columns, the vortex shedding from the columns is interrupted in a manner that 'lock-in' oscillations do not O C C u r .

Statistical values

If only the mean values and the standard deviations are wanted, eqn (1) can be solved in the frequency domain when the Dj-terms are linearized and the Bjk-terms are time-independent. Pinkster 14 has shown how to do this for slow drift response of a one degree of freedom linear system with second-order slow drift excitation loads. Relative to a time-domain solution this is a very robust and time-efficient way to obtain mean values and standard deviations of the motions.

Prediction of extreme values depends strongly on what mathematical model is used for the slow drift motions. Zhao and Faltinsen is studied the influence of a slowly varying wave-drift damping. They showed that the slowly varying wave-drift damping had little influence on the standard deviations of the motions while it had a significant effect on the extreme values. The results from the simulations showed that the extreme values of the motions tended to follow a Rayleigh distribution if slowly varying wave-drift damping was included. (No nonlinearities due to the mooring system were included.) This means the most probable largest value Xmax in a storm of time duration t can be written as

The standard deviations are nearly the same, while it is obvious that the extreme values differ. About 20 realizations of the same sea state were used to get good estimates of the most probable largest slow-drift motion amplitude.

Slow-drift excitation loads

Newman's approximation 16 is frequently used in calculating slow-drift excitation loads. The important building blocks in the expressions are the mean wave forces and moments in regular waves. There may be cases where Newman's approximation does not yield good results. First of all it is based on potential flow, and wave radiation and diffraction have to be important. This may not be the case in design wave conditions. Newman's approximation can be thought of as a Taylor expansion of the second-order transfer function for slow-drift excitation loads about zero difference frequency. The smaller the difference fre- quency is, the better the approximation is. This means that the approximation is better the larger the natural periods of the slowdrift oscillations are. To exemplify this we can consider the D D F presented in Fig. 1. The natural periods in heave, pitch and surge are, respect- ively, 35.3, 46.5 and 176s. This means that the approximation is expected to be better for surge motion than for the heave and pitch motion. If the second-order slow-drift transfer functions have pronounced maxima or minima for zero difference frequency, Newman's approximation is less good. Pronounced maxima occur for instance when the frequency is close to the natural frequency of the heave motion, and the heave damping is low. The reason is that the second-order transfer functions depend strongly on the linear wave-induced motions.

t ,~1/2 Xmax -- trx 2 log -~---~} (2)

where tr x and TN are respectively the standard deviation and the natural period of the slow-drift response variable. For a storm of duration 10h and T N = 100s this means Xmax = 3"43Crx. However, more work is needed both hydrodynamically and statistically to achieve more accurate and reliable estimates of extreme values for slow-drift motions. Equation (2) should be understood as a rough estimate.

To get good estimates of extreme values of slow-drift motions from model tests or numerical simulations a long simulation time is needed. Figure 4 presents results from numerical simulations. (Transient effects have been excluded.) Each record simulates the response of the same system in the same sea state. The differences in the results from one time series to another are due to random selection of phase angles and wave amplitudes.

Mean wave loads

In the following we will discuss mean wave loads according to potential theory. They can be calculated either by a direct pressure integration method or by using conservation of momentum in the fluid. We will refer to a method presented by Zhao et al. 17 and Zhao and Faltinsen. is It is a boundary element method using Rankine singularities in the near field and a coupling with an analytical expression at larger distances from the body. Interaction with current is accounted for and it is important for blunt bodies to consider the interaction with the local steady flow.

The interaction between waves and current can have an important effect on the mean wave loads. The reason is that the presence of the current changes the wave picture around the structure and that the mean wave loads according to potential theory are connected with the structure's ability to create waves. We will

Wave and current induced motions of floating production systems 357

-,,, ,' .v,..-,,, "T "

t - ._o

E

I.. " o

_o ( / )

I | I I 0 2501 '0 5002"0 7503 '0 1 0 0 0 4 ' 0

T i m e (s ) ~__

Fig. 4. Identical simulations of slow-drift motions of a moored 2-D body. The differences in the results are due to random selection of phase angles and wave amplitude.

illustrate this with an example with a hemisphere floating in the free-surface in incident regular waves. In calm water the centre of the hemisphere is at still water level. Numerical results for drift forces and heave motions are presented in Figs. 5, 6 and 7. Drift forces are presented for both a restrained model and a hemisphere that is free to oscillate in surge and heave. We note a significant influence of the current speed. Increasing current speed in the wave propagation direction implies both higher drift force and heave motion. Current velocities in the opposite direction imply lower response values than without current. We may also note that a strong amplification of the heave motion occurs at heave resonance. For U ~ = 0.064, the maximum heave motion is 2.4 times the incident wave amplitude. Large drift forces occur in the vicinity of heave resonance for the free floating model. A partial explanation is that large heave motions imply large radiated wave amplitudes due to the body. I f we translate the results into physical quantities and choose a diameter D = 100m, U V ~ = 0.064 corresponds to U = 2m/s . At resonance in heave this implies about 50% higher drift force relative to that with zero current velocity. If the flow does not separate around the

structure in combined current and waves, the mean current loads based on wave free conditions should not be added to the mean wave drift forces. When flow separation occurs, it is more uncertain what to do.

Zhao et al. 17 presented experimental values for the hemisphere. The tests were done in one of the towing tanks in the ship model basin in Trondheim. A towing carriage was used to simulate the current. The towing tank had a breadth of 10-5 m and the hemisphere had a diameter of 1 m. Wall interference effects were unavoidable and the numerical method was generalized to include the effect of the tank walls by using images about the tank walls. Figure 8 presents results for zero current velocity and U = 0.2 m/s. The effect of reflec- tions from the tank walls is very important and the results show clearly the limitations of using a conven- tional towing tank to evaluate wave-current loads on large-volume structures. An ocean basin with the breadth and length of the same order of magnitude would be preferable. The agreement between theory and experiments is reasonable, but the experiments show some scatter. This is mainly due to inaccuracies in the measurement of the incident wave amplitudes. This is more important for mean wave loads than for linear

358 Odd Faltinsen

1.60-

120-

0.80

040-

000 , - - O0

j . . ,~ ,,,,,,.~' ~_ .~........o

f 0'5 I/0 1.'5

U 0.000 ..o-o..o- ~ :

,-o,.c),.o. ~ : 0.032

u .-,-,.,.-+. - -~ - -= o,06z~

-,w.. x.--w- U .......... 0.032

210 w__2 R g

Fig. 5. Numerically calculated horizontal drift force $'1 on a restrained hemisphere in current and regular waves, p = mass density of water, g = acceleration of gravity, G = incidimt wave amplitude, D = diameter, R = radius, U = current velocity. (Positive value means current direction coincides with propagation direction of the incident waves), u~ = tv 0 + kU (w0 = incident circular wave

frequency, k = incident wave number). 17

wave effects, since the mean wave load is proport ional to the square of the incident wave amplitude.

The hemisphere is a relatively easy structure to study by a numerical method when the tank wall effects are not accounted for. Zhao and Faltinsen Is presented results for a vertical circular cylinder penetrating the free surface and having a draught-radius ratio of 0.25. This particular structure gave problems when the direct pressure integration method was used to calculate mean wave loads. An example of results for horizontal

mean wave loads is presented in Fig. 9 as a function of ~v~R/g, where w0 is the circular frequency of oscillation of the incident waves without current present. Both for zero and non-zero current speed we note an important difference in the calculations based on direct pressure integration and the results based on the equations for conservation of momentum in the fluid. The panel dimensions on the body were of nearly constant equal length. The reason for the differences in the results is that the direct pressure integration method is sensitive to

~ 1.60

1.20

080

0.40

~ ' - ~ . + j .+

000 --" i 04 2.0

~ . R 0

U - ~ . ~ --- 0.000

- 4 3 - 0 0 . U : 0.032

-+.+-.I-. U = 0064

-w- w.-w. - ~ - - --- 0 032

Fig. 6. Numerically calculated horizontal drift force $'1 on a hemisphere in current and regular waves. The hemisphere is restrained from oscillating in pitch. Symbols explained in Fig. 5.17

Wave and current induced motions of floating production systems 359

1~131 ~o 2.40-

1.80 -

120

0,60

000

U ~ : oooo

--O-O-[3. U : 0.032

. U

1 : - * - * - ' + " ~ 0 . 0 6 / .

% ,.-% O0 0.5 110 1'.5 2.0

tO..~2 R g

Fig. 7. Numerically calculated first-order heave amphtude [rt3[ of a hemisphere in current and regular waves. The hemisphere is 1 - - " ~- 17 restrained from oscillating in pitch. Symbols explained "n Fig. J.

the distribution of the elements in the vicinity of the corner at the bottom of the cylinder. Small elements had to be used around the corners to get the direct pressure integration method to agree with the conservation of momentum method. The most important parameter in the calculations by the direct pressure integration method was the distri- bution of the elements on the vertical side close to the corner. The reason was associated with the contribution from the velocity square term in Bernoulli's equation, which is singular, but integrable a t the corner.

In Fig. 10 are shown numerical results for vertical drift forces on a vertical cylinder that is free to oscillate in surge and heave and restrained from oscillating in pitch. The incident wave propagation direction is in the positive x-direction. The draught of the cylinder is equal to the cylinder radius. The current velocity is zero. The panel dimensions of the body are of nearly equal length. The largest difference between the two different methods occurs in the vicinity of heave resonance. The reason for the differences is again that the direct pressure integration method is sensitive to the distribution of the elements in the vicinity of the corner between the bot tom and the vertical side of the cylinder. When the direct pressure integration method is used to calculate the vertical mean wave force around heave resonance, the contribution from the velocity square term in the pressure is large and of opposite sign to the other contributions in the integral over the body surface. The absolute values of these terms are nearly equal to the velocity square term. This means a high accuracy is needed in the integration over the body surface.

SLOW-DRIFT DAMPING

The hydrodynamic damping can be divided into wave- drift damping, anchor line damping and viscous damping from the hull. In principle there is also a damping contribution from the risers. Anchor line damping will not be dealt with here, but can be substantial} 9 In model tests it is important to model the anchor lines properly. Geometrically, this sets requirements on the transverse dimensions and the depth of an ocean basin.

Wave-drift damping

The wave-drift damping has the same physical explana- tion as wave drift forces and slowly varying excitation forces due to wave radiation and diffraction. It is connected with a structure's ability to create waves. This means it is important for a large-volume structure. No interactions with viscous effects are assumed in the calculations of wave-drift damping. The wave- drift damping is linear with respect to the slow- drift oscillation amplitude and proportional to the square of the wave amplitude for moderate sea conditions. In principle one may say that wave- drift damping matters for all modes of motions, but in the present state of the art it is only evaluated for the translatory horizontal motions. The presence of wave-drift damping can be seen comparing free decay model tests of a large-volume structure in still water and in regular waves. We can explain the wave-drift damping in surge or sway by interpreting the slow- drift surge or sway motion as a quasi-steady forward and backward speed. It is well known that mean wave forces on a structure are speed dependent (see, e.g.

360 Odd Faltinsen

~- Pg~a

1.00-

075

0.50

0 2 5 -

x

x

0.00 , , , 080 1.00 120 1.Z,0 1.60

(,,0 2 --T-R

U = 0000

& THEORY (13 IMAGE)

- - - THEORY (WITHOUT TANK WALL INTERFERENCE)

x EXPERIMENT

1.00

075

0.50 -

025 -

x

x x x x

x

x

THEORY (13 iMAGE)

- - - THEORY (WITHOUT TANK

EXPERIMENT

0.00 ' '2 i/~ ' 080 1.00 1. 0 1 0 160

Wz R g

Fig. 8. Comparison between experimental and numerical results for horizontal drift force El on a restrained hemisphere in current and regular waves. Symbols explained in Fig. 5.17

Fig. 5). We can interpret the term in the mean wave force that is proportional to the speed as a damping term.

Figure 11 presents numerical and experimental wave-drift damping coefficients B~l D in the surge of a TLP in regular head sea waves. The model tests were performed in the ocean basin in Trondheim, which has a length, breadth and maximum depth, of, respectively, 80, 50 and 10m. The TLP is pre- sented in Fig. 12 and Table 2. There is good agree- ment between experiments and theory. The results show that negative wave-drift damping can occur. This is due to wave interaction effects between the columns. The wave-drift damping goes to zero for long periods. The reason is that the wavelengths

are so large relative to the cross-dimensions of the columns that the incident waves are unaffected by the structure. Viscous effects will then matter. This will be discussed in the section on viscous damping.

Faltinsen and Zhao 2 argued that there also ought to be a slowly varying wave-drift damping in irregular sea if there is a slowly varying excitation load. The physical reasons are the same for the slowly varying wave-drift damping and the slowly varying excitation loads.

Mean and slowly varying wave-drift damping in irregular sea can be calculated in the same way as mean wave loads and slowly varying excitation loads in irregular sea.

Wave and current induced motions of floating production systems 361

1 2 -T-pgl~o (2R}

~;-

%~000

/

/ i i i /

/ X X X

/~ -KI-D- / . /

,j 01500 01600 01SO0 t ',2o0 "~'Z' R

-~= 0.000 -~ based on momentum and

u 0.0479J energy relations

= 0.0000 "~ direct pressure integration

- , ~ = 0.0479 ...i

Viscous damping from the hull

Viscous damping is mainly a consequence of a flow separation. The main contributions come from pressure forces. However, viscous friction forces may matter, for instance, at low Keulegan-Carpenter number and

Viscous friction damping is important for slowly varying surge of ships in small sea states.

We will concentrate on eddy making damping, which is normally expressed in terms of drag coefficients. The drag coefficient is, for instance, a function of

Reynolds number when the boundary layer flow is laminar. The latter condition may occur in model tests.

4.500 -

3.000

1. 500

0.000

-1.500 I I I i 10.000 0.300 0.600 0900 1.200

Reynolds number. Surface roughness number k i d (k = characteristic

Direct pressure integration

/ ~ Based on momen!um ergy relations

(1)o 2 ~ ' R g

Fig. 10. Numerical results of vertical mean wave force t)3 with direct pressure integration method and a method based on conservation of momentum and energy. The body is a vertical cylinder that is free to oscillate in surge and heave and restrained from oscillating in roll. The draught-radius ratio is 1"0. Element length on the body is nearly constant. Zero current velocity. Symbols

explained in Fig. 5) 8

Fig. 9. Numerical results of horizontal drift force F2 for U / ~ = 0.000, U ~ = 0.0479 with direct pressure integration method and a method based on conservation of momentum and energy. The body is a fixed vertical cylinder with draught-radius ratio of

0.25. Symbols explained in fig. 5) s

362 Odd Faltinsen

8, 2 p ~,,',EI~,

10.0

5.0

0.0

-2.5

,~ :'- -"- THEORY (ZHAO & FALTINSEN)

EXPERIMENTS

Fig. 11. Wave-drift damping B~I in surge for the TLP presented in Fig. 12 as a function of wave period T. (a =

incident wave amplitude, D c = column diameter. 1

cross-sectional dimension of the roughness on the body surface; D =characteristic length of the body).

Keulegan-Carpenter number K C = UMT/D (UM =ampli tude of oscillatory relative velocity between the body and the ambient flow; T = oscil- lation period).

The nature of the oscillatory relative velocity between the body and the ambient flow.

The ratio between the current velocity U and UM (relative current number).

Body form. Free surface effects (or other boundary effects).

The reader is referred to Sarpkaya and Isaacson 21 and Faltinsen s for more details on how Co depends on the parameters mentioned above.

The magnitude of the drag force depends upon whether flow separation occurs. For a body with a sharp corner, flow separation occurs for any KC number. For ambient planar oscillatory flow past a

circular cylinder separation occurs for KC > 1-2. In combined oscillatory flow and current past a body without sharp corners, flow separation will depend on both KC and U/UM. When U/UM > 1, the flow will always separate. This corresponds to the situation where the free stream velocity component in the current direction does not change direction with time. We will illustrate this by visual observations of the flow around a hemisphere in regular waves. The current was simulated by towing the model against the wave propagation direction. The results on occurrence of separated flow are presented in Fig. 13 with KC on one axis and U/U~ on the other axis. In reality, we have to account for the local flow around the hemisphere. This will be a function of KC, ~ v ~ and U/UM. The results presented in Fig. 13 are for different values offaL'Dig and do not indicate a clear dependence on wv/D/g. On the other hand, it should be realized that the variation of w ~ in the tests may be too small. For instance, if we had tested a condition with very high frequency waves, we would have expected negligible waves on the downstream side of the sphere. This implies that the flow would separate for all values of U~ UM. Even if the flow never separated when U/UM < 1-0 in our tests, we cannot rule out that this can happen for other values of ~vv/-D/g. It is likely that the Reynolds number and roughness ratio will also influence separation. We have not systematically investigated these effects. In our case the sphere was hydraulically smooth. More work is needed to quantify when flow separation occurs in combined current and waves. On the other hand, the results in Fig. 13 and the previous discussion imply that the flow will not separate in many practical situations involving wave-current interaction effects on large-volume structures.

If the boundary layer flow is laminar and no flow separation occurs, the viscous forces are linear. It means that, if a drag force formulation is used, CD goes to infinity when KC ---, 0.

For elongated structural elements it is common to use strip theory in combination with the 'cross-flow principle'. In the case of current only it is known that the current direction should not be too close to the longitudinal direction of the structural element for the

'I

g

Fig. 12. Main dimensions of the hull of a TLP (Cf. Table 2).

Wave and current induced motions o f floating production systems

Table 2. The main parameters in full scale and model scale of the TLP presented in Fig. 12

Full scale Model scale

TLP displacement 106.520 tonnes 852 kg Platform weight 76.676 tonnes 613 kg Draught 37.50 m 0.75 m Column diameter (Dc) 25.0 m 0.5 m Pontoon height (H) 11.30 m 0.226 m Tether diameter (Dr) 0.81 m 0.016 m Stiffness in surge (k) 985 kN/m 394N/m

'cross-flow principle' to be valid. However, one cannot directly use the information from steady incident flow for unsteady ambient flow and oscillatory motions of the structure. I t is likely that the 'cross-flow principle' can be used for a broader variation of incident flow directions relative to the structure. This discussion has relevance, for instance, for a turret-moored ship in bow seas. For small wave headings the surge motion may be larger than the sway motion (see Table 1). It is an open question if the 'cross-flow principle' and strip theory can be used to formulate the drag damping in sway and yaw.

In formulating viscous slow-drift damping it is not common to account for the first-order velocities. We will show an example where it mattered to account for the first-order velocities in analysing the slow-drift damping. This was necessary in order to understand free decay model tests of a TLP in regular incident waves with a period of 15s (full scale) and wave height 12.4m (full scale). The TLP is presented in Fig. 12. The horizontal viscous force per unit length F1 on a strip of a column was written as

PCDD(ClI2)+rIR )it/1 + I (3) Ff = ~ .(1),, .(2) //~)

363

where//~) is the relative first-order horizontal velocity between the column and the incident waves. An equivalent linear slow-drift damping in surge Bfl can be found from energy considerations. It follows that

B~aTrwlr/12) 12 = I : ' F~T(712) dt (4)

where Ts is the period of the slow-drift oscillation, 1012>1 is the amplitude of 7712), and Fl r is the total horizontal viscous force on the TLP written in a similar way as eqn (3). The importance of viscous damping due to the linear wave effects in the free decay tests will increase from the short waves and low amplitudes to the large waves and large amplitudes. In the case of high wave periods and wave amplitudes, the relative horizontal motion between the first-order platform motions and the wave particle motions can dominate the slow-drift motion and have a significant influence on the prediction of viscous drag forces and viscous slow-drift damping. Figure 14 shows a comparison between using eqn (4) as damping in a simulation model and results from free decay model tests in regular head sea waves. A CD value of 0"7 was used for the columns and 1.3 was used for the pontoons. These CD values can be questioned, but the main purpose here is to stress that one cannot always neglect the influence of first-order velocities in formulating slow-drift viscous damping. Figure 14 shows a qualita- tively satisfactory agreement between theory and experiments.

CD values can be obtained experimentally by either free decay tests or U-tube experiments. Obviously there are scale effects, in particular when flow separation does not occur from sharp corners.

There have been many attempts to solve numerically separated flow around marine structures. Examples of

KC= 211:~a O

1.0

0.5

O FLOW SEPARATION CLEARLY OBSERVED

NO FLOW SEPARATION x DIFFICULT TO DECIDE

x x O0

~ A ~ / X x n 0 0 0 O

i ds 1'0 ts u ..... t

Fig. 13. Occurrence of separated flow around a hemisphere in regular waves and current (U = current velocity, KC = 21ra/D, Ca = incident wave amplitude, D = diameter, UM = 21r(a/T, T = incident wave period). Current and wave propagation directions

coincide) 7

364 Odd Faltinsen

methods used are:

Vortex sheet model 22. Discrete vortex method 2~. Combination of Chorin's method and vortex-in-

cell method 24-26. Navier-Stokes solvers 27.

A general description of the state of the art is that the methods are generally limited to two-dimensional (2-D) flow, and that the methods have documented satis- factory agreement in some cases, but that they are presently not robust enough to be applied with confidence in completely new problems, where there is no guidance from experiments. Due to lack of proper turbulence modelling of wakes it is difficult to simulate the flow around a cylinder in the near-wake of another cylinder. The vortex sheet model has a clear advantage in simulating separation from sharp corners, in particular for small Keulegan-Carpenter numbers. The vortex-in-cell method and Navier-Stokes solvers can more correctly handle separation from continuously curved surfaces.

Figure 15 shows simulation of oscillatory ambient p lana r flow past a rectangular cross-section by the vortex-in-cell method. The ambient flow is harmonic and the initial velocity is zero. The sharp corners represented numerical difficulties that were handled by the approximate method described by Scolan and Faltinsen 26. The drag coefficient was found to be about 4.0 for a Reynolds number of 104 and KC = 2.0. These results will be used later in the discussion of viscous damping of the D D F and TPS presented in Figs. 1 and 2.

Comparative numerical studies of slow-drift damping

As a part of the Norwegian research programme FPS 2000 a comparative study of numerical computation of slow drift damping for a TPS and a D D F was performed. The D D F and the TPS have been presented in Figs. 1 and 2. Some results on extreme values of motions have been presented in Table 1.

The comparative study concentrated on slow-drift

Surge (m) - - Numerical simulations :~ Experiments

~ i \~'- / " ' 1

~rno (s)

Fig. 14. Free decay tests of a TLP in regular head sea waves. Only viscous damping included in numerical simulations. Full scale wave period= 15s. Full scale wave height= 12.4m. The TLP is presented in Fig. 12 and Table 2. Comparisons with

model tests. )

damping contributions from the hull. Anchorline damping was neglected. In practice, this may have an important effect. The hull effect can be divided into viscous and potential flow effects. The most important potential flow effect is wave-drift damping. Strictly speaking there is an interaction between viscous and potential flow effects.

We will discuss the results for the D D F and the TPS separately. The methods used by the participants will be

.~.~,,~ . . . . .

, - - -

"%i,'L~

Fig. 15. Simulation of oscillatory ambient flow past a rectangular cross-section by the vortex-in-cell method presented by Scolan and Faltinsen. 26 T = oscillation period; t = 0 is initial time; ratio between the length and the height of the rectangular is 1.4; KC = 2.0. Ambient flow direction is in

the length direction of the rectangle.

Wave and current induced motions of floating production systems 365

described. It will be demonstrated that the predicted damping level can vary significantly. Recommendations for how the damping shall be predicted by the present state of the art will be discussed. A general comment is that the state of the art in predicting slow-drift damping is far less advanced than prediction of slow-drift excitation forces. Both effects are of equal importance in predicting slow-drift motions.

DDF results

The procedures used by the different organizations are presented in Table 3. In some cases we do not have sufficient information to state what methods have been used. More detailed explanation of the table will be given in the following text. An important viscous damping source is due to pressure drag. The horizontal viscous damping force per unit length on a strip of a column of the DDF can be written a s

P CDD( (2 + t :+

Tab

le

3.

Slow

-dri

ft

dam

ping

in

su

rge

and

pitc

h fo

r D

DF

(s

ee

Fig

. 1)

. C

ompa

riso

n of

m

etho

ds

(Hi/

3 =

6"0m

, T2

= g.

5s,

U =

0-5

m/s

),

Des

=

Des

ign

cond

itio

n (H

i~3

= 15

.5m

, T

2 =

13,5

s, U

=

l-0m

/s))

Org

aniz

atio

n nu

mbe

r H

as o

nly

drag

-dam

ping

pe

rcen

tage

eC

nA(5

c(~)

+ U+

)(2

))13

(R1)

-[- U

+ ~

(2) I

of

cri

tica

l da

mpi

ng

Cur

rent

C

oupl

ed

Tim

e F

requ

ency

Co

E

ffec

t be

en

mod

es

dom

. do

mai

n/

of

used

? st

ocha

st.

1. o

rder

li

near

iz,

velo

c.

used

by

Wav

e ra

diat

ion

dam

ping

diff

eren

t or

gani

zati

ons.

(O

p=

op

erat

ion

al

Wav

e S

tand

ard

drif

t de

viat

ion

of

dam

ping

lo

w f

requ

ency

Mea

n T

ime

Sur

ge

Pit

ch

val.

de

p.

(m)

(deg

.)

cond

itio

n

Rel

ativ

e da

mpi

ng

leve

l su

rge

(%)

0-7

3 N

o Y

es

No

No

Yes

?

No

No

0"7

Col

4

No

Yes

Y

es

Yes

N

o N

o N

o 1.

0 P

ont

0"7

Col

. 5

No

No

No

No

Yes

N

o N

o 2"

05 P

ont

0"7

Col

6

No

No

No

Yes

N

o Y

es

No

1"1

Pon

t 0-

7 C

ol

12

Yes

N

o 2-

0 P

ont

14

Yes

N

o

0-7

Col

18

N

o Y

es

Yes

N

o Y

es

No

No

0"9

Pon

t

20

Yes

0-7

Col

24

N

o Y

es

No

No

Yes

N

o N

o 2"

0 P

ont

1.3

Sur

ge

25

No

No

Yes

Y

es

No

No

Yes

2"

0 P

itch

1-

0 C

ol.

27

No

No

Yes

N

o Y

es

No

Yes

1 "

3 P

ont

Op.

0-

69

0.17

N

o N

o D

es.

0.82

0.

21

Op.

0.

55

0.2

No

No

Des

. 1.

5 0-

45

Op.

1.

69

0.52

N

o N

o D

es.

2.4

1-13

O

p.

1.48

?

?

Des

. 8.

13

Op.

2.

02

0.18

Des

. 2.

24

0.38

O

p.

0"8

Des

. 1-

08

Op.

1.

65

No

No

Des

. 2-

47

Op.

0.

64

Des

. 1-

13

Op.

0.

63

No

No

Des

. 0.

77

Op.

0.

96

0.37

Y

es

No

surg

e D

es.

1.62

0-

52

Op.

5.

64

0.08

3 N

o N

o D

es.

10-6

4 0-

3

15

30 1-2

1.7

1"0

1"0

10

Wave and current induced motions of floating production systems 367

reason for these large differences is that organization No. 4 has included the effect of current while organization No. 5 has not.

We have also listed the predicted values of standard deviation of low frequency surge and pitch motion in Table 3. The reason was that we wanted to see if there was any correlation between high damping level and low motion values. From the data in the table we cannot see a tendency like that. This means there are error sources in addition to the damping in the prediction of slow- drift motions.

TPS results

The procedure used by the different organizations are presented in Table 4. In some cases we do not have sufficient information to state what methods have been used.

If strip-theory and 'cross-flow principle' are used, we can write the horizontal (transverse) force per unit length on a strip of the TPS as

f ~ = 2 CDd(X)(y (2) + Vc +j'~))l) (2) + Ve +)(R)I (7)

where d(x) =local draught, )(2) = local slowdrift velo- city in the transverse direction, V = incident current velocity component in the transverse direction, and )~) = local first-order relative velocity in the transverse direction. The 'cross-flow principle' may be questionable when the incident velocity direction in a ship-fixed coordinate system is close to the longitudinal direction of the ship. This has been discussed earlier. 'Lifting' effects may have to be considered. This is a similar effect one would include in analysing ship manoeuvring or in directional stability analysis of a ship moored to a single point mooring system. However, one cannot directly use the results from ship manoeuvring to the slow-drift damping problem. From the table it appears that one organization has included quasi-steady lifting effects.

The drag coefficients that have been used vary from 0.6 to 1.0. If the 'cross-flow principle' applies, and the free surface effect is included by 'mirroring' the cross- section about the free surface, the flow is similar to that studied in Fig. 15. This suggests that a much higher Co value should be used.

Most of the organizations have included mean wave- drift damping in surge. None have included the effect in sway and yaw. Even if it is expected that the effect of wave-drift damping is larger for surge than for sway and yaw, it should not be neglected. Calculated results of wave-drift damping in the sway of the TPS indicate that. Only one of the organizations has included time- dependent wave-drift damping. It is consistent to do that.

Three of the organizations have simply set the damping as a percentage of the critical damping. This is too simplified.

We note that there are large variations in damping level. This is only documented for surge where it varies from 1 to 10% of critical damping.

From Table 4 we cannot see any correlation between the predicted damping level and the standard deviations of the low frequency surge, sway and yaw motions.

HYDRODYNAMIC ANALYSES OF RISERS

The motions of the platform influence directly the motions of the upper end points of the risers. In the following we will not discuss all aspects of hydrody- namic loads on the risers. We will focus on the effect of the platform on the flow around the risers and also try to see the similarities with the previous discussion of viscous damping. 'Lock-in' conditions will not be discussed.

The traditional way to estimate wave and current loads on risers is to use Morison's equation in combination with strip theory and the cross-flow principle. Lift-forces are not normally accounted for. Their importance is obvious in 'lock-in' situations, but they should also be studied in 'non-lock-in' conditions. The drag term in Morison's equation is similar in form to eqn (5). The parameter dependence of the drag term (and conse- quently also the mass term) has been discussed in the section on viscous damping from the hull. The Keulegan-Carpenter number is generally large in the riser analysis. Interaction between risers may influence the drag coefficient. Let us discuss this by considering an idealized example. If a cylinder is placed in the far wake behind another cylinder, it will experience a smaller incident velocity and, therefore, a smaller drag coeffi- cient if the free stream is used to normalize the drag coefficient 8. This procedure can be generalized to several risers, where the wake from one riser influences other risers. When a riser is in the near-wake of another riser, the analysis become far more complicated than the simple procedure outlined by Faltinsen 8. Zdravkovich 12 has given a survey of results for the interaction of pipe clusters in steady incident flow. For risers that can collide with each other, there is a need to study the hydrodynamic interaction and the correlation of vortex shedding between two adjacent risers.

When Morison's equation is used in the riser analysis, it seems necessary to account for the changes in fluid acceleration and velocities due to the presence of the platform, i.e. due to radiation and diffraction of the waves around the structure. However, this effect is normally not important in a '100-year design wave' condition.

In the drag term in Morison's equation it is normal to include the effect of current, but not the effect of the slow-drift velocity. These two effects may be of equal importance. For instance, consider the example with the TPS presented in Table 1. The extreme low frequency

Org

aniz

atio

n nu

mbe

r H

as o

nly

perc

enta

ge

of c

riti

cal

dam

ping

be

en

used

?

Tab

le 4

. Sl

ow-d

rift

dam

ping

in s

urge

, sw

ay a

nd y

aw f

or T

PS

(see

Fig

. 2)

Sw

ay-y

aw

drag

-dam

ping

eG, A

(P(~

) + V

c +

P(Z)

)l)~)

+ V

c +

P(2)

I

Cur

rent

Lif

ting

F

rict

ion

Wav

e ef

fect

s in

dr

ift

surg

e da

mpi

ng

Cou

pled

T

ime

Fre

quen

cy

C D

E

ffec

t S

urge

S

way

T

ime

mod

es

dom

. do

mai

n/

of

and

dep.

st

ocha

st.

1 st

orde

r ya

w

line

ariz

ve

loc.

Sta

ndar

d de

viat

ion

low

fre

quen

cy

Rel

ativ

e da

mpi

ng

leve

l

Sur

ge

Sw

ay

Yaw

S

urge

S

way

(m

) (m

) (d

eg.)

(%

) Y

aw

(%)

14

18

19

20

24

28

No

No

Yes

No

Yes

No

No

Yes

No

No

Yes

Yes

No

Yes

Yes

Yes

No

No

Yes

9

No

No

Yes

Y

es

No

No

Yes

Y

es

No

0"8(

?)

No

Yes

Y

es

Yes

N

o N

o

No

Yes

N

o 9

Yes

?

Yes

Y

es

? Y

es

Yes

N

o Y

es

0.6

No

No

Yes

Y

es

No

No

? Y

es

No

? Y

es

? Y

es

Yes

N

o N

o

Yes

N

o Y

es

1.0

No

No

Yes

N

o N

o N

o

0.8

No

Yes

N

o N

o N

o Y

es

No

No

No

(sw

ay)

Op.

2"

51

1-64

2'

93

Des

. 5.

31

2-8

5.09

O

p.

2-1

0"85

1"

5

Des

. 3-

0 1.

6 2'

9 O

p.

1"99

0"

92

2-05

Des

. 7-

6 1.

91

5-92

O

p.

1-94

N

o N

o D

es.

8-94

O

p.

1.09

N

o N

o D

es.

2.23

O

p.

2-37

1.

21

1.01

Des

. 3"

77

2"57

3"

16

Op.

2.

42

2.24

2"

58

Op.

0-

69

No

No

Des

. 1"

3 O

p.

3-91

1"

3 1.

36

Des

. 7"

45

2"47

O

p.

1"01

0'

8 N

o D

es.

3.48

0'

68

2 4 2 1

10

5

Wave and current induced motions of floating production systems 369

surge amplitude is 17-52 m in the design condition, The natural surge period is l18s. This means velocity amplitudes of the order of magnitude of 1 m/s, i.e. the same as current velocities. I f only current is accounted for and not the slow-drift velocities, the consequence is an increased damping in the riser analyses relative to a non-current situation. Since the slow drift velocity is time dependent, the consequence is a damping that is changing with time and can be larger or smaller than in a constant current situation. Including slow-drift velocities in the analysis may mean larger extreme loads on the risers.

CONCLUSIONS

Slow-drift motions dominate the horizontal motions of a floating production system in design conditions. It is also more important than wave frequency motions for heave, pitch and roll of a DDF.

The most important error source in numerical predictions of slow-drift motions is the damping. Results from a comparative study of numerical predic- tions of slow-drift damping for a TPS and a D D F show considerable scatter in the results. The same is true for the numerical predictions of slow-drift motions. Important damping contributions come from wave-drift damping, viscous damping from the hull and anchorline damping. As long as the flow does not separate from the structure, the wave-drift damping can be calculated with good accuracy. However, it is not always accounted for in numerical predictions. Viscous damping can partly be obtained by numerical methods, but the methods are generally not robust enough to be trusted in all cases. Further, the commonly used methods are restricted to 2- D flows. Free-decay tests or O-tube experiments can give valuable information on viscous damping. How- ever, scale effects, in particular for structures without sharp corners, need to be studied.

The normal way to calculate slow-drift motions does not recognize that the second-order motions can affect the linear wave frequency motions, and that instabilities and 'lock-in' can occur in special cases.

It is pointed out that interaction between waves and current can have an important effect on mean wave loads and slow-drift excitation loads. It is not known how to account for viscous effects in the prediction of combined mean and slowly varying wave and current loads. Model tests of floating production systems in waves must be performed in an ocean basin to avoid wall effects that can change the motions of the system significantly. It is also necessary with an ocean basin of sufficiently large dimensions to model geometrically the mooring system and account for anchorline damping. To get reliable estimates of extreme values of motions it is necessary to perform much longer simulations than are normal in commercial model tests. The statistical

distribution of extreme values and the effect of short- crested sea need further studies.

It seems necessary to account for changes in fluid acceleration and velocities due to the presence of the platform as well as slow-drift velocities of the platform when Morison's equation is used to calculate the loads on risers.

REFERENCES

1. Aanesland, V., Faltinsen, O., & Zhao, R., Wave drift damping of a TLP, Advances in Underwater Technology, Ocean Science and Offshore Engineering, 26, In. Environ- mental Forces on Offshore Structures and their Prediction, Kluwer Academic Publishers, Dordrecht/Boston/London, (Editors are not stated, but chairman of conference planning committee is P. Frieze), 1990, pp 383-400.

2. Beukelman, W., Added resistance and vertical hydro- dynamic coefficients of oscillating cylinders at speed. Report no. 510, Ship Hydromechanics Laboratory, Delft University of Technology, The Netherlands, 1980.

3. Beukelman, W., Vertical motions and added resistance of a rectangular and triangular cylinder in waves. Report no. 594, Ship Hydromechanics Laboratory, Delft University of Technology, The Netherlands, 1983.

4. Faltinsen, O., On seakeeping of conventional and high- speed vessels. 15th Georg Weinblum lecture. J. Ship Res. 37, (2) (1993) 87-101.

5. Herfjord, K., Nielsen, F.G., Motion response of floating production units, results from a comparative study on computer programs, Proc. lOth International Offshore Mechanics and Arctic Engineering (OMAE) Symposium, Eds. S.K. Chakrabarti, H. Maeda, G.E. Hearn, A.N. Williams, D.G. Morris and M. Hendrichsen, Book no. G00611, The American Society of Mechanical Engineers, New York, USA, Vol. 1, (part B) 1991, pp 629-40.

6. Hertford, K., Nielsen, F.G., A comparative study on computed motion response for floating production plat- forms. Discussion of practical procedures. Proceedings BOSS'92. London, eds M.H. PateU, R. Gibbins, BPP Technical Services Ltd, London, England, Vol. 1, 1992, pp. 19-37.

7. Nielsen, F.G. & Hert]ord, K., FPS 2000 Project 1.1 Comparative study. Evaluation of results and discussion of practical methods. E & P Research Centre, Norsk Hydro, Bergen, Norway, 1992.

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