Low-frequency damping predictions: Assessing the choices

ABSTRACT

This paper is concerned with the extension and application of both the far-field and near-field based added resistance gradi-eat (ARG) metbod(' of predicting the low-frequency damping of a floating Structure. The four methodS presented correspond to coipling two alternxtive full 3D forward-speed dependent fluid-structure interaction analyses with the two indicated mean second-order force prediction techniques. A simple 3D ellipsoid form is used in the reported comparative study. The fóur cal-culation procedures bave been standardised regarding the pro. cessing of hull geometry, the calculation of hydrostatic restora-tion coefficiCnts and the generarestora-tion of hydrodynamic quantities. Consequently it argued that Observed differences in the pro. dicted low-frequency damping coefficients of the ellipsoid are more attributable to the differences ¡n the second-order force prediction techniques than the different fluid-structure interac-tion analyses used. The Wholé tenant of the paper aimed at providing advice appropriate to making correct application de-cisions, so as to provide viable solutions without having to con-tinuously re-run any particular analysis method to appreciáte its sensitivitim and/or the veracity of the generated solutions.

INTRODUCTION

In sii earlier paper') two distinct full 3D fluid-structure interaction basedARG low-frequency damping calculation pro-cedures were presented and compared with an enhanced 2D strip theory based ARG prediction procedure2 and experinen-tal measurements of the low-frequency damping of the Wichers' 200,000 dwt tankez13. This was the first tme full 3D fluid-Structure interaction based calculations of low-frequency

damp-bouif UnIvrrsity of Tecurnooy

GtIp

Laboratory

Library

Mekweg 2-2628 CD

Deift

The Netherlands

PSII 5115 788873 - Fax 31(5781833

LOW-FREQUENCY DAMPING PREDICTIONS: ASSESSING THE CHOICES.

GRANT E. HEARN and PAUL GOODWIN

Hydromechanics Research Group, Department cf. Marine Technology, Armstrong Building, The University, Newcastle upon Tyne, NEL 7RU.

¡ng had been presented, although a method based on simplify-ing the translatsimplify-ing-pulsatsimplify-ing source Green function on the basis of low forward-speed had been developed at MAR1N(). Using this same simplification 01 the. Green function other researchers bave sought to improve low-frequency damping predictions by using more complete descriptions of the wetted-surface bound-ary condition'. Another Newcastle project is concerned With implementing a more complete free-surface boUndary condition to model the interactiOn between the steady (wave-making)

po-tential associatéd with pre forward-speed and the unsteady

radiation and diffraction velocity potentials.

The two 3D methods implemented previously') had been developed to analyse arbitrary shaped floating structures. Method I, designated Method lA bere, consisted of a simplified forward-speed dependent first-order fluid-strUcture interaction anó.lysis coupled with a direct pressure integration (near-field) method of predicting added resistance. The alternative 3D pro-cedure, Method 2, designated Method 2B here, used the more complex forward-speed dependent first-order fluid-structure in-teraction analysis coupled with the mOmentum conservation (far-field) added resistance prediction technique. Theother two possible combinations of 3D fluid-structure interaction analysis and second-order force predictor are:

the coupling of the simplified forward-speed fluid-structure interaction analysis with the [ar-field second-order force predictions, designated Method

2A,

the coupling of the complex forward-seed first-order analysis with the near-field added resistance calculation procedure, designated Method lB

That is, designatiom A and B indicate the fluid-structure anal-y.ie used, whereas designations i and 2 identifies the use of thç ear.field and the far-field second-order force predictors respec-lively.

All fourtssibie methods of predictingsecond-order forces and momenta, and hence low-frequency damping, have now been implemented and applied as part of an ongoing low-frequency damping research programme. Since the surge low-frequency damping of a floating moored vessel ¡s determined using the ARC method, the development of a robust added resistance pre-diction method is a useful by-prodúct of this on going research programmiS°1. Since reviews have already been published re-garding the development of predicting low-frequency damping(s) and the choices regarding the formulation of the first-order flùid-structure interactions(10) theminimum of equations used in the reported analysis will be stated, without explanatiòn. The no-tation med ¡e this paper is generally consistent with the earlier cited pab&atmon'.

The aime at the particular research project reported here

were:

To remove 'subjective' inconsistencies contained in the earllèr 3D analyses, Methods IA & 2B. To develop, implement and test alternative analyses, Methds IB & 2A.

To identify a robust second-order force prediction analysis for use in ship and mooring system design. To investigate bow best to undertake low-frequency damping analyses.

The 'subjective' choices cf our first aim, relate to computer implementation details of the different analyses. In particular there are different ways of generating discretisation details, and different levels and methods of correcting the first-order solution in readlnme kt the second-order force predictions. Similarly, the chokes rarding low-frequency damping prediction must, ultimately, depend es much upon the robustness and accuracy of the implemented methods, as the perceived improvements in

the different theoretical models availabk('°).

In the offshore design office the inclusion of forward-speed effects in the fluid-structure interaction analysis and the forward

-speed dependence of the wave excitation, through the generali-sation dthe second-order forces to include low-frequency damp-ing, ra a number of practical questions:

Bow can such second-order calculations be under-taken effectively?

What happens if existing zero-speed analyses are used to estimate the law-frequency damping? What are the consequences cf ignoring low- frequency damping in the analysis of moored structures? The last two questions have been addressed

The first question has only partially been answered in a recent

paper('). That is, the four indicated methodS of calculating

added resistance werS studied in depth for the Withers' tanker. In earlier publications() it was concluded that the near-field method of calculating second-order forces provided better en-timates of added resistance and low-frequency damping. Here this conclusion is reinvestigated in the context of law-frequency damping, in the light of the availability of the additional proce-dures lB and 2A.

The fluid-structure interaction analyses applied in this pa-per ignore any steady - unsteady potential interactions, and only the robustness aspects of the four cited alternative analyses will be considered. The basis of the four mathematical models to be applied are now briefly outlined.

INTEGRAL EQUATION FORMULATIONS.

The standard governing equations for an incompressible and inviscid fluid with a free-surface, under the assumptions cf irrotational flow, are transfo*med into an appropriate Fred-holth second-kind integral equation identity using Green's sec-ond identity. Here two integral equation identities are pre-sented for two different sets of assumptions; others are consid-ered elsewhere'°i The simplest possible forward-speed fluid-structure interaction model is that corresponding to the zero-speed model with the incident wave frequency dependence of the Green function, G, replaced by wave encounter

frequeflcy(I2.

That is, model A for the unknown source strength a requires application of the identity

--aa = i cdo - e5,

-

j

8G (A)

ISV

On

where w5 is the normal velocity of the wetted-surface of the structure, S,. For general forward-speed, rather than low Iòrward-speed, the Fredholm integral equation identity applied is

,-

oc

u',

oc

(B)

ac = i

ada + , n1ad.,

-JSw On

9JLw

On

-with fi corresponding to the forward-speed and n the outward unit normal. The mathematical details of G are quite different

in each case. In Formulation A the fluid singularity pulsates

- aithe wave encounter frequeñcy, wheÑaz in Formülation B the

fluid aou*ce pulsates and translates. In each case the required velocity potential, #, is calculable frOm

'JSW

(1)

FoTmulatlons A & B are treated as mathematical identities expressing the unknown source strength at a 'selected' point of S, Os a function cf other values of o over Se,. Determination of o first requires the partitioning of S into N surface elements.

Here

assumed. Thereafter application of one or other of these iden-'plane' rather than 'higher-order' boundary ekiflents are titles at the centróid of each plane element In turn, assuming invariance oto over each element, reduces determination of a to the solution of a linear matrix equation of order N.

SECOND.ORDER FORCE PREDICTORS.

Once a, and hence 4 have beOn deternnñed, hydrody-namic reactive coefficients and excitation forces are determined, änd the associated equations of motion formulated and solved. Forward-speed correction of the wave excitation, and the by-drodynamic reactive talma dérived from ,is then necmeary in

the case of applying Formulation A. With the motion responses and thé source strengths, or velocity potentials and derivatives, determined the mean second-order steady torces may be calcu-lated.

The mean near-field forwardspeed dependent second- or-der forces and moments are evaluated using

-J,,, jwa +

Iv#t¼s

-

!ReID;a . +

!,

_{f Jyy}

. ViJda

_PUJReI6e'1#z3

- 1pgAxjRevi4 "ei

wbe

4s)

_{- U-)#ids + pg(O,O,jsAW -}

_{5A,z1) ()}

and

Ñ= (fl1,nZ,n3)/(n

+,4)4.

(4)

is introduced to account for the slope of the wetted-surface, that

is, wall aidednuese is not assumed.

¡n Equations (2) & (3), is the resuttant velocity

(2)

potential.

Each of the six distinct components on the right

hand aidé of Equation (2) are readily Interpreted('1. In some earlier publications (rather than the associated computer cede) the 'iigns' In EquatiOns (2) and (3) have not always been cor-rectly assigned due to the casual switching from Inward to Out-ward normal randomly. Here this fault has been corrected The added resistance force, Rev, corresponds to the negated longi-tudinal force derived from Equation (2).

Derivation of the far-field added Tesistance is too involved so present here. The basis of the calculation is conservation of momentum (energy) and Newman's asymptotic limits for the radiation and wave scatter potentials expressed in terms of the

Kochin functions'5. That is,

Rj + R55

(5) with

R1 =2xpcAw.eosßReIH(ß,K.)I

. (6) =21rp[f 00

Jt/3

/31/3]

[H(o.K3)I'

Kcos9d# v/a

/1 - 4rcosß

p2r-ûo _K2cosD

+ 2rpJ

H(0,Ki)12 00

y1 - 4rcosß

= cos(1/4r) : r

Uwè/g> 1/4,

and is seto otherwise,

K0 =

K2

K.(1-2reose±'1-4Téò.39)

K1 -

2coa2ß

e

H=:q1H1+Hr

and

JI(ß, K)

_{¡f o1cxplKzo -}

iK(z,coa9 + y.sinø)Ids.

Here is the incident wave heading, is the incident wave frequency, and o is the source strength associated with the six radiation problems (j = 1,2, ...,6) and the diffraction problem

(j = 7). K0 is the zero-spew wave number, whereas K1 and K2 (defined slightly differently in References 111 & 15) corresponds to poles of the freesurface cOntribution of the translating- pul-sating Green 1unction' The body-body interactiön term,

RBB,is much more complicated and generally smaller than the

wave-body terifi, R15.

Since the characteristics of R15 are discussed later in the paper it IO worth noting that this term can be partitioned into four distinct components R151

i = 1,2,3,4. The calculations

undertaken are baSed upon direct evalùation of the following expressions:

Methods 2A & 2B, and general comparisons, may therefore

lead to differences Which are dependent Upon the options

imple-mented within theprocedures,rather than differences directly

attribUtable to 'sithple' versus 'complà' fluid-structure inter.

actionanalysesor, near-field versus far-field second-order force

analyses. To make sensible comparisons of the four different calcUlation procedures outlined, the following ratioflalisatións

of all the codes has been undertaken:

Geometry generation regarding formation of 'ele-mente' and calculàtion of: elethent área, Clement centroid, normal at centroid.

Evaluation of the simple soUrce part of G, 1/r, and

¡mage

I/i',

and their normals.

Evaluation of waterplane area and first md second moments of waterplane to generate hydrostatic restoratiòn coefficients.

Differences will also occur ¡h the processing of the hy-drodynarnic quantitim in readiness foe calculation of the

mo-tion responses and the second-order forces. ¡n particular

differ-ences willarise in the forward-speed dependent e'ahiations or fórward-speed corrections undertaken, that is:

Whereas the Froude-Krylov force/ moment evalua.

tifus are conáistent in allfourmethods, evalUation

of the wave scattering component of the wave

ex-citation iù MethodIA explicitly uses the integrand operator

_UntDdi/dz

with the other methods using the 'equivalent' operator(re)

Umy.

Reactive coefficient evaluations forall methods (ef-fectively) useUrn,4'e as equivalent to the operation

Umaad'1/ax, with_4, denoting the directly

calcu-lated forward-speed or the forward-speed corrected

radiation velocity potential').

Evaluation of relative motion ¡n component I of the nearfield added resistance predictor requires evalu-ation of the free-surface elevevalu-ation at the free-surface, not at the boundary element centroid nearest to the free-surface, and so evaluation of terms of the form

is reqwred in components I and V cf the near-field procedüre(H).

-Evaluation of the Kochin functions in the far-field

method requires consistency of evaluation of 8#1/On,. and in Méthod 2A this means' appropriate forward-speed correction of

=

Rin, =K.coaß>.t,jFk,,

(9)

= pKocoaßf5

ç%àda, and

=

_pK.eosßf#da.

.A superscript indicates the complex

conjugate of a

quan-tity, 4. the incident wave velocity potential, and the

Freude-Kvylôvfàice, in the

j'

direction,¡s defined by

=

t

a,,

₍₁₀₎

h is to be understoodthatonly real partsofthe expressions for

the RI,3.componentsarc Io be evaluated.

LOW-FREQUENCY DAMPING PREDICTOR.

The added resistance is calculated for a variety of

for-ward and reverse speeds for a range of incideflt wave frequen-cies. Thereafter the added resistance is treated as a function of ship speed, and the low-frequency damping evalUated for each

wave frequency *rsiñg.

1

(ôRw\

I.

Iu=o-The non-dimensional low-frequency damping coefficient is

de-fined as

=

_{6B!/fpV,J7Z)J,}

(12)

where Vis the displacedvolume(bere), Bis beam and L s_the length cf the structure.

PRACTICAL ASPECTS OF THE COMPUTER

IMPLEMZNT4TIONS.

Since the Newcastlelow-frequency damping researchpro.

grammehas existedfora numberof years,a number of

differ-est

reseárchErahave beeninvólved. Whereas anumber ofthe

generated codes have beenbased onthe indicated integral

equa-lions presented in an earlier section, this does not in practice mean that eachgeneratedcomputercode willbehavein exactly the same manner, ce use the same intermediate calculations!

Subjectit'ity, regarding implémentation and application of the

outlined fluid-structure interaction analyses will have occurred. Similarly, a number of options may be taken regarding the post processing of the n solution in readiness for solving the equa-tions of motion and the calculation of the second-order forces by eithér the near-field ur Ear-field methods.

I

That is the wetted surface boundary condition, as well as the ve-locity potential, is forward-speed corrected in the Kochin func-tion evaluafunc-tions. These sources of possible differenceshive been

discussed elsewbereD4).

ANALYS1SP AN ELUPsOm

The ellipsoid sekcted has an LIB ratio of 8

and a B/T

latio of 2, with L equal to 100m. To

reduce the computa-tional effoft required the geometric iymmetry is explOited and the discretisation employed in the calculationsis equivalent to 220 boundary elements (lacetS) representing the wetted surface, see Figure 1. Whilst erst-order hydrodynarnic results are not reproduced bere the predicted reáctive coefficients and cxci-tation forces and moments have been compared with the 100 facet based calculations of Inglií &

Price(').

Earlier predic-tions by these rmearchers used 3ust 52lacets! In fact their first-order analysis methods designated '(i)' and'(iii)' correspond to our Formulations A & B. The óberved differences between the 'Néwcastle' and the then 9JCL' predictions for both low and high forward-speeds are sufficiefltly negligible as

not Io be a

point of concern.

Figure 1. BOundary element representatiOn of ellipsoid.

In the anàlys nowreported the Froude numbersconsidered are relatively small, since only low speeds need be considered for the píediction cf the low-frequency damping. Due to a lack of both predictions and measurements of the second-order forces and lowfrequency damping dan ellipse, in the open literature, the comparisoñs presented have to be restricted to the alternative calculation techniques outlined earlier.

Since the low-frequency damping is dependent upon the Slope of the added resntance, treated as a function of forward-speed, for each frequency in turn, one might expect different combinations of forward-speeds to produce different estimates of b,. In Table i the nine different speed combinations used in this study Of the prediction of the low-frequency damping of the ellipsoid are specified as length based Froude numbers.

Table 1. DefinItions of Speed Còmblnatlons

The calculatiOn of b',, may be undertaken in different ways. Wichers1 in his towing tests of the famoUs 200,000 dwt tanker form used five speeds, namely -2.06 ms', -1.03 ms', 0.0 ms',

1.03 ms, 2.06 ms', and then produced the required added re-sistance gradient (at the speed origin) from the slope of a best fit straight line through the. measured added resistance data. see for example b'igure 11 of Reference 131. In the Newcastle

low-frequency damping calculations cubic splines are automati-catically fitted through a 5 speed set of added resistance values for each required frequency. Thereafter the slope is determined by direct differentiation within the cubic spline ràutine.

An obvious investigation to undertake is a sensitivity study with respect to

The range of speeds used in timating b',,,.

The symmetry or otherwise of the positive and neg-ative speed assignments Used in evaluating b,,. The dependency cf the predicted damping coeffi-cients upon changes to any one of the positive or negative speeds sélected, subject to others remain-ing fixed.

The Wicliers selected forward-speeds in fact correspond to com-bination 9 of Table 1, whereas speed comcom-bination I corresponds to the speed combinations used in earlier stUdies by the

Speed Combination

Froude Numbers (U/y'jL)

i

- 02 - 0.01 0.001 0.01 0.02

2 - - 0.02 - 0.01 0.001 0.01 0.04 3 - 0.02

0.01 0.001 0.02 0.04

4 -0.04 - 0.01 0.001 0.01 0.02 5 - - 0.04

- 0.01 0.001 0.01 0.04

6

- 0.04 - 0.01 0.001 0.02 0.04

i

- 0.04 - 0.02 0.001 0.01 0.02 8 - 0.04 - 0.02 0.001 0.01 0.04 9

- 0.04 - 0.02 0.001 0.02 0.04

Newcastlà research group". In Table 2 those speed

combi-natlofli ofTable Iwhose comparison might provide some Insight regarding the sensitivity of the low-Frequency damping to any one speed value change are identified. To understand Table 2 it must be appreciated that is moving from left to right in Table i the individual speeds aie designated 'second negative', 'first negative', 'zero', "first positive', and 'second positive' respec-tively, irrespective of actual numerical values assigned.

Forinu-laton A may be readily executed for zero-speed. However, in Formulation B the translating-pulsating source Creen function is used and numerical stability in all calculations necessitates that the 'zero-speed' condition correspond to a Froude nuñi-ber not less than 0.001. Hence this definition is ùsed in both formulations for the purposes of this comparitive study.

Table 2.

SensItivIties and Speed Combination Comparisons.

PRESENTATION OF RESULTS.

The equations of motion were formulated with respect to the centre of gravity, where LCC is assumed to coincide with amidships. Using poet processed solutions for c, the added resistance, Roi, is predicted using each method. The non-dimensionalised added resistance, incident wave frequency and wave encounter frequency are defined by

R.'w=Rw/IpgdB'/Ll, 10=

and

(13)

and eachofthe Roi components is non-dimensionalised in the saine maùner.

Having critically reassessed the Inconsistenci of Meth-ods IA & 2B of Reference (li, ú indicated in the previOus sec-tions, Methods IB and 2A were developed and tested. For all the forward-speed corrections to be implemented in Method iB, both first and second-order derivatives of the Green function, G, had to be evaluated and stored. However, initiai comparisons of Methods IA & IB added resistance predictions were very poor, and so investigation of the cause was necessary. Eventually nu-merical instabilities associated with certain second-order deriva-tives of the translating-pulsating Green fuñction were found. Identification of these numerical instabilities, peculiar to For-mulation B, necessitates definition of Method IA' as an imple-mentatiòn including all second-order derivatives of the pulsating source Creen function, whereas Methods IA and IB imply omission of the identified second-order derivatives') irrespec-tiveofwhether they aie, or are not, unstable.

3'Jcar-Fiehd A4ded..RiIstnnce

Since Methods JA' & IA differ in the inclùsion and exclu-sion respectivelyofthe second order derivatives associated with

the term V

in components I and Vof the near-field added resistance, thè predictions for the descnbed elflpsoid are pre-sented in Figures 2 & 3. They show very similar trends. In

fact there is only a 0.86% reduction in the largest peak value of Ri,, as a consequence ofomitting the second-order deriva-tives of the pulsating source Green function. If one studies the relativemagnitudes of the five components of R',, then

compo-nent J, the relative motion dependent term, has the dominant contribution as expected. Also a Simultaneous plot of added

resistance for Fn = 0.0 and Fn = 0.001 provides almost india-comible differences, as expected. Thus the useofFn =0.001 as the equivalent 'zero-speed' for Formulation B b of minor conse-quence.

The corresponding Method lB predictions of added resis-tance are presented in Figure 4. Comparison withMethod lA indicates a significant drop, nearly 30%, in the first and largest peak. In fact the differences between Methods IA & IB increase as the speed increases. The smallest change, corresponding to Fn = 0.04, represents a 27% drop in value of IB peak value relative to the corresponding Method IA value. In a compa-rable study fOr the Wichers' tanker form0) the corresponding differences were between 7% and 10%, indicating an obvious sensitivity to the geometry of the structure being analysed.

7ar-Field Added Registance

-The predictions of added resistance based on Methods 2A & 2B are presented in Figures 5 & 6. The characteristics Paired Speed Combination Sensitivity Investigated

land 4

2 and 5 3 and 6

second negative speed

4 and 7 5 and 8 6 and 9

first negative speed

2 and 3 5 and 6

Sand9

first positive speed

land 2

4 and 5

Yand8

I

£ C

j:

tO 1.5 21 2.5 3.0 3.5 40 4.5 5.

bderdWa'e Frety (No

f,= 0?/)

Figure 3. Non-dlmenslonaliBed added resistance using

Method IA' Le. derivatives of G Inclúded.

o

of the speed dependence of these predictions is quita distinct

on two counts.

Firet when comparing Figures 5 & 6 with

each other. Second when comparing these predictions With the near-field predictions of Figures 3 & 4. In Figure 5, For non-dimensionalised wave frequenci

f. < 3.0, approximate'y. the

added resistance increases almost monotonically with frequency. Thereafter for the higher Frequencies, f. > 3.0, the added resistance decreases in an oscillatory manner with distinctly different characteristics for the positive and negative forward-ipeeds. The high frequency oscillations of the added resistance is directly attributable to component R,5, which has a

magni-tude comparable with R8 Furthermor

R is the dominat-ing component of

R,

with the pitch contribution dominating the

and R

components. In contrast the R89 term is sthooth, without any oscillation.

The forward-speed dependency of added resistance.almost disappears in the Method 2B predictions of Figure 6. This change of characteristic is a direct resUlt of replacing Formula-tion A hydrodynamic data with FormulaFormula-tion B hydrodynamic

data. The characteristics of R9 and

R' can again be de-scribed (respectively) as oscillatory and smooth. However, the component magnitudes, the sensitivity of the R and R88

components to forward-speed, and the degrees of oscillation, are significantly less then the corresponding predictions of Method 2A, hence the differences in Figures 5 & 6.

ßummary of Added Resistance Characteristica

Changing the first-order ßuid-stnacture interaction analy-sis from Fôrmulation A to Formulation B modifies thecharacter of both the near-field and far-field added resistance predictors. Comparison of Figures 3 & 4 with Figures 5 & 6 indicate that the most significant changes iii the nature of the predicted added resistance are the rutt of changing from Method i toMethod 2. In fact comparing peak Method 2 values with peak Method I values indicates a factor of four change in magnitude, see Fig. ure 7. However, these noted characteristics appearmuch less extreme if the geometly is changed from an ellipsoid tothe

Wichers' tanker forin(') . In the case of a tanker R5 also

dominates R' by a factor of 4 to I.

7'ear-Field Law-Freauencv Dampini

The predictions of low-frequency damping based on Meth-ods IA' and IA are presented in Figures 8 & 9 as a function of incidént wave frequency and the different speed combinations defined in Table 1. The nature of each set of plots is very simi-lar with regard to the magnitudes predicted, and the displayed sensitivity to the selected speed combinations. Peaks in the

FroudeNumb

r..."'

F.U!

- - ..i,i

--F5.001

- Fil

S F. 001

I

:

'

L

il'

¡

-'

/IÍiL.

'

5.

-,

I

LO-

..-..'

liii liii

liii

ill-I

Jill

liii

FroudeNumb I

F5.004

b.

)

_F5.401

-- Fa.OWI --F5.001

.Fii.0

5.

F5004

-¡ ¡

¡4

i

i

i

ii

I

J-..

LO

r-...

i

I,,-.

lilt Juil -lili

liii

ilil-1.0 1.5 2.0 2.5 3.0 33 4.0 43 50

bderWae FÌiq.ierq (Noidñon) t

/QJg)

Plgure 3. Nön.dlrnensloualláed added resistance using

$ 1.5 C

2

e s s LO

s

E 0

(

C s

i

s, 'e s

I

E

(

0.0 1.0 1.5 2.0 2.5 30 3.5 4.0 4.5 .50

kdgd Wave Frequcy adnens I

FIgure 4. NmdIrnenslOnaIiSed

added resistance u.ng

Method lB Le secondorder derIvatives

of G excluded.

1.0 15 2.0 2.5 '3.0 3.5 40 4.5 50

bie Wave Feieiq meiOn) 1,.

Figure 5. Nóndlmeñsiona1Jsed added reslstaflcc using

Méthôd 2A.

o 10 a. C e o 6 e e s

I

s'

E E

(

O io

ii

2.0 .5 ao 3.5 4.0 4.5 50

dfl Wave Freq Noei)

(&.\'(L)

FIgure 6. Non-dIrnensonalised added resistance using

Method 2B.

-2

10 13 2.0 23 3.0 3.5 40 4.5 50 Icódeil Wave Freierq (Norsiona» I,

FIgure 7. Non-dimeflslonallsed added resistance

corn-púlng all five method..

FroudeNurnber

- F.. OI

F..4 --Fn%omI EE

,A',ç.

jf-

I,

r

Froude Nuni ber

::

-

--F..004

.

I - . .

I-2

-,..

O-

lull 111 liii iiui _i1ï.

lull

liii

UafldiForTed

=

IB - - 2Aft

1A1)OJI)

_,'-

,'

\

W)T)

SI

g'

I,

'I

g,

s

: /

ai

-\..._...._

\

t

I

-I,

s,

I,

r,

.

-FweNnber

- iOirUl, - - F..OXI F. .OJ1 B --Fo.0i4

-7...

-

'-\

-\ A

"R

H

JA

O-.5

lIli luir

lr,r- ui-ii

Iii-1 uil iris

1 C s o s o o s o C s s s s E

i

(

1.5 2.0 2.5 3.0 33 4.0 4.5

FIgure 8. NondInienslona1iBed low-frequency damping

using Method JA'.

s P n 1.

s

h

0

¡II, Jill liii

liii

Ill,

Jill

U

UU liii

50

added resistance plots of Figuree 2 & 3 lead to peaks in the

low-frequency damping at the same corresponding frequencies. Comparing Methods IA and IB, Figures 9 & 10, the reduction in the first 18 peak valUe of low-frequency damping relative so the Method IA prediction is now lesa then 18%. whereas a 30% reduction was observed in the corresponding added resistance predictions as assault of changing the fluid-structure interaction model used.

A common feature of the near-field predictions of Figures 8 to IO is the.3 distinct speed combination groupings of the 'first peal' values of the low-frequency damping. At other peaks and troughs, similar groupings exist but are less pronounced In

fact if one tabulates the peak values, in each case, for each ipeed combination, obvious (subjective) groupings prsaent themselves.

Using 'a', 'b', 'c' and 'd' to designate différent groupings of

similar peak values, With 'a' denoting the largest magnitude, the peaks of Figures 8 to 10 may be categorised as presented in Tablee 3. IS C o n o o C E s o IL 03 u e 0.0

1.013202.53.0334.0

43.50

1015202.53.03.5404.55.0

ødeFiepiøy No*nm)

1'

fr&nl Wave nS8f) l.Ui ø,'/

FIgure 9. Non-dlmenslonalised low-frequency dàrnping

Figure 10. Non-dhnensionaflsed low-frequency

damp-using Method lA..

Ing using Method lB.

Ii

À

5-

--3

--4

5

s

D-

--

s

A 1

(A

1/4

-I

Ul'Uv'

Table S. Relative ranking of peak values

as a function of speed combinations.

If one stUdies Tables I to 3, with Figures 8 to 10, thefollowing

observations may be made:

Methods IA' and IA predictions are very similar, with exactly the same rankings with respect to speed combinations.

Selection of tbe first positive and negative speeds have the greatest influence upon the predicted peak values of low-frequency damping.

Selection of tie secoñd positive and second negative speeds haw the least influence upon the predicted peak values of low-frequency damping.

Selection of speed combinations i and 9 (and per-haps ) are the more natural, but this does not nec-essarily lead to any consistency in the associated rankings of the associated peak value predictions. Observations two to fuer may be justified as follows

Each memb speed of the 'paired speed combina-tions' of Table 2 related to the 'first' positive and negative speeds lead to different rankings in Table 3

Each speed combination of the 'paired speed com-binations' of Table 2 related to the 'second' speeds lead to identical rankings in Table 3

Different peaks bave different sensitivities Io speed combinatioo 1, 5 and 9 and this cliaraCterisiicis

eloÖsensitive to the actual first-order fluid-structure

interaçtioñ analysis coupled with the near-field low-frequency damping predictor.

The ranking of any prediction, as a fuñction of the selected

speeds, has the following characteristics:

The 'highest' ranking is associated with using the 'larger' first positive speed, Fn = 0.02, in

combina-tion with the 'smaller' first negative speed, Fn =

-0.01.

A moderate first peak ranking (b) is always associ-ated with a symmetric first positive and first nega-tive speed, but differences in the second peak rank-ings (and c) correspond to a higher second positive speed leading to a higher moderate ranking.

The low st rankingis associated with using the lower first positive speed,

Fn = 0.01, in combination with the larger first neg-ative speed, Fa = O.02.

The variations in the low-frequency damping peak values, compared with the largest predicted peak value, is 12%, 12%, 8.5% for the first peak and 12.5%, 12.5%, 21% for the second peak with respect tò prOcedures

IA', lA

and

IB

respectively.

7ar-FleId Low-Frequency Damping

Figures 11 & 12 present the Method 2A and 2B predic-tions of low-frequency damping. Their characteristics are ini-tially a little surprising. First, the highly oscillatory nature of the Method 2A added resistance prediction curves provide low-frequency damping coefficients which are almost monotonic. In contrast the slightly oscillatory Method 2B added resistance curves provide highly oscillatory low-frequency damping

coeffi-cients. FurthermOr, the low-frequency predictions of Method 2B are more in keeping with Method IB, than the Method 2A predictions are with the Method lA values. The magnitudes as-sociated with the Method 2A predictions in Figure 11 are also unrealistically high. Cleajiy some explanation of these unex-pected characteristics is required. This we shall now attempt. Croupings.f peaks according to speed combinations

F&rst peak f=26

Second peak f,3 7

Method Ranking Speed

Combinations Ranking Speed Combinations IA' a b c

3,6

i,2,4,5,,9

7,8

a

b

c d

3,6

2,5

1,4,9

7,8

lA a b e.

3,6

1,2,4,5,9

7,8

a

b c d

3,6

2,5

1,4,9

7,8

IB a b é

3,6

1,2,4,5

7,8,9

a b e d

3,6,9

2,5

1!4

7,8

20 15

t

o..

i

E s

o

-J a 03

st

1.0 13 2.0 2.5 30 33 40 4.5 50

kident Wave Fiency (Nonckmensional) f, e

FIgure 11. Nozi-dirnen8loflállsed low-frequency

damp-ing usIng Method 2A.

.0.5

a

E u

o

ID O 1.0 43. .3 .2.0 1.0 13 2.0 23 3.0 35 4.0 4.5 5.0

t

Wave Freq mens f,

FIgure 13. Non-dimenalonallsed lów-frequency

damp-ing usdamp-ing Method 3B.

From Figure 5 one may observe that the differences be-tween the 'non-zero' speed dependent added ràistance predic-tions, relative to the 'zero-speed' values, are more significant for negative speeds than for positive speeds. Consequently there is a significant fanning out of the 'differences' for the negative speeds as the wave frequency increases. These characteristics are sufficient to explain the almosi monotonic increases in the Mèthod 2A low-frequency damping predictions of Figure 11. In contrast the oscillations in the low-frequency damping predic-tions of Figure 12 are attributable to the following factors:

.

For each frequency where the added resistances are effectively equal for all speeds, azero dampiñg value will occúr.

Between consecutive zero damping values the exis-tence of a peak or trough depends upon the added resistance alues increasing or decreasing with an increase of Speed, respectively.

Common to Figures 11 & 12 is a negative spike at f = 3.1. In Figure II, Method 2A, this is caused by an added resistance

value fòr Fn = -0.01 being out of character with the added

resistance at the other speeds. Consequently, for speed corñbi-nations 7, 8 and 9 which exclude this value, the low4requency damping values are consistent with values at other surrounding frequencies, and so the damping curve is smooth. For the other speed combinations the unrealistic and unexpected spike occurs. In fact, if one carefully studies the Method 2A added resistance curves of Figure 5 for Fn

-0001 and Fn = -0.01 the tWo

added resistance curves coincide at f. = 3.1, hince a kink oc-curs in the Fn = .0.01 curve immediately after the frequency.

A similar explanation may be applied to the f, = 3.1 spike

in the corresponding Method 2B results of Figure 12. In this case the 'rogue' added resistance value is associated with Fn = 0.01 and hence only speed combinatiòns 3, 6 and 9 provide low-frequency damping coefficients consistent With their neighbour-ing frequency values. If Figure 6 is replotted, as a function of wave encounter frequency, rather than incident wave frequency, see Figure 13, a kink due to the rogue added resistance value

can be óbserved at f = 3.2.

-The low-frequency damping peak values of Method 2B, Figure 12, are respectively 27% and 38% of the Method lA' peak values of Figure 8. In Figure 14 a comparison of the low-frequency damping coefficients for each of the five prediction techniques is presented for speed combination 9. As one con-sidera Methods IA' with IA, IB and 2B there is a gradual left shift of the frequency associated with the peû values of the lw-frequency damping.

s-"-2

---.3

I!N1Á__

11F J

I

ITI

s

s

10 1.5 2.0 23 3.0 3.5 40 43 50 55 60

Wait En Freiey (NomeiioritJ) f.

Figure 13. Non.dimensionallsed added resistance using

Method 2B plotted against wave encounter frequency

1.0 15 2.0 2.5 3.0 3.5 40 4.5 50

hddøi! Wave Fteq.aq (Nondai On) f

Figure 14. Non-dlmensionalised low-frequency

damp-ing compardamp-ing SII five rnet.110d8.

CONCLUSIONS AND FINAL REMARKS

The majorpoints to be noted from the reported sensitivity studies may be summarised as follows:

The near-field methodofpredicting added resistance remains superior to the far-field method irrespective

of the first-order fluid-strUcture interactiOn analyses

used iñ the calculations.

s The differences between the near-field and far-field predictionsofadded resistance are more extreme for the ellipsoidofthis study, than for the earlier tanker study('4).

s Use of Formulation B predictions in place of

Formu-lation A predictions does influence added resistance and hence low-frequency damping prediction.

Low-frequency damping predictions are affected by the speed cOmbinations selected, and differences in peak values may vary by as much as 21% in the near-field approach.

The influenceofany individual speed within a speed combination can be studied, and heÑ selection of 'fiat' speeds has been shown to be particularly im-portant.

Although aUtomated prediction of low-frequency damping is conveniènt, using cubic splines it can lead to spurious estimates of the damping. This is particularly true if a 'rogue' added resistance value corresponds to one of the 'first' èpeeds deployed in the selected speed combination.

In future work the cause of the 'rogue' added resistance value needs to be understood and, if possible, avoided through an appropriate analysis of the data used. Further confirmation

ofthe principal conclusion would be provided (maybe) if the neglected secondorder derivatives of the Formulation B Green function could be included in the analysis.

ACKNOWLEDGEMENTS

The financial support of the Science and Engineering

Re-search Council (SERC), through the Marine Technology

Direc-tOrate (MTD) .ofthis ongoing research programm. (1 2e-9.ii- Is), and the provision ofan SERC Research Studentship to support some of the Ñsearch reported in this paper is gratefully

ac-knowledged. Froude Number

---FssUI

_'',

₄

_,4..

o r

;ø;,,

,,,, ,,,,

ii 1II Il

l-Il-U

li,, lilt

À,

A

1_A

_{y '}

.. ¡1

:

\.

I

l4"'

\J

o-

Mean2ndCerForceTectnique-MAharMfl

AWmOJl)

uBpE.rawmon)

-.0

,,iui..iii,ir,i.ii,i.tii

lii hUt hII1

g

C

i

C

s

I

I

₍

REFERENCES

Hearu, G.E., Tong, K-C, and Lau, SM., 'Sensitivity of Wave Drift Damping Coefficient Predktiòns io the Hydro-dynamic Analysis Models used in the Added Resistance Gradient4ethod', Proceeding of Offshore Mechanics and Arctic Engineering Conference, February 1987, VoI. 2, pp.

213-235.

Ream, G.E. and Tong, KC.,'Evaluatiòn of Low-Freq nency Damping', Proceedings of Offshore Technology Con. ference,Paper 5176, May 1986, Vol. 1, pp. 229-241. Wkhers, J.E.W. and Huijsrnans, R.M.H., 'On the Low-Frequency Hydrodynamic Damping Forces Acting on off-shore Moored Vessels', Proceedings of Offoff-shore Technol-Ogy Conference, Paper 4813, May 1984, VoI. 3, pp.

315-324.

Huijsmans, R.H.M. and Hermana, A.J., 'A Fast Algo.

ruhm fas Computation of 3-D Ship Motions at

Moder-ate Forward Speed', Proceedings of Fourth International Conference on numüical Ship Hydrodynamics, Washing-ton, USA, Sept. 1985, Session I pp. 24-33.

Grue, J. and Palm E., 'Currents and Wave FOrces on

Ships and Marine Structur&, Proceedings of Dynamics of Marine Vehicles and StrUctures in Waves, Edited by

Prke, W.G., Ternarel, P. and Meant, A.J. Developments in Marine Techno1o', Vol. 7, Elsevier 1991, pp. 167180.

t

Beam, G.E. and Tong, K-C., 'Second-Ordèr Fluid Damp-ing', FinaiProgiess Report (or 1983-85 SERC MTD Com-pliant Structuré Cohesive Prograñmie, Vol. 1, October 1985, pp. 250.

Ream, G.E. and Tong, K-C., 'Second-Order Fluid Damp-ing'. Final Progress Report for 1985-87 SERC MTD Dynamics of Compliant Structures Managed Programme, Vol. II, December 1987, pp. 250.

Ream, G.E. and Lau, S-M., 'Low-Frequency Damping Predictions and Behaviour of Marine Structures in a Sea Way', Final Pmgreas Report for 1987-89 SERC MTD Floating Production Systeme Managed Programme, Vol. 1, January 1990, pp. 160.

Ream, Ò.E., 'Low-FrequEncy Damping; The Develop-ment of its Theoretical Prediction', Proceedings of Dynanucs of Marine Vehicles and Structures in Waves, Edited by Price, W.C., Temarel, P. and Kesse, A.J., De-velopments in Múiiie Technology, Vol. 7, Elsevier, 1991, pp. 237-252.

Hearn, G.E., 'Seakeeping Theories: Spoilt for Choice?', Trans. NECIES, March, 1991, Vol. 107, No. 2, pp. 45-66; June 1991, Vol. 107, NO. 3, . 109-112.

Heath, G.E. and Tong, K-C., 'A Comparative Study of Experimentally Measured and Theoretically Predicted Wave Drift Damping Coefficients', Proceedings of Off-ehore Technology Conference, May 1989, Paper 6136, Vol.

4, pp. 699-714.

Ream, G.E. and Tong, K-Ci, 'Added Resistance Gradient Versus Drift Force Gradient Based-Predictions of Wave Drift Damping', Proceedings of International Shipbuild-ing Progress,

1988, VOI. 35, No. 402, pp. 155-181.

Beam, G.E., Lau, S-M. and Tong, K-C., 'Wave Drift

Damping Influences Upon the Time Dòmain Simulations of Moored Structures', Proceedings of Offshore Technol-ogy Conference, May 1988, Paper 5632, Vol. 1, pp.

155-167.

Hearn G.E. and Goodwin, P., 'Far-Field and Ñèar-Field Investigations of Second-Order Force Prediction', In press: Proceedings of Practical Design of Ships and Mo-bile Units (PRADS), May 1992.

Beam, G.E., Tong, K-C. and Lau, S-M., 'Hydrodynamic 'Models and their Influence on Added Resistance Predic-tions', Proceedings of Practical Desigìi of Ships and Mo-bile Units (PRADS), JUne 1987, Vol. 1, pp. 302-316. Séivesen, N., 'Second- rder Steady-State Forces and Mo. rnents on Surface Ships in Oblique Regular Waves', Pro-ceedings of the Dynamics of Marine Vehicles and Struc-tures in Waves, University College London, April 1974, pp. 225-239.

lnglis, R.B. and Price, W.G., 'The Influence of Speed De-

-pendent Boundary Conditions in Three Dimensional Ship Motion Problems", Proceedings of Intern ationSi Shipbuilding Progress, February 1981, Vol. 28, No. 318, pp 22