Three dimensional second order hydrodynamic effects on ocean structures in waves

ARCHIEF

TePCDT1

MARINTEKNISK SENTER

MARINE TECHNOLOGY CENTRE TRONDHEIM, NORWAY

Lab.

v.

Schee,psbouwkunde

Tech nische Hogeschool

.0 R-86-54

Delft

Three dimensional second

order hydrodynamic

effects on ocean

structures in waves

BY

ARNE EDVIN LOKEN

by

ARNE EDVIN LOREN

TRONDHEIM., JANUARY 1906

DIVISION OF MARINE HYDRODYNAMICS THE NORWEGIAN INSTITUTE OF TECHNOLOGY THE UNIVERSITY OF TRONDHEIM

PAST DEVELOPMENT ON SECOND ORDER HYDRODYNAMIC EFFECTS

2.1 Mean forces

2-2 Difference frequency problem 2.3 Sum frequency problem

SECOND ORDER THEORY FOR FIXED OR FLOATING BODIES OF ARBITRARY SHAPES

IN WAVES AT FINITE WATER DEPTH

3.1 General outline of the development 3.1.1 Coordinate systems used

3.2 General second order boundary value

problem for sum and difference frequencies 3.2.1 Incident free waves

3.2.2 Derivation of second order forces from second order potential using Green's

theorem and an auxiliary radiation problem 3.2.3 Solution of the second order potential

directly using second order diffraction theory

3.2.4 Semianalytical solution of the second 27 order potential problem for a fixed vertical

cylinder using Green's theorem

3.2.5 Second order hydrodynamic boundary value 31 problem by using Green's functions and

integral equations

3.3 Second order contributions to forces and moments 34 from product of first order potentials

3.4 Total second Order forces 48

COMPUTER PROGRAMS AND TESTING 51

4-1 General 51

4.2 SECDIF4 A computer program for calculations 51

of first and second order wave forces on objects of arbitrary form

4 4 7 8 14 14 15 16 16 19 24

4.3 Testing of SECDIF .61 4.4. SECCyL, A computer program for calculation 62

of first and second order potential forces on vertical circular cylinders

4.5 Testing of SECCYL 63

NUMERICAL RESULTS - 64

5.1 Study of the free surface integral 66

5.2 Result for different body shapes 68

5.2.1 Fixed vertical cylinder 68

5.2.2 Floating sphere 71

5.2.3 Floating vertical cylinder 73

APPLICATION OF RESUAT5 IN ENGINEERING 112

ANALYSIS

DISCUSSIONS AND CONCLUSIONS 114

ACKNOWLEDGEMENT 116

NOMENCLATURE 117

APPENDICES APPENDIX Al APPENDIX A2 APPENDIX A3 APPENDIX A4 APPENDIX AS APPENDIX A6 APPENDIX A7 APPENDIX AS APPENDIX A9

-The first order hydrodynamic boundary value problem And, motions for the

Single and multlbody case

The Second order nonlinear inhomogeneolla free surface condition

The second order body boundary conditicin for a moving body with first order motions

The asymptotic behaviour of the second order potential and associated integrals using Green's aytmetry theorem

Green's function used in, the first and second order potential problem

DerivatiVesofha Green's function for

1/R point sOurce and integrated over a quadrilateral

Transformation-Of derivatives

of

potentials in local to'a global coordinate system

DeriVationofthe radiation potential for a vertical circular dyllnder in finite water depth

Rigid body dynamics of N vessels in the time domain

132

171

175

184

APPENDIX A1.0 Second order transfer functions for 186 different body Shapes

APPENDIX All - Irregular frequencies for different 207

body shapes 142 146 156 160 163

/. INTRODUCTION

Second order hydrodynamic effects: may in many cases be important for the dynamics of ocean structures in waves.

When a linear regular first order wave is interacting with an ocean platfOrM forces of different characters

are created. In addition to first order linear exciting forces, mean nonlinear second order forces and

non-linear forces varying in time with twice the first order Wave frequency act on the structure. . In the present

discussion effects of higher order than two are neglected.

Irregular waves are . assumed to be composed of an Infinite number of fundamental frequencies and amplitudes (a wave spectrum). In irregular sea the resulting second. order, exciting forces contain three components. These are the mean forces, forces Varying in time with the difference frequencies (Often called slow drift forces) and with the: sum frequen6ies of the wave spectrum (high frequency forces)

The difference frequency forces may in particular be important for design of mooring and dynamic positioning

of offshore structures at Well AS for offshore loading systems. For.large volume structures with a sMall waterplane area

the slow drift forces may create large vertical motions.

The sum frequency forces may become an important

excitation source in considering fatigue for certain Offshore platforms as for 'instance the tension leg Concept.

Chapter 2 of this report addresses the state of the' art in prediction of second Order hydrodymanic. forces. In

caldulate second order sum and difference frequency wave exciting forces in six degrees of freedom for large objects of arbitrary form. The theory is formulated within the framework of three-dimensional potential theory and one ore more structures may be present, taking into account the hydrodynamic interaction.

The second order potential. satisfies Laplace equation and inhomogeneous boundary conditions on the free surface and On the body. Both a second order diffraction theory and Green's theorem are used in the general three-dimensional case, to arrive at second order forces due to second

order-' potentials. All non-linear interaction terms are included.

A semianalytic case for the second order potential forces on,a vertical cylinder is also derived-.

The last part of chapter 3 contains the expressions for the mean second order forces which in turn are used to derive contributions from product. of first order potentials on time varying forces.

The results from detailed derivations are given in Appen-dices.

Chapter 4 explains some details of implementing the theory developed into computer programs.

Chapter 5 presents the numerical results in terms of second order transfer functions for the sum and difference frequency

forces for different body shapes. The different contri-butions to the second order forces are discussed.

In chapter 6.different aspects are discussed in determining the dynamic response of offshore Structure due to second order wave forces. These aspects may be

summarizedas the..importande-of.viscous-4amping and

excitation i slow drift damping, frequency or time domain

2: PAST DEVELOPMENT ON SECOND ORDER HYDRODYNAMIC EFFECTS

Over the past decade large efforts have been put into the re-search pertinent to second order wave effects on ocean struc-tures. The present discussion will be limited to the second order exciting forces on structures at zero forward speed.

Theories based on perturbation analysis and resulting com-puter programs

as

well as model experiments have attributed greatly to an.increased Understanding of these. effects.

This presentation on the state of the art is not meant to cover everything done in the Past.. The author rather wants

to discuss the content of certain characteristic papers showing the trend in the development of theories and computational methods leading to the present work.

A recent coherent and excellent state of the artpaper on second order hydrodynamic effects is given by Ogilvie

(1983).

A typical trend in the past is that some authors have Concentrated on the difference frequency problem, others on the sum, frequency effects. The present paper will try to tie these problems together.

2.1 Mean forces

Maruo (1960). was one of the first to give a coherent analysis of the different physical effects creating mean second order wave forces (i.e. interaction between incOming, diffracted and radiated waves). He considered a ship and his numerical analysis was restricted to the beam sea case.

In their analysis the three-dimensional sink-sourde Methodwasused.. A far field ekpansion. was .applied and contributions of second order in wave amplitude was,. re-tained-- The horizontal mean Second order forces and tOments werecomputed. through integration over a control cylinder oflarge radius. Their derivation of the expres-sions for the forces and mdments follow quite closely the method of Newnan (1967). The expressions for the forces were however generalized to finite watar depth by Faltihsen and Michelsen (1974).

An alternative Method to the far field method is the pro-cedure based on direct pressure integration along the body of second order pressure terms arising 'from perturbation expansion to second order. This approach has been used by Faltinsen and LOken (1978a) in the two-dimensional case and' by Pinkster (1980) and further by LOken (1982) for the three-dimensional case.

Faltinsen and LOken (1978b) computed mean second order forces by a varity

of

methods and compared with available model tests which at that time were rather sparse.

'Later Faltinsen et.al (1979) did extensive experiments on a tanker and compared with calculated data obtained .

by the method of Faitinsen and Michelsen (1974).

Pinkster (1980) presented extensive computer calculations based on the method of direct pressure integration and .compared with model tests for a varity of structures such

Karpinen (1979) gave an approach to compute mean second order forces on semisubmersible platforms by using KotChin functions. He displayed results from numerical calculation but did not

check the validity of his results against other methods and model tests. Further he did not include the hydro-' dynamic interaction between columns and pontoons etc. which may become important at smaller wave lengths. The method in its present form by Karppinen (1979) may also be of limited usefulness due to the fact that it predicts the mean forces growing to Infinity for large frequencies.

Pijfers and Brink (1977) calculated second order viscous

forces on two semisubmersible platform concepts. _{They included} Stokes drift in the velocities using Morison's equation in the second order force calculations. This part is usually included in wind driven current forces. Thus one should be careful such that the wind driven current is not accounted for twice in the calcu-lation of total mean forces.

LOken (1981) Computed first and second order effects on two floating bodies of arbitrary form influencing each other through hydrodynamic interaction. He applied the

far field Method used by Faltinaen and Michelsen (1174).. The comparison of computed results with model tests showed overall

good

agreement. The path of the integration along one Control surface of a single cylinder of large radius for the single biddy problem given by Newman (1967) May however not be strictly valid for the multibody problem. The work

of

LOken (1981) seems to show a good engineering applicability of Faltinsens and Michelsen's

formula when hydrodynamic Interaction effects are important. Some more work would however be preferable to see if

additional control surfaces are of practical importance in the problem. _{LOken (1982) later extended the computational} capabilities to include the Use of direct pressure inte-gration over each body to arrive at second order horizontal mean forces.

Pinkster (1980)- has recently given an excellent state of the art report on mean second orderhydrodynamic forces on ocean platforms.

In conclusion We can say that the computational capa-bilities are relative good for mean horizontal second order forces.

2.2 Difference frequency forces

The state of the art concerning slowly varying wave forces is somewhat less satisfactory than for mean forces. The most coMMOn procedure to day is to use the mean second order forces as a building brick in calculating slowly varying wave forces. This is a hypothesis which is only justified if. the more complete nature of the problem is known.

Faltinsen and 'Aiken (1978a), (1978b), (1979a)

and

1979b) presented a new method to Calculate Slow drift excitation forces on

an infinite long horizontal cylinder in. beam waves in deep water. Although limited to beam sea it is probably one of the most complete and consistent method existing to

day.

The hydrodynamic boundary value problem Was formulated and solved correctly to second order in wave amplitude. The second order problem contained all components (slowly varying) derivable from first order potentials and the second order'potential- The second order potential

re-sulting from a pair of incident Waves satisfied the Laplace equation with inhomogeneous boundary condition on the free

surface and On the body- Green's second identity was used to derive the formula for the slow drift excitation forces without actually having to solve for the second order

2.2. These figures are from Ogilvie (1983) which displayed some results from Faltinsen and LOken (1979a) in an easy understandable graphical-forie. Faltinsen and LOken (1979a) indidated, that the simplified way

by

using only the results from mean second order forces calculations could be a practical engineering way to calculate slow drift excita-tion forces. This was based on 4 qualitative comparison' of time histories of the force using complete and approxi-mate expressions in the. calculations. It should however be kept in mind that length of the time histories was probably 'too short: to conclude anything on the exstreme.values. Although

attempting these indications in the two-dimensional cases cannot be generalized to be valid in three-dimensional flow

cases-Pinkster 1980) gave .a thorough investigation of slow drift effects on three-dimensional bodies both computationally and experimentally. He did not however solve for the effect of the second order potential, but rather used and approximation which resulted in a direct scaling of the first order wave exciting forces. Definitely more work Was needed in solving the second order potential problem in the three-dimensional case. This is one of the topic of the present work.

2.3 Sum frequency problem

The major part of the work on the sum frequency problem has been devoted to the two-omega (2w) case in regular' waves. The research so far has further been restricted to

htri-zontal-cylinders of infinite length at infinite water depth in .the two-dimensional case and fixed vertical cylinders in the three-dimensional case.

I.2

0.74

"Ilan"

. 1

Fig. 2.1 Contours of constant values of Re{T22ij}/pg -all components included (data from Faltinsen and Ioken (1979a))

.2 1:0 LI 1.2

sifif71

Fig. 2.2 Contours of constant values of Pe{T22ij }/Pg -contribution of 02..omitted (data from Faltinsen

In the two-dimensional case Lee (1908) and parissis (1966) gave independently the solution -to the slim frequency problem for a heaving horizontal cylinder. Lee used the method of muitipOle expansion and Parissit the procedure of source distributions and integral equations. Potash (1171) included in addition to heave also the sway and roll

motions and their coupling using the method of integral

equations. Wing (1977) used Green's theorem to derive

Second order sum frequency forces on

a

horizontal cylinder -of infinite length. He concluded that the second order forces could be found:from integrals over the body and. the free surface.

Later Papanikolaou (1982), (1984) used the method of

Potash (1911) to the second order sum frequency diffraction and radiation problem.

lqrozlika (1982), (1984) applied the two-dimensional boundary element method on the same problem as considered by papani-kolaou. The motions of a horizontal cylinder in waves were determined by the solution of the equation of motions up to second order. Kyozuka (1982) also carried out model experiments for the radiation and diffraction problem in steep regular waves. Large deviations between numerical and experimental results could be observed.

A. large amount of work has been carried out for the wave sum frequency interaction with fixed three-dimensional objects. A long series of papers, soms.quite controversial, have been published concentrating the effort on second order

solutions for vertical circular cylinders in regular waves.

Charkrabarti (1972) made attemptof extending macCamy and Fuch's (1954) linear solution of the diffraction potential for a vertical cylinder to fifth order. He treated however the inhomogeneous free surface condition improperly and

surface piercing cylinder'. He argued that the solution given for instance by Raman et al (1976) was mathemati-cally Inconsistent. The inconsistencies being due to the fact thatboth the inhomogeneous free surface. Condition and the body boundary condition could not be met by a unique potential at the intersection line between the Cylinder and the free surface. Garrison (1978) and

Shen (1977) proposed that the irregularities might be removed

by

decomposing the second order diffraction potential into two components. One component would satisfy the inhomogeneous free surface condition, the other a homogeneous free surface condition. The sum of the two potentials would then have to satisfy the body boundary condition.

The views of Isaacson were strongly opposed and critisized by Mehausen (1980), (1981) pointing Out that the

in-consistency mentioned by Isaacson is

due

to the irregularity of the solution on the intersection of the undisturbed free surface and the vertical cylinder. Such. irregularities do not necessarily odour only in the second order problem but are also present whenever the free surface condition is

inhomogendous. This condition may also arise in a first order problem with a prescribed pressure distribution on the free surface. Miloh (1980) examined the nature of the irregularities. He showed that the solutions in the neigh-. bourhood of such irregular point or curve can be presented as the sum of a singular function of known behaviour plus a regular function.

Hunt and Baddour (1980) gave, the second order solution for non,..axisytmetriC standing waves bounded by a Circular cylinder. Chen and Hudspeth (1982) derived a second order theory

based

on

an eigenfunction expansion of the Green's function for ixisymmetric bodies subjected to two-dimensional sinus-oidal waves.

Garrison (1979) developed a consistent second order theory for a fixed object in finite water depth subjected to

regular gravity waves. The boundary value problem correct to second order was outlined based On solving the diffraction problem directly using Green's functions, His numerical computations were however limited to a fixed vertical cylinder.

One of the most coherent and complete analysis of the second order sum frequency (2W) problem so far, was given by

Molin (1979). This. seems also to be the opinion of Mei (1983). Molin used Green's second identity to arrive at expressions for second Order forces on fixed objects. He also addressed: the Convergence prOblam.of the free surface

integral which.has more or less been neglected by others.

Molin.s. numerical analysis-was confined to a vertical circular cylinder.

Recently Raman (1984) presented what he called "an exact sedpnd order theory". It appears as Also discussed by

Drake et Al (1985) and Molin (1985), that the paper contained many Mistakes and inconsistencies.

Sabuncu and GOren (1985) have contributed to one of the most recent development on second order sum frequency

forces. They have derived expressions for the second order vertical and horizontal wave forces on circular compOsite docks. Herfjord and Nielsen (1985) have considered the

sum frequency problem (wi + wi) in irregular waves for A fixed vertical circular cylinder. They did however, only

include contributions from product of first order potentials and thus excluded the difficulties inherent in establishing the solution .of the second order potential '

In conclusion one may indeed say-that the analysis of the second order sum frequency

problem

has been quite a contro-versial issue.

To the authors opinion the dirucial points in development Of a consistent second 'order sum frequency theory

can

be

summarized as follows:

-.The existence of a Second order potential .

Can the second order potential be said to obey the Sommerfeld'type radiation condition

The convergence Of the fide surface integre' (excluding vertical cdntrol surface at a large distance)

These

Points are equally important in deriving a consistent difference frequency theorY

3.

SECOND ORDER THEORY FOR FIXED OR FLOATING BODIES- OF .ARBITRATY SHAPES IN WAVES AT FINITE. WATER DEPTH

.3.1 General outline of the development

The hydrOdynamic theory which formS the basis for cal-. culations of second order sum and difference frequency

forces on objects in regular and irregular waves will be developed in this chapter. Most of the details emerging

from the derivations may be found in Appendices.

The theory is based on classical potential theory-. This means that the fluid surrounding the bodies it. inViscid, irrotational., 'homogeneous and incompressible. Thus the fluid motion state may be described by-a single scalar 0, which is the velocity potential. The velocity field can be obtained by the gradient of the potential:

- = m (3..1)

0 is generally a function of space and time and deter.-mined by imposing proper boundary conditions.

In the theory development it will be assumed that the velocity potential and deriveable quantities can be expanded using

perturbation analysii, i.e.:

= 41+ E202+0(E3)

(3.2)

where E is a. small quantity.

The Contributions of order greater than two Will be neg-lected

in

the analysis.

It should be noted that product of first Order potentials will result in second order effects.

A. general theory comprising both the sum and difference frequency second order effects will be developed. . One or More objects of arbitraty form, floating with first order. motions in six degrees of freedom, or fixed., will be considered. Thus taking into account hydrodynamic interaction effects in the case of the multibody case. The bodies may be submerged Or Surface piercing..

The second order forces will be derived for all six degrees. of freedom. Two methods will be used in deriving forces due to second order potentials on bodies of general form. One method is based on applying Green's second identity. The other method gives directly the solution to the Second order potential by development

of

a second order diffraction theory. The resulting boundary value problems are solved through the use of Green's functions and integral equa-tions. The knowledge of the first order velocity potential is of course a key stone

in

proceeding to a second order

solution. The first order theory for the general single and multibody case may be found In Appendix Al.

A second order potential theory for a single fixed Verti-cal surface piercing cylinder has also been formulated. Formulaes for the second order potential forces has been derived based on Green's symmetry theorem.

By including both the use of Green's theorem and second order diffraction theory as well as the semianalytical case for a cylinder., unique checking possibilities on numerical results are created.

3.1.1. Coordinate systems used

Four coordinate systems are used

in

this development, see Fig. A1.2. One earth-fixed .right handed system 0-X-Y-2, with origin in the mean free surface and 2-axis vertical and upwards. This coordinate system is used for reference

of phases and incident waves'. ThesecOnd coordinate system is a tight handed earth-fined bOdy coordinate system

ox9.-ycztwithorigin tight above the centre of gravity of the vessel 2. with origin

in

the mean free surface and .

Z.is pointinl Vertically upwards. In the case of one body the 0.-XY-Z and the

o-xt-yczt

usually coincide (the suffix 2. is sometimes deleted when it is considered un-important 'to

take

ahy distinction. The third coordinate system is a right handed body fixed system (o*-x*-y*-s*). When the body is at rest

o-xt-yczt

and

o-X4, y*,7z*

will coincide.. The last coordinate system used is a local element system

(20'iy',e)

Used in deriving the wave free potential In Appendices A6: and A7,

3.2. General second order boundary value problem for sum and difference frequencies

3.2.1 Incident free waves

The incident wave potential is expanded according to eq. 3.2 as:

I 2 I

0I = E ₀₁

t E

₀₂

(3.3)

For each wave i the first order inCidefit.4aVe potential may be written as:.

g cosh ki(h+z) 0

=k h

sin( - w t + c) (3.4)

1i

i

_wicosh

_i

i

Where 4t is the incident wave amplitude.; q the gravity con-stant. The wave frequency wi and wave number ki ate related through the dispersion relationship as:

w.2 = gktanhlitilb

27

1

ki. = (k cose, ki sine)

t

(X,Y)

(3.5)

6 is the wave heading angle Z is the vertical coordinate with Origin

in

still water line, positive Z is .pointing

upwards. The water depth is denoted by

h.

ei

is

a random phaSe angle and t is time.

The first order potential in eq. 3.4 satisfies the Laplace equation, a homogeneous free surface condition and normal velocity equal to zero on the bottOm.

Formulaes for the second order sum and difference frequency incident wave potential were given by 1,onquet-Higgins (1963) in the case of infinite water depth. Sharma. (1979) gene-' ralized the results by Lonquet-Higgins to be valid for finite waterdepth. Sowers (1975) gave expressions for the second order difference frequency incident wave poten-tial on a somewhat different form than Sharma.

The second order sum and difference frequency potentials for incident waves of frequendies wi and wj can be written

Is Is 02 = ij Is cosh klj(h+z)Bsij ((q+swj) o = 0.5 ljl(L)1-2ij i 3 v .. iv 3 cosh ks h sin(ei + sej) (3.9)

As an affix s" means the sum frequency problem and (3..8)

(3.6)

S

= denotes the difference frequency problem. In. the

corresponding algebraic expressions s = +I for the sum. fre-quency problem = -1 in the difference frequency' problem.

We impose the restriction, that > wj when s = -1 and

w 2 w when s = 1.

The other quantities not defined is eq. 3.9 are:

kijs = lki + ski(

= ti /41 - wit + ei = k tanh k h 2 2 - 2 2 (V) -VV'.)Vv Oc v.) 13 (/

_/))2-

k

tanh k

7 h 2(VVi - VVj)2(ti6tj + - 2

-i

(VV Vv.) - kj tanh h

i

2(VV. +

Vv.)4(t 4t.

- v v.)

+ i 3 i 3 J 1 Bii -

(/Vi +

VV..)2 - k + t anh k + h . 2 2 2 2 (iv + iv.){iv (lc-. - v. +Vv.(k ).

i

7 i 7 ) - v ) 7 3 i i 4., OA + vv/.)2 k; 4. t anh k

+ h

- i (3.14)

The second, order potential.eqt. 3.8 and 3.9 satisfies the Laplace equation, normal velocity equal to zero on the bottdm and an inhomogeneoue free surface condition.

The free surface condition may be written as:

smIs

ails

r'"2-

'

az:

at

1 aO11 a ao . 1 r- ---

(g,

-+--,g . at

az-

az

sIs

Ds 2 0

= 0

_{+ 02}

2 + 2 at

-

I 2

2 ' (T01) ls' at. on z = 0

Derivation of second order forces from -second .

order potential. using Green's theorem and an radiation,Rroblem

In the present section fOrmulaes for the sum and difference frequency forces are developed based on the use

of

Green's symmetry theorem. The bodies in waves may be restrained or moving with first order motions. All second order motions

are assumed to be -zero.

The total. first order potential is written as the sum of the incoming wave potential 0,, the diffracted potential

' R

01 and the. radiated potential 01 dile to first order motions.

(3.16)

In the general case of three

dimensional

bodies interacting With each other, 41 is found according to the procedure

outlined in Appendix Al.

The total second order potential 01 due to sum

(s

or difference

(s

= -) frequencies is written as:

(3.15)

The second order potential

02s

and it

components

has to satisfy the following

conditions.:

a(D2

az

'02

= 0 in the fluid domain (3.18)

a 2s +

-h s

at 2 ao1 a

ao1a2o1

a :L. 2 s + ) 2 --(v0 ) ] at az az- at at

on the fre surface

S(Z

=-0) (3.19)

an f2 (3.20)

on the 'body boundary 3.11

is a second order body boundary condition arising from first order motions (if a body

is

restrained f25 = 0),

az - 0 on the bottom S z = -h) (3.21)

Ds _{will contain products of incident first order waves and}

radiated (diffracted) waves, thus-02Ps

will

probably not meet exactly the usual radiation condition. This does not however prevent an exact solution using Green's theorem. The second order diffraction theory Chapt. 3.2.3 will however be an

approximation. 2 Ds 90 Ds a Is a2 is a o-2 _{o2 .} (D2 , + g 2 -h2 ,g m r25 on z=0 (3.22) at2 az7 az at

The body boundary Condition eq. 3.20; using eq. 3.17 is

where

Complex quantities. are introduced for the second order potential:

to

Is

° 2

Bn an

i=1 1=1

Big. 3.22 then becomes

Ds BO

.Ds

2ij z 4 s w .2 ,ij: g az r2ij (3.25)

In the-case s 4 it assumed that w

_i

and for s

-Wi> iii.

This assumption is of advantage .in the numerical. calculations.

s

The exact expressions

for

112s ( 11) .and f S(f;ij ) are given in Appendix A2 and Appendix A4.

The forces due td the second order sum or difference frequency potential can be split into two terms::

Is Ds

F . = F +

2ki3 -2ki3 F2kij (3.26)

sw-)t Is 3 1 :Cs Re(-ip(w sw.)ffo.; . n ds e F2kij 3 5,

ci3

(3.27)

Subscript k = 1, 2 -- 6denotes all six degrees of frdedom and' nkthe generalized outward normal vector.

+ (3.23) -1(-1) + s w,)t .2--= 2 e (3.24)

Ds.. Ds Re(-ib.(w- +-sw..)11421. i-

r

3 -1 (w,-i.sw.)t - ds e 3. ] (3.28)

In order to establish a solution

to.eq.

3.28 a ficticious. radiation potential

11;21scij is Introduced in Mode k

with

freguencY (wi + sty:

-i(w + sw.)t.

)

= e

2. 's

V- 4)2kij = 0 in the fluid

ail, s;

"2ki

an

,thi

If DS "2kij .2ij SB

2kiaops

Ds If 11) - =2-1-1; ds s'

"2ij

- j an SB SF +Sh +S 2kij = = 0 on the bottom ( z = -h) (3.29) (3.30) on §B (3.31) (3.32) 6 "2kij 2 s

47i-7- (Wi + sty tP2'= 0 on z = 0 (3.33)

A radiation condition is assumed -Using Green's second identity for cp

2ij

and

thr2kij gives:

(3.34)

The integral over Sh vanishes

by 'using

the bottom boundary condition

ps

$1so possible to show

that-the:intSgral.over:Sw

also vanishes. This

is

an important point in the development of the theory-and.the details may be found in Appendix A4.

The last part of the right hand

side,

of .eq. 3-34 can be .transformed by Using eq. 3.35 and -3.25 to:

2Ps

-s ij Ds "2kij If _akij-"--- -0

_az

_{2ij Bz} lds

sF

r

"2ij

(W .+

sw.)2

ps

i 3 s

-= "DP

az

2ij g akijAds

2kij Dd ao

fl

tps,_

[a

ADs

sF

4sii '

*21jwi sw12].ds3 1 s.

ikij.

ds .F

'Sy using eq. 3.23, the fiest part of the rigth hand side eq. 3.34 becomes: Is s 2.1.j - _ir ."21j fs

.) ds

(3.36) SB 2kijBm de = 'Y2kij* an 213

By using eqS.

3-23,

3.34., 3.35 and 3.36-, the second order diffraction force can. be expressed a$:

Ds

-s w2ij

F2kij = Refip(w, +sw.)( ( 2ij)ds

i - 2kij Bn sB sw.)t + 1- rf 2kii

rs

.ds ]e 3 1 i _SF 2i3 . (3.35) (3.37)

By applying Green's second identity to the total second order potential

OLj

and

Ips2kij,

an alternative formula to eqs. 3.26, 3.27 and 3.37, is obtained.

s

g

2kij =-Renio(col + swi) Lffi3_sB *2kij

, 1 trus ' g 2kij SF

0.003i.2kij

_pas

_e j an -.Y2kij an ' 1 (3.38) S,

In numerical computation eq. 3.38 would be much more difficult and time consuming to handle than eq. 3.37 etc. Thus in the further analysis

eq.

3.37 has been selected.

3.2;3 ' _Solution of the Second Order potential .

directly using second order diffraction theory

This derivation Will contain the theory for

the

solution

of

the sum and difference frequency second order diffraction potential directly. As before the bodies can be

of

arbi-trary Shape with

or

without the contribution from first order-motions on second order body boundary conditions. By solving for the second order potential directly one

is

able.

to calculate resulting pressures, wave elevations, velocities etc. in the fluid

in

addition to forces c6rrect to second order in wave amplitude. In the case of the muItibody problem it is also possible to calculate forces on each body. This makes the use of second order diffraction theory more attractive than applying Green's theorem.

The second order potential is split into the following parts

- s. Ds Is

2 = 02 + 02

The incident wave' part

0Ig

is given by

eq.

3.9.

The diffraction potential

Or

is further split into:

Hs (3.40)

The second order diffraction potential has to satisfy the inhomogeneous free surface condition eq. 3.22. The free surface problem is then split into two parts using eq. 3.40;

a2 Ps Ps o2_ ao2'

___7 + g---

= r2 at az a2 Hs_o

aHs

22 + g 2 - 0 (on SF) at az ( on S (3,41)

The body boUndary 'condition for the potential satisfying the homogeneous free sruface condition may be written as:

Hs a Pt a0 2. ao2 ao ;s + fs n an an 2 (3.42) (on g21) (3.43)

2 (h251 and f2 are displayed in Appendix A2 and Appendix A3.

The boundary value problem for 02Hs can now be solved in an analogous way to the first

ordei',

diffraction problem described in Appendix Al.

1,Ps-is at this, stage in the development still unknown. One

2

-approach to find .1)-52s might be based on distributing 1JR sources on the free surface and match the Solution to an Asymptotic expression at a large distance from the body. A probably more direct method was however Selected. This procedure is based on an analogous situation of having a pressure distribution,

periodic on time on the free surface, considered by Wehausen and Laitone (1950).

The potential OP is written:as:

2i3

PS Ps 7 + sw4)tJ

02ij = Re{024.j e

and r2ij as:

s -r 2J..] urs 2ij -i(w

+ syt

Ps

has to satisfy the following conditions:

2 Ps

4=0 (in the fluid)

(3.46)

4)2i3 Ps ao

Lki

az

, Ps 3

(wi + w

-s) 2, .422ij -J,is s -Dz. _g w2ij - r2ij _ (3.44) (3.45) (on z = -h) (3.47) (3.48)

In an addition a radiation condition is assumed at infinity. This may not be mathematically exact as discuSsed in chapt-It is also assumed that t is absolute integrable, i.e.:

Ifirl .Ids < (3.49)

SF

The velocity potential can then be expressed as folloWs:.

rs 0Ps = ff 2ij s

Gi ds

(3.50)

S_F 4y

2j

where Gs j is the Green's function for the sum or difference. 2i1

frequency problem. . Expressions for

i are given in Chapter

2j

FDe Re{io(w +

21ij

2

.5dmianalytiosolution of the

gotentialgroblem-fot-a fixed vertical cylinder

using Green's- theorem

The second order potential problem for a fixed vertical surface piercing cylinder is formulated both for the Sum and difference frequency case.

According to eq. 3.37 the horizontal force can be written as: Is ac0.5 sw 11 .)[ --1=1 adedz. 3 SB

2,3

al-1 -i(w

+ soy

+ ff

e

r dedile

gF

21ij

The total force from.second order potential is found from

eq. 3-.26. In the present case we have to rewrite the

non-'homogeneous free surface condition

in

cylindrical coordi nates. 2 ao a24) 1 301 a , act a 431 n 1 1 hs = 2

ig

at ar arat -2 1 94.1 1 1 a 4) a(131 a (1)1-s

7

37

7

aeat 2 az 3z3t1 1 , 1 a. 1 act act)

uyDy

a2 Dr ay r ae

This Implies that

if

we replace

2 2

a '01 a.°1

and

777E

by y

aeat

in eqs. A2.4 A2.8

of Appendix A2, the correct expressions in cylindrical coordinates are Obtained for h27j and r 3 .

2ij

The first order total velocity potential for a fixed vertical Cylinder can according to MacCarty

and

Fuchs (1954) bewritten as:

g.4i

cosh

ki

(z+h) 0

1j cosh

ht

CAT,

OC)

A7

(3.51)

-J.w4t

mm-

1 J.(k r)}cosmee '

where

Eth = 1, Eth =

2.foi m 1.

Am = Eth imJmi(kja)/HM(1)1(kja) (3.54)

.16(1)()(jr) = Jm(kjr) + i Ym(kjr) (3.55)

where Hm.(1) is the Henkel function.

In order to evaluate the r3ij(hlij) parameter the

following

derivatives are needed:

;,(1)

g

g. codh-k.(z+h) m

1

lc .0k IT (IIII1cr)

-arw.coOk3.h-i4orarri

-3 - j

mJmilkjr)1cosmee

La.=

z_li cosh kj(s+h)

{A H wi cosh kjh m m(1) k.r) .3M (ki F) 1-msinmee

g gj

einh(kj(z+h))c. = - 3 t {A H (1) z (4 .

cosh-

k . h --3 m=0 P m m Jm (k-3r)}cosmee g ;.

_{cosh(kj(s+h))ki2}

_' (1) W. coshk .3h ' Z AmHm m=0 EmillJ_(k.r)}cOsmee m m k . r ) (3.53) .96) (3.57) (3.58) (3.59)

In order to construct-the auxiliary forded surge motion velocity potential, used in finding the second order diff-raction potential, it. is neccessary to know the first order radiation potential.

The author was not able to find any work where the radiation potential for a vertical circular cylinder:

in

finite water depth

had

been developed from the basic boundary condition to a final close fotm solution.

The work published seems to have been mostly concentrated on the circular dock problem, see for instance Calisal and Sabuncu (1.984). SabUnct and CalisaI (19811 and 'Yeung (1981).

A doirect solution of the second order potential forces

acting on a vertical cylinder was considered very important in checking the results from the computer program for. arbitrary. body shapes.

In vied of these facts it was decided to work Out the formula for thezurgeradiation potential fro, basic assumptions. The details of

this

development'maY be found in Appendix A8: The first order radiation potential can be expressed as:

(1) w H1 (kr) coshk(Z+h). OR = aos61

2(1) '

-k- coshkh

v

(ka ) sin%)nh cosy (Z+h) + 4 n=1 v (stray h

+ 2v nh).H

(vna) n n -iwt n r) }e (3.60)

The Auxiliary forced surge velocity potential for the sum and difference frequency problem can now be constructed as:

(w, + sw.)(H, (1c1.3.; .r)coShk2is.j (Z+h)) 3 4 -a 2 (k7 )- cosh ks13 .h V, H1 (1);ks 2 12ij 21j a. s s

sin vn2ij h cosvn2ij(Z4-11) + 4 2, s- -s s

n=1(sin2Vs

2ij h + 2vn2ij h) n K1(v5 -r)le-i(u)1 wis1t n2ij_ .

where s = +1: means the sum frequency problem and s= 1 the difference frequency problem

k is. given by 2ij

(w + = gk2lj tanh khj h (3.62)

v_n2ij are the roots of:

2 S' (ed + sw.) = tan v h 7 gvn2ij n213 rs it defined by: (3.61) (1.63) (w4

+ sco)

10

Vs ' = 1 2ijAi (-- + ) (3.64) g2ij ks 2 _{sinh 2k213s-.h} 2ij'

H (1) is the Hankel function and KI the modified Hassel function

3.2.5 second order hydrodynamic boundary value_2roblem -using Gfeen's functions and integral eguatiOns

The boundary value problems for the sum and diffetence. frequency potentials with homogeneous free.surface condition

defineci.in eqs. 3.31 and 3.43 can now be determined using the integral equatioh approach. The. solution procedure is quite, analogue to the methods used in the past, see for

. instance Faltinsen.and Michelsen (1970 and Appendix Al

The solution to the radiation problem 4,2ijk (where the affik s represents the sum frequency problem if s + and the difference frequency problem if s = -) can be written as

rr s

*Ikij

=41' n$ 2ki7' G s .,...(x,y,z; ,n,)ds J.3 B ,

where k = 1,6 represents the six degrees of freedom.

The homogeneous part of the diffraction problem 011sij can be written as:

= rr nHs ns

-1A-

- (3.66)

Qs Hs -s

2kij and..;a2ijaretheunknown.soUrcedemtitievandq the

Green's function which can be written in alternative- ways,

see Weehausen and Laitone (1960):

s s

ds (x,y,21,n,c)

27((g21i,)2

-

(k)2 cosh Its

(Z+h) 21j

cosh k s

(ct

h)(Y (k P r1 - i J (k s r )

2ij o 2ij 1 o 2ij 1

w 2 s 2 + (K2i'l +. 4 E ₂ -s lc=1 u.k

h +. (K)h

.h -1K s 12 h + K. 2i3 2ij' 'where' K2ijs = k213- a. tanh ka213.h where for s = + (w., + w )2

K+ij2 = g = k2ij tanh k₂₁₃

and for (w. +w.)2 K =

-=213

7 tanh k2i3 h cos(uk(Z+h)) (3.68)

uk of

eq.

3.87 it the solution to the deviation,

tanukh + 21j (3.69) (3.70) (3..71) (3.72) cos(Uk(C+h)) Ko(likr1) (3.67) 1 1 ,G21i(x,Yiz:Fn.4) I t IT

co(U+glij)e-lihcosh klii(c+h)coshu(Z+h)_Jo(u )du

+ 2PV I. 0 usinhuh - cot/nth

i2ff((k211)27(1521i)2)cosh

k!ii(-4+h)cosh k2i1 s (Z+h)-: J (ks .r ) o 1

In eq. (3.68) PV indicates the principle Value Integral.:

The source densities of eqs. 3.65 and 3..66 are found when satisfying the body boundary cOndition. This

in

turn results 14 a Fredholm integral equation of the second kind. For the radiation potentials we get:

-274kii(x,y,z) +1104kii(E,n,,0-61p2sij (x,y,z;.E,n,c)ds

= nk k = 1,6 (3.76)

The corresponding- equation for the homogeneous diffraction potential Hs a 21102.(x,y,z)

+d,(22.

(xiy,zrE,n,c)ds a13: an 2ij Ps Is ao act) : lij 2i'

e

an an ' 2ij

The total Velocity potentials for the sum and difference frequency problems can now be written as:

6 Is Ps Hs s .s (1'21j = 4)21j (1)21' E *2kij.112k k=1 ( s = + = snt,s=-. = difference) (3.77) (3.78) R =IC(x + (y n)2 + 1 (3.73)

.R'

(sc. El 4- Car n 2 (z 2h

+ c)]

(3.74) r1 = (x - )2 + (Y n,,21 (3.75)

Ps Hs

0 is given by eq. 3..9, 0 by eq. 3.50. 41,2i. an

a!!) found from eqs: 3.76 and 3.77..

n (k .1,6) represent second order motions.

-As discussed in chapter 6 the second Order motions are

usually

found by time integration techniques since they are highly non-linear and dependent on viscous effects. In the .present analysis nlk isassumed to be zero,

6 E

k.1 /P2kij.2k = °

Second order contributions to forces and moments from product of first order potentials

In this chapter the second order contribution from product of first order potentials will be derived.

'we start With the complete Bernoulli's equation given by:

34/1 2 2

P = -Pgz - 13/21(137)

(77)

(77)

)

P.

(3:80)

where.po is the atmospheric pressure, 0/ is the total first order velocity potential comprising incidint waves and dis-turbance by the body.

Two coordinate systems are necessary, one intertial (space fixed axis) k,y,Z and one with body-fixed axis

x*, y*, z*.

z-axis is -vertical upwards with origo- in the.undistntbed free surface. _{Both cootdiate systems coincide when the body is} at rest.

In the perturbation expansion we want to keep order 2

in

the expansion paraMater E. For instance

,

s 'Y2kij

a1

_{t _1} 2

0(737 73) = 0(6 )

We will proceed by making a Taylor expansion of the pressure about theMean position of-the vessel correct to second, Order in wave, amplitude. P = pgz - p(77),-a ao P n (Iti)m )

a

"1

a

p(nx(xiyiz)727 (7.7)m)

p(n (xiy,z) 71

ao1 2 ao1 2

±(17)

)m + pm

a01,

at -111. (3.81) (3.82)

where m indicates the mean surface of the vessel and n , n

y

z

and is defined by

ny(x,y,z)

ni + n

-

b4z

= nj +

n4y 18* (3.83)

nm(x,y,z) = n

+ n5z - n6Y

We now want to retain contribution from pm which is of second order

in

Wave amplitude. These terms

will

consist of pr6duct of. first order terms along with terms arising from integration over' the exact Wetted body surface area as well as transformed first order forces into coordinate System

(k,y42)

of second order forces.

Now let us study the contribution- from the hydrostatic tetra pgz.. According to' Gauss theorem there will only act a Vertical force for the part of body .below z =

The possibility of

a

vertical second order term arising from the'pgz term will be discussed later in this chapter.

If waves are present we will get a contribution to the horizontal force and moment

(a)a

F2k = pg I fz nkdzols =I c2 n ds 2

o

where 4 IS the wave elevation, c denotes the integral along .maieriolane area and nk is' generalized outward normal: vector

in mean position.

.The generalized direction cosines are. defined by

= cos(n,x) 42 = cos(n,y) h3 = cot(n,2) n4 = n3y - n2z ns =

n12

- n3k n6 = n2x - nly ad)

The contribution from - p TE,.. is separated into two terms: One term is due to the integration over the exact wetted surface (b) F2k' = fPg"4- (-13 ' "5 ' Y71,4"hk4s c (3.84) (3.85) (3.86)

where c is the dynamic wave elevation at z = Oi and

- (n-3 --xns + yn) is the extension of the action of

c.

Then we have to integrate

plv

over the rest of the wetted area of the body- The integration of p 7E7 in the vessel coordinate system z* will give the total hydro-dynamic force from incoming waves added Mess and damping,

This force-PI must be equal to the rest of the equation of motion. Fitc must be transformed into the coordinate System x4y,z through

(c) + 7 = a x P* 2k where 41.4 = 1

The. different components of

F1V

follows then as:

(c) 2 F21 = w M(n - n4 z 2 F23 =14111:1

-

zgn5)n5 -+(c33n3 + c35n5)ti5 (c) Fi = (wMn3)n4 - (w2Mn - z n )n g 5 6 F2V' = 2_ w mn w 1z g -

(2

_{7-1-3 m)n5} n _{+ n42 )n4}

56

n )n + (w2_{I46n4 - wI66n6)n5} (3-87) (3,88) (3.89) -(c33n3 + c35n5)n4 (3.90) (3.91) (C3513 4 c5515."6 (3.92)

=-w2I46 n4 + w2I66n6In4. 25 ..., .-1' . 2T

In

"w2

"2

w -44n4 w '4616'"6 (c4414)n6

(2Mzn

F(d) = -w

_g2

+ w2 I 26 n4 - w2 146 n )n + (-w2 Mz n - W2 1115 ) g 1 55 (b3513 +c55n5)11 (c44n4)115

where m is the mass of the body. I, I

40'

-44' 155 and 166 are

the Mass tmoments of ytertia. The hydrostatic restoring coefficients are defined as:.

C33 = pg. !Ids Awp C35:: = ft xds Awp c44' = pgV(zB - zg) ptr.7(z8

-

z ) + pg Ifx2 s (3.98) Awp

We can simplify the contribution ofF(a) F(b)' F(c) by

2k ' 2k -2k

using the gradient

theorem:-ffilg dv =ff

114

ds (3.99)

Application of eq. 3.99 gives:

Pg Ify2 ds Amp -(3.93) (3.94) (3.95) (3.96) (3.97)

n ds

₁

c-nn +c nn

_{33 3 5}

_{35 5}

5.

2

n2ds =

(c22n2 + c 5n5)n4

- p

2

z

n6 ds =

A-pg if_Awia

x2 dFi5)n4

(+Pg ff Y dSn4) 5 -Awp

Where

nz

is given by eq. (3.83).

The relative wave amplitude t is introduced

Cr C nz

Substituting

ct, egs. 3.10Q, 3-101 and 3.102

into.tbe

expression for

Fl)F23(b)

and F,(Vc and adding the rest of the

4x

contribution from

ps,

eq.

3.82, we

can write ddWn the complete expressions for the second order forces-.

Each of the components in

six

degrees of freedom are:

f-

r211-1-- - (w2Mn3)15 c +(w2 M(n2

_{- zgn4))n6}

ao 3 3°1 , ,

1,

+p

LN,ri.x.(x.,y,z),I7(t77.) + n

(x,y,z, 171T-1

SB 3°1 1

+ nz(x,y,z) 1-1(7tr-)+7

3°1 2

_ax

(3.100)

.1 2 +

(-77)

)}n ds

(3.104)

cla -c 2

(3.102)

(3.103)

= 21. 24.t.454.(w2111.13)n4 w-- n zg,n ).)n6 a a01

+ plifyx,y,z) Trc

+n_y(x,Yit) --_ay

_at

a

"1

"r(x,Y'z) n (-7E)

1

1212

(4y

4. 3x YITI2ds

"1 2

ay r23 = (434MIn 2§151)n5 (w2m(n a a(1)1

Dif frix(x.Y.z) 1E(

7--1

a +ny(x,y,z) ( ) +riz(x,y,z) ao a , 1, az

at

1

41

2 "1 2:

1"

(-Ty)

n z 1)n_{4 g} 4 F24'.= (w2mn

'4

+ wm5n )n6 + (a)214 n4 w 6 n6)n6 -35n3 c5515)n6 a

+pLifnx(x,y,z)

isrc

117,

SB a01

. ny(xly,z)

_(T17) + 9 (x,y,z)-yi 1

.a12

"LI 2

"1 2

+ ₊ ( + (,-57) 014ds (3.105) (3.107)

az

)143ds (3.106)

2 -w214

46 6

4

+ w I

n Yn +(w2Mn z 2 g

a .,"i

+piffT1 (X,Y.fz) 1 x 'B a +ny(x,y,z)

,asl

) + n-(x,y,z) "

"t 2

'a01

1((tn0

( ay F _{2n ds} 26 2 r 6, c -, 444w2"4 pgV(zez 2 2,. 1,4F1.? n + w

,4

)115 off{nx(x,y,z) ny(x,y4z) + n

_(x,y,z)

:2 4n4 w2146116)116 c44n4 55115 + pgv(zB a fa01, az

at'

ao a -Tc, (177) a az

-at'

arp-a

--,---,

_az

_at

z )n )n

_{g 5 4} 1

_1 2

2 301,2, +

+ (-7)

_az , n (3.108) (3,1439)

The expressions in-eqs.: 3.104 -3-.109 will give the mean drift forces and moments if time averages are taken..

The method of derivation is similar to the one used by Faitinsen et al (19-80) for F-., F in the added

re-41 -22' F26

zistance case. The expressions will however differ somewhat due to the fact that FaltinSen et al (1980) considered strip theory approximations.

The formulaes forF

zi -22' F33 are identical to those derived by. Pinkster (1980). It should be noted that he used

a

coordinate system in which

zs

0.

Thus in order to compare

eqs.

3.104 - 3:109 with those of Pinksteri as mast be put. equal to zero and mass moments of inertia must be computed with- as as the origin.

The formalaes for F24, F25 and F26 contain contributions from

-hydrostatic restoring coeffiCients which seem to be omitted by Pinkster (1980)- The expressions for F2k (k 1,2,---6) except F23 seem to be identical with the results of Ogilvie

(1983) although his equations are not derived to the sable detailed level as in this report. _{The formula for F26 is the} same as found by Faltinsen et al (1980), except for their strip theory approximation.

Ogilvie (1983) has derived a vertical force component

in

addition to those given in

eq.

3.106 . According to

Ogilvie

this

component F2, may be written as

F

(d)

₌

-pgAWP(16(xe14 + yfn5))

23

Standing and Dacunha (1982) also include an additional vertical force component to eq. 3.106, but they do not reveal any details of its origin. This component was ex-pressed as

( 1

F2e)

₃ = - - cgzAwpfn42 + n52)₂

(3.110)

where Awpis.the water plane area, xf and yf the longitudinal and transverse position of the centre of floatation respectively.

4.

301.-

7,1)

+ =T

X = n + D x*

z is the vertical coordinate of the point where first order motions are defined.

In order to get the background for eq. 3.110 we need to include some equations for transformation of Coordinates

(and other variables)

from

one system to another. The following equations may be found

in

textbooks on rigid body dynamics. We Will however proceed based on Ogilvie, (1983)

We define two sett of axis:

oxyz

: inertial (space-fixed) axis

o*x*y*z*: body-fixed axis

These are the coordinate axes originally selected for the development.

The position vectors in space are denoted by:

x

0 (x,y,z)

0 (x*,y*,z*)

The position of o* with respect to o is given by

,n ,n

) (3,11.2) The position vectors are related by . an orthogonal

transformation matrix 3

(3.113)

cosn5cosn6

sinn5 -Cosn sinn6

5

where BT is unit matrix and

1r

transpose of B. If we assume infihitional rotations and displacement. and introduce:

=

we can .Write

x =

x= +

Ti. + a x x* (3.116)

Eq. 3.116 is equivalent to eq. 3.114, if we set

Eq. 3.116 is the. basis for all first order..development in which rotations can be considered as equivalent to vectors.

If we consider finite, rotations a certain sequence of the rotations has to be decided, Ogilvie used roll, pitch and yaw. In such cases the transfOrmation Matrix can be written

4s1

cosn4sinn5 +sinn4sinn5cosn5 cosn4cosn5 sinn4sinn5sinn5 sinn4cosn5 sinn4sinn6 -cosn4sihn5dosn6 tinn4Cosn5 -cosn4sinn-5sinn6 sn sn co 4co 5 (3.117) (3.118)

By using a Taylor expansion of cos and sin and retaining contributions 'in. up to

o(e)

in magnitude,

= 1 -

(n

1 +n 2)

2 32 16 1416

-n6

- (n

2 42 + 1516

Thus eq. 3.114 is rewritten as

-= Ti + [51, + Bp X* 1 2 -

7

13 /62) 0 -14 0 -

1(2

+ n62) 2 4

-n

+ 114116₆ 1 2 2' 1 -

1(n4

+ 15 ) (3.119)

Now the idea is to decompose eq. 3.119 into one. linear part

OL

and-one

perturbation .part Ope The linear part 8L is equal. to eq. 3.117. 1416 6 . 1 2 - 1(n4,2 + ns ) (3.120) (3.121) (3-122)

Ogilvie. found that the 4431* term gives rise to second Order forces if is for instance a force of 0(1) as the. bouyancy force. It is true that 3 is of order 0(621, The procedure

P

-.T

orthCogonal. The correct would eventually be to. keep (5)-131* term.

The author find it further questionable to include terms of the type described in eq. 3.110, since suddenly terms-of finite rotation are introduced. This again lead to the situation

of

formally keeping track of the sequence in which- rotations are computed -(Euler angles). The' complete.

solution procedure has-been based on handling rotations

as

a vector and not as a matrix

The authOr, has not found a good enough. justification at the moment to include any effect of the type as given by eq. 3.110.

It,

should

however be. noted that in the numerical calculation presented in chapter 5, the contribution from eq. 3,110 would-be zero.

Another investigator who has Included terms from the non-linearity of rotation is Triantefyllou (1982)- Pinkster and Gortmerssen (1977) did include eq. 3,122 in the context of _second order motions, but they did not incorporate the effect of eq. 3.122 on second order forces, neither did Pinkster (1980).

In order to get the time varying parts of eqs- 4.104 - 3.109 in irregular sea, the sum and difference frequency parts have to be extracted.

The time dependent part is written as:

. w.)mt

F2kij(t) 1.4j F2ki3 cos(wi +

S3

ss'

c+

s+

vc:'

Where.

1]

F

fer: functions for the sum and in..phase.,with cosine and sine. Spectively.

Fs,' aie Second order transt-2kij

difference frequency forces components of (wi + sty

re

-As an-example of derivation Of the product of first order potential resulting in second order time dependent forces, We consider the contribution 4r2 of eq. 3.104: The relative wave amplitude may be exprested

Cri(t) (3.124)

where eri is the phase angle of cri relative to the ilth incoming wave.

Using elementary relationship between cosine and sine to sum and difference of angles, the sum and difference frequency force due to.cxij.can-be written as:

1 F- - .(t)= j ri rj c c 1-,pgc

_cos(ri

e- + se. .)h t 21ij 4 ..cbs(w + sw/)t. I

Cici

T PgCriCrj

4M(Cri

PP

sin(wt + scyt (3.125)

Eq. 3.125 can be rewritten on the same form as eq. 3.123 with:

cs 1

F21ij /- T Pg4rirj c°s('r. + sEri)nids n ds

(3.126)

N

The other-contributions to FUli and F of eq. 3.123 can be obtained in.

a

similar fashion.

3.4 Total second order forces

The total -second order Sum.and.difference frequency fordes in irregular seas can now be expressed as:

*opt t (ei +

FP- At). = : Z

2kij

,E

CCi

, cosaW

i=3 i=

ss + t T .

sinfN

1=1 j=1 21d3 + Swj)t + (E

+ SEpl

(3.128)

where Ei and Ej are random phase angles. TItil is the second order transfer function derived from the second order potential forces and product of first order potential force.

Since we imposed the restriction on the contribution from the second order potential s = -1.wi > wj,.s = +1 wi wj

during, the practical procedure of solution, we have to derive some relations to get back the-complete matrix.

In this case we again separate the sum and difference frequendy problem.

Using only two frequency components in eq. 3.128, this can be written in difference frequency case es:

= 412

mc,-

_'2k11 c-tC.12

2k12221

k cosl(w1 - w2)t + -- t2)}

+C(Ts-

-1 2 2k12 C- c-T_2kji = T_.2kij , =s- 1

mS*

'2kij

7

s-57-T - = -2kji 2kij

Eq. 3.128 is then rewritten with two frequency components in the gum frequency problem:

,+ c+ s+ `2kij'`' = '1 ( .2klicos2mt + 2k114.n2wt) 2 -c+ C2 (T1422 cos2wt.+Tst-2k22 sin2wt) C C (1)+02 )4.'t21)cc)sl(w 'I" w2)t tC1;2(T11112 T31121)sin(w + + (c + c2)1

s-2k21)sinf(w' 'w2)t +- c - E2) } (3.129)

If Tc-*. and Ts-*. are the results from the second order 2kij 2kij

potential cO when (wi > wj) we get:

C-- 1

c*

T - T 2kij 2kij (.3.03.0 E' + ) (3..132)

s+

2kij

If is the

solution.,when.wi

wi And the resulting sedOnd order transfer function are

defined.

as

T:.t:j

and

TIZ:v

the complete matrlx of secondorder transfer function is:

Tet

=*

Tct

2kij

2 '2154i

(3.133)

c+ c+

T2kji =

2kij

ms+ 1 ms+*

* I '2kij:

(3.134)

4. COMPUTER PROGRAMS AND TESTING

4 1 General

The theory described

in

chapter 3 and Appendices has been implemented in two separate computer programs. SECDIF is a Computer program for calculations of first and second order Potential forces on objects of arbitrary form in waves. The program has been designed to handle hydrodynamic inter-action effects between several moored or free floating bodies.

SECCYL is a computer program for calculatiohof first and second order potential forces on a fixed vertical cylinder. SECCYL has been included for checking

of

the results from the general program SECDIF.

In this chapter a brief description will be given of the main program and subroutines developed. The procedure of testing the programs will further be outlined-. The checking

of

the computational result's from the Second Order theory is very difficult due to lack of reliable data. This fact makes the testing phase very important At every stage in developing the computer programs.

4.2 SECDIF, A computer program for caldulation of first and second order wave forces On objects of arbitrary

form

SECDIF has been based on further development of NV439B which is a computer program primarly designed to handle the multi-body Problem, see LOken (1981) NV4598 was in turn based on

the computer. program NV459 for the single body problem using three-dimensional sink source technique, see Faltinsen and Michelsen (1974).

The program structure and subroutines primarly developed to solve the general second order sum and difference frequency problem are displayed in Fig. 4.1. It should be noted that this only is _a part of the total program system.

The purpose of the different subroutines are explained in the following with reference to the theoretical part in chapter 3 and Appendices. Since it is important later on in describing the testing of the subroutines, they are .grouped In different levels according to Fig. 4.1.

MPSEC

This is the main program calling the main subrOutine MNSEC.

MNSEC, 1. level

This is the only first level subroutine administrating the second level subroutines.

WAVER 2. level

This subroutine reorganizes some of the body surface element data: its main task is however to administrate the cal-culation of GA and its first and second derivatives. GA is the value. of the frequency independent Green's function

integtatedOver a quadrilateral on the body and With the field point on the body.

The first and second derivatives of GR in a local coordinate system (element Coordinate system) are further tiansfOrmed to the global coordinate system Using the formulaes of .

Appendix A7.

The following values are transferred to mass storage-for later

'use: GR 2-3G an _{a GR} R -R r ax ay az2 a2G ac R r ayaz an GR 1 3x3y axaz a2GR ax az2

HESMP, 3. level

This subroutine calculates the frequency. independent Green's function GR integrated over source quadrilateral according to Faltinsen and Michelsen (1974).

HESPbT, 3,1evel

The subroutine, computes the same as SESMP but in an alter-native way.

GRVE1R, 3.1evel

The derivatives of the Green's function for a source inte-grated over a quadrilateral. are calculated according to Appendix A6i eqs A5.1 - 0.15.

SURFEL, 2,1evel

The computations of element data on the free surface are administrated by this subroutine.

GEOSUR; 3 level

This Subroutine calculate the free surface element data for a single valued surface piercing body.. Two regions are used to get a cylindrical outer boundary, see Fig. 4.2. For the multibody case the inner region has to be given as input.

WAVFR1, 2. level

Administrates the calculation of GR and its derivatives integrated over a source quadrilateral on the body with field point on the free surface. The following quantities are computed by calling subroutines GRVE1R and HESMP:

aG

ac

2

R R aG a G

GD,

ax ay az

WAVFR2., , level

The calculation GR and its first derivatives is administrated by this subroutine. The sourcequatriiaterals are on the free surface With field points on the body surface.

HESM 3 level

The integrated frequency independent Green's function and its first derivatives are evaluated in the local element coordi-nate system, see Faltinsem and Michelsen (1974) and Hess and Smith (1964)

* The same as in NV459 and NV459B

EQSYS1 2. level

The subroutine sets up the.linear algebraic equations re-sulting from the Fredholla integral equation of second kind, see Appendix Al. This is in order to solve the first order boun-dary value problem for the single and multibody case. The Influence matrices for the first order potential and its first and second derivatives necessary for the second order problem are calculated and stored on ale for later use.

GRFUN 3plevel

This subroutine administrates the computations of the total Green's function and its first and second derivatives for finite or infinite water depth by'dalling.subr. FGREEN. The frequency independent parts are added, see Appendix. A6.

FGREEN, 4. level

The frequency dependent part of the Green's functions are calculated, see Appendix A6. This is main subroutine of the subroutine package "FINGREEN" developed at M.I.T.

SOLVE1, 2. level

This subroutine solves the linear equation system set up by EQSYS1. The source densities for the first order single

or

multibody problem are found. The linear equation systems are solved

by

Crout'a Method.

FORDEl: 2,1evel

The first order wave induced forces andmotions are computed for the single: or multibody problem. as well as first' and

Second derivatives of potential necessary for the second order problem, see Appendix

* EQMOT, 3. level

The linear equations

of

motions Are solved for N moving bodies, see Appendix Al.

* DRIFTM, 3.1evel

The horizontal mean drift forces are comPuted according to the method of Faltinsen and Alchelsen (1974).

FREESU, 2.1evel

This subroutine administrates the computation of the first .order velocity potential And its first and second

deriva-tives on the free surface. These values are to be used for the second order hydrodynaMic boundary value problem, see Appendix

GRFUN1,

3.1evel

The total Greens functions and its first and second deri-vatives. On the free surface (field points) with sources

on the body surfaces are computed. the frequency independent part are added in this subroutine,

WAVE1, 3.41evel

The first and second derivatives of the first order incoming wave potential are computed, see Appendix

IdSPER, 2.1evel

This subroutine calculates the wave dispersion relationship for the second order problem.

FT2INH, 2. level

The second order inhomogeneous diffraction- potential is calculated see chapter

3-GRFUN2, 1. level

This subroutine administrates the Computation of the total Green's: function and its first derivatives used in the second order boundary value problem, see chapter 3.

SECFSC, 3. level

The inhordogeneous second order free surface condition

is-computed according to the forMulaes in AppehdiX A2.

21QSYS2, 2. level

This subroutine sets up the linear algebraic equation d from. the Fredholm integral equation of second kind in order to solve the second order hydrodynamic boundary Value problem.

The second order Green's functions arecalculated through the call Of GRFUN2. The tighthand sides of the linear equations for the second order- cliffraCtion problem and the ficticious,forced motion problem are computed, see chapter 3- This subroutine is designed to handle the single and multibody problem for the sum and difference frequency cased.

-SECSBC, 3. level

Second order body boundary conditions due to first order motions and derivatives of first order potentials are com-puted- for the sum and difference frequency case.

The formulaes used are given in Appendix A3'.

WAVE2 3. level

The sum and difference frequency second order incoming wave, potential and its first derivatives are calchlated, see chapter 3.

SATIFS, 2. level

The Green's function with source element on the body and field point on the free surface are calculated. The values are to be used.

in

calculating the free surface integral of the second order problem:

SOLVE2, 2. level

This subroutine solves the linear equation system resulting. from second order hydrodynamic boundary value problem.

SECTOR, 2. level

The second order diffrectionpotential forces are calculated for the sum and difference freqtency problem.

The computations are based on using Green's symmetry theorem or solving the second order diffractibn _problem directly, see chapter 3.

SECFIR, 2: level

The contributions

to

second order forces from product of first order potentials are calculated for

the

sum and diffe-rence frequency problem.

SECFKR, 3. level

The second order forces from the second order incoming wave potential are computed.

FORCE2, _2.1evel

The second order transfer functions for the sum and difference frequency problem are printed, see Chapter 3.

I WAVER

HESMP 5;;;71;

FGREEN ETC

EGREEN ETCI

Fig. 4.1 Main program structure of SECDIF WAVE1 MPSECI MNSEC IEIESMP MNSEC MNSECI-I.WAVER1 .GEVE1R WAVER2

REM

1

EQSYS1 SOLVE1 FORCE1I

.1DISPER

FGREEN ETC

MNSEC

SECFOR SECFIR FORCE2

SECFRRI

IWAVE2I

Fig. 4 . 1. (continue)

MNSEC

EQSYS2 SAI2FS SOLVE2

Outer region

ndro

Inner -region ndr

Radius vectors are constructed through end

points of upper layer of body surface elements

and origin a

drj

is specified and number of dri

ndri

ro

is specified and number of dro , ndro