A comparison of two methods for calculating wave drift forces

National Maritime Institute

A comparison of

two methods

for calculating

_{wave drift forces}

by

W J Brendling

Supported by the Department of Energy

Offshore Energy Technology Board

OT-R-8218

N M I

R 137

January 1982

National Maritime Institute

Feitham

Middlesex TW14 OLQ

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OT-R-82 18

This report is Crown Copyright but may be freely reproduced for all purposes other than advertising providing the _{source is acknowledged.}

A comparison of two methods for calculating wave drift forès

by

W.J. Berid1ing

Qepartmentof Energy

Offshore Energy Technology Board

National Maritime Institute

OT-R-8218

NMI Report R137

Summary

In recent years a huber of computer progras, including NMIWAVE, have been developed to calculate second order, slowly varying, wave drift forces. These have 11 been based on a Stokes series expansion of the wave forces. In the

resulting expresions for the second order drift forces, the first order pEoduct terms are evaluated exactly using the results of first order diffraction programs The second order potential term, however, is only approximated.

Atkins Research and Development has proposed a different approach. By making the rigid lid assumption, and using the technLques of Lagcangean mechanics, they derive computable expressions fOr the exact second order drift force subject to

the rigid lid condition. They argue that for freely floating strüctues with small waterplane areas, which only slightly disturb the surface of long waves, the Lagrangean method represents an advance over the Stokes expansion. The

principal problem with the Lagrangen method is that it provides no internal

warning of the failure of the rigid lid assumption..

This report attempts to assess the Lagrangean method in Oomparison with NNIWAVE. For the example considered, an Offshore Storage Terminal, the rigid lid

appioximation appears to be valid for the first order solution. The cotharison of the second order drift forces is somewhat inconclusive, because these forces are dominated by the second order potential which NMIWAVE only approximates The two

methods are probably complementary in the sense that their regimes of validity

-2

Introduction

The National Maritime Institute is at present engaged in a theoretical and experimental study on second order wave fOrces. [1] [2] _{This study has beeh}

based on a power series expansion of the hydrodynaxnic equations due to Pinkster. The first order terms in this expansion give rise to the classical linear

diffraction problem, which the NMIWAVE program suite was originally developed to

so1ve [4] _{The problem for the second order velocity potential is not yet}

computationally tractable. The expression for the second order drift force,

however, contains predominantly terms involving products of first order quantities The NMIWAVE program suite has been extended to calculate these terms. To

estimate the importance of the second order potential an approximation due to

Bowers [51 _{is used. The seàond order set-down potential of the incident waves}

is used to make a Froude - Krylov approximation of the second order potential

to the drift force.

Atkins Reseach and Dvelopmet Co. td. have recently proppsed a different

approach. They argue that sine many freely floating structures do not appear

to greatly disturb the free si.rface, t is reasonable to represent the water

surface by a rigid lid. This assumption, as a consequence of the uniqueness theorems for Laplace's equation, reduces the problem to one with a finite number of degrees of freedom. They then use the techfiques of Lagrangean machan-ics to

derive a finite set of non-linear ordinary differential equations for the motions, in terms of energy integrals Finally they expand these equations in a power

series in wave height.. The resulting second order forces are computable and exact for the flow with the rigid lid. It is not clear, however, how to evaluate

the effects of the rigid lid assumption.

The Department of Energy requested that, as part of the general research on wave drift forces they are funding at NMI, an attempt be made to compare the two

methods. Since the Atkins method has currently only been applied to,an Offhore Storage Terminal, shown in Fig 1, this model was adopted for the comparison.

Comparison of Assumptions of IWAVE programs and Atkins Lagrangean mehod

NMIWAVE Lagrangear méthód

Starting point of

analysis

Laplace's Equation.

Exact free surface

conditions. Boundary conditions applied in

displaced positions.

Rigid lid approximation

Lagrange's equations.

Finite set of non-linear ordinary differential

equations.

Expansion Stokes expansion.

Velocity potential,

displacements, forces expanded in terms of

wave height. Boundary

conditions transferred to mean positions. Perturbation. expansion. Displacements and Lagrangean energies expanded in terms of wave height.

First order -Exact (except for numerical errors)

Includes local added mass effect but neglects wave

diffraction and radiation. No damping. (Valid fOr

3.. Method of Calculation

NMI WAVE

('i) First order hydrodynamics

The NMIWAVE program suite calculates the first order radiation and diffraction

potentials by a Green's function technique..

tfl

The surface of the structure is divided into a finite number of plane area elements. Fltiid sources,

pulsating at wave frequency, are placed at the centre of each element, and their strengths and phases are calculated to satisfy the boundary condition of zero normal flow across each element centre. Having obtained the source strengths, NMIWAVE derives the pressure distribution over the structure

surface, hence the first order wave forces, added masses, and damping coefficients. It then solves the equations that determine the structure's first Order response.

(ii) Second order drift forces

There are, as already mentioned in the introduction, two trpes of contribution

to the drift force: one due to products of the first order terms, the other due to the second order potential The contributions due to the products of first order term are calculated exactly (to within the general numerical

limitations of NMIWAVE). The contributions due to the second order potential

are represented by the Froude - Krylov force due to the incident set dOwn wave.

Atkins Lagrangean Method

(i,) First order hydrodynamics

Atkins R & D used a finite element Laplace's equation solver (ASASHEAT) to obtain the local modification of the incident wave flow, and the local flow produced by the structure motions, using the rigid lid approximation of zero normal flow at the position of the surface of the undisturbed incident wave This requires filling the entire fluid volume with finite elements, and so a large outer boundary has to be put on the fluid. Having found the velocity potential they then evaluated a number of energy integrals of the form

NMIWAVE Lagrangean method

Second. order

potential

Incident set-down only. Not calculated explicitly. Implicitly exact for rigid lid. Modification for

free surface unknown.

Second order

forces

First order product

terms exact. Second order potential

approximated as above, (Valid for short waves?)

Exact fo rigid lid (except

for numerical errors).

Modification for free

surface unknown. (Valid

for long waves?)

Higher order Impractical! Possible. Also exact

theories of non-linear ordinary differential

equations. Is rigid lid

approximation valid for

-4-.

dv

These were then substituted into the first order Lagrangean equations which were solved to give fitst order motions.

(ii) Second order forces

The Lagrangean expresioti for th second order drift, forces =nvolves derivatives

of the energy integrals with respect to the positions of the structure and the

rigid lid. These were evaluated by rpeating the above first order calculations for a number of positons and then using numerical differentiation In order

to minimize the number of finite element calculations a number of ad-hoc approximations were used These could probably be rigorously justified but

need to be earnined carefully.

Numerical Models

NMIWAVE

The NMIWAVE model attempted to represent. the shape of the offshore

storage terminal as accurately as possible (see the facet drawing in Fig 2)

This resulted in the model having a larger displacement than the original This

is reasonable because the water between the inner and outer cylinders, andi close in between the outer cylinders can be expected to move with the storage terminal. The principal dimensions of the NMIWAVE model are

-PlaneS of symmetry x = 0

Number of facets (whole structure) 88

Numher of waterline points . 12

I

Maximum radius : 30.41 m

Maximum draft : . 76 m

Diplqinen volume

. : 1.192 x 1O5 m3

Centre of buoyancy (0.0, 0.0, -50.32.) rn

Centre Of gravity : (0.0, 0.0, -54.50) rn

Pitch gyradius : 22.73 m

Atkins Lagrangean Model

The Atkins Lagrãngean calculations were performed using a simpler axisymmet.ic model, with the lower portion replaced by a single cylinder whose radius was chosen to give the same displacement volume as the original The principal1 dimensions of the Atkins Model are.:

-Symmetry : Axial Maximum radius . 26.46 m Maximum draft 76 m

5.3

Displacement Volume . 1.059 x 10 m. Centre of buoyancy (0.0, 0.0,, 50.34) rn Centre of gravity ' : (0.0, 0.0, -54.50) m Pitch gyradius : 22.73 m

5. Results

Added Masses

The added masses obtained from the Lagrangean method are, as a consequence of the rigid lid assumption, frequency independent. They are compared with NMIWAVE results at one wave, and one group, period in the table below.

The pitch motions, in this table only, are about a nominal pitch axis through the point (0.0, 0.0, -21.9)m. This is used for the convenience of the finite

element calculations. From these results it can be shown that the surge and

pitch motions of the Lagrangeari model are decoupled about the point (0.0, 0.0, -48.0)m (the natural motion centre). All subsequent pitch terms will refer to an axis through this latter point.

First order wave forces

The table below gives the wave force on the Offshore Storage Terminal due to waves of 5m amplitude. The results from the Lagrangean method are compared with the Froude-Krylov force and the total wave force given by NMIWAVE.

Added Mass component Period NMIWAVE result Lagrangean result

Heave - Heave 50.0 secs

12.0 secs

8.44 x 10 8.09 x 10

kg

kg 4.81 kg

Surge - Surge 50.0 secs 12.0 secs 9.61 x 10 8.98 x 10 kg kg } 5.53 x io7 kg 50.0 secs 9.09 x 10kg m2 Pitch - Pitch 12.0 secs 8.50 x 10 kg m2 } -4.78 x 1010_{kg m2}

Surge - Pitch 50.0 secs

12.0 secs -2.61 x 10 -2.38 x 10 kg m kg m } -7.29 8 kg m

Pitch - Surge 50.0 secs

12.0 secs

-2.38 x 10 -2.37 x 10

kg m

kg m -7.29 kg m

Period Component Froude - Krylov NNIWAVE Lagrangean

Heave 2.93 x 10 N 5.36 x 107 N 3.95 x 107 N 12.0 secs Surge 4.30 x 107 N 7.97 x 107 N 5.62 x 107 N Pitch 3.63 x 10 Nm 21.01 x 10 Nm 20.45 x 10 Nm Heave 2.10 x 10 N 3.66 x 10 N 2.85 x 107 N 9.7 secs Surge 3.82 x 107 N 5.87 x 107 N 4.65 x 107 N Pitch 14.58 x 10 Nm 42.41 x 10 Nm 36.36 x 10 N

First order responses

The first order response amplitude operators calculated y NMIWAVE and At1ins Lagangean method are

Phase angles '--.

Atkins justify the use of the rigid lid assumption by supposing that the radiated

and diffracted waves :cancel and that therefore, the phase difference betwéén

the total diffracted wave force and the Froude-Krylo force, is cancelled by the phase difference between the response and the radiation force. This cancellation is investigated for the .NMIWAVE results in the table below

Period Component Froude-Krylov NMIWAVE -. Lagranqeax

Heave 1.54 x 107 N. 2.73 x 107 N . 2.17 x 107 N

8.8 secs surge 2.69 x 107 N 4.67 x 107 N 3.9.9 x 107 N

Pitch 20.08 x 10 Nm 51.00 x 10 Nm 43.80 x 10 Nm

Period Component NMIWAVE results Lagrangean res1ts

Heav,e 0.201 . . 0.195 12.0 secs Surge .0.294 . 0.249 Pitch 0.321°/rn 0.089°/rn Heave 0.093

0090

9.7.secs Surgë 0.155 . 0,135 Pitch 0.246 0/rn _{0.103 °/m} Heave 0.057 0.057 8.8 secs Surge .. 0.107 0.095 Pitch 0.210 0/rn . .0.097 °/m Period Component -Phase of Diffraction . force to F-K force Phase of Radiation . force to response Phase of Response to F-K force Heave _4.82: 1.1.62° . -0.08° 12.0 secs Surge 7..38 11.86° 1.60° Pitch 7.38° 0.40° 1.60? Heave 1.61° .

9490

_-1.03° 9.7 secs Surge 1.1.78° 11..30° . 5.60ff Pitch 11.78° 2.35° 5.60° Heave _1.990 6.59° -4.36 8.8 secs Surge 14.17° 9.10° 8.36° Pitch 14.17° 4.10' 8.36°

(v). Second order drift forces

The total slowly varying second order force in a pair of beating wave trains in open. water (ie. not including the group period free wave produced by a

wavemaker) calculated by the Lagrangean method is compared with that obtained by NMIWAVE The table also shows the magnitude of the second order potential,

set-down term used in the NMIWAVE calculation.

6. Discussion of results Added Masses

The added masses calculated by NMIWAVE are considerably larger than those obtaIned by the Lagrangean method This is probably due to the differences in the

numerical representation of the Offshore Storage Terminal by the two methods To provide a check on the results the added mass in surge can be estimated by segmentirg the Terminal The upper part of the Terminal consists of a cylinder of 9m radius and 30m length Since this cylinder is bounded by the free surface at one end and the tops of the lower cylinders at the other, the flow around this upper cylinder in surge will be approximately two dimensional. The added mass

of this segment of the terminal will, therefore, be approximately equal to its displacement mass

1029 x x 92 x 30 kg = 0.785 x

The lOwer portion Of the Terminal (as modelled by NMIWAVE) can be approximated by a cylinder of 30m radius and 46m length. Since this is a cylinder of finite

aspect ratio with no boundaries on the ends the added mass of this section will, therefore, be taken to be 0.7 of its displaced mass.

0.7 x 102,9 x Tr x 302 x 46 kg

9.368 x 1O7 kg

Thus this give an approximate added mass in surge çf 10.153 x 1O7 kg which

agrees reasonably well with the NMIWAVE value. First order wave forces

The wave forces calculated by NMIWAVE are larger than those calculated by the Lagrangean method This is to be expected since the NMIWAVE model is larger

than the Lagrangean one. They, appear, otherwise, to be in general agreement.

Periods Component Total NMIWAVE.

force NMIWAVE set-down . .. - -. force Total Lagrangean force 12 0 secs + 9.7 secs Heave Surge Pitch 37 10 kN m2 31.78 kN rn2 58.49 kN rn1 20 62 kN m2 40.42 kN m2 207.61 kN nr1 81 8 kN m2 32.7 kN m2 208.9 kN m1 12.0 secs + 8.8 secs Heave Surge Pitch S6.77 kN m 50.41 kN m2 75.74 kn m' 63.21 kN m2 41.95 kN m2 . 416.05 kN rn-1 140.5 kN m2 115.1 kN m 249.8 kN m1

First order responses

The heave and surge responses calculated by NMIWAVE and Atkins Lagrangeai method

are in good. agreement, the errors due to the differences in size having cancelled. The agreement in pitch is, however,, poor. _{There does not appear.}

to be a single direct cause of this difference, so it is probably due toa compounding of slight differences.

Phase Angles

The phase difference between the responses and the Froude- K±r1ov fOrce is

fairly small for the Offshore Storage Terminal in these relatively long waves.

This gives some validation of the rigid lid approximation used in the _Lagrangeaxi

method.

Second order drift forces

The agreelñent between the NMIWAVE and the Lagrangean values of the second order drift forces is poor. It can be seen from the NMIWAVE results that the scond order potential term, which is only approximate, is the dominant one.

7. Conclusions

-8

For cases where the rigid lid approximation is good the NMIWAVE and Lagrangean methods give similar results for the first order responses. _{The differenèes in} pitch are probably due to cumulative numerical errors; the NMIWAVE model in particular (for reasOns Of economy) has a fairly poor vertical resolution, Agreement of the first order responses is important since the second order

forces depend on the first order motions. This agreement can only be good, however, providing the rigid lid approximation is valid _{The Lagrangean method} unfortunately gives no direct indication of the validity of this approximatIon. The only possible test (other than comparing the results with a diffraction

program) would be to calculate the pressure exerted 'on the lid. _{This should}

be as a small as possible, however considerable research would be required to determine what values are acceptable.

The NMIWAVE drift force programs can not be expected to give good results 'in

cases, such as this, which are dominated by the second order potential. _The setdown term does, however, give a good indication of the magnitude Of th

total second order potential term, thus providing a warning of the problem,.

The Lagrangean method contains the full second order potential terms for the

r.igid lid, however it is very difficult to assess how this differs. from the

true free-surface, second order potential. Thus, while the Lagrangeari method

may give a better estimate of the second order forces _{for this type of structure,} it does nOt provide any indications of the breakdown of the assumptions.

The Lagrangean method is alo prone to numerical errors. To find the. first

order responses the Lagrangean method solves Laplaces equation by finite elements and then integrates the solution. The errors in this process are probably of the same order as those in NMIWAVE, which solves an integral

equation and then integrates the result. To find the second order forces, however, the Lagrangean method performs numerical differentiation of the

above integrals which greatly magnifies the errors. In contrast NMIWAVE performs further integrations which do not in general magnify errors.

Acknowledgements

This work was supported b _{the Department of Energy, through the Offshore Energy}

Technology Board, as part of an overall program of. research into fluid _loading of. offshore structures.

References

Standing R.G., Dacunha N.M.C. + Matten R.B. (1981) 'Mean wave drift forces: theory and experiments NMI report No. 124.

Standing R.G., Dacunha N.M.C. + Matten R.B. (1981) 'Slowly-varying second-,

order wave forces theory and experiments ' NNI report to be published

Pinkster J.A. (1979) 'Mean and low frequency wave drifting forces on

floating structures'. Ocean Engineering 6 pp593-615.

Standing R.G. (1.978). 'Applications of wave diffractiOn theory.' J. Num.

Methods in Engineering 13 pp49-72.

Bowers E.C. (1976) 'Long period oscillations of moored ships subject to shortwave, seas'. Trans. Roy. Inst.. Nay. Archit., London llSpplBl-191.

Cash D.G.F + Rainey R.C.T., (1981) _{'Design rules for the avoidance of}

subharmonic oscillations in large floating offshore structures' Atkins R' & P report ref. 20269/RCTR/DGFC.

S S W. L. OMPUTI T IQNAL ITCH AXIS NATURAL PITCH AXIS

41m

--i

28Om

76m SPLACEMENT

:109

.x kg

MENT OF INERTIA AUT

RIZ. IS 5,630 x 1O kg m2

ROUGH CENTRE OF G.

IIIIIIIIIIUIIIIIIIIUI

iIluuIuIuI.III..

SECTION A-A

OFFSHORE STOAGE TERMINAL.

26mØ 12 C 20rn a

i8

25rn 30 m B- CENTRE OF BUOYANCY

G CENTRE OF'GRAVITY

215m

BM= 109,000-005m

GM; 2566 -21-5= 416rn

WATERDEpTH122m

FIG. 1

2566m 46 m

NMI WAVE FACET MODEL

z