Wave and current induced motions of floating production systems

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E L S E V I E R

Wave and current induced motions of floating

production systems

Odd Faltinsen

Division of Marine Hydrodynamics, Norwegian Institute of 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 oscinatory 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 i n 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 \/D < c.lO(D = diameter).

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

I n 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 hnear wave loads and responses are, in general, possible to determine accurately by model tests or numerical codes like three-dimensional (3-D) diffracfion

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-fion of heave and pitch motions of ships with sharp corners hke bilge k e e l s . A p p l i c a t i o n of a 3-D diffraction programme requires that the hull is physi-cally modelled correctiy. This means for instance that the dimensions of the surface elements used i n 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 T L P , a turret moored production ship (TPS) and a moored deep draft floater ( D D F ) will also be presented. The D D F and the TPS are presented in Figs 1 and 2 and were used i n the Norwegian research programme FPS 2000. Twenty-three institutions f r o m all over the world participated in a comparative study of numerical predictions of wave frequency and low frequency oscillatory loads and responses o f the TPS and the D D F . Both an operational condition and a design condition were selected. The operational condition corresponded to significant wave height i f i / 3 = 6 m and mean wave period T 2 = 8-5s, while the design condition corresponded to JÏ1/3 = 15-5m and Tj = 1 3-5 s. Longcrested waves were assumed. The waves had a 10° heading for the TPS and head sea was used f o r the D D F . Current was accounted for. The mooring system was represented by hnear springs. Some results f r o m this study have been reported by H e r f j o r d and Nielsen^'* and Nielsen and Herfjord.^ They concluded that first-order quantities can be computed with a high level of

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z A

cc=70m

D=20m

Geometry of Deep Draft Floater. Fig. 1. Main dimensions of the DDF. accuracy, while second-order loads and motions show a

considerable amount of scatter. Some o f the scatter in the results can be explained by human errors due to input errors and how the structures were modelled. However, some o f the reasons for the differences in the

computed results are also due to differences in physical models. H e r f j o r d and Nielsen^ concluded that c. 80% of the scatter i n the low frequency motion response is due to the variation in damping. The remaining 20% is mostly due to variations i n d r i f t force estimates. We will,

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Wave and current induced motions of floating production systems

AFT SHIP z 1 FORE SHIP

353 z=9.0m z=-15.0m 7 6 5 4 e.g. 0 1 2 3 4 5 6 7 9 10 11 12 13 14 15 16 17 18 19 20 Section no. x=115.m AP (b) x=-115.m FP

Fig. 2. Profile and sections of the TPS; (a) Bodyplan distance between sections l T 5 m . Half breadth = 20-5 m. Draft = 15-Om. (b) Profile of ship. Definition of coordinate system, y-axis positive in starboard direction.

in the main text, discuss the damping formulation i n detail. M a i n contributions come f r o m wave d r i f t damping, anchorline damping and viscous damping f r o m the hull. The discussion of the viscous damping term has relevance f o r the hydrodynamic analysis o f risers. The formulation o f the viscous damping term is similar to the drag term i n Morison's equation. A difference is that the Keulegan-Carpenter number is small when studying viscous damping f r o m the hull, while i t can be large in the riser analyses. Hydrodynamic analyses o f risers are discussed more in the main text.

Table 1 gives numerical results f o r extreme low frequency motion amplitudes and extreme combined low frequency and wave frequency motion amplitudes f o r the D D F and the TPS i n the design and the operating condition. The translatory motions are f o r a point on the structure i n the still water plane. The horizontal coordinates f o r the motion reference point are the same as the horizontal coordinates for the centre of gravity of the structure. The results i n 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 f o r each motion

amplitude were typically of the order o f magnitude o f 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 i n Table 1. Table 1 shows that low frequency motions, in general, dominate the motion amplitudes i n surge, heave and pitch for the D D F and f o r surge, sway and yaw f o r the TPS. I n the design condition the yaw amphtude 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 o f the D D F . The extreme value of the surge amplitude of the D D F in the design condition is o f the order of magnitude o f the column radius.

I n the main text we will start with formulating the equations of slow drift motions and concentrate on the discussion o f mean wave loads and slow drift damping. For additional details and background on linear wave frequency and slow d r i f t motions the reader is referred to Faltinsen.* The work presented in this paper does not represent a state o f art survey o f all the contributions made by researchers i n this field. I t concentrates on earher work in which the author has been involved. We wiU only discuss results f o r long-crested waves. This does not mean that shortlong-crested seas

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Table 1. Estimates of extreme values of motions of a DDF and a TPS (see Figs 1 and 2) in operating condition Hiji = 6.0 m, = 8-5 s

and design condition H,/3 = 15 5m, = 13 5s. Head sea for DDF; 10° heading for TPS

Motion Platform Condition Extreme

low frequency amplitude

Extreme combined wave frequency and low frequency amplitude Surge DDF TPS Operating Design Operating Design 3.05m 5.05m 7.76 m 17.52 3.09 m 9.17m 8.64 m 22.15m Heave DDF Operating Design 0.084m 0.74 m 0.12m 0.95 m Pitch Operating Design 1.0° 2.1° 1.01° 2.89° Sway TPS Operating Design 3.55m 7.03 m 3.68 m 7.79 m Yaw TPS Operating Design 5.63° 13.57° 5.71° 13.72°

and widely spread crossing waves are not important. I t is an area where research is going on. Krokstad^ has given a state o f 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 o f a moored structure in irregular seas. Longcrested seas w i l l be assumed. By slow d r i f t 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 i n 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. A t 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 % and 773, respectively, so that ?7i is the surge, 772 is the sway, and 773 is the heave displacement. Furthermore, let the angular displacement of the rotational motion about the x,y and z axes be 774, ?75 and T/g, respectively, so that 774 is the roll, 775 is the phch and 775 is the yaw angle. The linear wave fre-quency motions and the slow d r i f t motions will be denoted 77]'' and 77^^, respectively.

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

following abbreviated form:

[{Mjk + Aj,)i)f'^ - f 5 , . , 7 ) f ) + q , 7 7 f ) - I - DJ + R;\

= m j = \ - 6 (1)

where Mj^ are the components of the generalized mass matrix of the structure; Aj^, Bj^. and Cj^. 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 d r i f t excitation force and moment components; F^, F2 and F^ refer to the surge, sway and heave exciting forces, while F4, F^ and F^ are the roll, pitch and yaw exciting moments. Dots stand for time derivatives; Aj^ 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. I n practice, many o f t h e coupling terms i n eqn (1) are zero. For instance, the slow drift surge motion of a ship is normally analysed as an uncoupled motion. Further, the slow d r i f t 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 i t is

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Wave and current induced motions of floating production systems 355

common to assume that the slow d r i f t motions do not influence the first-order motions. This may not always be true. Consider f o r instance a sea state w i t h a moored ship performing large amplitude r o l l reso-nance oscillations. The slow d r i f t sway motions cause a frequency of encounter effect, that changes the energy spectrum o f the hnear wave frequency excita-tion loads. This may influence a lightly damped roll response. This can be illustrated by Fig. 3, which shows results f r o m free decay tests o f slow d r i f t sway motions of a moored ship model in regular beam sea waves. The ship is described by Faltinsen et al}° The wave period was l-92s, and the natural r o l l period derived f r o m r o l l decay tests was 2-3 s. There is obviously a frequency o f encounter effect on the measured r o l l motion, but we cannot explain com-pletely the results by assuming a steady-state response o f the roll motions as a function of the frequency of encounter between the waves and the slow d r i f t motion. There seems to be a memory effect in the r o l l motion, due to small roll damping.

Resonance roll motion also causes other difficulties i n the analysis. Nonhnear viscous effects are significant i n the estimadon of rofl at resonance. This implies that a Hnear first-order rofl motion does not exist at resonance. This causes inaccuracies i n the prediction of second-order hydrodynamic forces.

The hnear wave frequency motions are calculated for the mean wave heading. I t was pointed out in the Introduction that the slow drift yaw motions can be large. The hnear wave frequency motions for the instantaneous wave heading due to the slow-drift yaw motion wifl differ f r o m the hnear motions in the mean wave heading. This may influence the results.

Instabihties of the lateral motions may occur in wind, waves and current o f a turret-moored ship. This depends on the longitudinal position of the turret and can be analysed i n a similar way as described by Faltinsen et al}^ 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 i n current can occur due to 'lock-in'. ' L o c k - i n ' may also occur i n small sea states when a current is present. The 'lock-in' oscillations are connected w i t h vortex shedding around the columns. Presently there exists no theoretical method that can analyse the phenomena. The 'lock-in' oscillations o f D D F s have been observed i n model tests and can be critical f o r the mooring lines and risers. The possibility of 'lock-in' f o r the D D F (see Fig. 1) can be examined by relating the vortex shedding period to the natural period o f surge, sway, r o l l , pitch and yaw motions of the D D F . can be determined f r o m the Strouhal number. We w i l l exemplify this by studying a case where the current is i n the x-direction. The uncoupled natural periods i n

SWAY( 30. 20. 10. 0. -10. -20. -30. 80. 100. 120. 140. 160. 180. 200. 220. 240. ROLL (DEG) TIME(S)

, .(Sill ill ll li III III,

ill Hi liii!!ilipiiiiiiiiiiiii!i!imiiii

1 i l iiilUiiliiiiiililiiliiii

illiiiiiiili

IP

'III 'II 1 " IF

TIIVIE(S) 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 al}'' Incident waves are regular and in beam sea direction. Natural roll period is in vicinity of wave

period.

sway and r o l l are, respectively, 176 and 46.5 s. T o our knowledge there does not exist any published results for the Strouhal number f o r a cluster of f o u r columns as in Fig. 1. However, we can use published data f o r two cylinders in tandem and side-by-side arrange-ments. We will denote the distance between the cylinder axes by s. A n important parameter is s/D, where D is the cylinder diameter. We w i f l first study a situation w i t h only the two f r o n t cylinders i n F i g . 1 present. According to data presented by Zdravko-vich,'^ the interaction between the cylinders w i l l not be strong when s/D^ 3-5. His data are f o r subcritical flow, but there is no reason to beheve that the interaction should be strong f o r transcritical flow. As a first-order approximation we can, therefore, neglect the interaction between the two rows o f cylinders in Fig. 1. Let us then study the interaction between two cylinders i n tandem configuration corresponding to the flow around the two cylinders i n one o f the rows o f cylinders i n Fig. 1. Also, i n this case s/D = 3-5. Okajima'^ has presented results f o r two circular cylinders o f equal diameter i n tandem i n transcritical flow. The f r o n t cylinder was rough. However, this is not believed to have a significant effect on the measured Strouhal number i n the transcritical flow regime. Okajima presented data f o r s/D = 3.0, which we can use. These data show a Strouhal number o f 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 win use 0.13 as the Strouhal number f o r the whole platform. The Strouhal number is defined as

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S,t—D/{TyU), where U is the current velocity. For instance when U=\m/s, 7\, = 54s. This illustrates that 'lock-in' oscillation i n sway are possible.

The amplitudes o f sway oscillations can be of the order of magnitude of the column diameter. I n reahty, it is important to study the influence of sheared current profiles on 'lock-in'. I n current and large waves, where the waves cause motions over substantial part of the columns, the vortex shedding f r o m the columns is interrupted i n a manner that 'lock-in' oscillations do not occur.

Statistical values

I f only the mean values and the standard deviations are wanted, eqn (1) can be solved in the frequency domain when the Z)^-terms are linearized and the 5y^-terms are time-independent. Pinkster^'' has shown how to do this for slow d r i f t response of a one degree of freedom linear system with second-order slow d r i f t 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'^ 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 o f the motions while it had a significant effect on the extreme values. The results f r o m the simulations showed that the extreme values of the motions tended to follow a Rayleigh distribution i f slowly varying wave-drift damping was included. (No nonlinearities due to the mooring system were included.) This means the most probable largest value x^ax in ^ storm of time duration t can be written as

f t

^max = a J 2 log — (2)

where and 7 ^ are respectively the standard deviation and the natural period of the slow-drift response variable. For a storm of duration 10 h and Tj^j = 100 s this means x^ax = SASa^- 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 f r o m model tests or numerical simulations a long simulation time is needed. Figure 4 presents results f r o m 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 f r o m one time series to another are due to random selection of phase angles and wave amplitudes.

The standard deviations are nearly the same, while i t 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'^ is frequently used in calculating slow-drift excUation loads. The important building blocks in the expressions are the mean wave forces and moments i n regular waves. There may be cases where Newman's approximation does not yield good results. First of ah it is based on potential flow, and wave radiation and diffraction have to be important. This may not be the case i n 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 exemphfy 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 176 s. This means that the approximation is expected to be better for surge motion than for the heave and pitch motion. I f 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.

Mean wave loads

I n 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 i n the fluid. We wifl refer to a method presented by Zhao et al}'' and Zhao and Faltinsen.'^ I t is a boundary element method using Rankine singularities i n the near field and a coupling with an analytical expression at larger distances f r o m the body. Interaction w i t h 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

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Wave and current mduced motions of floating production systems 357

1 0 0 0 4 0

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. I n 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. D r i f t 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 o f the heave mofion occurs at heave resonance. For U^/gD = 0.064, the maximum heave motion is 2.4 times the incident wave amplitude. Large d r i f t forces occur in the vicinity of heave resonance f o r the free floating model. A partial explanation is that large heave motions imply large radiated wave amphtudes due to the body. I f we translate the results into physical quantities and choose a diameter D = 100 m, V^JgD = 0.064 corresponds to

C / = 2 m / s . A t resonance in heave this implies about 50% higher drift force relative to that with zero current velocity. I f 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 d r i f t forces. When flow separation occurs, it is more uncertain what to do.

Zhao et alP presented experimental values for the hemisphere. The tests were done in one o f the towing tanks i n the ship model basin in Trondheim. A towing carriage was used to simulate the current. The towing tank had a breadth o f 10-5 m and the hemisphere had a diameter of 1 m. W a l l interference effects were unavoidable and the numerical method was generalized to include the effect o f the tank walls by using images about the tank walls. Figure 8 presents results f o r zero current velocity and C/=0-2m/s. The effect o f reflec-tions f r o m 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. A n 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 i n the measurement o f the incident wave amplitudes. This is more important for mean wave loads than f o r linear

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— 1.60 1.20H 0.80 H QiO 0.00 4 * s -Cy^s-Ci- = QOOO 1^ -O..Q-D- = 0,032 U u Q054 - « - x - » - ^ ^ = - 0 . 0 3 2

Fig. 5. Numerically calculated horizontal drift force on a restrained hemisphere in current and regular waves, p = mass density of

water, g = acceleration of gravity, Ca = incident wave amphtude, D = diameter, R = radius, U = current velocity. (Positive value means current direction coincides with propagation direction of the incident waves), w = OJQ + kU (wp = incident circular wave

frequency, A: = incident wave number)." wave effects, since the mean wave load is proportional 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'* presented results f o r 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. A n example of results for horizontal

mean wave loads is presented i n Fig. 9 as a function of

LOoR/g, where LUQ is the circular frequency of oscillation

of the incident waves without current present. Both f o r 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 o f momentum i n the fluid. The panel dimensions on the body were of nearly constant equal length. The reason f o r the differences in the results is that the direct pressure integration method is sensitive to

, '2 1-60 -, 120 0,80 O.W-i 0 0 0 » ^ ^ ^ w y - S — = QOOO Igö' - O - D - Q . _ y _ = 0.032 0 0 6 4 - ^ = - 0 . 0 3 2

Fig. 6. Numerically calculated horizontal drift force Fi on a hemisphere in current and regular waves. The hemisphere is restrained

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Wave and current induced motions of floating production systems ^ 2 . 4 0 -359 1.80 1.20 0.60 aoo - ^ W W > = QOOO - o o - o - = Q032 f ^ Q054 - = - 0.032 00 0 5 ~ 1 — 1.0 1.5 ~1 2.0

Fig. 7. Numerically calculated first-order heave amplitude \r]i,\ of a hemisphere in current and regular waves. The hemisphere is

restrained from osciUating in pitch. Symbols explained in Fig. 5." 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 o f momentum method. The most important parameter in the calculations by the direct pressure integration method was the distri-bution o f the elements on the vertical side close to the corner. The reason was associated with the contribution f r o m the velocity square term in Bernoulli's equation, which is singular, but integrable at the corner.

I n Fig. 10 are shown numerical results for vertical d r i f t forces on a vertical cylinder that is free to oscillate in surge and heave and restrained f r o m oscillating in pitch. The incident wave propagation direction is i n the positive x-direction. The draught o f 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 i n the vicinity of heave resonance. The reason for the differences is again that the direct pressure integration method is sensitive to the distribution o f the elements in the vicinity o f the corner between the bottom 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 f r o m the velocity square term i n 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 i n the integration over the body surface.

SLOW-DRIFT DAMPING

The hydrodynamic damping can be divided into wave-drift damping, anchor hne damping and viscous damping f r o m the hull. I n principle there is also a damping contribution f r o m the risers. Anchor line damping will not be dealt with here, but can be substantial.'^ I n 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 excitaexplana-tion forces due to wave radiation and diffraction. I t is connected with a structure's ability to create waves. This means i t is important for a large-volume structure. N o interactions with viscous effects are assumed i n the calculations of drift damping. The wave-d r i f t wave-damping is linear with respect to the slow-drift oscillation amplitude and proportional to the square of the wave amphtude f o r moderate sea conditions. I n principle one may say that wave-drift damping matters for all modes of motions, but in the present state o f the art it is only evaluated f o r the translatory horizontal motions. The presence of wave-drift damping can be seen comparing free decay model tests of a large-volume structure i n 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. I t is well known that mean wave forces on a structure are speed dependent (see, e.g.

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Fig. 8. Comparison between experimental and numerical results for horizontal drift force F-^ on a restrained hemisphere in current and regular waves. Symbols explained in Fig. 5."

Fig. 5). We can interpret the term i n 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^x" in the surge of a T L P 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 10 m. The T L P is pre-sented in Fig. 12 and Table 2. There is good agree-ment between experiagree-ments 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 reladve to the cross-dimensions o f 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^° argued that there also ought to be a slowly varying wave-drift damping in irregular sea i f there is a slowly varying excitation load. The physical reasons are the same f o r the slowly varying wave-drift damping and the slowly varying excitation loads.

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

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Wave and current induced tnotions of floating production systems 361 i - p g E ' l 2 R ) u VP = 0.000 u WR = 0.0479 u TgR = aoooo o-o- u = 0.0479 based on momentum and integration tü„2 ^ . 0 0 0

Fig. 9. Numerical results of horizontal drift force F2 for U/y/gR = 0-000, U\/gR = 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.

Viscous damping from the hull

Viscous damping is mainly a consequence of a flow separation. The main contributions come f r o m pressure forces. However, viscous friction forces may matter, f o r instance, at low Keulegan-Carpenter number and Reynolds number when the boundary layer flow is laminar. The latter condition may occur in model tests.

Viscous friction damping is important for slowly varying surge of ships i n smaU sea states.

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

• Reynolds number.

• Surface roughness number k/D [k •• characteristic

4 - p g e ' - i 2 R i 4.500 3.000 1.500 H 0000 H -1.500

Direct pressure integration Based on momentum and energy relations

1Q000 0.300 Q600 O900 1200 g

Fig. 10. Numerical results of vertical mean wave force F3 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

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Fig. 11. Wave-drift damping B^x in surge for the TLP

presented in Fig. 12 as a function of wave period T. = incident wave amplitude, = column diameter.'

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

• Keulegan-Carpenter number K C = U^T/D (J7^ = amphtude of osciUatory 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 (relative current number).

• Body form.

• Free surface effects (or other boundary effects).

The reader is referred to Sarpkaya and Isaacson^' and Faltinsen* for more details on how Cj) 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 f o r any K C number. For ambient planar oscillatory flow past a

circular cylinder separation occurs f o r K C > 1-2. I n combined oscillatory flow and current past a body without sharp corners, flow separation wih depend on both K C and U/U,,. When U/U^, > 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 o f separated flow are presented in Fig. 13 with K C on one axis and U/ (7^ on the other axis. I n reality, we have to account f o r the local flow around the hemisphere. This will be a function of K C , uj\fDjg and The resuhs presented in Fig. 13 are f o r different values o?uj^jD/g and do not indicate a clear dependence on ujyjDjg. On the other hand, i t should be realized that the variation o f oj^jD/g in the tests may be too small. For instance, i f we had tested a condition with very high frequency waves, we would have expected neghgible waves on the downstream side of the sphere. This implies that the flow would separate for ah values of i7/i7„. Even i f the flow never separated when C//C/„ < 1-0 in our tests, we cannot rule out that this can happen f o r other values of uJsjD/g. I t is likely that the Reynolds number and roughness ratio will also influence separation. We have not systematically investigated these effects. I n our case the sphere was hydrauhcally smooth. More work is needed to quantify when ffow separation occurs in combined current and waves. On the other hand, the results in Fig. 13 and the previous discussion imply that the flow wifl not separate in many practical situations involving wave-current interaction effects on large-volume structures.

I f the boundary layer flow is laminar and no ffow separation occurs, the viscous forces are linear. I t means that, i f a drag force formulation is used, C D goes to infinity when K C —)• 0.

For elongated structural elements it is common to use strip theory i n combination with the 'cross-fiow principle'. I n the case of current only h is known that the current direction should not be too close to the longitudinal direction of the structural element f o r the

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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

Platfonn weight 76.676 tonnes 613 kg

Draught 37.50 m 0.75m

Column diameter (DQ) 25.0 m 0.5 m

Pontoon height (H) 11.30m 0.226 m

Tether diameter (Dj) 0.81m 0.016m

Stiffness in surge (/c) 985 kN/m 394 N/m

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

I n formulating viscous slow-drift damping it is not common to account for the first-order velocides. We wifl 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 T L P in regular incident waves with a period of 15 s (fufl scale) and wave height 12.4 m (full scale). The T L P is presented in Fig. 12. The horizontal viscous force per unit length pf^ on a strip of a column was written as

where 77^^ is the relative first-order horizontal velocity between the column and the incident waves. A n equivalent linear slow-drift damping in surge Bfi can be found f r o m energy considerations. I t follows that

(2)|2

where is the period of the slow-drift oscillation, \ri'

(2) (2)

(4) ,(2) I is the amphtude of ri\ and is the total horizontal viscous force on the T L P 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 f r o m the short waves and low amplitudes to the large waves and large amplitudes. I n 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 resuhs f r o m free decay model tests i n regular head sea waves. A value o f 0-7 was used f o r the columns and 1-3 was used f o r the pontoons. These 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 quahta-tively satisfactory agreement between theory and experiments.

Co values can be obtained experimentally by either free decay tests or U-tube experiments. Obviously there are scale effects, in particular when ffow separation does not occur f r o m sharp corners.

There have been many attempts to solve numerically (3) separated flow around marine structures. Examples o f

1.0

0.5 H

0.5

• FLOW SEPARATION CLEARLY OBSERVED A NO FLOW SEPARATION

X DIFFICULT TO DECIDE

• • • •

1.0 1.5 Ucurrent

Fig. 13. Occurrence of separated flow around a hemisphere in regular waves and current (U = current velocity, K C = 1-KC,JD,

Ca = incident wave amplitude, D = diameter, C/m =1'ÏÏC,JT, r = incident wave period). Current and wave propagadon directions coincide.

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methods used are:

o Vortex sheet modeP^. • Discrete vortex method^^

• Combination of Chorin's method and vortex-in-ceh^method^''-^^

• Navier-Stokes solvers^^.

A general description of the state o f 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 f r o m experiments. Due to lack of proper turbulence modelhng of wakes i t 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 f r o m sharp corners, in particular for small Keulegan-Carpenter numbers. The vortex-in-cell method and Navier-Stokes solvers can more correctly handle separation f r o m continuously curved surfaces.

Figure 15 shows simulation of oscillatory ambient planar flow past a rectangular cross-section by the vortex-in-ceh 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^^. The drag coefficient was found to be about 4-0 f o r a Reynolds number of lO" and K C = 2.0. These results wifl be used later in the discussion of viscous damping of the D D F and TPS presented in Figs. 1 and 2.

Compai ative numerical studies of slow-drift damping

As a part of the Norwegian research programme FPS 2000 a comparative study of numerical computation of slow d r i f t 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

A Experiments V

\

_\ \

\

\ \

\

J » 50 (CD ISO !M M MO Time (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 = 15 s. Fuh scale wave height = 12-4 m. The TLP is presented in Fig. 12 and Table 2. Comparisons with

model tests.'

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

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

Fig. 15. Simulation of oscillatory ambient flow past a rectangular cross-section by the vortex-in-cell method presented by Scolan and Faltinsen.^* r = osciUation period; / = 0 is initial time; ratio between the length and the height of the rectangular is 1-4; K C = 2 0. Ambient flow direction is in

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described. I t will be demonstrated that the predicted damping level can vary significantly. Recommendations for how the damping shall be predicted by the present state o f 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 o f slow-drift excitation forces. Both effects are of equal importance in predicting slow-drift motions.

DDF resnlts

The procedures used by the different organizations are presented in Table 3. I n some cases we do not have sufficient information to state what methods have been used. More detailed explanation o f the table will be given i n the foUowing text. A n 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 D D F can be written as

Ff = P- C^D{x^'^ + U+ 4')) + U + 4 " I (5)

where p = mass density o f the water, CQ — drag coefficient, i ^ ^ ' = local slow-drift velocity due to surge and pitch motion, U = incident current velocity, 4 ' = first-order relative horizontal local velocity. Both 4 \ U and x^^^ have the same positive direction. From Table 3 we see that only one organization has included the effect o f the first-order velocities. We cannot state what the correct procedure is. On the one hand one could say that i t is consistent with a drag formula like eqn (5) to include the total relative velocity. On the other hand, theoretical or experimental justifi-cation is lacking.

Four organizations have included the effect of current. I t is correct to include this effect.

The drag coefficient depends on Reynolds number, K C number, roughness number, relative current number and structural f o r m . Most of the organizations have used C D = 0-7 f o r the columns. This is appropriate f o r one isolated smooth circular cylinder in steady incident flow in the transcritical flow regime. We cannot by simple arguments state that = 0-7 is a correct value. For each strip we should consider the combined effect of the first-order motions, waves, slow-drift motions and current and define relevant K C numbers and relative current numbers. The local value will depend on the local K C number and relative current numbers. Further, we should consider the interaction between the columns. This is particularly important for two columns in tandem position, especially at strips where the total ambient fiow velocity does not change direction with time.

I f the influence o f the vertical velocity on the pontoons can be neglected, we could write the horizontal viscous damping force per unit length on a

strip of the pontoons as

fB2-^CUD{X^'^ + t / + 4) ) | i ( 2 ) +

C

/+4)|

cos^ö (6) where D = height of the pontoon, and 9 is the angle between the x axis and the cross-sectional plane of the strip. This formula is based on the 'cross-flow principle' and is appropriate as long as 6 is not close to ±7r/2. The drag coefficient C D will depend on Reynolds number, K C number, roughness number and relative current number. However, the dependence is expected to be less than for the circular cylinder. Different organizations have used different C D values for the pontoons. I t varies between 1-0 and 2-05. Actually the C D value may be higher. The numerical simulations presented in Fig. 15 for K C = 2 resulted in a C D value of about 4-0. The rectangular cross-section studied in Fig. 15 is quite close to the cross-section of the pontoons o f the D D F . K C = 2 is a representative Keulegan-Carpenter num-ber. Bearman et al?^ presented experimental values for a square; C D was about 3.2 f o r K C = 2 according to their results.

I f the vertical velocities are comparable with the horizontal velocities at the pontoons, i t may not be appropriate to use eqn (6). We should then incorporate the effect o f the vertical velocities. There are different ways to do that.

When frequency domain solutions are used, the viscous damping is linearized by stochastic lineariza-tion. This assumes Gaussian response, which is not fully accepted by the research community. However, i t can be considered as an approximate solution procedure. I t was used by ah organizations.

Very few organizations have included wave radiation and wave-drift damping. Due to the long period o f oscillation o f the slow-drift motions it is appropriate to neglect the wave radiation damping. However, one should not neglect wave-drift damping. The physical basis for the wave-drift damping is' the same as mean and slowly varying drift forces (and moments). This means that both mean and time-dependent wave-drift damping should be included. Wave-drift damping depends on significant wave height and mean wave period. The mean wave-drift damp-ing will increase as the square of the significant wave height.

Two organizations have simply set the total slowdrift damping as a percentage of the critical damping. One o f them has set it equal to 1 % , the other one equal to 10%. This applies f o r both operational and design conditions. I t is too simphfied to use only a percentage of critical damping applicable f o r ah conditions. Two organiz-ations have calculated the relative damping level i n surge. One o f the organizations (No. 4) gets 15 and 30% for, respectively, operation and design condition. The other one (No. 5) gets 1.2% and 1.7%. One important

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Table 3. Slow-drift damping in surge and pitch for DDF (see Fig. 1). Comparison of methods (Hi/3 = 6 0m,T2 = 8-5s, U = 0 5m/s), Des = Design condition (Hys = 15-5m, T2 = 13,5s, U = l Om/s))

used by different organizations. (Op = operational condition

Organization Has only

number percentage of critical damping been used? drag-damping

Current Coupled Time modes dom. Frequency domain/ stochast. lineariz. Effect of 1. order veloc. Wave radiation damping Wave drift damping Mean Time val. dep. Standard deviation of low frequency Surge (m) Pitch (deg.) Relative damping level surge (%) 0-7 Op. 0-69 0-17 3 No Yes No No Yes ? 0-7 Col No No No No Des. Op. 0-82 0-55 0-21 0-2 15

4 No Yes Yes Yes No

10 Pont 0-7 Col. No No No No Des. Op. 1-5 1-69 0-45 30 0-52 1-2 5 No No No No Yes 2-05 Pont 0-7 Col No No No No Des. Op. 2-4 1-48 1-13 1-7 6 No No No Yes No 11 Pont 0-7 Col Yes No ? ? Des. Op. 8-13 2-02 0-18 12 Yes No 2 0 Pont Des. Op. 2-24 0-8 0-38 1-0 14 Yes No 0-7 Col Des. Op. 1-08 1-65 1-0

18 No Yes Yes No Yes

0-9 Pont No No No No Des. Op. 2-47 0-64 20 Yes 0-7 Col Des. Op. 1-13 0.63 iO 24 No Yes No No Yes 2 0 Pont 1-3 Surge No No No No Des. Op. 0-77 0-96 0-37 25 No No Yes Yes No 2-0 Pitch No Yes Yes surge No Des. 1-62 0-52 10 Col. Op. Des. 5-64 0-083 27 No No Yes No Yes 1-3 Pont No Yes No No Op. Des. 10-64 0-3 I : re

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reason f o r these large differences is that organization No. 4 has included the effect of current while organization N o . 5 has not.

We have also listed the predicted values o f standard deviation of low frequency surge and pitch motion i n Table 3. The reason was that we wanted to see i f there was any correlation between high damping level and low motion values. F r o m 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 resuhs

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

I f 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,^ = f C D < X ) ( ^ ( ^ ) - f Fe

+4))|^(^)

+ + .yy)| (7) where d(x) = local draught, j ' ^ ^ = local slowdrift velo-city i n the transverse direction, = incident current velocity component in the transverse direction, and y^^ = local first-order relative velocity in the transverse direction. The 'cross-fiow 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. ' L i f t i n g ' effects may have to be considered. This is a similar effect one would include i n analysing ship manoeuvring or i n directional stability analysis o f a ship moored to a single point mooring system. However, one cannot directly use the results f r o m ship manoeuvring to the slow-drift damping problem. From the table i t appears that one organization has included quasi-steady lifting effects.

The drag coefficients that have been used vary f r o m 0'6 to 1-0. I f 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 value should be used.

Most of the organizations have included mean wave-d r i f t wave-damping in surge. None have incluwave-dewave-d the effect i n sway and yaw. Even i f i t is expected that the effect o f wave-drift damping is larger f o r surge than f o r sway and yaw, i t should not be neglected. Calculated results o f wave-drift damping i n the sway of the TPS indicate that. Only one of the organizations has included time-dependent wave-drift damping. I t is consistent to do that.

Three o f 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 f r o m 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 inffuence directly the motions o f the upper end points o f the risers. I n the foflowing 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 i n 'lock-in' situations, but they should also be studied in 'non-lock-in' conditions. The drag term in Morison's equation is similar i n f o r m 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 f r o m 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. I f a cylinder is placed in the far wake behind another cyhnder, it will experience a smaller incident velocity and, therefore, a smaller drag coeffi-cient i f the free stream is used to normalize the drag coefficient*. This procedure can be generalized to several risers, where the wake f r o m 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*. Zdravkovich'^ has given a survey of resuhs f o r the interaction of pipe clusters i n steady incident flow. For risers that can collide with each other, there is a need to study the hydrodynamic interaction and the correlation o f vortex shedding between two adjacent risers.

When Morison's equation is used i n the riser analysis, it seems necessary to account f o r the changes i n 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 i n a '100-year design wave' condition.

I n the drag term in Morison's equation i t is normal to include the effect o f 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

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Organization Has only Sway-yaw drag-damping Lifting Friction Wave Standard Relative number percentage Ec^Ai/^^ + Vc + y^^^)\y'-^^ + Vc + y^^^\ effects in drift deviation damping

of critical 2 R R surge damping low frequency level uümpiiig been used? Current Coupled modes Time dom. Frequency domain/ stochast. lineariz Effect of 1st order veloc. Surge Sway and yaw Time dep. Surge Sway (m) (m) Yaw (deg.) Surge Sway (%) Yaw (%) Op. 2-51 1-64 2-93

3 No Yes No No Yes 9 _No _No _Yes _Yes _No _No

Des. 5-31 2-8 5-09

Op. 2-1 0-85 1-5 2

4 No Yes Yes Yes No 0-8(?) No Yes Yes Yes No No

Des. 3-0 1-6 29 4

Op. 1-99 0-92 2-05

5 Yes 2 5

Des. 7-6 1-91 5-92 Op. 1-94

6 No No No Yes No 1 Yes ?' Yes Yes ? _Yes _No _No

Des. 8-94 Op. 1-09

14 Yes No No 1

Des. 2-23

Op. 2-37 121 1-01

18 No Yes Yes No Yes 0-6 No No Yes Yes No No

Des. 3-77 2-57 3-16

19 No 9 ? _Yes _No 7 _Yes ? _Yes _Yes _No _No _{Op. 2-42 2-24} _2-58

Op. 0-69

20 Yes No No 10

Des. 1-3

Op. 3-91 1-3 1-36

24 No Yes Yes No Yes 1-0 No No Yes No No No

Des. 7-45 2-47

0-8 Op. 1-01 0-8

28 No Yes No Yes No No No Yes No No No No

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surge amplitude is 17'52m in the design condition. The natural surge period is 118 s. This means velocity amphtudes 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 d r i f t 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. I t is also more important than wave frequency motions for heave, pitch and roll of a D D F .

The most important error source in numerical predictions of slow-drift motions is the damping. Results f r o m a comparative study of numerical predic-tions of slow-drift damping for a TPS and a D D F show considerable scatter i n the results. The same is true for the numerical predictions of slow-drift motions. Important damping contributions come f r o m wave-drift damping, viscous damping f r o m the hull and anchorline damping. As long as the flow does not separate f r o m 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 i n all cases. Further, the commonly used methods are restricted to 2¬ D flows. Free-decay tests or U-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.

I t is pointed out that interaction between waves and current can have an important effect on mean wave loads and slow-drift excitation loads. I t is not known how to account for viscous effects in the prediction o f combined mean and slowly varying wave and current loads. Model tests of floating production systems i n waves must be performed in an ocean basin to avoid wall effects that can change the motions of the system significantly. I t 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 i t 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.

I t seems necessary to account f o r changes in fluid acceleration and velocities due to the presence of the platform as well as slow-drift velocities of the platfonn when Morison's equation is used to calculate the loads on risers.

REFERENCES

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