Far-field and near-field investigations of second-order force predictions

fl* Mir1ii of

Tec!uE I1 $fC82afl1CS Lab6r8t7 Ubrary M,keIweg 2 - 2628 CD Deift The Netherlands PMlS1 15Th6873-Fax31 15781836

FAR-FIELD AND NEAR-FIELD INVESTiGATIONS OF

SECOND-ORDER FORCE PREDICTIONS.

GRANT E. HEARN and PAUL GOODWIN

Hydromechanics Research Croup, Department of Mariné Technology, Armstrong Building, The University, Newcastle upon Tyne, NEl 7RU.

ABSTRACT

Two new alternative procedures for predicting added resistance are outlined and the results of their application are compared with two other alternative methods presented previously, 11]. The four methods correspond to coupling two alternative full 3D for aid-speed dependent fluid-structure interaction analyses with a far-field and a near-field mean second-order force prediction technique Wicher's 200,000 dwt tanker form is

used in the reported comparatie study Predictons arecompared with each other and

limitcd available added resistance measurements. The different possible interpretations and unpiementations of the outlined fluid-structure interaction and second-order force prediction procedures are highlighted Using a standardised set of options regarding the processing of hull geometry, hydrostatic and hydrodynamic quantities, it is argued that observed differences in the predicted added resistance of the selected tanker are more attributable to the differences in the second-order force prediction techniques than the different fluid-Structure interaction analyses used.

INTRODUCTION

tri 'PRADS 87', [1], two distinct full 3D fluid-structure interaction based added

re-sistance. calculation procedures were presented and compared with an enhanced 2D strip theory based added resistance prediction procedure, [2), an approximate

calcula-tion technique due to Gerritsma et al, [31, and various experimental measurements of ship added resistance. To the authors' knowledge this was the first timefull 3D fluid-stricture interaction based calculations of added resistance had been presented. The

two 3D methods implemented had been developed to analyse arbitrary shaped floating

forward-speed dependent first-order fluid-structure interaction analysis coupled with a direct pressure integration (near-field) method of predicting added resistance. The alterna-tive 3D proceduìe, Method 2, designated Method 2B here, used a much more complex forward-Speed dependent first-order fluid-structure interaction analysis coupled with the momentum conservation (far-field) added resistance prediction technique Clearly two other combinations of 3D fluid-structure interaction and sècondorder force prediction methods are possible, the coupling of the simplified forward-speed fluid-structure inter-action analysis with the far-field second-order force predictions, designated Method 2A, and the coupling of the complex forward'speed first-order analysis with the near-field added resistance calculation procedure, designated Method lB. That 'is, A and B indi-cates the fluid-structure analysis used, whereas 1 and 2 indiindi-cates use of the near-field

and far-field second-order force calculation procedures respectively.

All four possible methods of predicting added resistance methods have now been

implemented and applied as part of an ongoing research programme into methods of predicting low-frequency damping. Since, in particular, the surge lowrequency damp-iñg of a floating moored vessel can be determined directly from the gradient of the added resistance, as theadvancing velocity tends to zero, the development f a robust added

resistance prediction method could be a useful by-product of this on going research pro-gramme, f 4,5,6). Since reviews have already been publishedregarding the development of predicting low-frequency damping, [71, and the choices regarding the formulation of the first-order fluid-structure interactions, [8), the minimum of equations used in the reported analysis will be stated, without explanation. The notation used is generally

consistent with the earlier publication, (1j.

The aims of the current research programme reported here were:

s To remove 'subjective' inconsistencies contained in the earlier 3D analyses, Methods lA & 2B.

s To develop, implement and test alternative analyses, Methods lB & 2A. To address the problem of selecting preferentially the near-field or the faz-field methods of second-order force prediction for ship and mooring system

design.

To investigate 'how best to undertake low-frequency damping analysis for investigating moored structures.

imple-mentation 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 readiness for the second-order force predictions. Similarly, the

last cited aim is being investigated from two distinct points of view. Firstly, there is_the robustness and accuracy of the implemented 3D based computerised prediction tech-niques developed since 1984. Secondly, there is the consideration of the improvement of the mathematical formulation of the forward-speed dependent interaction problemin

terms of the interaction of the steady (wave-making) and unsteady (sea-keeping) veloc-ity potentials, [8]. In this paper the fluid-structure interaçtion analyses applied ignore any steady - unsteady potential interactions and only the robustness aspects of the four cited alternative analyses will be considered.

The analyses to be discussed are primarily being developed to assist with the improvement of the analysis of moored floating offshore structures. In particular, to

improve mooring System design, it is necessary to improve the predictions of the hori-zontal excursions of a moored structure. Horihori-zontal excursion predictions can be quite poor in comparison with experimental measurements of the horizontal cxcursions. The poor comparisons are often attributable to the use of zero.speed mean tdrift forces' in the associated time domain simulations of the motions of the moored structure in a random seaway. In reality more fluid damping exists than that attributablE to first-order fluid-structure interaction analyses, [2,4-6]. To overcome this practical difficulty a first-order Taylor's series expansion of the mean second-order forces is sufficient to appreciate why forward-speed interaction analyses are necessary and how the additional fluid damping required to improve the excursion predictions may be provided theoretically. Compa.r-isons of predicted and measured low-frequeñcy damping have demonstrated the validity of these rationalisations, [2, 9, 101.

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 generalisation of the second-order forces to include low-frequency damp-ing, raises a number of practical questions. How can such second-order calculations be undertaken effectively? What happens if existing zero-speed analyses are used to esti-mate the low-frequency damping? What are the cOnsequences o. ignoring low-frequency damping? The last two questions have been addressed elsewhere, [11, 12]. Here the first question is addressed through studying the four methods of calculating added resistance. Earlier publications [1, 9] concluded that the near-field method of calculating second-order forces provided better estimates of added resistance and low-frequency _damping. Here this conclusion is reinvestigated in the light of the availability of the additional procedures lB and 2A. Let us now briefly outline the four mathematical models applied.

MATHEMATICAL MODELS APPLIED.

Making the usual assumptionS of incompressibility, irrotatiónal flow, and water being adequately modelled as an inviscid fluid, Green's second identity can be used to pro-vide appropriate Second kind Fredhoim integral equations which are solved in place of the corresponding differential equation boundary value problem, [1., 8. The simplest possible forward-speed fluid-structure interaction model is that corresponding to the zero-speed model with the wave frequency dependence of the Green function, Ç, re

placed by the wave encounter frequency, [1-2,8]. That is, model A for the unknown source strength a requires application Of the identity

I. aG

cza= ¡

odsv.

(A)

.'sL,

On

For general forward-speed, rather than low forward-speed, the full forward-speed de. pendent Creen function based Fredhoim integral equation

r 8G

U2r

8G

i

cd.s + - n1cds - v,,

(B)

JS,,

On

gJL

On

is applied. The details of G in each case are quite diffçrent. In Formulation A the fluid singularity pulsates, whereas in Formulation B the fluid source pulsates and translates. In each case the required velocity potential, 4), is calculable froth.

(i)

Here v, the nrmal velocity of the wetted-surface of the structure, Se,, is quite different in each formulation, [8], U is the forward-speed

and n is the unit normal. Once

,

and hence 4), have been determined, reactive and excitation forces are determined and the associated equations of motion formulated and solved. Forward-speed correction of the the wave excitation arid the hydrodynamic reactive terms derived from solution

of Formulation A is necessary. With the motion responses an4 the source strengths,

or velocity potentials and derivatives, determined the mean second-order steady forces may be calculated!

The mean near-field forward-speed dependent second-order forces and moments are evaluated from

=

4!Lw IcI'd1 +

IV4)ds

s ¡ s

-- Re18i6

F ' + p

_j

Irr45 V4)]nds L 2 s, (2)

+ pU J Re[

. V4)ids

-

i76]k,

with the negated longitudinal force corresponding to added resistance, R, where

pf(

-

U-)ids - pg(O,O,3Aw

5AwXcj) (3)

and

= (fli,fl2,fl3)/(n + n.

(4)

Here1 W, has been introduced to account for the slope of the wetted-surface That is, wall sidedness is not assumed in Equations (2) & (3) is the resultant velocity

poten-tial. The components of Equation (2) are readily interpreted. Component I represents

the contribution from the changing wetted-surface area due to the relative free-surface elevation; component II gives the velocity squared effect of Bernoulli's equation;

corn-ponent III accounts for the effect of the first-order fluid force due to rotation of body axis; component W represents the change of the pressure field on the wetted-surface

due to the body displacement; component V corrects for the convective effect of the

pressu±e field variation due to the steady forward-speed and component VT accounts for

the second-order effect in rotation of the axis on the vertical force (which is zero for

pure head sea incidence).

Derivation of the far-field added resistance is too involved to present here. The basis of thecalculation is conservation of momentum (energy). and Newman's asymptotic

limits for the radiation and wave scatter potentials expressed in terms of the Kochin functions, [1. That is,

R R18 + RBB with

r ,o

r172 RBB=21TP1J

+J

_ao ÈJß = 27TPÇAWQCQSßReIH(ß, K0)] _jSW/2111H(0K2)12 dO - 4rcosO K2co.sO

+27rpJ

_IH(eK1)2,

ì dO, i 4rco.sO

a0 = cos1 (1/dr) : r = Uw/g > 1/4, and is zerO otherwise

L=

K2 K0(1 - 2rcos9 ±

/i- 4rcösO)

Ki 2cos2O

6

H=v,1H1+H7 and

H1(9,K)

¡f a1erp[Kz0 - iK(0cosO +

0siO)]ds.

Here ß is the incident Wave heading, w0 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). Ko is the zero speed wave number, whereas K1 and K2

(defined slightly differently in 111 ) cotresponds to poles of the free-surface contribution of the translating-pulsating Green function. Minor typographical errors associated with the sign of the lower limit of the second integra) and the sign preceding the third integral of RBB in fil have been corrected here. The body-body interaction term, RBB, is much more complicated and generally smaller than the wave-body term, RIB.

IMPLEMENTATION OF MATHEMATICAL MODELS.

Formülations A & B are treated as mathematical indentities expressing the unknown source strength at a point as a function of values of o over S,,. Partitioning S,, into N plane surface elements, application of these identities at the centroid of each element in turn, assuming invariance of o over each element, reduces determination of o in either formulation to the solution of a linear matrix equation of order N. Thus all computer codes based on plane boudary element solution of the indicated integral equations will

use this approach. However, this does not mean that each generated computer code

will behave exactly the same, or us the same intermediate calculations !

Subjectiv-ity, regarding implementation and application of the outlined fluid-structure interaction analyses will occur. Similarly, a number of options regarding the post processing of

the solution for o, in readiness for solving the equations of motion and the

calcula-tion of the second-order forces by either method will exist. Therefore comparison of predictions from Methods lA & lB and Methods 2A k 2B, and general comparisons, may lead to differences which are dependent upon the options implemented within the

procedures, rather than differences directly attributable to 'simple' versus 'complex'

fluid-structure interactiòn analyses in the near-field or far-field, or, near-field versus far-field second-order force analyses. Therefore in the context of the present study the following subjective chòices have been rationalised:

Geómetry generation regarding formation of elements and calculation of: element area, element centröid, nôrma! at ceritroid.

s Evaluation of the simpk source part of C, 1fr, and image i/r', and their

normals.

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

e o determined from Formulation A cannot be corrected directly for forward-speed and so o ¡n Method 2A should be replaced by the term

-i8/8n in the Kochin function, where is the fórward-speed çorrected

velocity potential.

Since o, rather than s, is determined in both Formulations A and B application of the kinematic free-surface boundary condition to determine , the wave elevation at

the free.surface, from , requires a first-order Taylor series expansion to evaluate çj. In particular, çj,, is computed from the near free-surface element potential using

1(04'

84'\

6!!,

p84'

faA)" VUT)

where 4' is the time dependent velocity potential, at the appropriate boundary element centroids, and includes both radiation and diffraction influences. To achieve this will require evaluation of the term V (as in component V), i.e. second-order derivatives of G are required.

For consistency 8/8n, the fluid velocity on S,, is replacedby y,1, the structure's velocity, of Formulation B, E8]. That is the boundary condition, as well as the velocity potential, is forwardspeed corrected in the Kochin function evaluations.

APPLICATION OF MATHEMATICAL MODELS TO A TANKER

As in earlier studies 11,2,9] the Wichers' tanker form ofTable i is investigated. Assuming geometric symmetry is to be exploited the 268 facet disçretisation of Figure 1 is used withboth the fiuid-structúre interaction analyses of FormülationsA & B.

TABLE i

Wichers' tanker 200k dwt, principal particulars..

(io)

The equations of motion are formulated with respect to the centre of gravity, where LCG is assumed at a.midships. Using post processed solutions for a the added resistance, R, is predicted using each method. The non-dimensionalised added resis-tance and wave encounter frequency are defined by

= Rw/1pgçB2/LppI and

f =

(ii)

L,

B T V D KG Cb

4*

FACET DISCRETISATION F WICHERS' TANKER

Figure 1. Plane boundary element representation of tanker.

Similarly, each cOmponent of Rw is non-dimensionalise4 in a similar manner Having investigated the preprocessing of Methods i & 2 of reference [I], as indicated in the previous section, Methods IB and 2A were developed and tested. In particular, áll the forward-speed corrections were implemented in the near-field, and so storage of both first and second-order derivatives of G Was necessary. However initial comparisons of Methods lA & lB were very poor and sò investigation of the cause was necessary. Eventually numerical instabilities associated with certain second-order derivatives of the Creen function, G, associated with Formulation B were found. Consequently here we introduce Method lA' as an implementation including all second-order derivatives of G, and now use Method lA and lB to imply omission of the unstable second-order derivatives. However, comparing Figures 2 and 3, the added resistance as a function of

forward speed is almost identical using Metho4s IA' an4 lA. In fact with or without

the second-order derivatives included components II', III' and iV' are exactly the same, component r dominates with negligible differences and the least significant contribution, component Y', has very small observable differences. Comparing Methods IA and iB, Figuies 3 & 4, the trends remain very similar with a 7% variation in the highest

peak and a 10% variation in the group of lower peak values of R. The generally

small differences between Methods lA & iB predictions suggest that the radiation and diffraction potential contributions to component V' are negligible in Comparison to the

incident potential and are therefore not too important. Whereas in component l'the associated implication is that the V1 term can be neglected. Since spatial correction

of the 84/ät term was included, because it only included first-order derivatives of G, it is not possible to conclude that boundary element centroidal evaluations of freesurface elevation are sufficient.

In fact the simple source and ¡mage source related quantities use an apprOpriate mix of numerical and analytic integrations according to separation of source point and field point, r. Here the waterplane area related quantities required for the hydrostatic coefficients use analytic integratiOns of appropriate Lagrangian polynomials fitted to the waterline.

Next we consider possible differences in the evaluation of the first-order

hydro-dynamic quantities required in the motion response analysis. This analysis, will be considered a first step in the second-order forcé calculation process, and therefore noted differences in this procedure can be identified in terms of the procedure designations lA through 2B. The differences in processing the hydrodynamic quantities for the motion respnses will arise from the forward-speed corrections, that is:

Whereas the Froude-Krylov force / moment evaluations are consistent in

all four methods, evaluation of the wave scattering component of the wave excitation in Method lA explicitly uses the integrand operator UflÍ8D/OZ

with the other methods using the 'equivalent' opérator Uin,0.

s Reactive coefficient evaluations for ail methods (effectively) use Urri, as

equivalent to the operation Um18/ôz, with denoting the forward-speed radiation velocity potential. From , generated fröm Formulation A, the

required . is determined using the usual strip-theory type corrections

=

: k = '1,2,3,4 and

_{4. =}

: k (9)

6

The post processing of the fluid-structure intetaçtion analysis to match a

par-ticular second-order force calculation prodecure also introduces differences associated with:

evaluation of relative motion in componént I and V4 in component V of thè near-field procedure.

.

evaluation of the Kochin functions in the' far-field method. These last differences can be particularised a.s follows:

the relative motion term of component I requires free-súrface elevation at the free-surface, rather than at the centroids of the boundary element adjacent to the free-surface. Also the free-surface elevation calculation should reflect

contributions due tö the diffraction and radiation potentials, not just the

-sa

to

I. S

fa

'.5 7.0 7.S 10 3.3 ..O S 5

Figure .2. Nondirnensionalised added resistance using Method i A' i.e second-order derivatives o G included.

te o. to s 0=-2.090m!. 0w-1. Ym/. u -o.sis _/. 0= 0.006/. u U w 0.51$ml. *

- °' 1

\ :

.

\

u= 2060m!. :

\

. \ ' \

:e '.

\ q 0'b I qO

' X

z '.3 METHOD lB LS 10 1.3 ..0 .3 33

A

METHOD lA i

e \°' r'

\, \.1,

f

.$o,.I '.

..

p ' C e S

L:va.'

Ie

CS fa 2.Ò 1.0 1.3 .b

Figure 3. Non dirnensionalised added resistance using Method i* i.e second-order derivatives of G excluded.

Figure 4. Non-dimensionalised added resistance using Method lB i.e second-order derivatives of G excluded. p.o e 0=-2.060m!. 0=-1.030m!. ° U 0.515 m/i 2.7. U 0.007m,. U = 0.515m,. £ 0= .1.030/. U = 2.060/. k a.. w

t.

The sensitivity to the differences in G associated with Formulation A and B is echoed in components II' and III', see Figures 7-10, whereas Figures 5 and 6 demonstrate an almost total insensitivity in component I'. Figures 7 & 8 and Figures 9 & IO show that Method lB produces larger negative values in components Il' & Ill' than Method lA. A slight reduction occurS in component 1V"s negative contribution when using Method

iB, whereas component V, has smaller positive values for forward-Speeds and larger

negative values for reverse-speeds. In general Method IB seems to produce smeller mean second-order forces in the low forward and reverse-speed regimes. Figure 1 provides a comparison of the Method IA predictions and measurements of added resistance by

Wichers for a speed of 1.030 in/s.

METHOD IA s U=-2.060/' u = =0.515ml. 0= 0.000m/. s U-= 0.515m,. a fJ= 1.030m!. ' U= 2.060m!. a' fi s S E'

z

hi

z

o C C.) z C hi C C METHOD lB

rs

L' = 2.060 =/s s = 0.515 /s (. = 0.007 ml. L.' = 0.515 mjs a _{= 1.030m!.} Ü= 2.060.m,', I. IS 0 &S IC

Figure 5. Non-dimensionalised near-field Figure 6. Non-dimensionalised near-field component I' using Method lA. component 1' using Method lB.

The near-field added resistance does reflect a Sensitivity to the fluid-structure interaction formulation used, but to complete the study the numerical instabilities iden tified will require replacement of the current quadrature techniques, capable of handling the cited numerical problems, if the conclusions dtawn are to be established absolutely. Comparing Methods 2A & 2B, Figures 12 & 13, the band of oscillation at the higher frequencies is significantly lower when applying Method 2B Component_{RB in} Figures 14 & 15, demonstrates that thé wave-body interaction term accounts for 80%

0=-1.030m,. 0-1.030m/,

2.5 '.0 2.3

4 2

Figure 7. Non-dirnensionalised near-field Figure 8. Non-dimensionalised near-field

component Ii' using Method lA. component II' using Method lB.

-0 S METHOD lA

'I

aS ' % U=-20.G0g%/s UI.030m/. U = 0.515 ;. U= 0.000ml, ! U 0.515 m/l 0= 1.030m/i .' = ?060m/' .1 z z o 8 Q

z

¡

f. METHOD lA U=-2.06O/. 0=-1.030m!. U = 0.515m!, 0= 0.000m/. U = 0.515m/. o 0= l.030/. 0= 2.060m/. S0

b

z z o 'Ç o Q 1.1 Q

z

UI -35 Q -)0 .5 -0.S :1 - $ * s os 05 .0 . 9 METHOD IB U=-2.060m/. U=-1.om/. = 0.515 mf. U 0.007/i u _{U= 0.515m,.} 0= 1.030w/i 0= 2.060/. 'V, f' s. f' METHOD IB U=-2.060m/. U--1.030m!, U = 0.515 m/. U= O.007/. ! 0= D.SlSm/. 0= i.030/i 0= 2.060m/i

Figure 9. Nondimensiona!ised near-field Figure lO Non-dimensionalised near-field component Ill' using Method lA. component lii' using Method lB

u u I U = 1030 /. icautk Prrdiétîo T U = 1.030 /s Wjth' Mrueit JI

Figure ii. Comparison of near-field predicted and experimentally _measured non-dirnensionalised added resistance.

of the mean second-order forces. The body-body interaction term _{RB of Method 2B} contrasts more starkly with that of Method 2A (cf Figures 16 & 17) with _{arger peaks} and a narrow oscillation band at t.he higher wave frequencies. The _{term RB can be.}

partitioned into four distinct, parts. The first part R1 is radiation dominated,

_the

second part R can be related to the Froude Krylöv force, whereas R,B3 arises from the scattering of the incident waves, and R'1B4 is negligibly small.

_{.RB1 and R2 are}

dominated by pitch, but it is R83 that is the dominant positive contribution to RBz4n both Methods 2A & 2B. However, thé nature of this dominance in. each case, Figures 18 & 19, are quite different and account for the differences in R8 presénted_{in Pigures 14 &} 15. In Figure 18 R}83 is. a smooth curve without oscillation until_{a larger reverse.speed} is considered. For Method 2B the larger reversespeed has the largest peak value and all subsequent speeds have progressively smaller vaues. In both methods the heave and

diffraction Kochin functions dominate the low f region of Rß and the_{pitch Kochin}

function contributes more significantly in the central

_{f range.}

Having described the sertsitiviti s of the near-fléld and far-field predictions 'to the adoption of fluid-structure interaction Formulations A & B one must compare the two sets of methods through comparison of Figures 3 & 4 With 12 & 13. This shows_{that the} peak far-field prediction is twice the peak nearfield added resistance prediction. The fa.rfield added resistance does not increase ¡h a straight forward manner with

co U.-2.CeO/.

u._).Qi.

u - -o.si _/. .0= OCO7/. U = O.$S W3/S 0(1= i.oeo/. ' (1= 2.$O/. Ii, 1/

-

G- G

-- -. 'p o0 S

(J=-2.0i/.

. 0.5- S U=-0.515/, - 0= 0.7m/. I (1= 0.S15/s 0.0 =. 1.03o/s, ' 0= 2.O6C/./ 2.5 o u Ñ ¡ z I.0 Q

I,

Q METHOD3A I. '.5 ¿0 2.3 0.0 0.0 0.? SS MnoD 3A f. 0.0 13 oO '5 30 S 3. P. J. Q METHOD 2B

z

Q Il Q Q S U =- -3.060 =1. O (1 à -0.55 _/. U 0.0Q7/s Ui= 0.SlSoe/s * (là ' 0= 2.OGOm/o fi

Figure 2. Non-dimensionaiised added Figure 13.. Non=dimenionalLsed added resistance using Method 2A.

i *0

resistance using Method 2B.

3.0

z

g Q O *34 _s

" METHOP2B

.';,a 'I. \¼»Iiia

. 'p..

I

;:",

-

0 --

0

I

s. V. U = 206O =15

u=-i.o/.

U = -.0.515=/s -U= 0.007/. U = 0.515 /. o 0= 1.030/à 0= 2.0601á f. I I S ¿S II 15 -0.0 45 0.0

Figure Non-dimensionalised far-field Figure 15. Non-dimen.sonallsed far-field

component R ing Method 2A. . component R5 using Method -2B.

-I/hid \ 1111 e T?; /':

,3T, / \4

3 J

Hi'.',

;tv' ¿

!

b P. 'b

'ti

T fr. L L _\

:

I _{\ '}

I.

''i

'\

\L..:'S

Q lia.' " h.., I I *.4

''

I

'

I.

J

I;i '\'

I

METHOD SA to as &b I.! S 'I I..

t.

1! to

t

o Q tay 4 I.' toi. 0.007 _{U 2.060 /i} Ï 2 O 0.515m,. _{u -ioeo/. t7t C} U = 1.030m!. _{U 0.535w/s} U- 2.060w/s

/

s'

METHOD 2A I U=-2.OEOm/s U=-1.030m/. U = 0.515wfi U 0.007m!. u = o.si5 /. S 0= 1.030w/s 0= 2.060/s I

t,

LS '.5 / L Ç"

..

U/ ¿

4

U. ¿

g';'

f,

o

f.

i.

I'

f a ?, ,. s,(' ,

1'

.15ß I J S '60

. 1

s

I.'

''j

Í:/

1.i.60 .

tIC

60 t. T 4 Lb 0=-2.060w!. 0=-1.030w!. U 0.515 rn/s 0= U = 0.515 rn/s U: 1.030w/s 0= 2.060m/. I. METHOD 2B

'

LI U = 2.060 rn/s U 1.030 _/s U = 0.515 rn/s 0= 0.007w/s U 0.515w/s

a u= 1./s

' 0= 2.060w/s s., f. as to

Figure 18. Non-dimensionalised far-field Figure 19. Non-dimensionalised fazefield

component.Rr53 using Method 2A. component R'VB3 using Method 2B.

ai io 2.3 t .3 30

Figure 16. Non-dimensionalised far-field Figure 17. Non-dimensionalised far-field

increases in forward-speed, as does the near-field approach. For the low forward-speeds considered, use of Formulation B, rather than A, generally reduces the predicted added resistance. The position of the peaks and observed differences in their location is due more to the differences in thesecond-order force calculation analyses than thé interaction formulations employed. The. near-field procedure remains the preferred approach to calculate the added resistance and hence owfrequency damping.

CONCLUDING REMARKS

Analyses are often described at length regarding the mathematical detail of the

for-mulations 11,2,8], but little is conveyed about the interpretation and subsequent

im-plementation of selected methods other than their ability to compare reasonably, or otherwise, with other analyses or experimental measurements. In this paper

mathe-matical detail has been minimal with more insight being offered on how subjectivity can enter the analyses echoed in computer codes. Here four analyses have been under-taken with differences minimised in handling geometric, hydrostatic and hydrodynamic details. Consequently, the differences reported can be attributed to differences in the mathematical formulations A & B and the second-order force procedures i & 2.

Having undertaken a simila.r study of an ellipsoid the authors. conclude that the near-field method is the more robust technique, providing results môre compatible with

experimental data (added resistance and low-frequency damping). The exclusion of

second order derivatives in component V of the near-field procedure makes this term independent of radiation and diffraction effects, whereas the influence upon component I is not too significant at the low speeds examined. in future research, steps to overcome numerical instábilities associated with higher order derivatives of G in Formulation B could. lead to an even more reliable and robust prediction procedure.

ACKNOWLEDGEMENTS

SERC MTD Support of this ongoing research programme (4,5,61 and the provision of a research studentship is gratefully acknowledged.

REFERENCES

}Iearn, G.E., Tong, K-C. and Lau, S-M., Hydrodynamic ModeI5 and their Influ-ence on Added Resistance Predictions, Proceedings of Practical Design of Ships and Mobi1e Units (PRADSI, June 1987, Vol. I, pp. 302-316.

Hearn, G.E. and Tong, K-C., Evaluation of Low-Frequency Damping,

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