Ocean platforms - Prediction of wave drift forces

Ocean Platforms

Prediction of Wave Drift Forces by

R.P. Browne

Introduction

Prediction of the second-order wave-induced forces and moments - the "drift

forcest

- on floating bodies is of considerable importance to the offshore industry in that it has a significant and sometimes governing effect upon the towing or self-propulsion of structures and the design and operation of mooring and dynamic positioning systems. In view of the current expansion in this field it is therefore not surprising that model testing basins are carrying out an

increasing amount of such work.

In the real environment, ocean platforms are subject to external forces arising from non-steady winds and currents and irregular multi-directional waves and it is this combined problem which must ultimately be solved by mathematical and/or physical modelling techniques.

The Present Position

Hydrodynamicists have formulated the wave problem mathematically and derived expressions for the wave drift force, (Maruo

_{[1], Newman}

[2], Ogawa [3], Verhagen and van Sluijs [Li], Kim and Chou

_[51)

but numerical solutions are so far limited to cases of

!thin?

ships [2] and two-dimensional sections for use within a strip theory approach [3], [5], all in regular oblique waves. Researchers have also tackled the associated problem of predicting drift forces and subsequent platform movement in irregular waves from responses in

simpler wave systems (Hsu and i3lenkarn[6], Rernery and Hermans

[7],

Pinkster [8],

Rye, Rynning and Moshagen [9], Newman [10]).

These studies, mostly involving model tests, have added significantly to our knowledge although it is not yet clear to what accuracy such predictions can be made from regular wave responses alone.

The salient features of the wave drift force problem arising from the above research are summarised below.

1. The hydrodynamic force and moment acting upon a body in

regular waves includes a second-order nonlinear

component - the drift force - proportional to the square of the wave amplitude.

Stheepahydromechanjc

Archlet

Mekelweg Z 2628 CD D&ft

2.

2. The drift force in regular waves includes a steady-state

component plus a second harmonic oscillatory term which is insignificant in relation to the first-order wave force. 3 The steady-state drift force in regular waves has

simply-calculated asymptotic limits as wave period tends to zero and the body behaves as a rigid barrier.

Except for very short wavelengths, the drift forces are caused by both diffraction effects and the relative

motions of the body, heaving, swaying and rolling motions generally being of greatest significance (also surge and pitch motions for bodies with similar length and beam).

5 The drift force in irregular waves is a varying quantity

with a non-zero mean, and if for simplicity one considers a discrete spectrum with many regular wave components, the drift force is composed of forces at both sum and

difference frequencies between all components. The slowly varying part of the drift force, occurring at difference frequencies between adjacent parts of the spectrum is of particular significance since it can excite low frequency resonance in platform mooring systems, etc.

The mean or time-averaged drift force in irregular waves may be calculated from response operators in regular waves. Defining the lateral (longitudinal) drift force coefficient

on a body of length or principal dimension L in regular waves of amplitude

A and frequency f, as

f lateral (longitudinal) drift force - pgL

The time-averaged drift force in irregular waves charac-tensed by a wave spectrum of energy density 3(f), is given by, Mean lateral drift force

= pgL

J S(f)cyw(f)dr (2)

The varying component of drift force in irregular waves can only be calculated exactly if the sum and difference frequency responses to all pairs of regular wave components in an irregular wave system are known. However, since

low frequency components of drift force are generally of greatest importance, the requirement may be reduced to a knowledge of difference frequency responses to adjacent components of the wave spectrum. The volume of work remaining to satisfy an 'exact' synthesis of irregular wave drift force response is however still extensive.

In order to avoid this complication, approximate methods of computing irregular wave drift forces from regular wave drift force operators have been investigated with reasonable success.

Working in the time domain, Hsu and Blenkarn

_[61

assume that an irregular wave train may be considered as a succession of -half-periods of regular waves, whereby, using a typical record of irregular waves, the wave drift force is calculated successively from the wave trace by means of (1) i.e.

Drift force =

PLC(f)

₍₃₎

Remery and Hermans

_[7J

used this method to calculate the surge motion of a barge and obtained good agreement with model tests.

Pinkster

[8]

working in the frequency domain uses (2) to calculate the mean drift force in irregular waves and calculates the spectral density SF of the slowly varying part at frequency f' from

co

22

= 2p g I

_{S(f)S(f+f')C(f +}

_)df (4) Jo

Rye, Rynning and Moshagen

[9]

used this method to predict and compare the surge motion of a semi-submersible moored platform model in irregular waves using drift force coefficients obtained

from regular wave and regular wave group tests. Although by no means conclusive, this work indicated that good agreement is obtained when drift force coefficients are determined from regular wave group experiments. This finding is substantiated by Newman [10] who demonstrates analytically that the accuracy

in predicting second-order slowly varying pressures in

irregular waves from regular wave transfer functions decreases with increasing depth below mean water level. The inference

from this is that (4) may well be suitable for predicting the slowly varying drift force on relatively shallow draft

structures such as drill ships and pipe-lay barges from regular wave tests, but prove less accurate when applied to deeper draft vessels such as semi-submersibles.

Clearly, much detailed research is currently required in order to determine the degree of accuracy possible through use of these approximate methods, in predicting the slowly varying drift force for difference platform geometries and irregular wave descriptions.

Regular or Irregular Waves

The first obvious approach to predicting the wave drift force on an ocean platform is to carry out model

experiments in irregular waves simulating the real environ-ment likely to be encountered by the structure. The broad requirements in this case in terms of experiment facilities are:

Accurate generation of the irregular waves - and since response is not merely a function of spectrum shape but also of relative phases between regular wave components

(these governing the occurrence of wave groups), both of these parameters must be satisfied. The ideal situation would be a filtered white noise spectrum having the correct shape and random phases of components.

A further requirement is to include as much of the spectrum's high frequency energy as is possible, since a

structure's response in terms of drift force tends to a finite, generally large limit as frequency tends to infinity.

For example, in waves generated by Beaufort force 5 winds (12th ITTO spectrum formulation), 7.14% of wave energy lies above a frequency of 0.214 Hz or period 4.l7 secs and wavelength 89 ft. This wave energy could account for more than twice its percentage in terms of mean response energy or drift force. In order therefore to remain within the limits of a possible 15% error in mean drift force it would be necessary, in the case of testing a J4QQ ft. drilling vessel, to include wave components of 0.22 times model length and

preferably smaller.

The second important factor is tank wall interference in relation to tank or basin size, model size and wavemaker capability.

Since a model structure under test will be virtually stationary and the experiments should typically simulate real time scales of from one to several hours, tank wall interference is inevitable. This can be minimised by using small models, but only within the limitations of short wavelength generation mentioned above, practical model manufacturing and balancing

capability and at the risk of obtaining a distortion between inertial and viscous forces acting on the model.

One is therefore led to the opinion that most cases of irregular wave tests on floating structures should be

carried out in seakeeping basins rather than conventional towing tanks and this is of course a necessity when complex mooring systems are to be modelled physically.

An experimenter's decision therefore of whether to test in regular or irregular waves is primarily restricted by

the fairly onerous conditions stated above, which can only be met in relatively few facilities worldwide. The limitations

are greater moreover if current and/or wind are to be simulated concurrently.

Provided that cost is not a serious limitation, the facility and required equipment are available and that the environmental conditions are known, the experimenter would

almost inevitably choose irregular wave testing as being direct and yielding responses immediately obvious to the observer. When facilities for irregular wave testing are not available or their cost is not justified, regular wave testing would be carried out. The resulting drift force coefficients would then be used in the prediction of drift forces in irregular waves and the design and analysis of dynamic positioning and mooring systems. As in the traditional ship motions field,

this approach has the benefits of both economy and flexibility. General Guidelines

From consideration of items l-1 above, the following guidelines should be followed in model tests.

Use a geometrically similar model of the structure under consideration including its underwater appendages and balanced to the correct displacement, vertical centre of gravity and radii of gyration in pitch and roll.

Select the scale of the model to suit the facility size and wavemaker capability, preferably such that tests may be carried out in waves as short as the smallest main dimension of the body, for example the beam of a drill ship. The model should also be sufficiently small to prevent significant tank wall interference without being subject to a distortion

between the inertial and viscous forces acting _{upon it.}

Attachments to the model for preventing drift motion and measuring the drift forces should be such that they impose negligible restraint upon oscillatory motions. _Attachments should therefore be made, wherever possible, to the principal horizontal axis of rotation of the model. In the case of a ship form this would be a horizontal fore and aft axis through the centre of gravity.

LI. The minimum number of attachments necessary and a

maximum of three should be used such that restoring forces and thence resonant frequencies are not built into the tethering system.

The attachments should be sufficiently soft in order that one may deduce accurately from visual observation when the body-wave system has reached a dynamically stable condition.

The geometry of attachments will depend upon the method used for measuring the forces.

The first recorded method of Suyehiro [ill shown in Figure 1 is still applicable.

The model displacement and

inertia is affected only slightly by the drift force weight, no

extraneous moments are applied to the model, no natural

fre-F

quencies are built into the ig.

system and the model position is self-centreing. On the other hand, the method is

labour intensive and rela-

0

tively inaccurate, especially I

since three sUch attachments w Drift force WsinO would in general be required

to measure the drift forces and moment on a body.

A second system would be to control the model from three independently-controlled constant-torque winches with horizontal line attachments. Figure 2.

4b4 VS

Fig. 2

/

6.

longitudinal drift force = F1-(F2+F3)cos3i lateral drift force = (F2+F3)sinp

Such a system could be controlled very effectively, but the manufacture of winches giving accurately controlled low torque capability and high frequency operation would be extremely difficult.

A preferred system is to attach the model to load cells using horizontal elastic lines, the length of which should be adjustable such that the heading of the model under test can be controlled. The spring constant of the lines and degree of restraint on the model are shown in the following simple example.

Take a drill ship model of length ₁₅ ft. and dis-placement 1500 lbs in beam waves of length 10 ft. and

height/length ratio 1/30. Assume sway/wave amplitude = 1 and added mass coefficient in sway =

0.5.

Then the maximum value of the first-order oscillatory sway force

= 236

lbs.

Assume a lateral drift force coefficient C = Then the lateral drift force =

_0.5

x 32 x

(1)2

x 15

= 6.7

lbs.

If for example, the lines had a spring constant of 1 inch per lb. load, this would mean a total line extension of

6.7

ins, and fluctuating load due to sway of ± 2 lb, that is less than 1% of the first-order wave force. In practice one would choose an even softer spring of perhaps 3 or tt ins.

per lb. load, thereby reducing the extraneous load to 1/4% of the first-order wave force and giving greater position sensitivity to the experiment. Care should of course be used to ensure that each elastic line does not approach its elastic limit in any test. If this occurs, then that line should be replaced by a stiffer one.

7.

Since some time will be required in a particular experiment for the model to drift to its position of dynamic equilibrium, a useful technique is to predict the probable drift from prior experiments and pretension the system to its

estimated equilibrium position. This can reduce run length and the associated interference problems likely to be

encountered.

(l500+70 1 2rr

=

In view of the fact that drift force is a function of wave amplitude squared and the drift force coefficient generally decreases with increasing wave length, testing in waves of nominally constant wave slope is recommended as yielding forces of the same order over the whole frequency range. Moreover, since the drift forces are generally small and the accuracy required in measurement of wave height is greater than that for normal seakeeping experiments, it can be advantageous to use relatively steep waves. Figure 3 shows the drift force coefficient obtained for a floating gravity structure of circular waterline shape. The majority of tests were carried out in waves of height/length ratio =

_1/30

with

a few check spots at height/length = 1/50.

Knowledge of the asymptotic value of the drift force coefficient as wavelength tends to zero is extremely valuable since it helps define the response curve in an important area which cannot be covered entirely by model tests.

As wavelength becomes progressively shorter, a floating body responds less and less to incident waves until, at the limit, it may be considered as fixed in space and wave diffraction forces only are present. Waves are scattered from the waterline facing the oncoming waves with a shadow of calm water behind the body.

The normal force per unit length of waterline at angle p to wave direction and facing the oncoming waves is then

1 2

.2

FN = pg sin p . .

. (5)

Applying

₍₅₎

to the case of a floating body of circular plan waterline of radius R, the drift force coefficient as

wave-length tends to zero C(c), is given by:

Fig. !

WAV

)71R7i7A/

2 C (co)

-

j

sin'pdL1 =

XW 2 0

For a ship-shaped form; Figure 5

2 3

1 1L/2 2

Cyw(') =

_j

sin i-i'

dx

-L/2

the longitudinal drift force coefficient is:

1 2 1 2 = _Pg J

sin psindS/pgç(2R)

(6)

ABC

where

dS = R dTi

₍₇₎

Therefore:

(8)

(10)

The wave drift force in Y-direction per length dx of vessel

is:

= FN

cos(L1'-p)

_cos(p'-p)

(9)

i 1L/2 2

(co) = _{- sin p'tan(p'-p)dx}

Lj

-L/2

and the drift moment may be obtained by taking moments of the normal force on each element of waterline about the origin.

Ocean Platforms

Measurement of Wave Drift Forces References

[1] H. Maruo

"The Drift ofa Body Floating on Waves.t' J.S.R. Dec.

1960.

[2]

[31

[141

[51

[61

[91

[10]

J.N. Newman

"The Drift Force and Moment on Ships in Waves." J.S.R. March,

_1967.

A. Ogawa

"The Drifting Force and Moment on a Ship in Oblique Regular Waves." Shipbuilding Laboratory Delft

Technical University Report, 155, Sept.

_1966.

J.H.G. Verhagen and M.F. van Sluijs

"The Low-frequency Drifting Force on a Floating Body in Waves." I.S.P., April,

_1970.

C.H. Kim and F. Chou

"Prediction of Drifting Force and Moment on an Ocean Platform Floating in Oblique Waves." I..S.P., October

1973.

R.H. Hsu and K.A. Blenkarn

"Analysis of Peak Mooring Forces Caused by Slow Vessel Drift Oscillations in Random Seas." OTC) Houston,

_1970.

G.F.M. Remery and A.J. Hermans

"The Slow Drift Oscillations of a Moored Object in Random Seas." OTC, Houston,

_1971.

J.A. Pinkster

"Low Frequency Phenomena Associated with Vessels Moored at Sea." Soc. of Petroleum Engineers, _19714.

H. Rye, S. Rynning and H. Moshagen

"On the Slow Drift Oscillations of Moored Structures." OTC, Houston,

_1975.

J.N. Newman

"Second Order, Slowly Varying Forces on Vessels in Irregular Waves." International Symposium on the Dynamics of Marine Vehicles and Structures in Waves, University College London, _19714.

[11] K. Suyehiro

"On the Drift of Ships Caused by Rolling Among Waves." Trans. I.NA. Vol.

66, 19214.

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