Non-linear wave induced second order motions of floating and compliant structures

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NON-LINEAR W4 VE INDUCED SECOND-ORDER

MOTIONS OF FLOATING AND

COMPLIANT STRUCTURES

by

GRANT E. HEARN

Advanced Design for Ships & Offshore Floating Structures

Nonlinear Wave Induced Second-Order

Motions öl Floating and Compliant Structures

Grant E. Hearn

University Ql Newcastle Upon Tyne.

introduct iòn.

The fluid-structure interaction analysis of a floating or compliant structure can

be viewed with different levels of complexity. The first task is to decide upon

the level of complexity to be. associated with the modelling of the wave system

responsible for the fluid-structure interaction. This so called 'incident' wave is

generally considered to be regular (in terms of its geometric form) and harmonic in time. The waves are also usually described as 'prögressing', as they travel from their region of generation to other parts of the seaway. Description of the fluid flow resulting from the wave motióñ will be linear or non-linear depending upon

the wave slope, and hence the amplitude of the fluid velociti induced by the progressing regular harmonic wave upon the water particles. Irrespective, of the nature of the associated induced fluid velocities the waves will ultimately interact

with some structure. The structure may be an' advancing ship, a fixed jacket

structure, a compliant sVructure or a floating offshore structure. In this case our attention is d±awn to the scale of the structure and how this might affect the wave structure interaction.

For slender structures or Structures made up of slender members, the viscous nature of the associated fluid flow can be considered more important that the

pressure loading resulting from the contact made by the incident wave system. In

fact from an 'inviscid' viewpoint the structure or structural members are often

considered to be transparent to the wave system. In this case the potential wave. loading is known as Froude-Krylov loading and the nonlinear aspects of the loading are attributed to viscous drag through the Morison formula.

For the larger displacement structures the concept of transparency is inappropriate and so it is necessary to recognise the development of a scattered wave field due to the incident wave structure interaction. The phenomenon of wave scattering is clearly to be. associated with the transfer of momentum from the fluid to the

structure. If the structure is floating the loading due to the reversal of momentum will lead to the structure moving. For each degree of freedom of motion additional so called 'radiation' wave systems will be Set up as a consequence of the water now being excited by the motions of the structure. Thus the structure will experience fluid loading as a result of the incident wave System and its own motions.

Irrespective of whethér the floating structure is steadily advancing or moored the impact of forward speed effects upon the modelling of the fluid structure interac-tions and hence the fluid loading and mointerac-tions is very critical, and a number of choices have to made regarding the sophistication of the analysis tó be undertaken. In the following sectjons we shall thereföre consider:

formulation of the zero speed and forward speed fluid structure inter-action problems,

alternative solutions of the first order problem, calculation of mean second order forces,

the impact of first order models upon second order force predictiöns, the prediction of low frequency damping and its impact upon the calculatión of low frequency motions of moored Structures.

2. Modelling Fluid-Structure Iìiteractiöns.

In this section we remind ourselves of the basic fluid equations and then generate a general fluid structure interaction analysis model with forward speed depen4ence. Thereafter different levels of simplification are discussed with respect to different situations of interest.

2.1 Some Basic Fluid Equations.

The fluid is assumed to be inviscid and incompressible and subject to irrotational flow. The condition of incompressibility reduces the continuity equation

(la)

to

(lb) Since irròtational flow means there is no circulation then

and so Equation (2) and the vector operator identity

VA(V)O,

(3)

applicable to any scalai function which is sufficiently continuously differentiable, suggests that the fluid velocity vector 1 may be treated as

(4)

In. this case we refer to as a velocity potential.

The Cartesian rectangular reference system Oxyz is assumed to advance with steady speed U, with z positive forward, z positive upwards, and the origin O

located in the undisturbed free surface. Assuming (u, y, w.) denote the fluid velocity components, the scalar form of Equation (la) is

äu 8v 8w

- ± - + -.- =,o.

_äx

ôy ôz

Also, noting that Equation (4) meas that the yelocity potential function (x, y, z, t)

and. the fluid velocity components (u, y, w) satisfy the basic relationships

u = -, y = --- and w = -,

ay 8z

we may combine Equations (lb) and (4) to produce the well known Laplace

equa-tion .

(7a)

The Laplace equation may also be expressed. in tEe form

2'

2 2

V

+;;-+;;-:=O.

âx äy

äz

upan combining Equations (5) and (6). This equation clearly applies everywhere in the fluid since it encompasses the conditions of incompressibilty and irrotational

flow.

2.2 Formulation of General Interaction Problem.

Jf the fluid is considered to be bounded by the surface S, theñ in the context of general offshore related fluid structure interaction analysis S may be considered

to comprise of the wetted surface, of the structure, S,,,, thê.free urface, Sj, and

(5)

(6)

the seabed, Sb. For reasons which will become clear a little later in the nòtes we also introduce a closed vertical arbitrarily shaped cylindrical surface, S,., which terminates at its intersections with S and 5b In the theoretical studies of models operating in a tank, Sb will become the tank floor and S,. will be partitioned into tank walls and appropriate control surfaces. Both the surfaces 5L and Sb are to be considered impermeable. The impermeability of the seabed, or the tank floór, and the assumption that the fluid is inviscid, means that

= O on z = d,

(9)

where d is the depth of the water.

Since the location of the free-surface is not known a 'priori we shall assume that it is described by an equation of the form

S1(z,y,z,t) =0.

(lo) Since fluid particles on the surface are not allowed to leave the surface their velocity relative to the free surfáce is zero. Equivalently, using the substantial r material derivative, to follow the particle on the surface this condition may be expressed as

DSÍ_

Dt

The operator takes into account the local changes due to the. passage of time and the changes brought about by changes in spatial location. If the structure

advances with speed u in the positive z direction, then relative to the steady advancing reference system Ox, y, z can be expressed as

D_8

_ô

Dtät

On the free-surface continuity of atmospheric pressure and fluid pressure is to be assumed. Since pressure gra.dient rather than absolute pressure is responsible for fluid flow, the atmospheric pressure cali be taken as zero without any loss of generality. Also the. Eulerian equations of motion of the. fluid, with respect to the steady translating reference system, can be reduced to the composite pressure equation of Bernoulli, namely

p =

_[( - u-) +(V)2 +z].

(13)

Here p is the hydro dynamic pressure.

Expressing. the free-surface Sj(x, y, z, t) = O in the form

S1(z,y,z,t) =zç(z,y,t)

(14)

a composite free-surface condition, applicable on the insta.ntaneous position of the free-surfacê, is expressible iñ the form

±U2U

8t2 8x2 Oxôt

at

ax

±V.V(V) ±g--=O.onz=ç.

(15)

This equation is etermined from the cóhdition O, and Equations (11) and (13), and some manipulation!

Qn the instantenous position of the wetted surface of the vessel, cóntinuty of the structural and fluid nörmal velocity components leads to the kinematic boundary. condition

V.(V-Ü)

ñ (i

±flAz),

(16)

where is the poSition vector of a wetted surface point with respect to the body fled frame, Ox', y', z', j is the velocity of the origin O' of the body fixedreference system relative to the steadily advancing reference system Ox, y, z and f2 is the angular velocity of the body fixed reference system, and hence of the vesSel. Since Equation (7) is to be satified at every point in the fluid, this equation may be considered the governing part ial differential equation. Since the form of the equation is elliptic, in terms of the classification of partial differential equations, its solution demandS specification of bounda.ry conditions for Qn all subsurfaces of a closed bounding surface S On the control surface S this can be quite difficult..

For the moment it is sufficient to note that an appropriate boundary condition must be app led on S.

At present both the rigid bódy motion and the fluid structure interaction problems must be solved simultaneously in. the time domain., because of the dependence on the instataneous positions of the free surface and the wetted surface of the vessel.

it is also necessary to note that the total veli,city potential in the case of the

speed problem, , and. the unsteady problem potential associated with the

interactions of the vessel änd the incident wave, system. That is, represents a perturbation of the steady wavemaldug problem. Hence we must consider the two different free surface elevations , assoçiated with the vessel steadily advancing in

otherwise calm water, and the final prâffle ç resulting from the perturbation of ç5 as a resiilt of introducing the incident ave system and hence the additiOnal fluid

structure interaction. -

-It is the implied, but as yet unquantífied, interactions of t)le normally two dis. tinct hydrodynamic problems of waveziaking and wave scattering, he obvious nonlinearities of Equation (15) and the hidden complexities of the wetted surface boundary condition of Equation (16) that make the forward speed dependent iii. teraction analysis complex. Likewise it is not possible at this stage to be precise about what we intend to mean by nonlinear, or even second order, forces.

Clearly, some simplifications are required to make the mathematical problem

tractable.

2.3 Linearisation öf General Interaction Problem.

A formal approach to identifying the simplifications is perturbation analysis. Thus both and ç are to be expanded as series of the fórm

I(x,y,z,t) = 3(z,y) + 4 (x,y,z,t) ± 2'2(z,y,.z,t) +

... (17)

and

ç(z,y,t)

ç3(z,y) + cçi(z,y,t) ± E2ç2(x,y,t) ± ... . (18)

That is, different order governing equation pairs [ç(z,y,t), (x,y,.z,t)1 may be obtained for the unsteady potential

_{given (,}

) are known.

The value of- on the instantaneous free surface in terms of its value on z = ç8 is. given by

(z,y,ç;t) = 5(z,y,ç3)+ E[i(z,y,ca;t) +

+

O(2) , (19)

this leads to the linearised (first order) unsteady potential i satisfying (') a3 / g _-2

2U(Va.V:8 +V

v--.9z2 8z8z äzäz äz

är

+

(V3

V(V42)]

[a2i.

± g- ±

U2881

2tT''}

V(VØ V8) +.V8 V(VS

= O onz

with çj the first order correction of the free surface satisfying

+Va!VlU

-

g ± V'3

-

uJ

Similarly, to satisfy the wetted surface continuity condition the outward normal

on the instantaneous location of S, is required and since this is not possible a'

priori the outward normal vetor: n., the origin velocity Vo and the angular vélócity fl are. also expressed as series of -the -forth

fi

ñ +

± 0(c3)

=EV01 + E2 V2 ± O(es)

and

fZ=ffl1±2fl2±-O(e3).

(21)

Hence, it follows that Equation (16) may be expressed in the form

V + e[ñ3 V1 + ti1 . 1T= fi

. [1 +

A?] + O(e2) (22) where V3, = - Ui is the fluid velocity caused entirely by the steady fOrward

motion -relative to the translating reference system. Next using Taylor series

ex-pansions of V and on the instantaneous surface, in terms of their values On the. steady forward speed surface, ç, leads- to

n3 Va. + En.s

(' V)V -f-e[n.3 V1 + n.1

V] = cfi3 [Vai +

fl

A T]. (23)

This equation is correct to first order. Since the zero-or4er term. vanishes identi-cally and the first-order outward normal component ti1 can be expressed in terms of n.e, further vector identity manipulation leads to.

-(24) (20a)

(?7j,,2,t73)exp

iwe('14,?5,t76)eXP(iLet) ,

where w, the encounter frequency, satisfies

2ir

='o - ---Ucosß.

iwèt

(26a)

(26b).

(27)

Here wo denotes the incident wave fequency. and ß defines the wave heading of the incident wave relative to the negative z axis; Hence awave heading of 180 degrees corresponds to a head sea. If the ftist-order unsteady potential is nOw partitioned.

as

6

i(z,y,z,t) = 1(z,y,z,t) + D(zyzt) +

>2

1k(Z,y,Z,t)

(28) k=1.

then Equation (24) reduces to

a

8D

6

8k

- = - +

+ > 7k

exp(zwt)

8m 8m _k=1

a.Öi+Ts[(1V)5i - (i .V)V] on SW.

Alteniatively, expressing the right-hand Side of the wetted :stirface boundary

con-dition as . , .

6 6

>2(iweflk)'1k.+ U >2 mkm (30)

k=1 k=1

(29)

where i is the first-order displacement

Öi e + O A F, (25a)

and the corresponding first order velocity is given by

Voi. + fl A F. (25b)

So far, the time dependence of the rigid boating or compliant structure, and the fluid motiOns resulting from the 'unsteady' motiöns have not been specified. From this point, on the incident waVes and hence the responses to their excitation will be considered harmonic, and so, typically, the fii-st.Grder translations and angular velocities will be expressed as

4

it can be shown, after some manipulation, that

E

mktlk

= -

1

[(8

.V)t8] ,and

k=1 ₍₃₁₎

E

-

[(fta

.V)(A3)]

where i3 V8/Ü corresponds to the unit forwar4 speed case. Hence, on the mean

wetted surface S, we have

a'k

_iwnk+Umk

_1,

and

8n (32)

However, we must note that after all this manipulation to simplify the problem by replacing calculations òn unknown surfaces to calculations on assumed known surfaces, and. the linearisation of the unsteady problem, now assumed to be har-monic, the oscillátory and steady forward speed potentials interact in a noilinear m.nnert I±i fact only if we .egléct the steady poténtial and its associated free. surface ç, will, the free surface and wetted surface boundary conditions simplify any furthèr.

Therefore neglecting the effect of the free surface condition, Equation (15),

reduces to the more commonly quoted form of the for d speed free surface, namely

82 ₈

₂₈₂₁

at2

+±U

82 -

=0 on z=0,

(33)

whereas in the corresponding wetted surface boundary condition the forward speed related m terms may be simplified to

Tnk=O.:

k=1,2,3,4,

7725 fl.3, ₍₃₄₎

m =

-The terms (ni,

2, ns) are the direction cosines associated with the first order

definitions of the outward normal i.

The. free surface condition expressed in Equation (33) can in sme instances still be considered too complex for the situation being analysed. In the case of moored

structures the large amplitude low frequency excursions of the structure cari be viewed as a low variable forward speed. On the otherhand for ships the orders of

the magnitude of the wave s associated with the forward speed and the oscillatory interactions may be considered comparable. Therefore further enhancement of the free sutface condition can provide a compromise between the intractable form Of

Equation (20a) and the much simplified form of Equation (33). 2.4 Different Forward Speed Formulations.

The first three formulations will presented assume no interaction between and and . The fourth formulation will assume a 'weakeñed' interáction between the steady and unsteady potentials on the basis of low forward speeds. In each case linearisation will have been performed. The results are simply quoted as they can. be deduced from earlier sections of the notes.

Formulation A

The simplest approach to treating low forward speed is to assume it enters the

calculations through the encounter frequency only á.t the basic interaction analysis.

hi this cse wê have

V2 =0

{a

g_+Ju=o.on S,

a

=o,on S

Z(IJeflk+Umk on SW.

Setting U = O obviously provides the corresponding zero speed forrnulatiön. Förmulation B

In this formulation the explicit dependence of the composite free-surface boundary condition upon U is recognized to the extent that Equation (33) is applied. That

is,

ra ía

a

TI

=0 on

(B)

with the remaining equations unchanged. Formulation C

O

In this formulatiön the Green function associated with Formulation A is either

expanded to remOve the U2 dependence(2_3) or the U2 dependence is removed from the composite free-surface condition on the basis of low forward speed(). In

either case the free-surface boundary condition is effectively reduced to

rO

82 82

[g; +.= 2U8

]=0

oi (C)

As before. the remaining equations of the formulation are as above for Formulation (A).

FormuM ion D

.

.

When includiag tie steady and unsteady wave interaction the dynà.mi free-surface condition derived by applying Bernoulli's .equation. leads to

'i

+

IvI2j,

(35)

as the Lorentz transform implies that bé rp1aced by

(36)

Using these equations and neglecting higher order terms of V3 the new prqposed free-surface boundary condition may be written(s) as

-

_u--)2]

+2(

-

u-)(v3

o Sic. (D)

2.5 The Generai. Zero Speed Pröblem

Having considered the forward speed problem in general terms the sithplef zero speed problem is now examined so that one can appreciate how the higher order representations of the incident wave aie derived. This is of relevance when deter-mirjg the nonlinear forces modelled by the Morison formula used to estimate the fluid loading on the slender structural members of offshore jackets and platforms Introducing, for convenience, the shorthand notation

with the normal derivative denoted by , the nonlinearised set of governing

equations are: - :

-V2 =0 everywhere in the fluid,,

= O on z = 4,

onzç anci

(38)

zÇz + yy + 'zÇz ± Çt =0 on z In this case we may use the Taylor's series

(z,y,,t) =

(x,y,0,t) +

+

ç2(Ç)

+ .... (39)

to reduce the 'exact' kinematic and dynamic free-surface conditions of Equations (11) and (13) to provide the linear equation pair

gçi±it=0

it iz 0, (40)

and the second-order equations pair

9Ç2 + 2t = Çi''itz -

+

y + z]

(41) Ç2t - 2z + Çiizz - 171z'1Z ÇIyiy.

Both sets of equations now apply on the un4isturbed free surface i = 0. Ii fact, in general one can generáte the nth order equations pair with the generai form(6)

gÇn + Lt =F_1

nt - 1'nz (42)

The expressions denoted by Fa_1 and G_1 refer to combinations of k and k

with k <n

1. As n increases so the complexity of the equations increases quite

significantly,

2.6 Specifying the Incident Wave Velocity Potential

Solution of Laplace's equation, Equation (7), usmg the boundary conditions

ex-pressed in Equations (9) and (40) for the first order problem, and subject to

Equations (9) and (41) for the second Order problem, leads to

agacòshk(z+d)

and

. a3a2 cosh.2k(z ± d)_exp(2kx)

(44)

the first and second-or4er incident wave potentials. The first order waveis assumed to have an amplitude 'a'. pence. for k the wave nuznber = ¡ça corresponds to the wave slope1 which is assumed small. The coefficient a simply deterrni.es the actual wave form characteristic at the origin at time, zero. Strictly speaking the pressure

equation, Equation (13), is only unique, regarding the pressure p to withuii an

additive constant. So recognising the constant is different for each different order of solutiön it is possible to show that

= ?'R[/4ezp(inwt) + C,(t)},

(45)

with Re denoting the real part of the quantity in pa.exithesis The C(t) term is

there to eiiminate the Bernoulli equation constant. In particular

C1(ì) = O and C2(t) = g2t ' (46)

where

2sinh(2kd) (47)

is the wave set-down.(), so called because it represents the time average, of the second-order elevation, and this corresponds to a negative offset from the. undis-turbed free-stirface. Other higher order contributions are 'available89.

2.7 The. General Diffraction Problem.

Having seen how the complexity of the incident wave models may be generalised the next natural step is 'to introduce the presence Ql a structure of sufficient

dis-placement to ensure wave scattering takes place. The structure is assumed to be fixed whether it is a gravity type offshore sttuture or some general floating

structure. As already noted earlier the introduction of the structure necessitates two additional boundary conditions. The first represents continuity of the normal velocity across the wetted-surface 'of the structure (assumed impermeable). The second condition must be imposed to ensure uniqueness of D and to control the far-field nature of the solution.

This second condition is imposed because of the elliptic nature of the . governing equation expressed in Laplace's equationL Since in these notes steady-state bóund-ary value problem formulations are to be favoured in preference to an initial value

problem approach, we have two choices regarding the determination of the

bound-ary condition on S. We may either iitroduçe a real Or fictitious damping term

or use physical insight. When introducing additiònal damping it is normal to re-quire the harmonic solution satisfying the other boundary conditions to vanish at infinity. When the additional artificial damping is allowed to tend to Zero the final result will satisfy the appropriate radiation condition. The boundary S,. is often referred to as the radiation surface, and hence the condition satisfied by the fluid flow On this stirface is designated the radiation condition. The artificial damping approach was suggested by Rayleigh. Sómetimes the physical situation suggests the form of the radiation condition, but this intuitive approach may also lead to incorrect formulations.

The radiation condition is a condition imposed On the scattering potential

,D

not This scattering potential exists to model the interaction of the incident waves with the structure. Treating 'I as the sum of the potentials and the perturbation analysis leads to the following first order and second order diffraction analyses. In particular the time independent potential q5f must satisfy

V2Ç5 =0 everywhere in the fluid,

=0 on z = d,

- mo

=0 on z = 0,

+ =0 on the wetted surface of the structure,

whereas _4' must satify

V2 =0 etrerywhere lii the fluid,

on z=d,

q5

-

4u4

=c4 on z

= O, and

+

0 on wetted surface of structure. Here mo is the wavenumber defined by m0 =

The free-surface forcing function a may be expressed as

SI_ SI

SS

= -

Q5j

(ZE

-IWAIIAS J.5 jV'1 W1zz -111z

+

₊

-ss_._

iw.sLs

a2 - -

¡W1 W1zz

-[(s)

()

2(5)

For the first-order diffraction problem intuition suggests that the character of the

disturbance at some distance from the source of the disturbance, tie structure,

should correspond to a progressing wave moving away from the structure. Cer-tainly, there is no obvious reason for the scattered waves to travel out to 'infinity'

and then suddenly start travelling back towards the struçture. Mathematically

this can be expressed as

-

izìf]

O in tue far - field. (53)

The subscript r denoting differenting radially in the horizontal plane. The second-order diffraction problem is less stiaight forward. Ogilvie and Tuck('°) dicussed the difficulties of providing higher-order solutions in the context of strip-theory some twenty years ago. The three-dimensional próblem has oriy ielatively recently been dealt with satisfáctorily("). Here the formulation used by Çlark et al(1213) has been presented.

To define the radiation condition at second-order the general forcing function

a2

=

II (54)

and and its far-field behaviour must be considered carefully. The incident wave forcing term c4r is similar to

.S

with components replacing those of «. The

disturbances due to a have been designated 'locked waves' by Moli", whereas the third disturbance associated with a is called a wave Whereas the

free-wave part of the disturbance does satisfy a Sommerfeld radiatión condition. and a double frequency dispersion equation the asymptotic behaviOur of the locked waves is the key to identifying the ftill radiation condition. The associated analysis is

)

with

not reported here for reasons of space and complexity, but the interested reader should consult cited references(7,10_14) for further details -and other appropriate bibliographic details

2.9 Summarising the Results.

Analysis consists of choices. With each choice there is a different level of computa-tional effort and a different degree of 'correctness' regarding the modelling of the selected fluid structure interactions of interest. In practical offshore engineering the trick is to try to minimise the computational effort, whilst maintaining a level of modelling appropriate to the situation being analysed. Having looked at the

) choices of modelling the interactions, we. now consider the selection of different

solution techniques.

In the literature many papers on finite element, boundary element and hydrid

methods.of Éolution are to be found. In this discourse only the different

bound-ary element technique based procedures will be considered and applied in later sections.

3. Sòlution of Linearized Formulations.

Since is harmonic the fluid flow may be described using either the time inde-pendent velocity potential or a corresponding distrIbution of source strengths o. The solution. domain in either case epenäs upon whether the selected Green function automatically satisfies the free-surface condition, the sea-bed condition and an appropriate radiation conditión or some. subset of these boundary con-ditions. Since the formulations now presented are primarily for deep water, the impermeability condition and the behaviour of the Green function G will usually remove Sb from the integral equation formulations generated from Green's second identity('). The only exception considered here is in the eigen.function matching technique described.

3.1 Low Order Boundary Element (LOBE) Formulations.

By LOBE we. mean the elements used to discretise the surfaces, are plane and the variation of the nrikn own velocity or source strength is assumed invariant over each element.

3.1.1 Zero Speed Open Water Formulation

Fredhohn integral equations are(15)

a ± f

4ds

=

fc

v,1ds

(5)

for a pulsating source Green fuictiön satisfying all the required boundary condi-tions except the wetted-surface. boundary condition, whereas

_a+f!ds+f'[2G]ds

Sw 8m Sr 8m 8n

trÔG

w2 i r

± I

_JS1

_--ç5q]ds= ¡

Gv,ds.,

. n g JSw

if G correspOndjs to the simpler Rankine source defined by G = 3 where r is the distance between the fluid singularity point . a.nd some generic point in the fluid. The term. on Sr is now replaced using either the Sommerfeld radiatión condition expressed in the form

çlI

-+f-ds-

Sw 8m

u2r

8G

±.

_{g JL0}

_äz

(56) (57)

where r indicates a radial derivative n some conveniently selected ylindirical radiation boundary. Alternatively' if wé knos the far-field behaviour of the velocity potential then. matching of the inner and the otiter solutions can be undertaken to satisfy(16_17)

f

8\

(ä\

d-j1

(.58)

with IZ' and f2 indicaiing inner and Outer sö'lutions. Using the full pulsating source Green function the corresponding integral equation fOr theiinkioWn sOurce

strength is

,

ac

o= j

_{JSw 8m} (59)

3.1.2 Forward Speed Open Water Formulation

The corresponding translating-pulsating source Green function based integral equa-tions are

2i

_{f Gdy}

_G]dyfGds,

(60)

for determining the velocity 'potential directly and

t 8G

U2'

8G'

aa

I ads +

n1cdy -

V,1,, (61)

for determining the unknown source strength, with recoverable frQm

f

aGds,

JSw

The contour Lo is the undisturbed free-surface waterline('). 3.1.3 Slow Forward Speed Open Water Formulation

If the pulsating-translating sour e Green function is expanded in terms of the

parameter r = Uw/g, and terms involving r2 and higher orders are

consid-ered negligible because of the low forwarl speed, then we may assume that G G0 + rG1 and similarly o = CO + Tal. Consequently the single second-kind Fred-holm integral equation (61) may be reduced tç the coupled second-kind Fredhoim equations(' -32)

ao0

=

'5w and ¡ 0GO _ç 8G

ao1 = i

c,1ds

-

cods

-JSw On JSw On

with recoverable from

_{= f}

a0G0ds and

4h

₌

c1G0ds + a0G1ds.

Go is in fact the zero speed Green function and Gi is a forward speed correction(s) which is also expressible in terms of G0. Any computer code capable of solving the zero speed problem can be immediately used to solve this forward speed problem. This is readily justified upon observing that the first integral equation has exactly

the same form as the zero speed integral equation, and once Go is known, the

second integral equation also reduces to a zero speed form. with G0 replaced by

G1.

3.1.4 Eigenfunction Matching Technique for Zero or Low Forward Speeds

In this case the inner problem is formulatéd by applying Green's second identity. The resulting integral equation is

+

J

«4ds

+

Js

8Go

a0ds

-where G is the steady velocity potential associated with the Rankine source. Im-plicit in this formulation is the assumption that U is sufficiently small that it does

G]ds

G]ds=f vGds

SW

not occur explicitly in the free.surface condition. When U is -zero this integral equation corresponds to the finite depth form of Equation (56)

Here the stirfaces 5b,c represent the tank bottom and the tätik walls and 5b i the seabed or tank bottom in water of depth d. The linearised freesurface

con-ditions, the. sea-bed r tañk bottom conditions and the ta.nJ wall cbnditiona are implicitly satisfied by the above formulation. The usual wetted-surface diffracton and radiation boundary conditions are embedded in v,. On the radiation surface, S, continuity of velocity potential. and normal velocity are facilitated thì'ough the eigenfunction expansions in the sense of Equation (58).

For the open water case the polar form of theeigenfunction(16_17) ma.y be expressed as

(r,O,z)=E{Alok,m0coshmo(z+d)

±EAlJ'(',)cosml(z +

d)}exP(ilO)

whereas in the in-tank problem(95) we usé

i

«z, y, z) E Aiöezpz(iDioz)cösh rno(z + d) cos[()y']

¡=0

w

± E Ajoezp±(D10z)coshmo(z ± d)ços[()y'}

(66)

lLc+1

+

Aj1czp±(Djyz)cosm5(z + d) cos[(.)y']

1=Oj=1

with W the tank width and

w

(67)

The A11 eigenfunction coefficients are determined with the "urknown potentials on

the surfaces S, Sj and Sbc or Sb' The functions denoted by H1 and K1 are the

Hankel function of the first kind and the modified Bessel function of the second kind respectively with:

motanh(mod) -, rn1 tan(rn1d)

TW,

(68)

g

(65)

D10 =[m

(fr)2j

and D11. =[m}+

()2]

(69)

The in-tank eigenfunction expansion can have imaginary values of D10 when mo is less than . Hence L denotes the highest falues of ¿ for whic.h D10 is real. The

wave numbers corresponding to m0W lr are the resonant modes of the tank.

For i > L the exponential teribs associated with the imaginary values of D10

are modified to correspond to decaying contributions as z becomes very large in either direction. The double summation, involving the coefficients D11, represents the evanescent modes. The upper limits L and J ìn4icate that a finite number of terms are used iii. the applications.

3.1.5 Green Function Matchin Technique

Zero Speed Problem

Heré the inner problem is exactly the same as for the zero speed open water

formulation i.e.

_a+fSw

än

Sr 8n ân

i

[

_G]ds=f

S 8n g S,,

(70)

but now rather than specify on Sr using a Sommerfeld radiation condition, or some other matching condition, the requfred radiation boudáry derivative is determined using the first-kind Fredholin integral equation(18_19).

ai' +J 4/-ds

₌

_{f Gds.}

(71)

s,

än

Srân

The negative sign correctly takes into account the difference in the sign of the outward normal on S for the inner and outer problems and ' is the other solution. Forward Speed Problem

The inner Rankine source based fOrmulation is now given by

_a5±f q5ds±f {G]

Sw äfl Sr

an

an

1 [c

_{s1 ön}

![we +iU-.]q5G}ds=f G

_{g 8x} sU,

(72)

whereas the outer integraÏ equation 19)

_cx+J2d.s-

s a

U2'r

,âG

±'p [-

_gJLr

äx

(73)

Again the latter equation is treated as a first-kind integral equation in terms of

the normal derivative of the velocity potential on the radiation boundary. 3.2 The Higher Order Boundarg Element (HOBE,) Formulation

In the formulations presented the solutions are usually generated using flat panels and the assumption that the sought unknown function (velocity potential or source

strength) is invariant over the individual panels. In the higher-order boundary

element methòd(2D_21) curved panels and higher order funçtional representations of the irnknown velocity potential, or source strength, are used. We therefore require appropriate interpolation functions within each element. Serendipity elements, in the terminology of finite elements have been used at Newcastle University. in particular both quadratic and cubic representations of the unknown function over quadrilateral and triangular elements have been considered. The number of nodes

per element are therefore 8 and 6, and 12 and 10 for the quadratic and cubic

representations respectively. For plane boundary elements it is normal to have one node only at the element centroid.

1fr (, 'i, ç) denotes the position vector of a generic point of an element then

= E,r(Ck,77k,çk)Nk(u,v)

and , (74)

k=1

where n. and m indicate the degree of the representation of»the element geometry and velocity potential respectively for the selected element geometry, and Nk and

Mk are the shape functions evaluated at the kth node of' the element located at (Ck, 'm, çk). Thus (, 77, ç) represents the coordinates of a generic point on an

element and (,

, k) denotes the specific points used to define the element

representation. In terms of the local element coordinate system (u, y) the elemental area ds will satisfy

ds = J(u,v)du dv, (75)

where J(u, y) is the Jacöbian. It may be shown(') that the discretised form of

the higher-order boundary elerrient form

jMk(uv)(fle

j=lk=1

i

e

ç-w

ac

(76)

+rii)J(u,v)du4v

_{=EffGvJ(u,v)dudv,}

where N is the total number of elements used tó model the wetted-surface boundary Sn,. We iiay also approximate normal derivatives of the velocity potential to the same degree m, thus

m

¡c=1

an.

(77)

may be used in the zero speed Green Function. Matching formulation of Equation

(16). Other HOBE formulations fqr zero and forward speed have been imple-mented and tested(18).

3.3 Some General Comments.

In Section 2 the parameter a was used to control the profile of the incident wave, and for this purpose may assume the value ±1 or ±1.

In all of the integral equation formulations the parameter a is the solid angle

subtended at the node where the integral equation is applied. The consequence of using the incorrect solid angle at the free-surface has been demonstrated. for the 2D HOBE method (21) For the .3D lOBE approach ita assignment is even more critical and care in its evaluation is very, important. For example, for the corner point of a rectangular box. form of S the solid angle is ir/2 for the inner problem formulation and 7ir/2 for the outer problem formulation!

Since a is the angle subtended at a point by a surface S then it only has the value

2r

, in 3D formulations, when S is smooth at the point of ¡nterest Otherwise a

scheme investigated by Hearn and Donati (20) _{is used. For nodal points on the}

intersection of the wetted-sirface, S, and the free surface, S1, file solid angle

definition must also take into account the contribution from the image part of the Green function. For more details of the HOBE method see Hearn (18-19)

Completeness of the formulations requires explicit definitions of the zero speed and. forward speed Green functions. These may be found in reference(i) for both finite and infinite water depths for both i) and 3D formulations. In SectÏon 4 they

will be simply quoted without derivation since the calculation of nonlinearforces

is critically dependent upon the ability of the software to robustly determine the. value of second derivatives of the Green functions.

In strip-theory the wetted-surface of the vessel SL is simply replaced by the sec-tional contours CIL,, with a corresponding reduction in the dirnensionality of the associated integrals and a different value for thè solid angle a.

4. Evaluation of First and Second Order Forces

In this section the differences between evaluation of first and second order forces is considered. In general the first order forces are derived from application of the linearized form of Bernoúllï equation given in Equation (13). The first order forces are generally classified as active and reactive. The active or wave excitation forces are determined from knowledge of the incident potential f and the scattering potential f. The reactive forces are determined from the radiation potentials and these forces are usually partitioned into components in phase with the

acceleration and velocity associated with a particular mode..

Second order forces can be seen as being geiï&1ited from the linearized Bernoulli equation using a second order incident and diffraction potential, or, they may be determined from application of the full nonlinear Bernoulli equation, using all the available first order potentials. Here we shall only consider the latter case since the former case is still in a state of development and most applications relate to regular simple bottom standing geometric forms for water f finite depth. In facl if the second order diffraction potential is not required explicitly the second order wave excitation forces can be found using Green's second identity to provide a

eqiuvalent second order force Haskind relationship(22).

The two particular methods of evaluating second order forces discussed here are generally known as the near-field and far-field approaches. Having described each procedure an. outline and provided some. indication of the associated difficulties of calculation we shall discuss the impact of the calculation of first order quantities upon the calculation of second order forces,

The fluid force and moments acting upon the structure are obtained through

integration of the resolved, pressure over the wetted surface using the relationships

M=jpñAds,

Sw

where ñ is the unit outward normal. 4.1 The Near-Field Approach.

The unsteady pressure and force are assumed to he expressible in the form

p =

+

O(EN+i),

1=0

=

P() + Q(N+1) (81)

1=0

The bracketed supersc:ript on p an4 F denot the 'order' of the term. substitution of these perturbatioñ expansions into E(uation (78) and (79), and the collection of all terms of like order lead to the following expressions fòr the zeroth first and V second-order forces: (9) =

f4)

p(°)i(0) ds, (82)

-

fs)

(i)ço)ds

-fs

(°(')

ds

-

fs)

(o)(o) (83)

-

f(p)

p(°)it(2) ¿s _fg(1)

-

fsM (2)(0) W W W (84)

-

p(°)n(')

-fs)

p(')il(°) ¿9

-

f)

(Q)(Q) V V

The Bernoulli fluid pressure equation, Eqüation (13), coupled with the perturba-tion expansión for p in Equaperturba-tion (80) upon rearrangement leads to

(0)

=

(85) (i)

₌

_--

u-)

(86) and (2)

=

_p{(v)2

+

(

-

(87)

The zeroth order forces are the hydrostatic buoyancy forces

pg fe(o) 5. (88)

(7g)

The first order force consists of the reactive and wave excitatio4 forces, together with hydrostatic restoration forces, that is

Here A and Zcf are the waterpla.ne area and the longitudinal coordinate of the centre of flotation respectively.

The corresponding second order force is extremely complex and provid steady and slowly varyiñg force components. Considering the average value of the: second order force over a wave period it may be shown that

P9 JLW IÇaI2TÍ

± P

15w

II

-

F')} +

PWe f Im[t5a Vq5]ids

-

puf Re[öa Vds

-

.pgAxRe[rj 776]k

with

(pJu)qids

+ pg(0, 0, 3A

--

_{N = (mi,n2,n3)/(n1 +Th2)2.}2

21

(90)

Here, N, has been introduced to accôunt for the slope of the wetted-surface.

That is, wall sidednes is not assumed. The abövè expressions and notation are consistent with Heárn and Tong (i), but diffei slightly tó the zero forward speed expressiòns presented by Standing et al (23) and Pinkster(24). V

The components of. Equation (90) are readily interpreted. V

component Ï represents the contribution from the changing

wetted-surface area' due to the relative free stirface elevation,.

component II gives the velocity squared effect of Bernoulli's equation, component m accounts for the effeçt of the first order fluid force. due to rotation of the body axis, V

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 pressure field variation due to the steady forward speed and

component VI accounts for the second order effect in rotation of the axis on the vertical force and is zero for pure head sea incidence. Evaluatión of E4uation (90) is critically:, dependent on the solution of the first. order potentials. and the motion responses, in particular the diffraction potential is required explicitly.

The evaluation of higher order derivatives is important in both the evaluation

of the second-order forces(16,17,25) and in the lOBE fluid-structure interaction

formulations('.

In particular in the HOBE formulation the speed associated

term UVç has required the development of special transformations to allow

eval-uation of Eqeval-uation (78). These transformations have been derived (1819) and implemented.

It is clear that evaluation of the second order wave loads can only occur after the first order interaction analysis for regular harmonic waves and the corresponding motion analysis has been undertaken in the frequency domain.

In more general terms Standing et ai (23) noted that wave drift forces have a mean and low-frequency component, which are the combined result of the second-order pressures and interactions associated with first-order wave field, and the pressures in the second-order wave (e.g. 'set-down').

Salvesen (26) demonstrated that the second-order wave gives rise to ow-frequency forces,. but not to mean drift fórce components, The 'set-down' wave is one partic-ular effect of non-linear interactions within a Wave field, whereby the Water surface is depressed beneath the wave group surface but elevated between such groups. This is due to the orbital velocity of the fluid particles being larger within wave groups of large waves than those occurring due to the interaction between such groups. it is the variation in pressure between these two regions which cause this

effect.

4.2 The Far.Field Approach.

Rather than integrate directly the pressure over the wetted surface of structure

is the volume of fluid bounded by the surface S and the total momeñtum, Mtum is given by

Mtu,nPfqdV

(93)

where q is the velocity of any poiflt in the fluid. The rate of liflear momentum is deterrniied from

DÍtum =D_

_I

dV Dt DtJV

=P f

dV + p 'VdS (94)

where V is defined in Equation (16). After considerable manipulation it may be shown that the longitudinal component of the second order force in the negative x-direction, the added resistance, is expressible in the form

Rw:=RIBtRBB

(95) with K1 2còs2O 6

H=>jHj+H7 aiid

i=1 H1(O,K)

=

J a5ezp[Kzo - 1K(z0co.sO + y0sinO)JcLs.

Here is the sourcé strength associated with the six radiation problems (J

1,2, ...,6) and the diffraction problem (j 7). rn0 is the zero-speed wave number,

RIB = 2rpçAw0cosßRe[H(ß, mo)] (96)

RBB =

2lrp[f

00±1/2

-

_f372]

[lH(o,1c2)I2

Kcos9

dO

,r/2

/1- 4rcosO

2rao _{K2 cosO}

+ 2irp L

IH(0, K') 2 :

The Other parameters are:

a0

cos(1/4r)

K2

o(1-2rcoso±V1-4rcoso)

whereas K1 and K2 correspond to poles of the free-surface. contribution of the. translating-pulsating Greèn function(27). The derivation of the final expressions is dependent upon Newman's asymptotic limits, for the radiation and scattering potentials expressed in terms of the Kochin functions(28) H5. The body-body in-teraction term, RBB, is much more complicated and generally smaller than the wave-body term, RIB. The RIB term can. be partitioned into four distinct com-ponents RIB : .1 = 1,2,3,4. The calculations undertaken are based upon direct evaluation of the following expressions:

RIB1 Rj RIB3 RIB4

=

_prn0cosß Tìj = _jmocosß

=

_pmocosßf

4ds,

and

= pm0cosß

_{' q5ds.}

A superscript * indicates the complex conjugate of a quantity, is the. incident wave velocity potential, and the Froude-Krylov force, in the5th direction, is defined by

f

811)1

FKJ

=

Pi5

It is tO be understood that nly real parts of the. expressions for the RIB nents are to be evaluated.

(99)

(loo)

compo-5.0 First Order Models and Second Order Forces.

The second order forces are not trivial calculations and the predictions generated through computer software are greatly affected by differences in implementation. Assimiing Formulations A and B of Section 3 are to be used with the near-field and far-field second order force predictors, the fOur methods may be designated lA,

lB, 2Â and 2B where the designators 1 and 2 identify the near-field and far-field predictôrs.

Typically differences in software can occur with regard tO:

e the discretisation process and the geometric handling

I _{evaluation of the Rankine source} - each Creen function has the form

eraluation of the hydrostatic restoration terms in the sene of standard waterplane calculators versus wetted surface integrals.

determination of the active and reactive forces.

Each is now cozsidered in terms of different techniques actually implemented at

Newcastle.

5.1 Boundarg element generation.

The discretisation process can be described as a simple partitioning of S,, and any other subsurface of S into a number of plane surface elements. From a applica-tions viewpoint this means selecting coordinates on the actual surface and indicat-ing how the points are to be used to make up individual triangular, quadrilateral and polygonal. boundary elements. To indicate how differences in implementation evolve the associated details for a quadrilateral element (facet) are presented.. In this part of the notes íis defined as the position vector of the 1th node of a facet, assuming clockwise numbering of nodes. Since Geometric Method i includes all the steps of Geometric Method 2 their details are presented in reverse order. 5.1.1 Geometric Method 2

Defining vectors and 12 as t = i - and 12 Li - r2, thewetted-surface outvard normal, , and the area of the quadrilateral element are given respectively by

.r=

_-

and _I11A1I. (101)

Clearly and 12 are the diagonals of the quadrilateral element. Treating the

'mean point' of the associated nodes as the 'centroid', , Of the element it follows

that

-4

Ei

i=1 (102) 5.1.2 Geometric Method i

This method includes all the previous steps of Method 2. However, the definition of the element centroid is refined. The logic behind the modification is that specifying four points in space does not necessarily mean they all lay on a flat plane. In the steps now indicated ari 'equivalent' mean plane is generated. The modified nodes,

are defined by

Defining the additional Vectors

j_m

m.

1')Q

_{and the areas A : i = 1,2, (104)}

the centroid of the equivalent plane element can be shown to be given by

i .- A m A m

i_

im

m z11!4

-r'2L2

Ai+42

5.2 Rankine source evaluation

The Rankine source, its associated image singularity, and their derivatives, can

be treated separately because they are not frequency dependent. huitially very

simple strategies were developed for evaluating these quantities. However, realising that predictions could be affected by the strategy used, more refinements were developed(29). To piesent the different strategies adopted LS and /5; are defined as the areas of the 'influenced' and 'influencing' facets respectively. The following terms are then introduced to -indicate, different procedures for evaluating terms o the form f ds and their associated derivatives:

'Simple pole' implies evaluation of the integrand at the centroid of the 'influenced' facet, and weighting this value by the facet area.

'Numerical' integration implies product Gaussian quadrature using 1 to 64 points, depending upon the ratio 3. (ÌS

S5).

'Analytic' integration means direct application of the Hess and Smith(°) equiva-lent line integration procedures.

'Extended Analytic' integration treats the special case of 'r = 0' using extensions of Hogben & Standing(31) analogous to those of Hess and Sm.th.

5.2.1 Integration Strategy i

This strategy was used in Method 1 of Reference [1]. Defining a distance scale,

r3 = (IS L S

the integrandsof the fundamental source, the image source and their derivatives are. evaluated as follows:

Simple pole method when r 3r3.

Numerical integration when 2r3 < r < 3r3.

Analytic integration when r 2r3 and r 0.

5.2.2 Integration Strategy 2

The second simpler strategy was used with Method 2 of Reference [1]. lii this case the integrátion of the Rankine source and its derivatives are evaluated according to:

Simple pole method when r > Analytic integration when r

Extended analytic integration when r = O.

The image source and its derivatjves are evaluated only for non-self-induced cases using the simple pole method.

5.3 Hydrostatic restoration terns

The fact that the wetted su±face óf the. structure is defined explicitly means that the hydrOstatic terms may be evaluated by direct application of A chimedes

prin-ciple, that is, direct integratiöñ of the hydrostatic pressure, or, mathematically equivalent, from a knowledge of the waterplane area, an the appropriate first

or second óments of the waterplaie aréa

hÌiydrostatk force and moment

components are derivable from

EH pg

L

s,

and MH = pgf [(j -

_{) A nids,} (106)

where ,. is the locatiön of the assumed centre of rotation. Existence of port-starboard Symetr will remove the need to calculate some of the forces and move-ments. Numerically the evaluations can be carrièd out using eIther of the following

pro ce4ures.

5.3.1 Hydrosta.tic Method i

In Method 1 of Reference [1] knowledge of the surface area and the normal direction

to the individual facets was used. Thus, givei the N facet discretisation of S

say, the implied necessary integrals were evaluated using

N N C33

- g

E

_nz S C3 pg

-i=1 i=1 N R

=

Yc

L S ±pg(yc - ycr)(Zc -

A !$ (107) i=1

N N

C55 = pg

-

r)

-

-

r)flzj

L S

i=1

where the i boundary element centroid is (zu, Z1), and

ny, n)

denotes the outward. norma.l. Effectively all integration is carried out using the mid-point value based argument of the 'simple pole' procedure of Section 5.2. 5.3.2 Hydrostatics Method 2

Knowledge of the indicated mathematical equivalences for the case of

!r =

also allows evaluation of the hydrostatic restoration coefficients using

= pgA C35 = C53 = pgA

C44 = pg[(zb

-

z9)V + AL?;1 C55 =pg[(zb - zg)V + (108)

where denotes waterplane area, and Abf,, and denotes i longitudinal and transverse moment of the waterplane area respectively. Evaluation of these terms, in Method 2 of Reference [.1], was, undertaken ising analytic integrations of the Lagrangian polynomials fitted to the waterline. Here zb, Zg and V denote

location of vert ical centre of buoyancy, vertical centre of gravity and the displaced volume of the vessel.

5.4 Rationalisation of Prediction Methods.

In these notes, and other recent publications(273637), rationalisation of the imple-mentations means selecting Geometric Method 1, with Integration Strategy i and Hydrostatic Method 2. This is to remove subjéct differences of computer coding and ensure differences are only due to differences in model or second order force predictor.

55 Differences in first-order hydrodynamic quantities.

Considering the motion analysis as a first step in the seçonçi-order force calculation process, any noted differences in the motion re,pqnse. procedure can be identified in terms of the procedure designations lA, through 2B. The, differences In pro-cessing the first-order hydrodynamic quantities will arise from the availability of derivatives of G and the forward-speed corrections applied.

Availability of 8i/8z (:j .= 1,2,

...,7), in Method lA, means direct calculation

of terms of the form Unk84'/8z, whereas in the other methods an equivalent operator UrnS is used iii the integration associated with the reactive and wave

In Methods lA and 2A, treating as the directly calculated from Formãla-tian A, the equivalent forward-speed velocity potentials are deteripined using the corrections

and

=q5±4-q5

:j=

with k=..

(lO9)

5.6 Second-order force evaluations.

The differences introduced in the post processing of the fluid-structure interaction analysis to match a particular second-order force calculation procedure are con-cerned with the evaluatiOn of V in the near-field procedure, and the evaluation of the Kochin fùnctions in the far-field method.

The need to calculate V arises for two reasons. The first is associated with

determination of the free-surface elevation. Since u, rather than 4, is determined in both Formulations A and B, application of the kinematic free-surface boundary condition to determine

_{f ,}

the wave elevation at the free-surface, from

at the

facet centroids, requires a first-order Taylor series correction, that is

1f ä'

Ô\ öf

(8

VI--UT),

(110)

where ' is the time dependent total velocity potential including both radiation and diffraction influences. The second reason is the need. for direct evaluation of V in component V of Equation (90).

o determined from Formulation A cannot be corrected directly for forward-speed

and so in Method 2A u5 should be replaced by the 'equivalent' term 84"/8rt

-in the .Koch-in function, where 81/8n and

_j

forward-speed corrected, as discussed above. That is the boundary condition, as well as the velocity potential, is forward-speed corrected in the Kochin function evaluations.

5.7 Evaluation of harmonic parts of Green's function.

For the pulsating source Green's function of Formulation the. harmonic part is also evaluated using the same. 'numerical integration' procedure described in the. discussion of the Rankine singularity. However., the finite integral, representation. of Hearn for G and its first der.ivatives(32) have been extended to include evaluation of the required second denivatives(29).,

For the translating-pulsating source Green function, the derivatives were not

had to be derived analytica.11y and then determined numerically. Consistent with the free-surface integral of the Green function, the derivatives are expressed as one-fold integrals of the standard complex Exponential Integral. The latter func-tiOn is evaluated using a series approximation for small arguments and a rational

function approximation° for the larger arguments.The one-fold cylindrical co-ordinate based integrai representation of G is found to be computationally less demanding than the corresponding Cartesian two-fold integral form and is under-taken using procedures based on Patterson's method

5.8 Green's function derivative e valuation.

The purely pulsating source Green's function is numerically stable in terms of function and first and second derivatÏve evaluation. In contrast the translating-pulsating source is singular for both the subcritical (r < 1/4) and supercritical (r> 1/4) flow regimes. Whereas the singular nature is not always clearly evident in evaulations of G the instabilities are very much in evident in the derivatives

82G 32G

and

äzOz ôzäy

The singular nature is dependent upon the type of cubature procedures used and the particular combinations of field and source points, which are a function of the discretisation process. In the subcritical regime the ce.' = 7r/2 value, causes K2 to become singular. For supercritical conditions both endpoint singularities and a singular K2 cause numerical problems. Hence we. introduceMethod lA' as Method lA with all second derivative contributions included. A robust Method 1W has yet to be implemented, but the nature of the associated numerical problems are gradually being understood;

To give some appreciation of the complexity of the evaluating the derivatives con-sider two alternative mathematically equivaleit descriptions of the Green function for the simpler pu1ating source case. The Green function may be written as

G(p,q) =

. + +

2moPvf

e1C(Z0) dK + i2rmoem0(0)Jo(moR), or

G(p,q) = +

-

m0[Ho(m0R) + Yo(moR) i210(moR)]emo(2+z0)

r° e_m0t dt - 2m °emo(z*z0)

+

34

Equivalently we may write where

11

G(p,q) = - ±

+ 2rnIi +sImi,

,cceK(z+Zo)Jo(KRJ dk

1=

I Jo

Krn0

Imi

2lmoem0+z0)Jo(fl2oR).

Derivatives with respect to z and y will, lead to differentiation of J0 and hence

the derivatives (fl.rst and second) can

be expressed in terms of I, Irni, and the

aditiona.1 functions

pooK(z+z0)J1(KR) dk I.

12PV1

, 116

- JO

Km0

Im2 2irrn2é0(z+zo)J1(m0R), The alternative Green fuiction representatión has the form

G(p,q) = +

.

-

0em)(irHYo(rnoR) + 21F1) + ¡mi.,

'(118)

IF2

¡

e_mot IR2 + t2dt,

Z+Z0

(117)

O

e_tdt

IF1 =j+..,/22'

(119)

HY,I(X) = H(X) ± Y,,(X), (120)

and the additional auixilary' function

(121)

The principal value (P.V.) of the derivatives of the pulsating (zero-speed) Green function necessary for the near-field added resistance computatinn can be written

as follows(29):

ÔG(p,q)

( -

)[]

₊

2mo(i +

o)) + I2 ± iIm2], (122)

8Gp,q)

-Yo)'[ +'fri)3

+'-2(1

+

(z + zo))

_{+ I2 +}

_{um2], (123)} 8G(p,q (z - z0) (z+ z0)

+

+ 2rnIi + 1m01m1, -. r3 (r')3

r'

(124) where and

82G(p,q) 1 [

3(xzo)2]

i

[

3(zz0)

8x2

- -

L -

r i (,.l)3 I. (r1)2

2m0'

[moI2+ j(1

+ (z

(xzo)21

_tI2

\

i f

+2m0 R2

i

f(z+ze)

+(,)

mer

jj

[(z

=

z0)2 (mImi - 1m2 OG(p,q) e3C(p,q) ay aC(p,q)

(zz0)

(z+z0)

r3 (r')

r'

+im0lmi,

(z

z0) [, + (,)3

+

2 (mor' + (z + zo))

e'b0(0) (irHYi(rnoR)

+

0IF2)

+

11m2],

11 1

2m0f

, (z+zo) (r')3

±-Imor +

r'

-,T10fr) (irHYi(moR) +

+

11m2], r1

z ± z)

(125)

h a similar manner introduction of auxiliary functions for the alternative form

Green function lead to the equations(29)

(128)

em0+0) [irHYo(moR)

₊

2ÏFi]

(130) (129) 82G(p, q) zo)(Y Yo)

1 fz+z,

i

_\

2m + .(,J)5)

-(mIi

-,\

2 1

(z+z0)

212

= (z

8z3y 2m01

(z+z0)

(126)

+-r[(l)2(

ri

OT)+1+

r'

2Im2)]

13(zzo)

3(z+z0)

2m0 82G(p,q)

8z8z

(z - z0)

mI2.

r5 + _{(,J)5 (r')3} R2 't.. r' (127).

2

_R

sm0im2

82G(p,q)

i

az2

-82G(p,q) 37

1[13(zz,)

L ' (r')31 (r)2

em0z{irto(z

- z

)2(7rHY(mR)+ 21F1)

(y yo) (z

z)2

_{(irHYi(moi) ±}

_1IF2)J

ii)2 [(z +z)

+

mor']

2m0(zzo)2[ i

((z+zo)

..,\

1 ((z±z0)

±

R2 1(r')2 \' r'

mor)

R2 r' [(z -!zO)2(mOIml 1m2) ± (y yo)2ITn2], 82G(p,q)

fi

88

= (Z - z0)(y - Yo) +.

()5

+

_{mo(0) [r(moHYófrrzoR)' jirYi(moR))}

+

2m0(IF1

-

IF2)]

₊

[z +,z)

mor']

+

[(z+Zo)

±

mor']

j (mimi

2irn2)]

r '(zz)

(z±zo)

2m

z)13

_r5

+'3frI)5

_(r')'3

+

m mo(z+zo)(ffy(mR)

2m

IF )

-2m ((z + Z

R2 _ri imoIm2].

+

+

man) (133)

6. Analysis of the Wichers' Tanker

Having beeñ given some insight regarding the complexity ofthe calçulations and the choices regarding the mplementatiôn we wiU now look at the application of some of the different analyses to a moored 200,000 dwt tanker. This example is selected because of the large amount of theoretical data and eperimental nea-surernents available. All of which are public, domain.

Exploiting geometric symrnetry the Withers' tanker specified in Table 1 is analysed using a 268 facet discretisatiôti of Figiire 1.

The equations of motion used were formula.ted with. respect to the centre of ro-tation and LCG. is assumed amidships. Using post processed solutions for the saurée strength, o, the added resistance, Rw, is calculatedfor a variety of forward

Table 1. Principal partiçula.rs Wichers' 200k dwt tanker.

Figure 1. Boun4ary Element Discretisatiçn of Withers' Tanker.

and reverse speeds for a range of incident wave frequencies. Thereafter the added resistance is treated as a function of shIp speed, and the low-frequency damping

is evaluated, for each method, for each, wave-frequency frOm

1 f8Rw\

b_

)

U=o (134) The non-dimensionalised incident wave-frequençy, added resistance a d low-frequency

damping are deßïied by

fo = ço,/(tipp)g),

=

Rw/[pgçB2/Lpp] and bL,

bB2/[pV/(g/L)J,

(135)

where B is beam and L is the length of the Structure.

The low frequency damping is of importance in designing mooring system. In some cases the low frequency damping is so important that If time domain analyses are run without inclusion of its effect the difference between predicted excursions and measured excursions is 100 % in error due to over prediction. Using such informa-tion in a design scenario could be an expensive errOr. In this section we consider

B T V D KG Cò

39

the impact of coupling different first order solvers with the two second order force predictore. It is also worth noting that since the low-frequency damping requires evaluation of the velocity derivative of Rw its prediction(27) is dependent upon the particular speed combinations selected. In the application now reported the speed-combinations and analysis combinations, defined in Tables 2 and 3, are con-sidered. Speed combination 9 of Tablç 2 corresponds to the selected forward-speeds

of Wichers' study(), whereas speed combination i corresponds to that used in

the earlier Newcastle studies [1,35,36,29]. In developing prediction Methods lA', lA, lB, 2A and 2B the default combination of analyses adopted, is combination i of Table 2, both here and in References [37,38].

Table 2.

Table 3.

Definitions of Speed Combinations. Definition of Analysis Combinations

In Figures 2 & 3 the Method lA' and lA added resistance predictions, together with the experimental measurements(), for the Wiçhers' tanker are presented as a function of non-dimensional incident wave-frequency

f.

The predicted added resistances are almost identical. Comparing Figures 3 & 4 the Method lA and lB predictions also show very similar trends, although there is a .7% variation in the highest peak and a 10% varIation in the group of lower peak values of Ru,. The generally small differences between Methods lA &. lB suggest that the radiation and diffraction potential contributions to V of Equation (90) are neglegible in

Speed Comb.

Frou4e Numbers (tJ/v"E)

i

- c.02 - 0.01 0.001 0.01 0.02 2 - 0.02 - 0.01 0.001 .0.01 0.04 3 - Ó.02 - 0.01 0.001 0.02 004 4 -0.04 - O.bi 0.001 0.01 0.02 5 - 0.04 - 0.01 .0.001 0.01 0.04 - 0.04 - 0.Ól 0.001 0.02 0.04 7 - 0.04 -, 0.02 0.001. o.O]. 0.02 8 - 0.04 = 0.02 0.001 0.01 0.04 - 0.04 - 0.02 0.001 0.02 0.04 Analysis 1

2 3 4 5 6 7 8

Comb. Geometry 1

1 2

1 2

1 2 2

Method Integration 1 1

1 2 2 2

1 2

Strategy Hydrostatic 2 1

2 2 2

1 1

i

Method

C, s o s o o s .0 C I u I s .5 s .5 ( 10 23 13 Figure 2. Non-dimensionalised 'added resistance using Method lA'

i.e.. second derivatives of G included..

2.

o

4

l.a 13 2.0 2.5 3.0 .33 4.0 43 5.0

kthrd Wate Facy rdmef , s I(tJg)

tO 15 2.0 2.5 3.0 3.5 4.0 4.5 5.0

k±ij Wave Rqiercy (mei) t, s oI(L)

Figure 4. Non-dimensionalised added resistance using Method iB i.e. second derivatives of G excluded.

3.0 2.5 C s o s o o s o C I s s s 'O_I 0 .5

(

.03 .1.0 10 13 2.0 2.5 3.0 3.5 4.0 4.5 LO dWawe Fraty f,. Figure 3. Non-dimensk'nalised added resistance using Method lA i.e. second derivatives of G excluded.

I It t . --i ¡ ' I I _fi¡ -I,, J 1/, it _it

i

/,'//4

ti ,' ¿'.

/ i,/ :

-e..

-U.4M

-- -- J.IIIn

-u.s

-u.'I*

-u.a e t ---

-t'

_- -¡ t' ¡ .1 i

/_lì

_{i I}

//'..'

,'

I ' I /

itI I

:

I /I.. --u.z - V.1SuI - -u.suI -u.0 e e V -u.-ao t- - u.t.ie Speed . -

I

- - - - -. Y \

--/ / - -O

-.1

ìi.i nu

Jill Jill

liii

Pill -lili

lIli

u --. --. -ft.'

I'

¡ t

It

-- - -bi

iA;'

-. ¡ - . ! -u.t e e EIII4C.uu '3 -.0 r-rr-r ir ii 10 1.5 2.0 2.5 3.0 3.5 40 4.5 5.0

kient Wave Freqacy nensiut) t,. I(Ug)

Figure 5. Non-dimensionalised added resistance using Method 2A.

i