Second-order theory and calculations of motions and loads of arbitrarily shaped three-dimensional bodies in waves

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to be published at the Journal Marine Strustures

SECOND-ORDER THEORY AND CALCuLATIONS OF MOTIONS

AND LOADS OF ARBITRARILY SHAPED 3D BODIES IN WAVES

TECHNISCHE UNIVERSITEIT

Laboratorium voor

Scheepshydromechj National Technical Univercity of Athens Archief

Department of Naval Architecture and Marine Fnginweg 2, 2628 _{CD Deift}

Tel.: Oi' _{- 786873 - Fax:}

015-ABSTRACT

This paper deals with the development of a complete second-order theory for the evaluation of motions and loads of arbitrarily shaped 3D bodies in waves at finite water depth. The developed order potential theory leads to the solution of integral equations for the evaluation of second-order potentials and corresponding second-second-order pressures, forces, motions etc. The developed algorithm for the treatment of the second-order inhomogeneity of the free-surface boundary condition

allows the application of the related computer program to bodies of arbitrary shape, in difference to previous theories applicable only to axis ymmetric bodies. The paper includes typical numerical

results for the second-order motions and loads for various bodies of both axisymmetric and

nonaxisymmetric shape.

I INTRODUCTION

Nonlinear effects due to the interaction of floating bodies with incident waves are often very important for the design of offshore structures. In the last decade, great effort has been devoted to the evaluation of the second-order velocity potential and of second-order forces acting ori vertical cylinders and axisymmetric

bodies. The required solution of the second-order boundary value problem is, compared to the solution of the

first-order one, much more complicated due to an inhomogeneous term appearing in its free-surface boundary condition. Because of this complication, many early attempts failed to satisfy completely the second-order free-surface boundary condition. In a well known paper, Mohn9 describes a method for the calculation of the second-order forces on axisymmetric bodies subject to monochromatic incident waves, which avoids the calculation of the second-order potential explicitly. However, the evaluation of the second-order potential leading to second-order pressures, wave profiles and runup is by itself of great interest compared with the evaluation of integral values only, like forces and motions. In the last decade, many authors dealt with the second-order problem, some of them solving directly for the potential and others only for the second-order forces. Among them, Eatock-Taylor and Hung3 calculate directly the second-order wave force acting on a vertical bottom-mounted cylinder by monochromatic waves. Kim and

Yue6-7 calculate the second-order potential around axisymmetric bodies for monochromatic and bichromatic

waves. Loken8 calculates the second-order velocity potential for three-dimensional bodies of general shape, taking into account the influence of the free-surface inhomogeneity in an approximative way. In the following, an exact theoretical-numerical method for the calculation of the second-order velocity potential and all related physical quantities of interest for three-dimensional bodies of general shape subject to

monochromatic or bichromatic incident waves will be presented.

2 PROBLEM FORMULATION

We consider a rigid body of general shape, floating on the free surface of a fluid and being subject to an incident regular wave (see fig. i). Let Q be the domain of the fluid, SF the free surface of the fluid, 5B the

Georgios N. Zaraphonitis, Dr.-Eng. Apostolos D. Papanikolaou, Professor*

*

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bottom surface at constant depth H, and S the wetted surface of the body. We introduce an inertial coordinate system (OXYZ) with O a point on the free surface at rest and OZ vertical and positive upwards,

a second coordinate system (CX'Y'Z') parallel to the inertial one, where C is a point fixed on the body and a

body fixed coordinate system (Cxyz), which coincides with (CXY'Z) at rest (fig. 1). Let Ñ be the unit vector normal to the instantaneous wetted surface positive outwards expressed in the inertial coordinate system and ñ the same vector expresed in the body fixed coordinate system.

We assume the fluid to be incompressible and inviscid and the fluid motion to be irrotational so that it can be described by a velocity potential. In order to calculate this velocity potential, we have to solve a boundary value problem consisting of the Laplace equation in the fluid domain and appropriate conditions on the boundaries of the domain from the body up to infinity. The resulting boundary value problem is as well ki-iown nonlinear, thus we introduce a regular perturbation expansions method to decompose it into a series of linear boundary value problems. The velocity potential and all the unknown quantities of the problem are expanded in power series of a small parameter e being proportional to the incident wave

steepness:

= + + (1)

The boundary equations ori the instantaneous position of the free surface of the fluid SF and the wetted surface of the body Sr are expanded in Taylor series about the position of these surfaces at the state of rest

(SF0 and respectively). Collecting terms with the same order of e, we formulate a series of boundary

value problems of different order. Here we present the boundary value problems of first- and second-order: First-order boundary value problem:

(1) ' (1) + a2 +

a-

₌₀

ax-

ay

az-

-a1

a2 g + o

az

at2

-(1) (0) j) Vc1 Ñ =

az

=0

Radiation Condition (0) N

Second-order boundary value problem:

a22)

+

ax2 + ay-

az2

(2)

,

(2) (2)

ga

_=A

az

at2

(0) ₍₂₎ .

=B

(2)

az

=0

Radiation Condition

=0

(in Q) (at SF0)

(at Sw)

(at _SB) (at infinity) (in Q) (at SF0)

(at Swo)

(at SB) (at infinity)

(3)

where:

A2

-1)

a&'

a2(l)1

_aV(l))2

- g

at

aZ[

_az

_at2

_j

a and (2) (2) ... (0) (1) (1) (1) (1) B = N +

- N -VD

N \, _i=1 -. T

The incident wave potential at a point X = [X, Y,Z] in Q is equal to: ri

_{a cosh (k(Z + H))}

jk(Xcos +Ysin)

'- w

Y,Z) =

i _O) _cosh e

where w is the wave frequency, k the wave number, a., the wave amplitude and is the angle of wave

incidence (=18O0 for waves running in the negative X direction).

Let Gj(P,Q) be Green's function pulsating with frequency w, where p=[x,y,z1T is a field point in Q and

Q=[XQ,YQ,ZQ]T _{a source point (Wehausen and Laitone14). Respectively, we define for later use G2(P,Q) as}

Green's function pulsating with frequency 2w. For the evaluation of the diffraction and radiation

potentials, we express them through a distribution of pulsating sources over Swo with source density c(Q)

(Papanikolaou1 2):

(p(p) = --$$ a(Q) G1(PQ)dSQ

(5)

s

The source density ci is calculated from a Fredhoim integral equation of second type, so that cp(P) satisfies

the wetted surface boundary conti tion.

aG1(P,Q)

_()(p)

_dS

_{= 2}

2ir

H ₍₀₎

Ç0

Ñ' '(P)

'<

3 EVALUATION OF POTENTIALS - MONOCHROMATIC WAVES

3.1 First-Order Potential

(1) ₍₁₎

Let

.

(t) and

. , i=I,...,6 be the real and complex first-order motions representations of the body

(Yt)

=

Re{X1)e_t},

_{i = L... ,6).}

_{For the solution of the first-order problem we introduce} complex velocity potentials and we decompose the first-order potential in several components, namely, the incident wave, the wave diffraction and the six radiation potentials:

{r

6 (1) ₍₁₎ ₍₁₎ (t)

- Re

(p1 ₊ _D (Za) (Zb) (i) -.(0)

=v

-N

(6)

(4)

In practical situations the solution of (6) is obtained through discretization of the wetted surface by N triangular or quadrilateral elements of constant source dencity a over their area 8S and transformation of

(6) to a system of N linear equations with N unknowns. Thus, equation (5) turns to:

N

(pO)(p) ₌

n=1 3.2 Second-Order Potential

The second-order potential can be decomposed in the following way:

(2) ₁ (2)

_j2wt

(2)

(X,Y,Z,t)

=p2(X,Y,Z)+Re1p2

(Y,Z)e

t

(2)

a,gk

where

=

-2sinh(2kH)

In the following we will restrict our attention to the calculation of

p(X,Y,Z), since this is the only

(2) ₍₂₎

term of the second-order potential with a contribution to the second-order forces. Let

(t) and

the real and complex second-order motions of the body

(2(t)

+

_{Re{Xe_J20)t} ).}

We decompose into a term for the second-order incident wave potential, six radiation potentials and a second-order 'scattering' potential:

6

(2) (2) (2)

2 2I +

The second-order 'scattering' potential must not be confused with the diffraction of the second-order incident wave potential. Instead of, as can be seen from the boundary conditions on the water free surface and the body wetted surface, it contains the diffraction of the second-order incident wave potential, but also terms resulting from the interaction of all first-order potentials given in (3) among themselves.

The second-order incident wave potential is given by:

3j a

cosh (2k(Z + H))

_e20S

8 _{sinh (k.H)}4

The six second-order radiation potentials satisfy boundary value problems similar to the first-order ones

and they can be evaluated in the same way. Thus, it only remains to calculate

(5)

2 (2) (0)

VpÑ =b2

(2) 2 (2)

40 (p2+g

_az

(2)

o

wherem=stforP E Swüand'n=4tforP E Q

The main difficulty in (13) is the calculation of the last integral on the free surface, extending from the body's waterline to infinity. After this integral has been calculated, equation (13) can be solved in the same

way as we used to solve the integral equation of the first-order problem.

33 Calculation of the Free Surface Integral

It is already shown that, in order to evaluate the second-order 'scattering' potential we have to calculate first a series of integrals along the free surface S _FO'of the following form:

I(P.)

₌

5JG2(P.,S)a2(S)dS5

(14)

F0

for all P, i=1, 2,..., N (where N is the number of elements used for the discretization of the wetted surface).

For the evaluation of integral (14) we introduce a circle with center O and radius R0, so that the whole body

lies within that circle (fig. 2). In this way, we divide the free surface in two parts: one between the bodys waterline and the circle (Sp) and another one from that circle to infinity (SFext). The integral will be calculated numerically in the first part and analytically in the second one.

It can be proved that the second-order 'scatteing' potential satisfieS the following integral equation:

aG(P,Q)

m92(P)

±

f5(p2(Q)

dS= fJG2(PQ)b7(Q)dSQ

JJG2(P,S)a2(S)dS5 (13)

a SW0 SFO

The integration outside the circle will be performed in a polar coordinate system with center at O. For the analytical calculation of the integrals, a must be written in a more suitable form than that given in (11).

Radiation Condition (at infinity)

where:

(

(1) jO) ₍₁₎ _a

ap

N 2 0) (0) cp

+v

(I)

₍

(1)\

+j0)V(p

₎

()

(1)

N V(p

- g

1) (2) + 1 2 40) -(2)

(I)-'

(1) X X (1) (1) -N (0) (11) (12)

a2=--5--(p -\g

_az

and (2) (0) (2)

b2=V(p21 N

₂₁ N (in Q)

(at So)

= a2

(at SF0) (at SB)

(6)

Let be the sum of the first-order diffraction and radiation potentials. The first-order potential then takes the following form:

(1) (1) (1)

P P

(15)

Substituting (15) in (11) we decompose a2(S) as following:

a2(S) =

aBB(S) + am(S)

(16)

The term aBB comes from the interaction of

with itself, whereas the term am stems from the

interaction of with .The corresponding a11 term disappears trivially.

For the representation of Green's functions appearing in (7) and (13) we use John's series expansion (John'5). For large horizontal distance r between source and field point we keep only the first term of that series, since the modified Bessel functions appearing in the higher order terms tend rapidly to zero.

G(P,Q)

2jit

cash k(Z + H) cash k(Z

+ H)H0(I)

k2H

-

v2H + y

-

k2

(17)

We express the Hankel function appearing in (17) in polar coordinates using the so-called Graf's theorem (Abramowitz and Stegun1. Substituting the result thereof in (7) we can finally express in the form (Zaraphonitis16):

(S) ₌

_{H(k1R)(Acos(nO)+A5sin(ne))}

(18)

n=O

Substituting (18) in the equations for am and aBB and using Graf's theorem to express G2(P,Q) in polar

coordinates, we can prove the following equations:

and

7C

JG2(P,S)am (S)dO

AmHn(kR)Hm(k2R)Jnm(kR) ±

-t

n=Om=O

+

_{AmHn(kR)Hm(k2R)Jinmi(kR)}

(19)

(7)

J G2(P, S)a(S)dO = h0 C 0H0(kR)H0(kR)H0(k2 R) ±

TC

+

ith0C0H0(kR)H0(kR)H0(k2R) +

00 00 +

_{m(P)Hn(kR)Hm(1)Hn+m(l(2R) +}

n=O m=O +

_{1tEm(P)Hn(kR)Hm(kR)Hinmi(k2R)}

(20) n=O mO

where k, is the wave number corresponding to wave frequency 2w. It remains now to calculate a series of

I integrals in the axial direction, of the form:

J Hn(kR)Hm(kR)Hi(k2R)RdR

and

JH0(kR)Hm(k2R)Ji(kR) RdR

(21)

R0 R0

where n, m, I = 0, 1, 2,...., c. For the expression of the Bessel and Hankel functions in (21) we use recurrence

relations and 1-lankel asymptotic expansions (Abramowitz and Stegun1l. and finally we reduce integrals

(21) into a series of integrals of the form:

jaR

L0= J

_RR

dR, n=0,1,2,3

o

For the calculation of Ln, n=O,1,2 we further employ Fresnel integrals and recurrence relations. Finally, for n=3,4,5,... we express En by Gamma functions, which we calculate using appropriate asymptotic expansions and recurrence relations.

4 BICHROMATIC WAVES

Let the body be subject to an incident wave train, consisting of two harmonic waves with frequencies Wa and w (Wa > wb). The time dependent part of the second-order velocity potential includes harmonic terms with

frequencies 2wa, 2Wb, Wa+Wb and WaWb. In this paragraph we will describe briefly a method for the

evaluation of the second-order potential oscillating with frequencies Ws=Wa+Wb andWd=Wa-Wb (sum- and difference-frequency problems).

We decompose these second-order potentials in a way analogous to equation (9)

6

(2) (2) (2) (2)

s/d

PIs/d+Ss/d_i(0) a± Ob)Xi2S/djS/d

i =1

The second-order sum- and difference-frequency incident wave potentials are already known (see e.g. Bowers2), while the radiation potentials can be evaluated in a way completely analogous to that of the first-order problem. It remains then to evaluate the second-order sum- and difference-frequency 'scattering'

(2)

potential

Ss/d

The procedure for the evaluation of the second-order sum- and difference-frequency 'scattering' potential is

completely analogous to that of the case for monochromatic incident waves. An integral equation similar to (13) is formulated. The integral over the free surface is calculated numerically between the water line and a

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circle surrounding the body, and analytically outside this circle. To perform the analytical integration, Green's functions, first-order potential and the free surface inhomogeneity are expressed in polar

coordinates as in (18). The integration in the circumferential and axial direction, is completely analogous to the case of monochromatic incident wave. The details of this procedure are given by Zaraphonitis16.

5 CALCULATION OF PRESSURES, FORCES AND MOTIONS 5.1 Cakulation of Pressures

Orce the velocity potential has been calculated, the fluid pressure is given by Bernoulli's equation:

p = - pgz -

-

--pVl2

(24)

According to the employed perturbation procedure (1), the pressure p can be expanded into power series of e:

= ₍₂₅₎

5.2 Calculation of Forces

The hydrodynamic force P and the moment M exerted on the body by the fluid can be calculated by

integration of the fluid pressure over the wetted surface of the body:

P= 5Jp

dS and

M JJpCxÑcIs

(26)

SW SW

Expanding equations (26) into power series of e, we derive the following expressions (Papanikolau &

Zaraphoni tis1 ): 1 (1) (1) (1) \_ (I) = p gA

-

₊

-4 Y

+ p J5

dS (27a) S Wo

/

2

2\\

(2) (2) (2)

()

_(') (1) P

= pgAW\- Ç3 +5 XF4 YF+ d4 +

_{,,)k +MR}

₊ (I) (2)

+ p 5f

dS +

_{55 V}

dS + p

J5(Ñ

v1))

dS pg 2 (2Th) cosO S S S WL Wo Wo WO T

where M is the mass of the body, A the waterplane area,

= [x y, ZF]

the coordinates of the center

of floatation in the body-fixed coordinate system,

G the motion of the center of mass of the body in the

inertial coordinate system, d the distance of the free surface from C at state of rest, R the relative motion in

the vertical direction of the free surface with respect to the body's waterline, B the angle of inclination of the wetted surface to the vertical direction at the waterline, R is the transformation matrix from (Cxyz) to (CXYZ') coordinate system and k = [0,0, 1]T is the unit vector in the vertical direction.

Let _{be the total moment exerted to the body by the fluid and gravity action, expressed in the} (CX'Y'Z') coordinate system. It can be proved that:

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(1) (1) M =

pgA3

(1)j

+ pg4 V

and (2) (2)

-1=pgA3

XF o (2)1 - O

- L12

+pg5

1V

ZG_ZB +

L11

M..

= 'J

-XF

o_

-

-'B

.G o o (1)

+ pg5

_y

+ L12 o

+ 4pgAw d

o

-

(zB - z0)

o (2) ±

pg4

wo M O O O

Mz0 My0

O M O

Mz0

0

Mi0

O O M

MyG Mx0

O O

MZ

_MyG

_'CII

'C12 'C13 MzG O

_'hG

'C21 'C22 'C23

My0

Mx0 O 'C31 'C32 'C33 (28a)

+p

552(f

x

)dS±p

$5

V(fx )dS±p

jj().

_V')(f

x )dS SW0 SW0 SW0 (1)2 (f _x ) (1)( (1) (1) (1)

pg

R _cosO

dl +R

x

)R (Ith, )

(28b)

In (28a,b) V is the displaced volume of thc body at rest, I is the inertia matrix, Z) the vector of angular velocity with respect to the inertial coordinate system, fß _[xB,

B' ZBI

and f

=

[x0,

1T

the coordinates of the center of buoyancy and mass in the body-fixed coordinate system and the waterplane area moments of inertia with respect to the i and j axes in the body-fixed coordinate system too.

53 Equations of Motion

The following equations of motion can be proved to be valid to the second-order for a body free to move in waves: + F ,i =

2,...

6

_(29a) 6 ₍₂₎ 6 (2) (2) (2) (2)

ZM

ii _J

±F +F +F

i Hi Mi

,i=1,2,...,6

(sb) j::1

j1

where [M] the generalized body inertia matrix:

and [H] the hydrostatic coefficients matrix. It can be proved that = O except for:

L 12

+ L11

o (1)2 (1)2'\ 4

+5

J

-o O

I

(10)

r-H = - pgA W'

H44 = - pg(L 22+V(zB - z0)),

H55 = - pg(L11+V(z - z0))

(2) ₍₂₎

= p

JJc

itdS + -p Jf

SW0 SW0 (2) ₍₂₎ ₁₎₁2

¡

(1)

=p5J

(fxit)dS++p5$V

(fx it)ds+pff

Vfx

it)dS-S S S WO WO WO 2 (1) ₍ X it)

4pg

T

()2

_(1)2'\

r (2)

(2) (2)

LFHl,FH2,...,FH

=pgAd

)[O,O,L

_YF,XF,OJ

(2)

r (2)

(2) (2) T (1) (1)

d2(R11í)

LFM1,FM2,FM3I = R

M dt2 r (2) (2) (2) T (1) (2) 2 (2)

LFM4,FMS,FM6I =-

+XGX

(R(1U)

f0x

Md (R21ÍG)

dt2 .(l) (1) (2) b

=[-6

5

(l)

(1)

+6

2 2

(,(l)

(1) (1) _j

itdS+p JJx

. v

),dspg

_R

_coS8&

SW0 T (1)

_5

4 6 DISCUSSION OF RESULTS

The method described in the previous paragraphs has been used for the evaluation of the second-order potentials, pressures, forces and motions for a number of bodies floating freely on the free surface or being

fixed in space and excìted by monochromatic or bichromatic waves. For the calculation of Green's functions

we used a method presented earlier by the second of the authors (Papanikolaou12) and the well known program FINGREEN (see Newman11). A comparison with other theoretical or experimental results has

been done, in all cases where such results were available. The agreement with these results was

satisfactory to excellent. Some typical results thereof will be shown next.

The second-order forces and moments (equations (27) and (28)) consist of second-order terms due to double

products of first-order quantities (Part I), Froude-Krylov forces due to the second-order incident wave potential (Part II), and second-order terms due to the 'scattering' potential

p. In order to compare

specific terms thereof, these last terms due to cp are decomposed further into terms due to the wetted

surface forcing (Part III), and terms due to the free surface forcing (Part W).

(1)

= p

x it) dS

H34= H43

pgA

_F'

H35 =H53 =pgAWx,

H45 =H54 =L12

(1) ₍₁₎

P JJ

il'dS

and SW0 SW0

(11)

In figures 3a and 3b we present theoretical results for the real and imaginary part of the second-order horizontal force exerted by monochromatic waves on a bottom-mounted vertical cylinder, for a fluid depth equal to 1.16 times the cylinders radius (H=1.16Rj). The dashed and solid lines represent the (numerical) results of Mohn & Marion10, while the symbols are the results of the present method. The dashed line and the symbol + give the total second-order force, while the solid line and symbol Ê stand for the specific part

of the second-order force due to the free surface inhomogeneity (Part IV). As can be seen from these figures, the agreement between the two methods is very good and the importance of Part IV very significant.

Figures 4a and 4b show the real and imaginary part of the second-order horizontal force acting on a rectangular barge (so-called 'Faltinsens barge') fixed on the free surface, being excited by a monochromatic

wave. The dimensions of the barge are L=B=90m, D=40m, H=lOOm (L: length, B: beam, D: draft, H: fluid depth). The dashed line gives the total second-order force, while the solid line gives the specific part of the second-order force due to the free surface inhomogeneity (Part IV). Figure 4c shows the specific part of the second-order difference-frequency horizontal force corresponding to the excitation of the barge by the free surface inhomogeneity only and for the case of a bichromatic incident wave with VaL=3.2 and vi,L=2

(v=w2/g). The wave amplitudes are herein assumed equal to 1m. This figure shows the force calculated as a

function of the radius of the circle separating the areas of numerical and analytical integration on the free surface. The solid line gives the results for the force when the analytical integration is included, while the dashed line gives the results for the case of omission of the analytical integration. From this figure, the

rather quick convergence of the analytical integration can be concluded.

In figures 5a, 5b, 5c results for the first- and second-order motions in surge, heave and pitch of a vertical floating cylinder, subject to a monochromatic incident wave are presented. The cylinder has a draft to radius ratio equal to 7.9:4 and the ratio of the depth of the fluid to the radius of the cylinder is equal to 45:4. The dashed and solid lines represent the (numerical) results of Mohn & Marion10, while the symbols correspond to the results of the present method. The dashed line and the symbol * give the first-order

motions of the cylinder, while the solid line and symbol o give the second-order motions. As can be seen from these figures, the agreement between the two theories is very good for first-order motions, not that good for

the second-order surge motion, but much better for the second-order heave and pitch motions. It is obvious

from the theoretical point of view, that the second-order motion responces will exhibit manifold resonances due to various mode interactions in the first- and second-order motions.

In table I we finally present results of (total) second-order forces and moments for a bottom-mounted vertical

cylinder of radius Rü=lOOm and depth H=lOOm, subject to bichromatic waves of amplitude a;y=lm. The

results are given for the difference-frequency and sum-frequency problem. A comparison of these results with those by Kim and Yue7 shows satisfactory agreement.

7 CONCLUSIONS

A method for the evaluation of the second-order potential, pressures, forces and motions for bodies of

general shape, subject to monochromatic or bichroma tic waves in fluid of finite depth is presented. Typical results of second-order forces and motions for three different body shapes, including a rectangular barge, are given. Comparisons with other available results show in general a very good agreement.

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John, F.: On the motion of floating bodies: li. Simple Harmonic Motions. Communications on Pure and Applied Mathematics, 3: 45-1 01, 1950.

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197-202, 1979

Mohn, B & Marion A.: Etude au dexieine ordre du comportement des corps floattanfs en houle reguliere. 1FF Report 33031 1985

Newman, J.N.: Algorithms for the free surface Green function. Journal of Engineering Mathematics, 19: 57-67, 1985.

Papanikolaou, A.: On integral-equation-methods for the evaluation of motions and loads of arbitrary

bodies in waves. Ingenieur-Archiv 55 17-29, 1985.

13, Papanikolaou, A. & Zaraphonitis, G.: On an improved method for the evaluation of second-order

motions and loads on 3D floating bodies in waves. Schiffstechnik, Nov. 1987

Wehausen, J.V. & Laitone, E.: Surface Waves. Handbuch der Physik. Springer Verlag, Berlin,

447-778,1960

Zaraphonitis, G. & Papanikolaou,

A.: On

the evaluation of the second-order free-surface inhomogeneity for 3D s7lip motion problems. Third International Workshop on Water Waves and

Floating Bodies, Woods Hole, Massachusetts, 1988

Zaraphonitis, G.: Second-order theory of motions and loads for three-dimensional bodies of general

shape in waves. PhD Thesis, NTU Athens, 1990.

Table 1: Complex second-order sum and difference frequency horizontal force and pitch moment acting on a bottom-mounted cylinder by bichromatic waves of unit amplitude

Bottom-mounted Cylinder, R=H=1COm, a=1m

Difference Freauency Problem Sum Frequency Problem

av1 av2 F1 F5 F1 F5

1.0 1.0 -.922e3, .)00e0 -923es, .000cO -.275c3, .900e3 347eS, .109e6 1.0 1.2 -.835e3,-.522e3 -874eS, -.177c5 -.476e3, .648e3 303eS, 881eS 1.0 1.4 -.720e3,-.920e3 -.807e5,-.298e5 -.684e3, .446e3 246eS, 690eS 1.0 1.6 -.574e3,-.122e4 -.717e5,-.382c5 -.863e3, .290e3 183eS, 526eS 1.0 1.8 -.412e3,-.144e4 -.608e5,-.438e5 -.962e3, .163e3 140eS, .376e5 1.0 2.0 -.244e3,-.156e4 -.483e5,-.461e5 -.967e3, .921e2 119eS, 252eS 1.2 1.2 -.826e3, .000eO -.870e5, .000eO -.675e3, .430e3 239eS, .683e5 1.2 1.4 -.758e3,-.425e3 -.835e5,-.130e5 -.8le3, .280e3 .173e5, 512eS 1.2 1.6 -.667e3,-.766e3 -.780e5,-.226e5 -.101e4, .173e3 .113e5, .366e5 1.2 1.8 -.549e3,-.103c4 -.701e5,-.295e5 -.109e4, .902e2 .686e4, .233e5 1.2 2.0 -.412e3,-.123e4 -.601e5,-.342e5 -.104e4, .785e2 .723e4,.144e5 1.4 1.4 -.769e3, .000eO -846eS, .00000 -.101e4, .198e3 .113e5,.3&4e5 1.4 1.6 -.719e3,-.369e3 -.820e5,-.168e4 -.113e4, .182e3 .509e4, .268e5 1.4 1.8 -.636e3,-.668e3 -.767e5,-.183e5 -.117e4, .175e3 .174e4, .169e5 1.4 2.0 -.535e3,-.911e3 -.692e5,-.239e5 -.111e4, .208e3 .144e3,.975e4 1.6 1.6 -.740e3, .000eO -.838e5, .000eO -.121e4, .232e3 -.199e3, .183e5 1.6 1.8 -.695e3,-.333e3 -.812e5,-.895e4 -.122e4, .307e3 -.332e4, .116e5 1.6 2.0 -.614e3,-.602e3 -.755e5,-.151e5 -.104e4, .538e3 .369e4,.183e5 1.8 1.8 -.717e3, .000eO -.827e5, .000eO -.113e4, .623e3 .719e3, .215e5

1.8 2.0 -.667e3,-.305e4 -.794e5,-.746e4 -.104e4, .817e3 -.468e3, .219e5 2.0 2.0 -.684e3, .000eO -.802e5, .000eO -.912e3, .111e4 .196e4, .2e5

(13)

Fig. I: Geometry of Boundary-Value-Problem

Re{}

pgR0a 0.0 2.0 4.0 6.0 a.o .0

Fig. 3b: Second-order horizontal wave force amplitude of bottom-mounted

cylinder, Imaginary Part

s

Fext

Fig. 2: Definition of

Free-Surface Domains present method 4oiin Part (t'') + - - - Totol Force pre.ient method 4oi Part (N) + - - - Total Force -+ + -- I

t

-.+--.-, . +\ A

.,.--.-r

I A A : I = .

-:

_:

ii

it

I i 1,111

ii

ti

III.?

0.0 1.0 2.0 3.0

Fig. 3a: _{Second-order horizontal wave force amplitude of bottom-mounted}

cylinder, Real Part

to 2.0 3.0 Im

{F}2.0

pgR0a 0.0 2.0 4.0 6.0

(14)

0.0 Re { _} pgLa -1.0 -2.0 -3.0 2.0 1.0 0.0 100.0

F(R)

90.0 80.0 70.0 60.0 50.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0 500.0

Fig. 4c: _{Second-order difference frequency horizontal wave force}

amplitude on a floating barge in bichromatic waves as a

function of integration radius Rcj, L=B=90m, D=40m

Part (IV)

- - -

Totol Force Port (IV)

- - -

Total Force -- -S -5_

--S-I

/

--. _'5 \

:::','::

':

1.0 1 .5 2.0 2.5

Fig. 4a: Second-order horizontal wave force amplitude on a floating barge, L=B=90m, D=40m, Real Part

LO 1.5 2.0 2.5

Fig. 4b: _{Second-order horizontal wave force amplitude on a floating} barge, L=B=90m, D=40m, Imaginary Part

Im {F}

pgLa

(15)

(2) R0E,3 1-O 0.5 0.0 3.0 2.0 1.0 0.0 0.5 0.0 kR0 0.75

Fig. 5b: First- and Second-order heave motion amplitudes of a floating cylinder in monochromatic waves

present method Mohn

o 2 ort motion

- - - I

ord. motion

Fig. 5c: First- and Second-order pitch motion amplitudes of a floating cylinder in monochromatic waves

pre3ent method Mohn o 2 ort motion s - - - i ort motion

:(1\

_a

j-

s

,-. - (1) s

Jo//2

- (1) - R0,5

'

'L

0.00 0.25 0.50 0.75

Fig. 5a: First- and Second-order surge motion amplitudes of a floating cylinder in monochromatic waves

0.00 0.25 0.50 0.75 present method Mohn e 2 ortI, motion

- - -

ortI, motion 0.00 0.25 0.50 1.5 2 (2) 2 1.0