A singularity distribution method for free surface flow problems with an oscillating body

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r 1f TECHNISCHE IJNIVERSffEIT Laboratorium voor Scheepshydromechafllca Archief Mekelweg 2, 2628 CD Deift

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A SINGULARITY-DISTRIBUTION METHOD FOR FREE-SURFACE FLOW PROBLEMS WITH AN OSCILLATING BODY

by

Ronald Wai-Chun Yeung

Supported by

The Office of Naval Research

Report No. NA 73-6

Contract N00014-69-A-0200-1023 NR 062 181

August 1973

Technkcche Hoaeschool, De{t DCCUMENTA TIE

DATUM: 18

COLLEGE OF ENGINEERING

by

Ronald Wai-Chun Yeung

Supported by the Office of Naval Research

under

Contract No. N00014-69-A-0200-1023 NR 062 181

Reproduction in whole or in part is permitted for any purpose of the United States Government.

Approved for public release; distribution unlimited.

College of Engineering University of California Berkeley August 1973

Tehn-

_--'o,

DOCU:, -DA TU M t A SINGULARITY-DISTRIBUTION METHOD

FOR FREE-SURFACE FLOW PROBLEMS WITH AN OSCILLATING BODY

TECHN!SCHE _UWVERSITEIT

Laboratorium voor

$cheepshydromochazca

Archief

MekIweg 2, 228CD De!ft

Boundary-value problems associated with the forced oscillation of a body of general shape in the free sur-face of an inviscid. fluid are considered. The originally external boundary-value problems are first converted to internal ones by applying the radiation condition at a finite distance from the source of disturbances. Next,

application of Green's theorem, using the source function for an unbounded fluid, reduces the problem to the solu-tion of an integral equasolu-tion with the unknown funcsolu-tion being the velocity potential along the entire boundary

of the fluid. No restrictions on the body shape nor the bottom geometry are necessary. A modified method to

deal with the case of an infinitely deep fluid is also presented.

The method is first applied to two-dimensional problems. Hydrodynarnic force coefficients for a cir-cular cylinder oscillating in a fluid of finite depth and for a bulbous section in a fluid of infinite depth are computed. Agreement with results obtained by others

is very good.

The procedure is then extended to three-dimensional bodies oscillating in infinitely deep fluid. The computer program developed can handle any body with two planes of symmetry and all six degrees of freedom. Test

computa-tions were carried out for a sphere and two ellipsoids. The very good agreement with computations performed by others validates again the proposed method of solution.

The method studied herein can also be applied to other first-order time-harmonic free-surface flow

Page ABSTRACT TABLE OF CONTENTS ii LIST OF FIGURES iv LIST OF TABLES vi I. INTRODUCTION 1

Survey of Past Work 2

Present Approach 6

II. MATHEMATICAL FORMULATION 9

1. Governing Equations 10

2. Linearized Equations 12

3. Steady-State, Time-Harmonic Problems 14

4. Pressure, Forces, Moments, and

Waveheights 18

III. SOLUTION TO THE TWO-DIMENSIONAL PROBLEM 25

1. Application of Green's Theorem 26

2. Discretization of the Integral Equation 31

3. Problems with Symmetric Geometry and

Motion 36

4. The Infinite-Depth Case 38

5. Test Cases and Results 40

IV. SOLUTION TO THE THREE-DIMENSIONAL PROBLEM 58

1. Local Disturbance of the

Three-Dimensional Green Function 60

2. Discretization of the Integral Equation 63

V. REMARKS AND CONCLUSIONS 84

ACKNOWLEDGEMENTS 90

REFERENCES 91

APPENDIX A. Radiation Integrals for

Two-Dimensional Problems 95

APPENDIX B. The Cauchy Principal-Value Integral of the Two-Dimensional Pulsating

Source 98

APPENDIX C. The Potential and Normal-Derivative Integrals of a Triangular Patch

Element 100

APPENDIX D. The Potential and Normal-Derivative Integrals of the Radiation Element 109

LIST OF FIGUPES

Fig. No. Page

i Coordinate Systems 9

2 Cylinder in Oscillation 25

3 Subdivision of Contour ,S 31

4 Matrix for

_9:Jr

5 Potentials on Body Surface with Various

Radiation Boundaries for Circular Cylinder

in Heave, K=0.9 41

6 Potentials on Body Surface with Various Radiation Boundaries for Circular Cylinder

in Heave, K=1.5 42

7 Comparison of Potentials on Body Surface

as Obtained by Three Methods, K=0.9 44

8 Comparison of Potentials on Free Surface

as Obtained by Three Methods, K0.9 45

9 Added-Mass and Damping Coefficients for

Circular Cylinder in Heave 47

10 Pressure Amplitude on Body for Circular

Cylinder in Heave 48

11 Phase Angle of Pressure on Body for

Circular Cylinder in Heave 49

12 Added-Mass and Damping Coefficients for

Circular Cylinder in Sway 50

13 Pressure Amplitude on Body for Circular

Cylinder in Sway 51

14 Phase Angle of Pressure on Body for

Cir-cular Cylinder in Sway 52

15 The Decay Factor (v,,O) 62

Fig. No. Page

17 Subdivision of the Surfaces So,

65

18 Quadrant Definition 67

19 Element and its Images 69

20 Comparison of Potentials on Body Surface

For a Sphere in Heave Motion, K0.4 76

21 Potentials on the Free Surface for a

Sphere in Heave, KO.4 78

22 Approximation to a Nonlinearly Varying

LIST OF TABLES

Table No. _Page

i Added-Mass and Damping Coefficients for a Bulbous Section in Heave, by

Three Methods 57

2 _{Summary of Results of all Test Cases 81}

3 Added-Mass and Damping Coefficients of a 1:1/8:1/8 Ellipsoid at Three

The problem of the forced harmonic oscillation of a rigid body in an ideal fluid with a free surface is of fundamental interest to researchers in the field of ship

motions. The primary products of such a problem are the

so-called added-mass and damping coefficients. The

added-mass coefficient is associated with the component of fluid reaction in phase with the acceleration, whereas the damping coefficient is associated with the component in phase with the velocity. In the realm of linearized theory, the solution of the forced-oscillation problem also provides sufficient information for predicting the diffraction forces on the body, if the body is held fixed in an incoming-wave system [see Newman (1962)J.

The subject matter has a long history of development. Most of the work revolved around two methods of solution, which will be reviewed briefly below. In this work, a

new approach, based on the application of Green's theorem, to the solution of the first-order problem with no forward speed is considered. The method has enough versatility that, practically, no restrictions have to be imposed on the geometry of the body and of the bottom of the fluid.

1.1 Survey of Past Work

Ursell's (l949a) work on a semi-submerged circular

cylinder heaving in a free surface can, perhaps, be considered as the beginning of the modern history of theoretical work on forced-oscillation problems. _The velocity potential was constructed by an infinite series of non-orthogonal polynomials (the multipoles) and a

suitable wave source at the origin, on the physical ground that the forced oscillation of the body produces a standing wave in its vicinity and a propagating wave at a large distance from the body. These singularities satisfied Laplace's equation and the boundary condition on the free surface. Their strengths were determined by the boundary condition on the body. This resulted in an infinite system of linear equations to be solved. The procedure was later commonly referred to as the method of multipoles. With the additional assumption that the waves were long, Ursell (1949b) further developed the method to give results for rolling cylinders with more

ship-like sections. A rather complicated isogonal transformation was used in this instance. Using the technique of conformal mapping,

+ -f- , real,

where is the complex variable of the physical plane,

and that of the mapped plane, Tasai (1959) and later Porter (1960) extended the multipole-expansion method to

a wide variety of symmetric sections. Shapes obtained by this transformation have the property that they

inter-sect the water-surface perpendicularly, a case for which John (1950) has proved the existence of a solution. It

may be added that all work discussed so far is for deep water, with the exception of Porters (1960) , in which case the theory corresponding to finite water depth was also presented. C. H. Kim (1969) later extended this

work of finite water depth to a set of Lewis forms, a

two-parameter transformation, and presented results of added-mass and damping coefficients for all three modes of motion.

An alternate approach to the solution of the problem is the method of integral equations. This amounts to the application of Green's third identity to two potential functions,

(P)

, the source function which contains

a singularity (or l/j. in the case of three

dimensions) , )?, being the distance between points P and

G.,

and , the unknown velocity potential. The

Green function is so constructed that the free-surface, bottom, and radiation conditions are all satisfied.

Accordingly, only the integral over the body contour, £d,, remains, and for both P and Q. on , one then gets

the following equation for (f

'ìr(p)

+

where f(G..) is the normal velocity on the body, caused

by the prescribed motion. This is a Fredhoim integral

equation of the second kind with a singular and fairly

complicated kernel. The method had been invaluable for

obtaining qualitative answers, but unattractive for generating solutions of the general kind before the advent of the computer. Nevertheless, Ursell (1953) had made an ingenious application of the method in

obtaining a high-frequency solution of a circular cylin-der in heave. With the additional assumption that the

cylinder was "flat", the shallow-draft approximation, MacCamy (1961, 1964) was apparently the first person to apply the method successfully. The integral equation was considerably simpler in this case. Meanwhile, Smith

and Hess (1958, and in a series of reports that followed) had demonstrated the successful application of the integr equation method, consisting of distributing the appropria sources on the body surface, in various types of flow and accoustic problems. Frank's (1967a) work appeared to be

in line with this trend, and restrictions on body geometr

were finally completely removed. It should be noted in

passing that Frank had been able to produce solutions for a number of body shapes to which John's (1950) existence proof does not apply. The above summary of works on the two-dimensional problem is by no means exhaustive. A recent review by Wehausen (1971) is more

complete.

al-te

The scope of the three-dimensional problem had been

quite limited. Here we do not have a mapping technique available, but rather, rely on the use of an appropriate curvilinear-coordinate system. The first piece of work was contributed by Havelock (1955), who applied Ursell's

method to the problem of a semi-submerged sphere heaving

in infinitely deep water. Earlier, MacCamy (1954) also

considered this problem, including aspects of free

motion, but no calculations were presented. Havelock's

work was extended by S. Wang (1966) to water of finite

depth. The integral-equation method was used by MacCamy

(1961) and W. D. Kim (1963a,b) for disks, circular and elliptic, undergoing forced motion on the free surface. Later, Kim (1965) , using ellipsoidal polar coordinates

and the integral-equation method, calculated the added-mass and damping coefficients of several spheroids and ellipsoids for surge, heave, and pitch motion. These

are, by far, the most complete results for the three-dimensional problem. Up to now, no results for bodies of arbitrary shape are available. Here it is worthwhile to note that Hess and

_Wilcox

(1969) had attempted the three-dimensional problem of an arbitrary body, and had gone only as far as calculating the complicated Green function that occurs as the kernel of the integral

equa-tion. It is expected that the scheme will be successfully

1.2 Present Approach

Through the history of the subject, one sees a

trend towards obtaining solutions for bodies of arbitrary

shape. In this regard, the integral-equation method is

superior to the multipole-expansion method, for it is known that a rather large number of terms is necessary

(for two-dimensions) for mapping a bulbous shape into

a circle. On the other hand, progress in the

integral-equation method was hampered by the complicated _Green

function. _{Besides, a Green function will not be}

available if the bottom of the fluid is not flat.

In view of the difficulties mentioned above, it is of interest to approach the problem from a different

angle. _{First, one should recall that the problem being}

considered will have a unique solution only if a condition at infinity (the radiation condition) is imposed.

Recently, Bai (1972), using a finite-element approach, has shown that this condition could be applied with

good approximation at a finite distance from the body. The approach here is to apply Green's theorem to the

unknown potential,

¶, and to the source function for an unbounded fluid over a fluid domain bounded by the body, the free surface, the bottom and a control surface over which the radiation condition will be applied. For the problem at hand, the normal derivative of on the

boundary is either known or expressible in terms of One gets, as a result, an integral equation for over the "entire" fluid boundary. It is evident here that

we have traded away the rather complicated Green function

in return for a larger boundary. It is not obvious that the integral equation in this case has a unique solution;

however, it is found that it does give the correct

solution for special cases that have been considered

earlier.

In this work, we focus our attention on examining the feasibility, the advantages, and the disadvantages of the method of solution mentioned briefly above. We

are thus interested more in comparing our computed results

with available existing solutions than in generating

solutions for a wide class of body shapes. The computer program developed for two dimensions can handle any

two-dimensional body, symmetric about a vertical axis

or not. The program developed for three dimensions can

handle any hull form with symmetry about a centerplane and a midship section. The program could be extended to hulls without fore-and-aft symmetry, but at a considerable

increase in computer

time. We note that the "John phenomenon" encountered by Frank (1967a) does not occur in this method.

In what follows, a fairly standard mathematical formulation of the problem is given in Chapter II.

Chapter III considers the solution of the two-dimensional problem. This serves as a test of the idea behind the

method. Solutions for a circular and a bulbous section

with the three-dimensional problem. _{Computations were} carried out using the program developed for three

different body shapes, a sphere and two ellipsoids. Agreement with earlier computations is generally very

good. We emphasize that these forms _{were chosen only}

because computations based on a different procedure were available for comparison.

II. MATHEMATICAL FORMULATION

We assume that a body of some general shape is

oscillating near the free surface of an inviscid fluid.

Referring to Figure 1, we have the coordinate system

O mounted on the body, which oscillates with

amplitude about the equilibrium point O of the

inertial system

Op6.

Here, denotes a smallness

parameter. The two systems coincide when the body is at

rest, and the undisturbed free surface corresponds to the plane = O. Rotational oscillations, denoted by

oftL) . and

'L)

about the center of rotation

) are also permissible.

o

Bottom

Oscillating body

Figure 1. Coordinate Systems

Center of

or

Yt

7for

-(«}t)Y

+

-

=.

(2-3)

Let the surface of the body be described in the

OU

system by

F(i,t)

= O. If the translatory and

angular oscillations are both small, the relation between

the two systems is as follows:

=

+ ' - + /

=

+ - + /

(2-l)

=

-

-11.1 Governing Equations

We will now assume that the motion of the fluid is

irrotational and that the effect of surface tension can be neglected. Hence there exists a velocity potential

, such that the velocity field

is given by V . From the continuity equation, must satisfy Laplace's equation

=

(2-2)

Boundary Condition on the Free Surface. Let

= Y?4,)t)

be the equation of the free surface. The kinematic boundary condition states that the normal velo-city of the fluid must be the same as that of the free surface. Accordingly, we have

pressure on the free surface must be constant, can be

obtained from Bernoulli's equation:

-+ +

= o.

(2-4)

Explicit dependence of these two equations on

Y

may be eliminated, but the argument of will still contain "1'

Boundary Condition on the Body. The kinematic boundary

condition on the body is

c)=o

(vr!

for

or

=1(3). t1

±

(2-5)

where the dependence of r on

t

comes from the

transfor-mation relating the system to the system. We

see that (2-3) to (2-5) are boundary conditions applied on surfaces that change with time. The body boundary

condition is linear since the motion is prescribed; that of the free surface is nonlinear, however.

Other Conditions. If a bottom exists and is given

by the equation ₌ , we have the kinematic

boundary condition

?r

=0

(2-6)

Alternatively,

Liwt

0,

(2-7)

Jo

In order to have a well-posed problem and a unique

solution, a condition at infinity must be imposed (John, 1950, P. 74). The radiation condition states that at a

large distance from the body, the waves must be propagating

outwards. We will return to this condition later.

11.2 Linearized Equations

The problem as formulated is nonlinear and the

domain of definition in which Laplace's equation is

satisfied changes with time. This is rather intractable.

We will now proceed to linearize the problem, and, as a

result, all boundary conditions will be linear and satis-fied on a surface fixed in space at all times. We assume the potential can be expanded in a perturbation series

in

2.) 1

= E

(,5,l)

+

...

(2-S)

A similar expansion will be assumed for the free surface

Y('x,t)

z

=

eY -±)

e _{I +}

-substituting into (2-3) and (2-4) and expanding

in a Taylor series about 0, we get

(1) (s)

é (-i, 0,3 )

- E

Y

C-x,

t)

=

OC6z)

ti)

t

i- 0(62.)

The first-order condition on the free surface is then

(2-9)

--

+ (9-)

(3-ceJ

F,)[-e1(±

+

C()Ei]

(,)[e,(-) + (ç)ß -(fr)J +

=

n,k1&)

+ fl

+ so

c)J

cç(.-L)

n(--;ç)] ,&)

where = O is the equation of the surface in the

Oj-

system when the body is at rest. The left-hand side

of (2-5) , to the first order, is

= EF%)) +

+ .4.

Equating these, and denoting the submerged portion of the

surface, = O, by 1S,, we obtain

(2-lla)

where

(nrì»i)

are components of the unit normal on

pointing outwards from the fluid. It is interesting to note that the procedure results in a boundary condition that is satisfied on the mean (or rest) position of the

body and the right-hand side of (2-lla) consists of _products

= o.

(2-lob)

For the condition on the body, we expand in

terms of using (2-1). The right-hand side of

of functions that depend on space and ones that depend on time.

Henceforth, we will discard the superscripts of

Y

, etc., and consider only the first-order problem. It will also be convenient to write

(-xck), )

=

(ott /

")

(c41o5,oÇ)

(2- lib)

where O

, = 1 , 2,. . , 6, correspond to sway, heave, surge,

pitch, yaw, and roll motions, respectively.

11.3 Steady-State, Time-Harmonic Problems

Neglecting the transient response, we will now seek

only a solution which depends harmonically on time. Let

_.O-:1

cX,(-L)=

e[amJ= LCPe.

_J,

t.=1

(2-l2a)

where C = C,,11 + _{j C2,} . being the imaginary number

associated with time, and the angular frequency of

oscillation. c2() is a complex-valued function containing

information on both amplitude and phase of the motion.

Next, in view of the nature of the boundary condition

(2-lia), we will employ the following decomposition of

, one used by Lamb (1932), following Kirchhof f:

à.()J

with (It) (nl) (nl)

P

-

+

t

(ni) Here, 9)

is a function depending on space only. The superscript n. denotes the mode of motion involved. Substituting this definition of into (2-l) , (2-lOb)

(2-lia) and (2-6) , we obtain the following set of

boundary-value problems for the space function

f(t)(711,)

z

7T

(»1)= o

(1;Lj,)outside ofSo (2-13a) a) _f4) (i,o,)-2fl

= o,

(2-13b)

where Y is the unit normal with components

(t1,r1,Y),

and ) . Evidently the decomposition in (2-12b)

allows us to precipitate the time-dependent terms of the body condition out of the problem. It is of interest to note that the boundary conditions for are all homo-geneous, the only non-homogeneous boundary condition for

is that associated with the body. would have U)

=

(2-13c)

?fl0

- o

(2-13d)

with S being the bottom surface: . In writing

(2-13c), we have used the notation

(n,

ri

) = (ri, r, rh.)

an indeterminant solution if a radiation condition at

infinity were not imposed.

We will now derive a radiation condition for this

purpose. Let _{R be the horizontal distance from the origin,}

i.e.,

_Ç=

, and & be the polar angle in the

('n, ) -plane. Let the bottom of the fluid

L =

be a constant beyond some minimal value of _R Then a set of elementary solutions satisfying Laplace's

equation and the free-surface condition can be obtained by the method of separation of variables (see Wehausen,

1960, pp. 472-475). With little difficulty, it may be

shown that only the following combinations in the set will give a solution corresponding to waves propagating

out-wards to infinity*:

oC Co't

i')

0R)Js(i&+,

_n=integer,

r

1)

=

cok

,l0(-f)

(i,i)J cn#),

where ZJ , are Bessel functions of the first and second

(i)

kinds (Watson's notation) , and is a Hankel function.

a separation constant, is the root of the following

equation:

','tOJ Ot

=

(2-17)

Physically, it0 is a wave numnber.For the limit case that

&

= 00 , 1t0

=

= 2)

. The function (ii0R) has an

* Alternatively, _{see Thorne,1953, pp. 711-12 for} derivation of the infinite-depth case.

asymptotic expansion of the following form for large

values of R (Watson, 1966, p. 212)

k0R)

_{'iriR' 2: 4 L}

+ (2-18)

Using expression (2-18) , one can see that

_L-t1

=

R E

C(-L,91)e

J

(2-19)

cask

'i

pÇ-aJ iLdir)].f(o)

yrPI

I

does represent an outgoing wave system. A few algebraic steps also yield the fact that satisfies the

following differential operator: (fl

+

(_i__ ii0)L

which implies that

(-I + ₊

'ZR..

(2-20)

Equation (2-20) is the radiation condition we shall use.

Bai (1972) has recently been successful in applying the condition at a finite value of _R. for a few axi-sylnrnetric

cases. This, of course, was based on the assumption that

the 'local disturbance', which is the difference between the total potential and the propagating-wave potential, dies out fast enough to be 'negligible' at the place where

more classical form

b)lt 4(a7

_(2-21)

given, for instance, by John (1950, _P. 54). The fact that the waves decay at a rate of can also be derived

from the standpoint of energy conservation; viz, wave energy

is proportional to the square of the amplitude, and the area over which energy

flux occurs

is proportional to R

[see Equation (2-36)].

Finally, it should be remarked that condition (2-20)

must be included when solutions of the boundary-value

problems (2-13a) - (2-13d) are sought.

11.4 Pressure, Forces, Moments, and Wave Heights

The items of primary interest in solving the problem at hand are the hydrodynamic forces and moments acting on

the body. The pressure

f

can be obtained directly from

Bernoulli 's equation

--çt

-i-=

_{_t -r}

[ 9(-)

(2-22)

in which only the first-order terms have been kept, and is of order . The forces and moments (about O)

acting on the body with respect to the system, are

-ft)=i;(

rid

/ -t)

-Me=JJ

(2-23) (2-24)

Here SC- is the actual wetted surface at time

t, dS

the

area of an infinitesimal surface element. These integrals can be linearized by expanding the integral in a Taylor's series about = O (using, of course, Leibniz's rule since both range of integration and integrand depend on ).

The net result is that S(-E.) is replaced by $, . Some

straightforward calculations then yield, for the static portion of (2-22),

=

V+)

+

{VAwr

k).) +'í+j)}

-

Vci+ J

(2-25)

hisTA.. {_v[8+(8+)]

+ (AwFF), + Q

(AwÇi +

(2-26)

where (6,,)denote the coordinates of the center of

buoyancy, ('X.,O,

_3F)

the center of flotation, V the

dis-placement, the waterplane area, and _1L, 'r, ix the

longitudinal, transverse and product moment of inertia of the waterplane about the origin O, respectively.

The hydrodynaniic forces, which are our primary con-cern, come from

the -rt term in (2-22).

Substituting

(2-12b) into the linearized form of

(2-2

3) and (2-24), we obtain [using the notation introduced in (2-14)}

= _t

nd=

_{j E}

j]) -ç2I

L

2jj ¿K

so o i =1,2,3 (2-27) Ci:)

'L

(o 4 f

+cJc

5

....jo.-[Ç

Is,, j=(t23. _(2-28)

Let us introduce the notation

_(M1,M2,

_M3) and define

+

=

?if(ftJd,

t1,a,.,é;

_jt2.e.

₍₂₂₉₎

then the force (or moment) component in the i-th

direction can be written as

= ht

-

(2-30)

where repeated indices are summed from 1 to 6. _{The liii}

are called 'added masses', 'moment of added mass', or 'moments of inertia of added masses', depending on their dimensions, and each represents the fluid reaction in _the j-th direction due to an acceleration of the body in the

i-th direction. The are called 'damping coefficients', and correspond to fluid reactions in phase with velocities. A simple proof using Green's Theorem shows that

t'i=Pì'

í=

These coefficients are functions of the geometry of the

fluid boundaries and the frequency of oscillation. Hence solutions of the boundary-value problems defined by Equa-tions (2-13a) - (2-l3d) and (2-20) yield values of these coefficients. Following a nondimensionalization convention used by Kim (1965), we may define the nondimensionalized

Fi

+ 4j

- i for .=i,Zs3; j=1Z13.

+ .

for i1,2,5 j&'5;

yfor

t,2,3

¿

___

for _=(,Z,3

ça5 )

(2-31)

where U, is a typical length scale of the body, for instance, the half-length of the body.

The free-surface elevation

Ycç-.,t)

is given by (2-lOa) and (2-12b)

(m)

Y

complex in ¿.

= -(K,o,,t)

=--

(232)

where we have introduced the dimensional free-surface

M)

space function

Y (X,)

due to mode '»1 . Evidently,

V(m) 6-z (sn)

-Cf) Cpy _(2-33)

with an asymptotic behavior for large R. given by (2-19)

if the bottom approaches a constant depth. The

asymptotic free-surface behavior is intimately related

to the . To see this, we follow a derivation used

by Wehausen (1971) . We write

rift1

J r1

so

The resulting relation, after substituting (2-35) into (2-34) and

integrating the subsequent

_{expression, is;} with the horizontal bar denoting the complex conjugate

of the quantity, and apply Green's theorem _{to a fluid}

boundary defined by o, the free surface, ,

the bottom and

_SR

_{a vertical cylinder center at the} origin with radius R. , enclosing the body. Upon using the appropriate boundary conditions on these surfaces, including (2-20), we obtain:

Aft

iJf{()

= yn0 J

_c.?E

1$

(2-34)

Next we write _{in terms of}

using (2-33) and (2-19),

o) =

[

0k

'(R,&

r

c0k j

c=

Kv.J

z

o (2-35) (2-36)

Pct0k)

approaches as , and approaches

j, _as where

D(i0&)

4Jt2m0

Zt0,

We give an alternate derivation based on energy

consideration for the analogous relation in two dimensions. For simplicity, we consider only the diagonal terms, i.e.

We assume that the bottom

=

(1') approaches two different depths on the left and on the right,

denoted by

_&L

and respectively. The work done

on the body per period of oscillation ¿W is given

by air

LW

=

=

Or7tjjCk(1

(k)

(/)=_

+L)

_[

eR+R]

(2-37)

This energy expenditure must balance the energy flux per period across two imaginary vertical sections established at distances, sayXg)>Q and , from the body. At

these distances from the body, the potentials are those of progressive waves; for example, for the right side:

i

(2-3 8)

where is the asymptotic wave amplitude on the

right, '}vj, the corresponding wave number, and

''

a phase angle. The energy flux across a surface in

one period of time is given by (see Stoker, 1957, _{p. 45)}

tZ1¼-rr

it p1 (2-39)

we obtain

Zr

lcI

A

fl1

+

'L

tLk)]

Equations (2-36) and (2-40) provide a means of checking the consistency of the potentials at the body and those in the far field.

In this chapter, solution to the problem of an arbitrarily-shaped cylinder oscillating in a fluid of an arbitrary depth will be discussed. An alternate method to deal with the infinite-depth case will also be proposed.

In analogy to equations (2-13), the boundary-value problems for the complex-valued functions

in two dimensions are as follows (see Figure ₂₎

,) = o _{1,2,and é (X.,) outside} (3-la) of

.S,

'v1. o

= o,

4

tL

IXL

Fa)' ,

-k

III.

SOLUTION TO THE TWO-DIMENSIONAL PROBLEM

Figure 2. Cylinder in Oscillation

,

(ifi.)

(';<O) =

(,t)

('?L0)

=

C1'tL9)

where, as indicated in (2-14)

tt1=fl,

Vt=1t,,and

Yt = fl)-V?)((-j. To this set of conditions, we

must also add the two-dimensional version of the radiation condition (2-20)

where and

'L

_{correspond to wave numbers, based}

on first-order theory of progressive waves (Equation 2-17) ,

on the right and the left sides of the body. We recognize that (3-lb) and (3-2) are boundary conditions of the mixed type, while (3-lc) and (3-id) are Neumann conditions. In

all cases, they fall into the category that the normal

derivative of is either known or expressible in

terms of

111.1 Application of Green's Theorem

Let 14)('X) be a scalar function with continuous

first derivatives in a closed regular region and

continuous second-order derivatives in the interior of R

Let be the curve that bounds ; then application

* _{A regular region is bounded by a simple closed regular} curve, one that consists of a finite sum of smooth

arcs (see Kellogg, 1929, p. 97).

» o /

(3-2a)

where

-

(

/

being variables of integration on A . Here we will choose ,S to be the body contour _{., , the free}

surface , the bottom , and two control surfaces

and located at the horizontal distances

2'rrCfkP) =

of Green's second identity yields the following well-known theorem:*

= -

Jjv2()A

+

I

(3-3)

where ,& is the distance between a point ('Xi) in and

a point on the boundary, and cL/i an arc length

element. We assume that the function 92('X-) that we

are seeking is continuous in

_1?t.jS

and that its derivatives exist and are continuous in ¿Z. . (This latter

assump-tion is reasonable since the velocities have no dis-continuity in the flow field, i.e., vortex sheets are

absent). By (3-la) and (3-3), the following equation,

expressing the potential (f)

at any arbitrary field point

_P

in the fluid in terms of its potential and normal

derivative on the boundary, is valid:

en)

9r1

cL)

(3-4)

*

Petrovskii, I. G., Partial Differential Equations, Saunders (1967), p. 251.

and

L respectively. Our radiation conditions at ,Sp

and 5L are applicable only if = constant and L) = constant and only if the motion in these regions is essentially that of progressive waves moving to the right and left, respectively. We observe that (3-4)

et)

and

a double distribution of strength . The potential

contributed by the first distribution is continuous throughout space, and exists and is continuous on

S

itself if 1' is

piecewise contlnuous*.

Hence, a

ft

discontinuous normal-derivative boundary condition does not contradict our original assumption on used in arriving at (3-4) . Next, we make use of the boundary conditions of (f»") on

S

and rewrite (3-4) in terms of

(y (qn)

only. We then let the observation point _F' approach

(,t)

,S and obtain the following integral equation for

9'

on

=

f f't

&t)+ j

{h _.))Li]

+

J

)[ (35) 'SRi

J

(s)

-SLY1

tft?tL7LJ

consists of a simple distribution of strength

* _{See Kellogg, op. cit.,}

p.

160, § 5.

t It is of interest to note that if one had started

with

_{= Jo-}

,

'(')

being an unknown

distri-bution on the boundary, one would find that the

resulting integral equation for OC) , obtained

using the same boundary conditions, does not yield the correct solution.

Solution of this integral equation yields the value of

Cf along S without further calculations. Note there are basically two kinds of terms that contribute to the

kernel of the equation:

k

and CJ'L. . Note also

that the nonhomogeneous term of the equation, which is real, is contributed by the body boundary condition, and that the imaginary

and real parts of 9interact only

through the radiation boundaries.

In this work we will not be concerned with the

uniqueness of solution of the integral equation (3-5) , but will, rather, concentrate on the verification of the

numerical solution obtained by using (3-5) with those obtained by other methods. In other words, the investi-gation is focused on the feasibility of this approach to solution, granting the solution thus obtained is unique. It should be clear here that we have placed no restrictions whatsoever on the geometry and that we only have to deal with the boundary of the fluid, rather than the entire

fluid domain. Hence, if successful, this approach can be used effectively to tackle problems of very general geometry, subject, of course, to the limitations that the boundary conditions are of similar type, and that the cost of its application is not too overwhelming.

Furthermore, we should add that the formulations (3-l) to (3-2) have been successfully applied to two-dimensional and axi-symxnetric bodies by Bai (1972) using a finite-element scheme. The success of the present approach also

relies on the ability to apply the radiation conditions at a finite distance from the body. Otherwise the function

extends over a boundary of infinite length; numerical solution of would then clearly be impossible. Evidently, the approach given here can be extended to the more

trivial case of a fluid domain with finite extent, for example, a canal or a basin. The radiation condition is then simply replaced by = O and no outgoing waves

occur. For this case, Chang (1972) started with an

unknown distribution, 0'(5), of

₍₎

along the boundary

of the fluid, and had shown rigorously that the resulting integral equation for Y(S) has a unique solution if the

forcing frequency does not coincide with the natural frequencies of the basin.

It is also worthwhile to point out that if we had

chosen, instead of , the more traditional Green

function,

T(3)

±

_(3-6)

where is so constructed as to satisfy both free-surface and bottom (if it is flat) boundary conditions, the

resulting integral equation contains only an integral over .S0 , [see, for instance, Frank (1967a)J. We would, of course, have to work with a more complicated kernel

111.2 Discretization of the Integral Equation

We proceed to describe a method for obtaining an approximate solution to the integral equation (3-5) when the body and bottom geometry are given. The method of discretization will be used. This reduces the integral equation to a set of linear algebraic equations, with the unknowns being the values of the potential,

at a discrete set of control points along the boundary. The contour is first subdivided into a number of segments (see notations in Figure 3). Each segment is

2

3

= number of intervals on body,

N = number of intervals on right free surface,

= number of intervals on right radiation boundary,

etc.

Figure 3. Subdivision of Contour 5

then defined by a straight line joining two pairs of

coordinates

r)

('(

Consider a typical

integral of (3-5) along a sub-boundary of the contour .5, for instance

,5:

with

jYz _with

9:(t;)= _o,

N

=

where is broken up into a number of small

straight-line segments

S.

Inside each segment, , may be

approximated by some distribution function. In the pre-sent case we follow a scheme used by Frank (1967) and later by Potash (1970) ; is assumed to be constant

and represented hY

'j4t1)

*. _{it is}

conceivable that this is reasonably accurate if the divisions are fine. Whence it follows that

V2

[(_)Z

-V]

r(+1, 7+i)

I

()Z]

1=

_J

where ('t, ) are the coordinates of the -th observation

point. A system of linear equations for can then be

obtained, if we choose the observation point _P in (3-5)

to be at the mid-point of each element. Accordingly, the discretized form of the integral equation (3-5) is:

*

Evidently, improvement can be made by assuming a

higher-order distribution function for within . This

will in fact be carried out later in the three-dimen-sional problem.

('nt) -+

[Q

-=N+i where (,jjj+t,j+t)

=

[(-

(-]d

+. C

{. -

j

J2*i

r

LI.

o

- _L

-y _J

T -f

e,

j)

= 1,2,...,

_T'

=

s

-)

-+ I

-(-

?+l

[(-XI)2 ()]

+

4jj

1')

[-+

)j1fl

+

[ç-+ (-7]}

(3-7) -(3-8a)

Q43(j1)7+l)

=

Ji

4 (3-8b)

1(-?i)

x-1i

Here, we recall that the quantity is complex and observe that the majority of the elements of the matrix

for has zero imaginary part. It is worthwhile noting that for the straight-line sub-boundaries, such

as S,

_'L

(or SB), Q4_o if both

and are on the same boundary. In practice, spacing between grid points on the radiation boundary can be bigger as one goes deeper (for example, a esfi1 spacing function

is used) . A fairly high density of points, however, has

to be maintained on the entire free surface, since the

potentials remain oscillatory even when one is far away from the body.

One might like to decouple (3-7) into a larger set of real equations, but no particular advantages will

result. In fact, the present form takes less storage

space and less time to solve for the system. _{In more} compact notation, (3-7) can be written as:

lLJi

i

21

₊

_¿ji K

where is the Kronecker delta,

and

c

K=if.

F.

r'

(3-9) for _{l,...,?J5; ==?J+l,..,)J4P} for for

The matrix (7ç

±

) , in general, is neither

diagonally dominant nor symmetric. Its general nature is sketched in Figure 4, where one can see that elements with non-zero imaginary parts are those in columns

associated with the radiation boundary (henceforth termed 'radiation columns') , and that because of (3-9)

the magnitude of the columns changes when one moves from one sub-boundary to another. The lack of diagonal

2I pJs

= real elements X = complex elements Figure 4. Matrix for 9j11

dominance makes application of matrix solution schemes such as the Gauss-Seidel iteration method* difficult. For some cases, slow convergence can be made to occur if a proper relaxation parameter is used. For the general situation, choice of this parameter is not known before-hand and is difficult to determine. A finite-step direct method, such as Gauss-elimination with partial pivoting, is more reliable and predictable. In practice, some use can be made of the fact that most of the matrix elements have zero imaginary part. This could be accomplished by performing a column interchange to shift all radiation columns to the right-most side of the matrix; real arith-metic in the usual Gaussian reduction process can then be used until the complex columns are encountered.

,JT P4

* _{See, for example, A. Ralston, A First Coursein Numerical} Analysis, McGraw-Hill (1965) , pp. 429-438. XX$.X..

.4.

. s. X . . .

111.3

Problems with Symmetric Geometry and Motion

If the geometry of both body and bottom is symmetric

with respect to.=O, and if )= O, the solution of

the

problem, will then be either an odd _{or even function}

of % _{, depending on the oddness or evenness of the body} boundary condition -fk). Here (i) ) denotes values

of the set of points that satisfy _{= O, the equation}

of the body.

Let be an integer that can take the value i _or

1,

and has the following properties:

-

if

i,t=

1 i . for '»t= ,2,3 I if i

if (._$)'

j. for

45

(3-]O)

/ if ('n-3)#

where = i corresponds to symmetry about the plane

= O (and = 3 _{corresponds to symmetry about the}

plane _{O, if we are dealing with a three-dimensional} problem). An examination of the definition of

-f c')

reveals that

f(-),) =

'wt= 1, 2 and

6.

(3-li)

Hence,

_X,)=

_(1,t)?&,U),

_{441= 1, 2 and 6}

_(3-12)

and it is then only necessary to seek the solution of the right side of the cylinder. The integral equation for

79

, , is then

9(4p)

₊

_(1,k]4

+

js

[(

-

Jo h)

+

(&)(

i

+

Cf)[( Ç1k_

t4?R

h) -i.

6f'(

1.J91(-

4')v

ic)]

dA

+J/)[

_{+ %(n)}

(3-13) R. where

J

_s

indicates integration is performed only on the right side,

n-=

A

apt-with

Yt,

r1 taking values on the right side, and

[()2 (Vh]/2

[v

cpZ].

(314)

The discretized form of this integral equation,

corres-(3-15) fo r ponding to (3-9), is then

T, [

-

+ c

_-

K with S L

/ Q.

+

'(i,t)

_Q9

_for

(Q4-

)+ _- for

(9

-4tR+ l)"-.'vt

_{. 4} L IJs

fA=

+

In these equations, the first point (

= 1)

and the last

point _{J) are on the line of symmetry,} X _{= O, and}

are given by (3-8a), (3-8b), and

=

3

Q =-Q(-i? _1, 7j)

(3-16)

Solution of (3-15) gives the potential on the right side

('o).

That on the left side is then given by (3-12).

111.4 The Infinite-Depth Case

The case of an infinitely deep fluid can be approxi-mated usually by a 'large' depth. _{What is important here} is the depth/wavelength _{ratio, which should be at least}

0.5. _{For long waves, the approach given in 111.2 and}

111.3 becomes quite impractical, since an unusually large number of points will be necessary to _{represent the}

radia-tion boundaries. We offer here another approach to deal with (3-5).

If the fluid is infinitely deep, we expect the disturbance to vanish as A.j1 approaches -oc . Hence, the

third term of (3-5) can be discarded. _{Consider now the}

term associated with 5g. Let us replace

Cf &,')

by

¶

, i.e., let us assume the distribution of the

potential is an exponentially decaying function of the vertical distance from the free surface. Then [for

P =

L-where we have defined the integrals

(i

=

=

Je

('M)

j

e.i[

Lit-.v

h]d4 = C Co)

Can)

o

/

(,1)Z (?)Jd

(3-17) (3-18)

Thus, by this procedure, we have, in essence, replaced a

sequence of unknowns on this radiation boundary by a

single unknown on the free-surface. Note that this

assumed distribution of

9) at is valid to the saine

extent that the radiation condition is valid. This can easily be shown by the method of separation of variables

jsee Wehausen (1960) , pp. 472-4741. Details concerning

the evaluation of the integrals and are discussed in Appendix A.

What is said above for _{applies, of course, to}

ZL

also. We give below the discretized form of the

integral equation for the case of a symmetric body oscillating in an infinitely deep fluid:

where with

-+

U,4't) Q for J=

i).Jpjs

+ for (r1.

+ t-.)+

4

(4,) ( 1

+ for _l;J.= jJ2+1 L

and

_{for 4=J+i}

(<o)

_{is the observation}

point on the radiation boundary. In this notation is defined to be

111.5 Test Cases and Results

The methods discussed in the previous sections were applied to two body shapes, _{a circular section in water} of finite depth and a bulbous section in _{water of infinite}

depth.

Circular Cylinder in Heave and Sway

The problem of a circular cylinder in heave was first solved by Ursell (1949a) . It has become a common

test problem for later methods. Here the body is symmetric and formulations given in 111.3 are applicable. Numerical solutions for the heaving motion were first examined for two frequencies,

K L

_{= 0.9, and 1.5, with a being} the radius of the circle, and _/0., = 5.0. Figures 5 and

(2) Z)

6 present polar plots of the potentials 9j and on

the body for a number of values of 2, the distance at

POLAR PLOT OF

POTENTIALS ON BODY SURFACE WITH VARIOUS RADIATION BOUNDARIES

FOR CIRCULAR CYLINDER IN HEAVE

PHI (1)

-.25

0 .25

.50

.75

-.50

.50

cx

-a) /t =(.35

(X. - a) iL 0.70

cx-a),L

1.40 C X. -O-) /I

2.0

-.25

o .25 .50 .75 1.00

Figure 5.

PHI (2)

-.25

O .25

.50

POLAR PLOT OF

POTENTIALS ON BODY SURFACE WITH VARIOUS RADIATION

BOUNDARIES

FOR CIRCULAR CYLINDER IN HEAVE

PH I(I) 0 .25 .50 .75 -.50 .50 -.25 CXO-)/-9 23 (Xt-O)/P 416 (X1-o)/I.= 37 C X,-a) íD .1l.7c 1.00 -.25 O .25 .50 .75 Figure 6 PH 1(2) -.25 0 .25 .50

the dotted circle corresponds to values of that are identically zero; the radial distance from the reference circle represents the value of 92 . The potentials as

shown evidently depend on where ' is applied, but

con-verge to a unique solution as is moved further and further away from the body. Bai (1972, Chapter II) examined the magnitude of the local disturbance for the

finite-depth case in some detail, and suggested that the 'decay factor'

for all V (3-20)

could be used as a measure of the importance of the local disturbance. One may see that, for a given depth, the method proposed here will eventually lose its attractive-ness if

ì

is too high, because in such cases maintaining

a constant (ZR-

ct)/k

ratio would mean a sizeable number

of wavelengths from the body, and it is not wise to use less than 12 points per wavelength to approximate the potential on the free surface.

Figures 7 and 8 compare the potentials obtained by this method with those obtained by using the finite-element method

(-&/a

= 5.0 also) and by using Frank's method ( -k

=

00 ). Comparison with the last method is

pertinent since at this frequency

.&/

= 0.72, 2. being the wavelength, and the fluid could be considered as

infinitely deep. It can be seen that the agreement is very

- .50 -.50 -.25 .25 .50 15 1.00 -1 I I I I .50 ?/a= 6. 9 7 9 6

-FRANK'S METHOD PRESENT METHOD FINITE ELEMENT METHOD

Figure 7.

COMPARISON OF POTENTIALS ON BODY SURFACE AS OBTAINED BY THREE METHODS

(CIRCULAR CYLINDER IN HEAVE, K=O.)

PHI(1) PHr(2) - .25 0 . 25 .50 .75 - .50 - .25 0 .25 .50

COMPARISON OF POTENTIALS ON FREE SURFACE AS OBTAINED BY THREE METHODS

(CIRCULAR CYLINDER TN HEAVE, K=O.9)

1.0

.8

.8 4 2 o 2 Q-4

-.8

-.8

_-1.0

1.0

.8

.6

.4

-2

:°

2 Q-4 6

.8

_-1.0

/eLa 6. 9796 5.o FRANK S METHOD -. . X....

PRESENT METHOD

FINITE ELEMENTI1L

i i E i i I i I I i i i I I I i i o .5

1.0

1.5

DIMENSIONLESS DISTANCE.

Xix.

Figure 8.

Computations for this geometry were next carried out for a range of frequencies. Added-mass and damping

coefficients for the heave and sway motions are shown in Figures 9 and 12. The dimensionless pressure coefficient,

on the body and the corresponding phase angles, are also plotted in Figures 10, 11, 13, and 14. Here,

for easier comparison with other authors, the dimensionless

-

(J.) (J.)

and are defined as follows:

quantities , .',

=

/j/(

n,2) .

/

(í2)

-4 (4)/ (4)1 (i)

i

- Z

_{tait [j7A/}

j

4=1,Z

L4

+''%rJ

_()

-ui)

where , are the dimensionless pressure coefficients

in phase with acceleration and velocity respectively; the former includes also contribution from the static

component:

- ¿) (ii)

4'A

_)(fi

₊

frt)

In these graphs, results obtained by Frank's method

(i=oo)

are indicated by a solid line. Parenthetical numbers for and 6 indicate the segment index used

in previous sections. Whence .(4) represents pressure at the midpoint of the keel segment, (i), the phase angle at the midpoint of the segment just below the free

surface. Agreement iS in general good, for both heave and

sway, for moderate and high frequencies. Discrepencies in the low-frequency end do not, however, imply a failure of the present method; this is because for low frequencies

(3-21)

2.0

1.8

1.6

1.4

1.2

.8

.6

.4

.2

o

ADDED-MASS AND DAMPING COEFFICIENTS FOR CIRCLE

IN HEAVE

-./a= 10. 0

.5

i2

1.0

1.5

DIMENSIONLESS FREQUENCY, K

Figure 9

Ci-L I<IM (i97) FRANK'.$

ET1OD (.$t=OQ) o. . . . 5.0 T PRÑT MTH0P

X.... -°-.

=15.oj

2.0

I

z

W

01

LL u-

Uil

El

u

Uil

c

D

U) (f) W û- U) co

w

-J

z

D

(J)

z

W -1

o

PRESSURE AMPLITUOE UN

0DY FOR CIRCLE IN HEAVE

.5

1.0

1.5

2.0

DIMENSIONLESS FREOUENCY, K

Figure 10.

-180. 0

-170.0

-160.0

N - 1 50 . O ..---1 40. 0 (f)

-13Q. O

-120. 0

w

0110.0

Z -100.0

J.

-90.0

-80.0

-70.0

-60.0

--50.0

--40.0

-30.0

-20.0

-10.0

o o

PHASE ANGLE 0F PRESSURE ON BODY FOR CIRCLE IN HEAVE

I-

Figure 11.

.5

1.0

1.5

2.0

DIMENSIONLESS FREIUENC"?, K

2.0

ADDED-MASS ANO DAMPING COEFFICIENTS

FOR CIRCLE IN

S.JAY FRANK'S tlETHOfl

o....

io- -.O

X.... &/a.

-5.O

Figure 12.

.5

1.0

1.5

2.0

DIMENSIONLESS FREQUENCY, K

H-'

8

z

Lii

(Ji

6 p-4 U- U-

UJj

4

D

(-J Ui1 2 (J) 'J) LU 0 8 (J) Ui -J

z

G

D

(J)

z

4 LU

D .2

PRESSURE AMPLITUDE ON BODY FOR CIRCLE IN SWAY

Figure 13.

.5 1.0 1.5

2.0

DIMENSIONLESS FREQUENC

K

180.

170.0

160.0

1 5 0 . O

'140. O

130.0

120.0

w

0110.0

Z

ioo.o

b-1

PHASE ANGLE OF PRESSURE

ON BODY FOR CIRCLE IN SWAY

6s

()

Figure 14.

-. .

. . FRANK'S MTHDD

Pi'a

-5.0

X....

Rda.. -I').o

-J

(D

z

a

w

(J) ci:

90.0

80.0

70.0

i 50.0

Q-40.0

30.0

20.0

10.0

0 o

.5

1.0

1.5

2.0

DIMENSIONLESS FREQUENCY,

K

the waves are long and the effect of finite depth begins

to be felt. The behavior of the added-mass and damping

coefficient curves in the low-frequency range as a function of the fluid depth,

&

, is quite intricate, in general.

For a detailed exposition, C. H. Kim's (1969) article should be referred to. Here, Kim's results are indicated by dotted lines in the graph of added mass and damping coefficients for heave (Figure 9) . The present calcula-tions are consistent with Kim's.

Bulbous Section in Heave (t=oo)

The bulbous section being considered consists of a

fully submerged circle with a narrow vertical section

mounted on top of it. This shape is of particular interest for two reasons. First, Frank (1967b) has found that for a

'bulgy' shape of this type, there exists a small range of frequency in which the damping for heave motion is

vanishingly small. Second, this shape does not satisfy the convexity requirement that John (1950) has imposed on

the surface of

the body in his existence proof. Therefore it seems possible that different methods of solution may yield different results.

The appropriate equations for the present situation

are given by (3-19) . Estimate of effects of local

dis-turbance based on (3-20) , however, ceases to be useful

as . We take an alternate approach here to obtain

a similar relation for the infinite-depth case. First,

we recognize that the present problem can be solved by distributing pulsating wave-sources on the body. Hence,

examination of the asymptotic behavior of the local dis-turbance of the wave source will provide us with some insight about the choice of The source function of unit strength for the problem (available from Thorne,1953,

p. 710) is: ('x,,t)

={

(Xj2]/2

_±Ç

e.

ok(-VcI1}

_Coo'

Lvt4

2.j

i

-

, (3-2 3)

where () is the location of the source. For a large

distance from the source, compared, say, with the depth

of the body, could be treated as approximately zero; and the term would be vanishingly small. For the Cauchy principal-value integral, Appendix B shows that

-V k

=

-(+)

f

f(1)J

J_°

(3-24)

The first term combines with the term in (3-23) to give the propagating wave potential; the second term can be bounded [see Equation (B-3)]:

+ íosf

_{(t7)J 4}

o

+

()]

LI] +

...,

for all

I

+

Q((y2)

2) [X - I

Hence, in contrast to (3-20), for the infinite-depth case, the local disturbance decays as the inverse of the distance from the body, rather than exponentially. (3-25) is,

, o,

however, deceptive, because it would seem that the

governing parameter for deciding the importance of the local disturbance for all frequencies is

-g/,

'X'2

being the wavelength. What happens here is that for high frequencies,

when the wavelengths are short, maintaining the same

ratio does not necessarily mean the local disturbance is negligible at the ieg term becomes important. If

denotes the largest coordinate and the deepest coordinate of the body, then for a given value of

some simple calculations yield the following approximate

equation for the term:

j-L-J

2

< i.

(3-26)

which suggests that for all 1' , in order to have the

term small, (xi) must be small, i.e., there is a minimum bound on . We see, therefore, at high frequencies,

(3-26) forces one to increase the value of V% , contrary to the result given by (3-25) . Typically, a value of

0.2 will be necessary to yield consistent results. The above illustrates the type of analysis one usually has to perform in order to decide whether a certain choice

of %

is proper. An improper choice of X could usually

be detected from the solution, such as the lack of proper

phase relation between the real and imaginary part of C2 on the free surface in the neighborhood of . For

the problem with a finite-depth approximation _described in the earlier sub-section.

Table 1 presents numerical results of the _application

of the present method for the aforementioned bulbous

section. _{This work was carried out in a joint}

investiga-tion among this author, K. J. Bai, C. M. Lee, and A. Steen (1973). Added-mass and damping coefficients _{were calculated}

by three methods, the present method denoted _{by 'Y'. the} finite-element method by 'B', and Frank's method by _'L'.

The results are consistent and certainly _{support the} conjecture that a unique solution does exist for bulgy sections of this type. _{Note that all three predict the} existence of a radiationless frequency.

No te s

i) B, Y, L represent finite-element method, _present

method, and Frank's method, respectively.

2) _{Parenthetical numbers indicate the number of points}

used to represent the body. _{Quadratic element was} used in finite-element method; the others, _constant potentials assumed within each segment.

d- 2. -?

5.ßa.

Table i

Added-Mass and Damping Coefficients

for a Bulbous Section in Heave, by Three Methods V== 20a3 2

/2

»2a/V

22

22/V

2 ₂₂ K B (13) Y (21) L (28) B (13) Y (21) L(28) .01 .4349 .461 .474 .1172 .138 .147 .02 .4153 .422 .432 .0973 .101 .107 .03 .4045 .408 .417 .0716 .073 .077 .05 .4003 .405 .414 .0347 .035 .038 .07 .4090 .413 .422 .0140 .014 .016 .10 .4242 .428 .438 .0002 .002 .002 .15 .4396 .442 .454 .0017 .001 .001 .20 .4433 .446 .459 .0063 .005 .006 .30 .4403 .444 .457 .0091 .008 .009

IV. SOLUTION TO THE THREE-DIMENSIONAL PROBLEM

We consider in this chapter the solutions to the boundary-value problems defined by Equations (2-13) and

(2-20). _{We shall apply techniques similar to those used}

successfully in the two-dimensional problem. _{We shall} start with a three-dimensional body of arbitrary shape and with an arbitrary bottom, but later, owing to

practical limitations, we shall seek solutions for bodies with two planes of symmetry and oscillating in infinitely deep water.

In principle, extension of the method used in

Chapter III to three dimensions is straightforward; one

simply replaces, for instance in Equation (3-4) , the

J.&ç

k

function by and the line integrals by surface integrals. _{In practice, the complexity involved is an} order of magnitude higher, because storage and computa-tional requiremenets go up at least as the square of the size of the matrix. Some of these points will be

apparent later.

We first rewrite Green's theorem in three dimensions as follows:

4'c')=

_jf

_Ic

pS (4-l)

& _s

where S consists of the body surface , the free surface

SF between the body and a vertical cylinder of radius

_R

*

See, for example, W. D. Macmillan, The Theory of the Potential, McGraw-Hill, N.Y., 1930, p. 288.

, lying between the free surface and the bott9m, and

the bottom surface 5. Applying the appropriate boundary conditions and letting F approach , we obtain the

following integral equation for fcp) on the entire

boundary surface, S

r

Jj

1c.)

}Tj4) L,(&) +

Jft(

-I

d.A so

()

+

JJ (L)

tl)h

(4-2) j h'/7.

r

= I

+

&J .- _o

where _{(>o), in general, takes a value of 21r if f' is on} a planar surface, but otherwise is the solid angle inside the fluid in the neighborhood of _f7 if ' happens to be

a vertex

point*.

Here is the boundary condition

on the body given by (2-13c) and fl'tR the wave number associated with the asymptotic bottom depth.

If the depth is infinite, we expect that as

and that the integral for ,SB can be discarded.

Also, for this case, we may replace _{on the}

radiation boundary _R by an exponentially decaying

dis-tribution suggested in 111.4, i.e.,

0. (4-3)

[f ('ei?

_fi

(Q(1t) ' _i

(L)

+

+ JJP -

ti

JJsr.J

_j]cL

o r211

J dgl

I _ti

+jlê.(

(R,o,e)

Li

o

-=

JJ

çC'Pt))

_I5:'I

which requires the determination of 92 _{on the body and}

the free surface S only.

IV.l _{Local Disturbance of the Three-Dimensional Green}

Function

* R

is treated as a space variable in this section.

(4-4)

We digress temporarily from the solution _{of (4-4)} to examine the behavior of the local disturbance in three dimensions in order to gain some insight into the choice of the value of R . We adopt the same scheme as that used in 111.5. _{The Green function,available from} Wehausen (1960, p. 477), is

o,o1o,t)

=

a

₊

-

_Y0(g)7

J

*

(4-5)

where we have chosen to locate the source _{at the origin.} The Bessel functions, of course, combine to give divergent waves as discussed in 11.4. _{With error at most of}

we can express the potential

_4LCR,)

_{corresponding to} the local disturbance as

L 2(R1+

±

_dt

J _{4 Rz}

2e?'['y,(,.*)

-

IT

=

2{

(R2t2)

-

v? [

_U-06'P)

+

Jo _R2+3

with A-L being the Struve function of order zero*. The

rate of decay is evidently 2/ t, unless substantial cancellation is made possible by the second and third

terms. A measure of the importance of at

relative to the propagating-wave potential could be defined as

(v1R,O) -

LI/2rvJi.

irR

Ji()4 IJ_(s)h/z{4

_"r'C

L

is plotted as a function of vR in Figure 15. It is evident that for vR greater than 13.0, this ratio is less

than one percent.

The neglected integral of (4-5b) should also be

con-sidered for . We define

AR)=

_yív

J

-

)I(1,vR)

_(4-7)

(4-5b)

* _{See Gradshteyn and Ryzhik,}

(1960, p. 322) for the integral from = O to oo

t _{An alternate derivation is available from John (1950,}

THE DECAY FACTOR

In o o

A

-\

o s

io

is

VR

20

25

Figure 15.

with

I(9,vR.)

=

(4-8)

This very interesting integral is shown parametrically in Figure 16. For a given

vR,

iR'(o

, it has an upper bound corresponding to a value of Tvr= -4.0. The

envelope is simply the following equation:

which implies

'i'R

Therefore, we again run into a situation similar to that discussed in Chapter III, section 5; the radiation condi-tion must be applied at sorne minimal distance away from the body, regardless of 2/

IV.2 _{Discretization of the Integral Equation}

Let us subdivide the surface cJSr_ by a grid system as shown in Figure 17, the body by transverse sections

and waterline sections as commonly used by naval architects, the free surface by quasi-radial lines and circumferential

lines. To each grid point we assign an index, t , and

(2r thus denotes the potential _{of the t-th mode at grid} point t . For the sake of notational simplicity, the

superscript * will be discarded in this section; the

implicit dependence of on the mode

tt

is understood.