H. (Hajime) Maruo. # Challenge to better agreement between theoretical computations and measurements in ship hydrodynamics online

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order of 6, the wave amplitude is of the order of 6 Â£ . The velocity of

the orbital motion of wave is 0(6), so that the order of <})-. is 6 Â£. In the

boundary condition on the hull surface for the radiation potential, namely

-r â€” = i w(z -xifj) -^r - U ty -~r - U(z -x\J0 -5â€” r ( -5 â€” ) (136)

3n g 3n' dn' g dn' \9z '

the first term is of the order of 6, while other terms which are related to

1/2

the forward speed are of the order of 6 Â£ .On the other hand, the

1/2

lowest order of the free surface condition is 6 and the next is 6 Â£ . If

we take the lowest order terms only in both boundary conditions, the so-

lution is in the case of zero forward speed. We can discuss the effect of

1/2

the forward speed by taking terms of order 6 Â£ .In this case, the free

surface condition for the radiation potential becomes

*\

9 *R

d \

+

g

+

V

II

^ 2

dz

3t3x

dt

2 2

3cj) 9 cj> 3 <J> 3c}>

+ 2 1)^^-11^-^=0 (137)

9y 9t3y 3z 2 3t

Because of the terms except the first and second ones, we cannot solve the

boundary value problem in two dimensions by ordinary methods. Ogilvie and

29

Tuck employed a successive method by which the solution for the above

52

boundary condition was derived from the solution for zero forward speed.

The solution obtained involves integrals over the plane z = 0. It seems

of some interest that the outer solution for the radiation potential takes

a degenerated form like

l> R ~-â€” m(x)+2ifi(z-i|y |) ^T~J ^'i' 1 < ' I >'! â– 1 ^ :

The first term in parentheses means a simple harmonic plane wave propagat-

ing outwards in the y direction, while the second term is related to the

variation of the source density along the x-axis which means the inter-

ference between different sections. An extensive discussion of this case

is given by Ogilvie and Tuck and will not be reproduced here.

Next we consider the case of high frequency with low forward speed,

-1/2 1/2

that is co = 0(e ), U = 0(e ). This is the case of short waves such

that the wavelength is of the same order as that of the breadth of the ship.

3/2

In this case, the order of magnitude of 1 is Â£ . The free surface

condition becomes

9 2 <J>

1 + R -z-Â± = (139)

at 2 6 3z

1/2 3/2

up to the order of 6 e . The next term is of the order of 6 Â£ , which

includes the effect of the forward speed, but has quadratic forms. There-

fore, the ordinary linearized treatment can be applied to the first order

only, by which the effect of the forward speed cannot be taken into

account. The asymptotic form for the outer solution for the radiation

potential takes the form

4iri icot + v(z-ilyl) , s / - i/rw

d> ss - e ' â– " ' m(x) (140) â–

R g

that means outward-going plane waves. There are no three-dimensional terms

involved and the solution is purely two-dimensional. The strip theory is

53

then valid but without any effect of the forward speed. It should be noted

that the diffraction problem cannot be treated in the same way, because the

assumption of the slow variation along the x-axis is no longer valid. The

diffraction in short waves will be discussed in another section.

Hydrodynamic Forces in Heaving and

Pitching

It is widely known that the hydrodynamic forces and moments acting on

oscillating ships are predicted by strip theory with fairly good accuracy.

However, the reliability of the strip theory is still open to doubt as

discrepancies between computed and measured results are observed occa-

sionally. These discrepancies may be attributed to the effect of the

forward speed and fluid motion in three dimensions. In the preceding

section, we have observed that both the forward speed and the three-

dimensionality can be taken into account within a plausible approximation

1/2

of the perturbation scheme, if the forward speed is of the order of Â£

where Â£ is the beam-to-length ratio.

The hydrodynamic forces are obtained by the integration of pressure

over the hull surface. The fluid pressure in the near field is given by

a . sa(2D) a ^(2D) a .(2D) a *(2D)

3cj> 3<j) 3<j>. 3c() 3<j>-

- - (p-p n ) ^mMd^ + d^ â€¢ t^ + u ir^ *ir- ( 141 )

p r r 1 dx 9y dy dz dz

up to the order of 8e. The hydrostatic pressure is omitted because it is

simply determined by the geometrical relations. Although the third and

fourth terms are omitted in the usual linearized theory for oscillating

ships, they appear in the same order of magnitude as the second term, so

that they have to be retained if one wishes to take account of the effect

of the forward speed. The vertical force is given by the integral

â– If.

F 9 = (P- Pn ) |f dS (142)

2 ^ r 0' 3n

54

Although the quadratic terms in Equation (141) seem to be troublesome in

30

evaluating the above integral, a theorem which has been proven by Tuck

becomes a powerful aid. The theorem is proved in Appendix B. Consider a

velocity vector V of an irrotational motion of an inviscid fluid outside

the hull, which satisfies the boundary condition on the hull surface such

as

V â€¢ n = (143)

where n is a unit vector along the outward normal to the hull surface.

Next we define the vector m by the relation

9V

m = - 3- (144)

Further we define vectors n and m by

n = r x n

\ (145)

5 " - to ( ^ Xn) J

where r is the position vector (x,y,z). Then, the following relations are

valid.

If

[mcjH-n(V'Vcj>)] dS = -

n d> w d s

S L o

(146)

[m (J)+n (V-

VV.) | dS - ! n (j) w d s

L

where w is the z-component of V and L is the still waterline of the hull.

The line integral on the right-hand side is omitted when the body is

55

slender and wall-sided at the water plane. For the vertical force, we take

the z-component of the first equation, while for the pitching moment, the

y^-component of the second equation is taken. In the present case, we put

VMU, U^,U^) (147)

Next we define vector functions ){, |, )( , and ip , which are two-dimensional

harmonic functions in the lower half space outside the ship and satisfy the

boundary conditions on the hull surface, such as

3X W

= m

(148)

ttâ€” = n, U -râ€” = m

9n 3n -

ax* * a$* ,

-7Tâ€” = n , U -râ€” = m

3n - 3n -

Furthermore, these functions are assumed to satisfy the free surface

condition

|i _ v <j> = atz=0 (149)

dz

We have expressed the near-field expression of the radiation potential for

a slender ship by

^N = <j)(2D) + (1+VZ) g l (x) (150)

Then, the vertical force is written in the form

(2D) 3z

p dx i

J C(x)

. T dc

3n'

(2D) 3<J>< 2D) (2D) 3^ 2D) (2D))

C(x) I ' (cont ->

56

+ dx {i(jJ g;L (x)+Ug^(x)} f (l+Vz) |^r dc

J ^C(x)

(151)

where C(x) is the contour of each section along which the integral with

respect to c is taken. On applying the above mentioned theorem to the

second term, we obtain

,(2D) 3z ,

Fâ€ž = p|dx| i w <Jr y -Â»â€” r dc

,< 2D > dc

J J C(x)

J ^C(x)

+ p j {ico gl (x)+Ug^(x)} {-B(x)+VS(x)} dx (152)

The last term is derived by the application of Gauss' theorem in which

B(x) and S(x) are breadth at the waterline and sectional area of each

transverse section, respectively. If we write the z-components of X and iÂ£

by x an( i *P Â» respectively, the two-dimensional part of the radiation

potential is expressed as

(J) (2D) = {iw(z -xij;)-lty} x + U(z -aap) ip (153)

g z g z

Then, the vertical force is written as

3X

r 2 f dx z

= - p I dx {(*) (z -xty)+iwlfy} x TT dc

j C(x)

r r 3x z

+ i p a) U dx (z -xifO t\) tjâ€” r dc

J J c ,_

U dx (z -xlJO

U 2 jdx (z g -x*) f ^

J C(x)

- i p to U | dx (z^-xip) | x "a - r dc

X z

C(x)

- p U | dx (zâ€ž-xi|0 I ^ _-Â£. dc (154)

(cont. )

57

- p {itog 1

(x)+Ug^(x)} {-B(x)+VS(x)} dx

(154)

Because of Green's reciprocal relation, which can apply to x and ip , such

as

f X z 3^ dc = I

J C(x) J C(x)

i 2

' a - i

z dn

dc

the second and third terms are cancelled. Therefore, we obtain

F_ = - p

dx {to (z^-xiJO+iwUip} | X "T~r dc

- 2 J

dx (z -xlji)

C(x)

\p -r-r dc

z dn

C(x)

+ P

-iO){B(x)-.VS(x)} +U 2- (B(x)-vS(x)}

dx

g-^x) dx

(155)

1/2,

The last term is derived by integration by parts. If we assume U = 0(e )

2

and omit the term of 0(6e ), the second term drops out. A similar ex-

pression is obtained for the hydrodynamic moment about the y axis. The

first term indicates the result obtained by the strip theory, and the other

terms give the effect of the forward speed and the three-dimensional motion.

Numerical Results of Radiation Problem

at Zero Forward Speed

Although numerical analysis works when the forward speed and the

effect of three-dimensionality are present, no published result of this

kind is known so far. There is a rather comprehensive result, on the other

hand, for the case of zero forward speed by means of a similar formu-

31

lation. It is obtained simply by letting U = in the original formula

for finite speed with no substantial difference in the method of numerical

58

calculation. The result can well illustrate the effect of three-

dimensionality of the fluid motion which is remarkable at lower frequencies.

In the first place, let us put Â§ = e $ and express the two-

dimensional solution for a heaving cylinder in the form as

i â€” Â°Â°

(2D)

' - - a.

kz cos ky j, . Kz

e â€” J dk-7Tie cos Ky

_

y a 2m

^L/ (2m-l) !

m=l

,2m-l

v 2m-2

+K

â€ž 2m-l 2 2 J . 2m-2 I 2^ 2

dz \ z +y / dz \ z +y

(156)

It is readily shown that this expression satisfies the free surface

condition

_3$

dz

- K $ = atz=0

(157)

where K = V = co /g in the present case. Since the inner expansion of the

above function is

,(2D)

= a n [Â£nKr+Y+Kr cos (l-Â£nKr-Y)+Kr9 sin 6+TTi]

Z

m=l

,2m

2m

Kt

cos 2m6+ -= â€” r- cos (2m-l)6

zm-1

(158)

where we have employed the cylindrical coordinates

z = - r cos b, y = r sin

(159)

and Y is Euler's constant, 0.5772157.

In order to make the inner expansion of the far field potential match the

above, we employ the expression in the near field as

59

,(2D)

- a Q (l+Kz) (y+TTi)

I *Â«'

- -| (1+Kz) I aA( x ') sgn(x-x') Â£n(2K|x-x'|) dx'

"-Z

it

JK(1+Kz) J a (x') [H Q

-I

(K|x-x' |)+Y (K|x-x* |)

+2iJ (K|x-x' |)] dx'

(160)

where x = +Â£ is the x coordinate at each end of the ship. The second term

on the right-hand side is added to conform to the expansion of the two-

dimensional solution. We can rewrite the above expression as

-I

> (2D) + i (1+K; n,\u; ;m(I<! ,''\) wgu(^-y') dx 1 (161)

where

N(u) =-y-Â£n2u+

Hq(u') duÂ« + f Y Q (u') du' - TT i

+ IT i

Jq(u') du'

(162)

It can be shown that

Mm N(K|x-x' |) =

K-*=Â°

60

so that the three-dimensional part vanishes when the frequency becomes

infinite, and the fluid motion becomes purely two-dimensional for which

the strip theory holds exactly.

Now let us consider the boundary value problem. When a ship is in

heaving and pitching oscillations, each transverse section has a vertical

velocity V(x) e , where V(x) is related to the mode of the oscillation.

When the slender body approximation is employed, the boundary condition

satisfied by the velocity potential $ e at the surface of the body is

to 7 " " V(x) 3n"'

(163)

If we introduce the expression for $, given by Equation (161), the boundary

condition can be written in the form

,(2D)

3n'

(x)- -| K I a^(x')N(K|x-x' |)sgn(x-x , )dx'

-I

dz

3n'

(164)

Here we introduce the solution of the two-dimensional problem of a heaving

cylinder for which the boundary condition at the body surface is

,(2D)

3n f

dz

9n'

(165)

With this solution, we put the coefficient of the source term as

a " A o

(166)

Then the coefficient a n for the boundary condition of Equation (164)

satisfies the equation

a Q (x) =

(x)- - K aJ(x')N(K|x-x' | )sgn(x-x' )dx'

-I

A Q (x)

(167)

61

The coefficient A n can be determined by the method well known in the two-

dimensional theory of a heaving cylinder. Then we can determine a n by

solving the above equation. However, we need not solve the equation

exactly. The reason is as follows. If the frequency is very low or to the

contrary very high, the integral on the right-hand side can be neglected.

So we can put

a (x) = V(x) A Q (x) (168)

It may be assumed that the deviation from the above relation at inter-

mediate frequencies is not large. Therefore, we approximate

a Q (x) =

J6

V(x)- \ K {V'Cx'HqCx^+VCx'^Cx')} N(K|x-x* | ) sgn(x-x' )dx'

-I

x A Q (x) (169)

In order to determine other coefficients, we substitute V(x)A(x) for a Q in

the boundary condition and put

W(x) = -| K {V , (x')A (x , )+V(x , )A^(x')} N(K|x-x' |)sgn(x-x')dx' (170)

-I

Then the boundary condition becomes

â„¢(2D) .

fr = [V(x)-W(x)] Â£r (171)

The solution of the two-dimensional problem with this boundary condition

determines other coefficients aâ€ž . We can rewrite

2m

W(x) in the following form for numerical purposes.

determines other coefficients aâ€ž . We can rewrite the function aâ€ž(x) and

2m u

62

[V(x')A (x')-V(x)A (x)] N'(K|x-x' |)dx' (172)

( (x) = V(x) A Q (x) Tl- | KA (x){N(K|Â£+x|)+N(K|il-x|)}l

W(x) = y K V(x) A n (x) {N(K|Jl+x|)+N(K|Â£-x|)}

â™¦1*J

I

[V(x , )A (x')-V(x)A (x)] N'CKlx-x'Ddx' (173)

where

N'(K|x-x'|) = - f^j + fH (K|x-x'|) + yY (K|x-x'|)

+ iri J (K|x-x'|) (174)

The vertical component of the force acting on the ship by the fluid

pressure is given by

-J/fe

F z = ~ J I "^ P ds < 175 >

" S

and the moment about the y-axis is

M

y

â– IS (n - 1 ') ? ds <i76)

The second term in the parentheses can be omitted because of the slender

body assumption. Then the vertical force and the pitching moment on the

slender ship are given by

63

J dx j it^-J" f Â» d *

M y = " P ] X dX J fÂ£ I^T dS = " j f Â« X dX

-S. C(x) -Â£

where C(x) is the contour of each transverse section. The function f(x)

gives the force per unit length at the section. If we divide the velocity

potential in two-dimensional and three-dimensional portions as

= e ($ v y +$ v ') (178)

we can write

,(2D) 3z , , . .(3D) 9z ,

f (x) = 1 p w | $ -5â€” r ds + i p (o $ -râ€” r ds

C(x) J C(x)

= f x (x) + f 2 (x) (179)

Now take the added mass m and the damping coefficient N for a heaving

cylinder of infinite length. Then the two-dimensional portion of the

sectional force is expressed by

f , (x) = - (iOJm +N ) [V(x)-W(x)] (180)

i z z

while the three-dimensional portion has the expression

f 2 (x) = i p co S(x) - i Â£& B(x) W(x) (181)

where S(x) is the area and B(x) is the waterline width of each section.

64

In pure heaving, we put V(x) = iooz , and we can write

f(x)e

io)t

- (m +m') (-0) z ) - (N +N') i w z

z z g z z g

(182)

where m 1 and N' indicate the three-dimensional effect to the added mass

z z

and damping coefficient, respectively. In the case of pure pitching, we

put V(x) = - icoxij; and we can write

f(x)e = - (m x+m") to Tp + (N x+N") i u) \p

(183)

Here we put

a=m+m' b=N+N'

d = m x+m" e = N x + N"

(184)

and define the following integrals.

-r

*

d x dx

b = b dx

= a x dx E =

e = | e dx

-%

I

B = 1 e x dx

Â£

I; = b x dx

-Â£

> (185)

65

These constants give hydrodynamic coefficients in the coupled equation of

heaving and pitching as follows.

(a+pV) z + b z o + c z - dip - e$ - gl^J = F

(A+k Z pV) i> + BT|> + OJÂ» - D z - E z - G z - M

yy & & &

(186)

where V is the displacement volume and k is the radius of gyration. On

yy

account of Haskind's relation, we have the relation

D = d E = e

(187)

For numerical example, the hull form of Series 60, C = 0.7 is employed

because reliable data of model experiments have been available for

comparison.

A it â€¢& &

The coefficients a , b , d , e , and also a, b, A, B, d, and e are

calculated. They are compared with results computed by means of the strip

theory and the measured results, as shown in Figures 5 through 7. One can

X 10

-2

2.0

1.0

PRESENT CALCULATION

ORIGINAL SLENDER

BODY THEORY

STRIP THEORY

O EXPERIMENT BY GERRITSMA

AND BEUKELMAN

â– Q

X 10

5.0,

KL/2

Figure 5 - Hydrodynamic Coefficients a and b

66

X 10

Figure 6 - Hydrodynamic Coefficients A and B

X 10

Figure 7 - Hydrodynamic Coefficients d and e

67

observe a remarkable improvement in agreement with measured results by the

calculation including the three-dimensional effect, especially at lower

frequencies. One of the reasons of deviation with the results by the

strip theory may be attributed to the logarithmic term, Â£nKr, in the two-

dimensional solution which makes the added mass infinite at zero frequency

limit. The three-dimensional part of the velocity potential has a term

which cancels the above logarithmic singularity.

The application of the simple slender body theory, which has been

described in an earlier section, presents such an unrealistic result as

negative value of added mass and infinite increase in damping coefficient

at higher frequencies. Computed results for the source term a n (x) are

shown in Figure 8. The result by the simple slender body theory which

determines the near-field solution by the condition for a double body

shows much deviation from other theories. This fact may be the main reason

of the ill behavior of the slender body calculation. The fair agreement

between strip theory and the present calculation suggests that the source

term can be determined by the two-dimensional calculation without regard

for the three-dimensional effect, or a n (x) = V(x)A n (x).

X 10

_ -100 â€”

-150 â€”

-180

Figure 8 - Source Distribution

68

As explained in the preceding section, the present theory is a conse-

quence of the inclusion of the second term in the slender body expansion.

Nevertheless it is by no means a second order theory, but still a first

order theory. The formal perturbation procedure, taking successive

approximations, starting from the lowest order term and proceeding to the

second order approximation, never leads to the same result as the above.

This fact is a peculiar feature of the asymptotic expansion of the singular

solution such as the present problem.

WAVE PRESSURE ON SLENDER SHIPS

Boundary Value Problem for the

Diffraction Potential

Although the strip theory is employed in the usual practice of pre-

dicting wave exciting forces on ships, the diffraction problem does not

admit the use of the strip theory in longitudinal waves. Although the

strip theory is an acceptable approximation in high frequencies for the

radiation problem, the short wavelength associated with the high frequency

invalidates the condition of slow variation along the ship's axis. In the

case of long waves, on the other hand, the frequency becomes low and the

three-dimensional effect comes in as in the radiation problem.

Now we consider first the case of a ship with forward speed in long

waves, so that we assume U and (0 are both of order of unity. The boundary

value problem for the diffraction potential has been given previously, but

its solution needs some contrivance. The boundary value problem in the

near field is the two-dimensional Laplace equation

3 2 * D 3 2 * D

D + â€” r^ = (188)

2 2

9y 3z

with the hull surface boundary condition of Equation (104)

3^ " " 3^ < 189)

69

and the free surface condition of Equation (114)

3 *D ^0

-5- - = - U ? â€” - at z = (190)

dZ W n. 2

dz

If the incident wave propagates along the direction of x, Â£ is constant in

the transverse plane. Then the function

1' = <J>_ + U 5 -^ (191)

D w dz

is a plane harmonic function in the lower half space. The boundary

condition for it is

3n* 3nÂ« C w 3n' \3z / K }

on the hull surface, and

|^- = at z = (193)

dz

If we take only the first order terms, we can write the hull surface

boundary condition as

lil = _:!wlz_ _3_ r!o.

3n ? 3z 3n' + U C w 3n' V 3z '

|^- = - i a) C |V + U c X U-^ (195)

dn w dn w dn \dz '

70

This is the same form as the hull boundary condition of the radiation

potential for heaving, because the latter is

" T R . 8z TT 9 f " r

^â€” r = 1 CO Z ir-f - U Z -r-r

dn g on g dn

dz

(196)

Therefore, the diffraction is taken into account in the hull boundary

condition by replacing the vertical movement of the section by the relative

displacement to the surface of the incident wave. This relation holds in

1/2

the case of U = 0(e ) too. Then we can take up to the next term, so that

the relation is to be modified as

|^ = - iu { (1+Kz) |Â£r+u c AIt^

dn w dn w dn \ dz

(197)

Kz

The added term Kz comes from the exponential factor e in the incident

wave potential and indicates the Smith correction.

The boundary value problem for the diffraction potential is now re-

duced to a similar form to that for the radiation problem. The field

equation for the potential <J>' is the two-dimensional Laplace equation

-0> ' + IV =

9y

dz'

(198)

with the hull boundary condition given above. The free surface condition

at z = is

dz

iil_^_ A ." -

)z g

when U = 0(1),

1/2,

w = 0(1) \

= when U = 0(e ' ) , to = 0(1) '

(199)

71

Therefore, the inner solution for the diffraction potential in the case of

co = 0(1) is expressed by the form

<|, = (K (2D) "U ^g-O + ^l+iiLz) ^ ,

where <J) is the solution of the two-dimensional problem with the above

2

mentioned boundary conditions. The factor CO z/g in the third term on the

1/2

right-hand side can be added only when U = 0(e ) and should be omitted

-1/2

in the case of U = 0(1). Another case to be considered is CO = 0(e ).

This is the short wave case, but some complication appears if we apply the

slender body theory. The basic idea of the slender body is that the field

equation in the near field can be reduced to the two-dimensional Laplace

equation. However, the short wavelength hinders the above possibility. In

-1/2

the case of CO = 0(e ) with U = 0(1), the ratio of the wavelength to the

1/2

ship's length is A/Â£ = 0(e ) and the variation of the flow field along

the x-axis is related to the wavelength. If the order of the diffraction

potential is 6 Â£, the order of magnitude of each term in the Laplace

equation in the near field is

^D ^\ 3 \

2 2 2

8x dy 3z

(6) (6e _1 ) (Se -1 )

Therefore, omitting the term of higher order than e, we get a two-

dimensional Laplace equation. However, the omitted term in the long wave

2

case has been of higher order, Â£ . Therefore, the validity of the two-

dimensional equation becomes much weaker in comparison with the long wave

case. This may damage the accuracy appreciably. The free surface

condition in this case is

3 V 3 *Â° + , â€ž ^ + , â€ž 3 *Â° ^ ^O 3 *D + . â€ž , 3 V

^â€” + g ^ + 2 U . ,. + 2 U -r -râ€” 5 râ€” -5â€” + g U X, x~ =

â€ž 2 b 3z dtdx dy dtdy _ 2 dt Â° w â€ž 2

dt dz dz

at z = (201)

72

1/2

if terms of order 6 Â£ are retained. The terms involving the effect of

steady forward potential prevents the straightforward solution, so that

this case should not be used for the practical purpose of prediction.

1/2

If the forward speed is low, namely U = 0(e ), the ratio of wave

length to ship's length is of the order of e. Since each term in the

three-dimensional Laplace equation has the same order of magnitude, its

two-dimensional version is no longer valid. Therefore, the strip theory is

not applicable to the diffraction problem in the longitudinal waves. An

alternative method for the diffraction problem in short waves will be

discussed later.

Wave Pressure and Hydrodynamic Forces

As was mentioned before, the diffraction problem requires not only the

integrated total force, but some local quantities such as wave pressure at

each point on the hull surface and the distribution of forces along the x-

axis. If we write

<b = A + d> (202)

T y D T w

the periodical pressure on the hull surface is given by

-i(p-P )-i.* + u|i + U3/|4 + U^|f (203,

up to the order of 6 Â£, if U and a) are both of the order of unity. Al-

order of 6, the wave amplitude is of the order of 6 Â£ . The velocity of

the orbital motion of wave is 0(6), so that the order of <})-. is 6 Â£. In the

boundary condition on the hull surface for the radiation potential, namely

-r â€” = i w(z -xifj) -^r - U ty -~r - U(z -x\J0 -5â€” r ( -5 â€” ) (136)

3n g 3n' dn' g dn' \9z '

the first term is of the order of 6, while other terms which are related to

1/2

the forward speed are of the order of 6 Â£ .On the other hand, the

1/2

lowest order of the free surface condition is 6 and the next is 6 Â£ . If

we take the lowest order terms only in both boundary conditions, the so-

lution is in the case of zero forward speed. We can discuss the effect of

1/2

the forward speed by taking terms of order 6 Â£ .In this case, the free

surface condition for the radiation potential becomes

*\

9 *R

d \

+

g

+

V

II

^ 2

dz

3t3x

dt

2 2

3cj) 9 cj> 3 <J> 3c}>

+ 2 1)^^-11^-^=0 (137)

9y 9t3y 3z 2 3t

Because of the terms except the first and second ones, we cannot solve the

boundary value problem in two dimensions by ordinary methods. Ogilvie and

29

Tuck employed a successive method by which the solution for the above

52

boundary condition was derived from the solution for zero forward speed.

The solution obtained involves integrals over the plane z = 0. It seems

of some interest that the outer solution for the radiation potential takes

a degenerated form like

l> R ~-â€” m(x)+2ifi(z-i|y |) ^T~J ^'i' 1 < ' I >'! â– 1 ^ :

The first term in parentheses means a simple harmonic plane wave propagat-

ing outwards in the y direction, while the second term is related to the

variation of the source density along the x-axis which means the inter-

ference between different sections. An extensive discussion of this case

is given by Ogilvie and Tuck and will not be reproduced here.

Next we consider the case of high frequency with low forward speed,

-1/2 1/2

that is co = 0(e ), U = 0(e ). This is the case of short waves such

that the wavelength is of the same order as that of the breadth of the ship.

3/2

In this case, the order of magnitude of 1 is Â£ . The free surface

condition becomes

9 2 <J>

1 + R -z-Â± = (139)

at 2 6 3z

1/2 3/2

up to the order of 6 e . The next term is of the order of 6 Â£ , which

includes the effect of the forward speed, but has quadratic forms. There-

fore, the ordinary linearized treatment can be applied to the first order

only, by which the effect of the forward speed cannot be taken into

account. The asymptotic form for the outer solution for the radiation

potential takes the form

4iri icot + v(z-ilyl) , s / - i/rw

d> ss - e ' â– " ' m(x) (140) â–

R g

that means outward-going plane waves. There are no three-dimensional terms

involved and the solution is purely two-dimensional. The strip theory is

53

then valid but without any effect of the forward speed. It should be noted

that the diffraction problem cannot be treated in the same way, because the

assumption of the slow variation along the x-axis is no longer valid. The

diffraction in short waves will be discussed in another section.

Hydrodynamic Forces in Heaving and

Pitching

It is widely known that the hydrodynamic forces and moments acting on

oscillating ships are predicted by strip theory with fairly good accuracy.

However, the reliability of the strip theory is still open to doubt as

discrepancies between computed and measured results are observed occa-

sionally. These discrepancies may be attributed to the effect of the

forward speed and fluid motion in three dimensions. In the preceding

section, we have observed that both the forward speed and the three-

dimensionality can be taken into account within a plausible approximation

1/2

of the perturbation scheme, if the forward speed is of the order of Â£

where Â£ is the beam-to-length ratio.

The hydrodynamic forces are obtained by the integration of pressure

over the hull surface. The fluid pressure in the near field is given by

a . sa(2D) a ^(2D) a .(2D) a *(2D)

3cj> 3<j) 3<j>. 3c() 3<j>-

- - (p-p n ) ^mMd^ + d^ â€¢ t^ + u ir^ *ir- ( 141 )

p r r 1 dx 9y dy dz dz

up to the order of 8e. The hydrostatic pressure is omitted because it is

simply determined by the geometrical relations. Although the third and

fourth terms are omitted in the usual linearized theory for oscillating

ships, they appear in the same order of magnitude as the second term, so

that they have to be retained if one wishes to take account of the effect

of the forward speed. The vertical force is given by the integral

â– If.

F 9 = (P- Pn ) |f dS (142)

2 ^ r 0' 3n

54

Although the quadratic terms in Equation (141) seem to be troublesome in

30

evaluating the above integral, a theorem which has been proven by Tuck

becomes a powerful aid. The theorem is proved in Appendix B. Consider a

velocity vector V of an irrotational motion of an inviscid fluid outside

the hull, which satisfies the boundary condition on the hull surface such

as

V â€¢ n = (143)

where n is a unit vector along the outward normal to the hull surface.

Next we define the vector m by the relation

9V

m = - 3- (144)

Further we define vectors n and m by

n = r x n

\ (145)

5 " - to ( ^ Xn) J

where r is the position vector (x,y,z). Then, the following relations are

valid.

If

[mcjH-n(V'Vcj>)] dS = -

n d> w d s

S L o

(146)

[m (J)+n (V-

VV.) | dS - ! n (j) w d s

L

where w is the z-component of V and L is the still waterline of the hull.

The line integral on the right-hand side is omitted when the body is

55

slender and wall-sided at the water plane. For the vertical force, we take

the z-component of the first equation, while for the pitching moment, the

y^-component of the second equation is taken. In the present case, we put

VMU, U^,U^) (147)

Next we define vector functions ){, |, )( , and ip , which are two-dimensional

harmonic functions in the lower half space outside the ship and satisfy the

boundary conditions on the hull surface, such as

3X W

= m

(148)

ttâ€” = n, U -râ€” = m

9n 3n -

ax* * a$* ,

-7Tâ€” = n , U -râ€” = m

3n - 3n -

Furthermore, these functions are assumed to satisfy the free surface

condition

|i _ v <j> = atz=0 (149)

dz

We have expressed the near-field expression of the radiation potential for

a slender ship by

^N = <j)(2D) + (1+VZ) g l (x) (150)

Then, the vertical force is written in the form

(2D) 3z

p dx i

J C(x)

. T dc

3n'

(2D) 3<J>< 2D) (2D) 3^ 2D) (2D))

C(x) I ' (cont ->

56

+ dx {i(jJ g;L (x)+Ug^(x)} f (l+Vz) |^r dc

J ^C(x)

(151)

where C(x) is the contour of each section along which the integral with

respect to c is taken. On applying the above mentioned theorem to the

second term, we obtain

,(2D) 3z ,

Fâ€ž = p|dx| i w <Jr y -Â»â€” r dc

,< 2D > dc

J J C(x)

J ^C(x)

+ p j {ico gl (x)+Ug^(x)} {-B(x)+VS(x)} dx (152)

The last term is derived by the application of Gauss' theorem in which

B(x) and S(x) are breadth at the waterline and sectional area of each

transverse section, respectively. If we write the z-components of X and iÂ£

by x an( i *P Â» respectively, the two-dimensional part of the radiation

potential is expressed as

(J) (2D) = {iw(z -xij;)-lty} x + U(z -aap) ip (153)

g z g z

Then, the vertical force is written as

3X

r 2 f dx z

= - p I dx {(*) (z -xty)+iwlfy} x TT dc

j C(x)

r r 3x z

+ i p a) U dx (z -xifO t\) tjâ€” r dc

J J c ,_

U dx (z -xlJO

U 2 jdx (z g -x*) f ^

J C(x)

- i p to U | dx (z^-xip) | x "a - r dc

X z

C(x)

- p U | dx (zâ€ž-xi|0 I ^ _-Â£. dc (154)

(cont. )

57

- p {itog 1

(x)+Ug^(x)} {-B(x)+VS(x)} dx

(154)

Because of Green's reciprocal relation, which can apply to x and ip , such

as

f X z 3^ dc = I

J C(x) J C(x)

i 2

' a - i

z dn

dc

the second and third terms are cancelled. Therefore, we obtain

F_ = - p

dx {to (z^-xiJO+iwUip} | X "T~r dc

- 2 J

dx (z -xlji)

C(x)

\p -r-r dc

z dn

C(x)

+ P

-iO){B(x)-.VS(x)} +U 2- (B(x)-vS(x)}

dx

g-^x) dx

(155)

1/2,

The last term is derived by integration by parts. If we assume U = 0(e )

2

and omit the term of 0(6e ), the second term drops out. A similar ex-

pression is obtained for the hydrodynamic moment about the y axis. The

first term indicates the result obtained by the strip theory, and the other

terms give the effect of the forward speed and the three-dimensional motion.

Numerical Results of Radiation Problem

at Zero Forward Speed

Although numerical analysis works when the forward speed and the

effect of three-dimensionality are present, no published result of this

kind is known so far. There is a rather comprehensive result, on the other

hand, for the case of zero forward speed by means of a similar formu-

31

lation. It is obtained simply by letting U = in the original formula

for finite speed with no substantial difference in the method of numerical

58

calculation. The result can well illustrate the effect of three-

dimensionality of the fluid motion which is remarkable at lower frequencies.

In the first place, let us put Â§ = e $ and express the two-

dimensional solution for a heaving cylinder in the form as

i â€” Â°Â°

(2D)

' - - a.

kz cos ky j, . Kz

e â€” J dk-7Tie cos Ky

_

y a 2m

^L/ (2m-l) !

m=l

,2m-l

v 2m-2

+K

â€ž 2m-l 2 2 J . 2m-2 I 2^ 2

dz \ z +y / dz \ z +y

(156)

It is readily shown that this expression satisfies the free surface

condition

_3$

dz

- K $ = atz=0

(157)

where K = V = co /g in the present case. Since the inner expansion of the

above function is

,(2D)

= a n [Â£nKr+Y+Kr cos (l-Â£nKr-Y)+Kr9 sin 6+TTi]

Z

m=l

,2m

2m

Kt

cos 2m6+ -= â€” r- cos (2m-l)6

zm-1

(158)

where we have employed the cylindrical coordinates

z = - r cos b, y = r sin

(159)

and Y is Euler's constant, 0.5772157.

In order to make the inner expansion of the far field potential match the

above, we employ the expression in the near field as

59

,(2D)

- a Q (l+Kz) (y+TTi)

I *Â«'

- -| (1+Kz) I aA( x ') sgn(x-x') Â£n(2K|x-x'|) dx'

"-Z

it

JK(1+Kz) J a (x') [H Q

-I

(K|x-x' |)+Y (K|x-x* |)

+2iJ (K|x-x' |)] dx'

(160)

where x = +Â£ is the x coordinate at each end of the ship. The second term

on the right-hand side is added to conform to the expansion of the two-

dimensional solution. We can rewrite the above expression as

-I

> (2D) + i (1+K; n,\u; ;m(I<! ,''\) wgu(^-y') dx 1 (161)

where

N(u) =-y-Â£n2u+

Hq(u') duÂ« + f Y Q (u') du' - TT i

+ IT i

Jq(u') du'

(162)

It can be shown that

Mm N(K|x-x' |) =

K-*=Â°

60

so that the three-dimensional part vanishes when the frequency becomes

infinite, and the fluid motion becomes purely two-dimensional for which

the strip theory holds exactly.

Now let us consider the boundary value problem. When a ship is in

heaving and pitching oscillations, each transverse section has a vertical

velocity V(x) e , where V(x) is related to the mode of the oscillation.

When the slender body approximation is employed, the boundary condition

satisfied by the velocity potential $ e at the surface of the body is

to 7 " " V(x) 3n"'

(163)

If we introduce the expression for $, given by Equation (161), the boundary

condition can be written in the form

,(2D)

3n'

(x)- -| K I a^(x')N(K|x-x' |)sgn(x-x , )dx'

-I

dz

3n'

(164)

Here we introduce the solution of the two-dimensional problem of a heaving

cylinder for which the boundary condition at the body surface is

,(2D)

3n f

dz

9n'

(165)

With this solution, we put the coefficient of the source term as

a " A o

(166)

Then the coefficient a n for the boundary condition of Equation (164)

satisfies the equation

a Q (x) =

(x)- - K aJ(x')N(K|x-x' | )sgn(x-x' )dx'

-I

A Q (x)

(167)

61

The coefficient A n can be determined by the method well known in the two-

dimensional theory of a heaving cylinder. Then we can determine a n by

solving the above equation. However, we need not solve the equation

exactly. The reason is as follows. If the frequency is very low or to the

contrary very high, the integral on the right-hand side can be neglected.

So we can put

a (x) = V(x) A Q (x) (168)

It may be assumed that the deviation from the above relation at inter-

mediate frequencies is not large. Therefore, we approximate

a Q (x) =

J6

V(x)- \ K {V'Cx'HqCx^+VCx'^Cx')} N(K|x-x* | ) sgn(x-x' )dx'

-I

x A Q (x) (169)

In order to determine other coefficients, we substitute V(x)A(x) for a Q in

the boundary condition and put

W(x) = -| K {V , (x')A (x , )+V(x , )A^(x')} N(K|x-x' |)sgn(x-x')dx' (170)

-I

Then the boundary condition becomes

â„¢(2D) .

fr = [V(x)-W(x)] Â£r (171)

The solution of the two-dimensional problem with this boundary condition

determines other coefficients aâ€ž . We can rewrite

2m

W(x) in the following form for numerical purposes.

determines other coefficients aâ€ž . We can rewrite the function aâ€ž(x) and

2m u

62

[V(x')A (x')-V(x)A (x)] N'(K|x-x' |)dx' (172)

( (x) = V(x) A Q (x) Tl- | KA (x){N(K|Â£+x|)+N(K|il-x|)}l

W(x) = y K V(x) A n (x) {N(K|Jl+x|)+N(K|Â£-x|)}

â™¦1*J

I

[V(x , )A (x')-V(x)A (x)] N'CKlx-x'Ddx' (173)

where

N'(K|x-x'|) = - f^j + fH (K|x-x'|) + yY (K|x-x'|)

+ iri J (K|x-x'|) (174)

The vertical component of the force acting on the ship by the fluid

pressure is given by

-J/fe

F z = ~ J I "^ P ds < 175 >

" S

and the moment about the y-axis is

M

y

â– IS (n - 1 ') ? ds <i76)

The second term in the parentheses can be omitted because of the slender

body assumption. Then the vertical force and the pitching moment on the

slender ship are given by

63

J dx j it^-J" f Â» d *

M y = " P ] X dX J fÂ£ I^T dS = " j f Â« X dX

-S. C(x) -Â£

where C(x) is the contour of each transverse section. The function f(x)

gives the force per unit length at the section. If we divide the velocity

potential in two-dimensional and three-dimensional portions as

= e ($ v y +$ v ') (178)

we can write

,(2D) 3z , , . .(3D) 9z ,

f (x) = 1 p w | $ -5â€” r ds + i p (o $ -râ€” r ds

C(x) J C(x)

= f x (x) + f 2 (x) (179)

Now take the added mass m and the damping coefficient N for a heaving

cylinder of infinite length. Then the two-dimensional portion of the

sectional force is expressed by

f , (x) = - (iOJm +N ) [V(x)-W(x)] (180)

i z z

while the three-dimensional portion has the expression

f 2 (x) = i p co S(x) - i Â£& B(x) W(x) (181)

where S(x) is the area and B(x) is the waterline width of each section.

64

In pure heaving, we put V(x) = iooz , and we can write

f(x)e

io)t

- (m +m') (-0) z ) - (N +N') i w z

z z g z z g

(182)

where m 1 and N' indicate the three-dimensional effect to the added mass

z z

and damping coefficient, respectively. In the case of pure pitching, we

put V(x) = - icoxij; and we can write

f(x)e = - (m x+m") to Tp + (N x+N") i u) \p

(183)

Here we put

a=m+m' b=N+N'

d = m x+m" e = N x + N"

(184)

and define the following integrals.

-r

*

d x dx

b = b dx

= a x dx E =

e = | e dx

-%

I

B = 1 e x dx

Â£

I; = b x dx

-Â£

> (185)

65

These constants give hydrodynamic coefficients in the coupled equation of

heaving and pitching as follows.

(a+pV) z + b z o + c z - dip - e$ - gl^J = F

(A+k Z pV) i> + BT|> + OJÂ» - D z - E z - G z - M

yy & & &

(186)

where V is the displacement volume and k is the radius of gyration. On

yy

account of Haskind's relation, we have the relation

D = d E = e

(187)

For numerical example, the hull form of Series 60, C = 0.7 is employed

because reliable data of model experiments have been available for

comparison.

A it â€¢& &

The coefficients a , b , d , e , and also a, b, A, B, d, and e are

calculated. They are compared with results computed by means of the strip

theory and the measured results, as shown in Figures 5 through 7. One can

X 10

-2

2.0

1.0

PRESENT CALCULATION

ORIGINAL SLENDER

BODY THEORY

STRIP THEORY

O EXPERIMENT BY GERRITSMA

AND BEUKELMAN

â– Q

X 10

5.0,

KL/2

Figure 5 - Hydrodynamic Coefficients a and b

66

X 10

Figure 6 - Hydrodynamic Coefficients A and B

X 10

Figure 7 - Hydrodynamic Coefficients d and e

67

observe a remarkable improvement in agreement with measured results by the

calculation including the three-dimensional effect, especially at lower

frequencies. One of the reasons of deviation with the results by the

strip theory may be attributed to the logarithmic term, Â£nKr, in the two-

dimensional solution which makes the added mass infinite at zero frequency

limit. The three-dimensional part of the velocity potential has a term

which cancels the above logarithmic singularity.

The application of the simple slender body theory, which has been

described in an earlier section, presents such an unrealistic result as

negative value of added mass and infinite increase in damping coefficient

at higher frequencies. Computed results for the source term a n (x) are

shown in Figure 8. The result by the simple slender body theory which

determines the near-field solution by the condition for a double body

shows much deviation from other theories. This fact may be the main reason

of the ill behavior of the slender body calculation. The fair agreement

between strip theory and the present calculation suggests that the source

term can be determined by the two-dimensional calculation without regard

for the three-dimensional effect, or a n (x) = V(x)A n (x).

X 10

_ -100 â€”

-150 â€”

-180

Figure 8 - Source Distribution

68

As explained in the preceding section, the present theory is a conse-

quence of the inclusion of the second term in the slender body expansion.

Nevertheless it is by no means a second order theory, but still a first

order theory. The formal perturbation procedure, taking successive

approximations, starting from the lowest order term and proceeding to the

second order approximation, never leads to the same result as the above.

This fact is a peculiar feature of the asymptotic expansion of the singular

solution such as the present problem.

WAVE PRESSURE ON SLENDER SHIPS

Boundary Value Problem for the

Diffraction Potential

Although the strip theory is employed in the usual practice of pre-

dicting wave exciting forces on ships, the diffraction problem does not

admit the use of the strip theory in longitudinal waves. Although the

strip theory is an acceptable approximation in high frequencies for the

radiation problem, the short wavelength associated with the high frequency

invalidates the condition of slow variation along the ship's axis. In the

case of long waves, on the other hand, the frequency becomes low and the

three-dimensional effect comes in as in the radiation problem.

Now we consider first the case of a ship with forward speed in long

waves, so that we assume U and (0 are both of order of unity. The boundary

value problem for the diffraction potential has been given previously, but

its solution needs some contrivance. The boundary value problem in the

near field is the two-dimensional Laplace equation

3 2 * D 3 2 * D

D + â€” r^ = (188)

2 2

9y 3z

with the hull surface boundary condition of Equation (104)

3^ " " 3^ < 189)

69

and the free surface condition of Equation (114)

3 *D ^0

-5- - = - U ? â€” - at z = (190)

dZ W n. 2

dz

If the incident wave propagates along the direction of x, Â£ is constant in

the transverse plane. Then the function

1' = <J>_ + U 5 -^ (191)

D w dz

is a plane harmonic function in the lower half space. The boundary

condition for it is

3n* 3nÂ« C w 3n' \3z / K }

on the hull surface, and

|^- = at z = (193)

dz

If we take only the first order terms, we can write the hull surface

boundary condition as

lil = _:!wlz_ _3_ r!o.

3n ? 3z 3n' + U C w 3n' V 3z '

|^- = - i a) C |V + U c X U-^ (195)

dn w dn w dn \dz '

70

This is the same form as the hull boundary condition of the radiation

potential for heaving, because the latter is

" T R . 8z TT 9 f " r

^â€” r = 1 CO Z ir-f - U Z -r-r

dn g on g dn

dz

(196)

Therefore, the diffraction is taken into account in the hull boundary

condition by replacing the vertical movement of the section by the relative

displacement to the surface of the incident wave. This relation holds in

1/2

the case of U = 0(e ) too. Then we can take up to the next term, so that

the relation is to be modified as

|^ = - iu { (1+Kz) |Â£r+u c AIt^

dn w dn w dn \ dz

(197)

Kz

The added term Kz comes from the exponential factor e in the incident

wave potential and indicates the Smith correction.

The boundary value problem for the diffraction potential is now re-

duced to a similar form to that for the radiation problem. The field

equation for the potential <J>' is the two-dimensional Laplace equation

-0> ' + IV =

9y

dz'

(198)

with the hull boundary condition given above. The free surface condition

at z = is

dz

iil_^_ A ." -

)z g

when U = 0(1),

1/2,

w = 0(1) \

= when U = 0(e ' ) , to = 0(1) '

(199)

71

Therefore, the inner solution for the diffraction potential in the case of

co = 0(1) is expressed by the form

<|, = (K (2D) "U ^g-O + ^l+iiLz) ^ ,

where <J) is the solution of the two-dimensional problem with the above

2

mentioned boundary conditions. The factor CO z/g in the third term on the

1/2

right-hand side can be added only when U = 0(e ) and should be omitted

-1/2

in the case of U = 0(1). Another case to be considered is CO = 0(e ).

This is the short wave case, but some complication appears if we apply the

slender body theory. The basic idea of the slender body is that the field

equation in the near field can be reduced to the two-dimensional Laplace

equation. However, the short wavelength hinders the above possibility. In

-1/2

the case of CO = 0(e ) with U = 0(1), the ratio of the wavelength to the

1/2

ship's length is A/Â£ = 0(e ) and the variation of the flow field along

the x-axis is related to the wavelength. If the order of the diffraction

potential is 6 Â£, the order of magnitude of each term in the Laplace

equation in the near field is

^D ^\ 3 \

2 2 2

8x dy 3z

(6) (6e _1 ) (Se -1 )

Therefore, omitting the term of higher order than e, we get a two-

dimensional Laplace equation. However, the omitted term in the long wave

2

case has been of higher order, Â£ . Therefore, the validity of the two-

dimensional equation becomes much weaker in comparison with the long wave

case. This may damage the accuracy appreciably. The free surface

condition in this case is

3 V 3 *Â° + , â€ž ^ + , â€ž 3 *Â° ^ ^O 3 *D + . â€ž , 3 V

^â€” + g ^ + 2 U . ,. + 2 U -r -râ€” 5 râ€” -5â€” + g U X, x~ =

â€ž 2 b 3z dtdx dy dtdy _ 2 dt Â° w â€ž 2

dt dz dz

at z = (201)

72

1/2

if terms of order 6 Â£ are retained. The terms involving the effect of

steady forward potential prevents the straightforward solution, so that

this case should not be used for the practical purpose of prediction.

1/2

If the forward speed is low, namely U = 0(e ), the ratio of wave

length to ship's length is of the order of e. Since each term in the

three-dimensional Laplace equation has the same order of magnitude, its

two-dimensional version is no longer valid. Therefore, the strip theory is

not applicable to the diffraction problem in the longitudinal waves. An

alternative method for the diffraction problem in short waves will be

discussed later.

Wave Pressure and Hydrodynamic Forces

As was mentioned before, the diffraction problem requires not only the

integrated total force, but some local quantities such as wave pressure at

each point on the hull surface and the distribution of forces along the x-

axis. If we write

<b = A + d> (202)

T y D T w

the periodical pressure on the hull surface is given by

-i(p-P )-i.* + u|i + U3/|4 + U^|f (203,

up to the order of 6 Â£, if U and a) are both of the order of unity. Al-

Online Library → H. (Hajime) Maruo → Challenge to better agreement between theoretical computations and measurements in ship hydrodynamics → online text (page 4 of 6)