A Theoretical Study on Free-Surface Flow around Slowly

(Read at the Autumn Meeting of The Society of Naval Architects of Japan, Nov. 1991) 143

A Theoretical Study on Free-Surface Flow around

Slowly Moving Full Hull Forms in Short Waves

by Hironori Yasukawa*, Member

Toshinobu Sakamoto*, Member

Summary

A theory is described on the free-surface flow slowly moving full hull forms in short waves. Under

the assumption of small perturbation over the double body flow, a new function is derived which satisfies

the free-surface condition including the effect of deformed flow due to the presence of the ship hull. By

using the function, an expression of the velocity potential is introduced. In order to know the basic

feature of the present theory, calculations are made of wave pattern generated by a pulsating source

moving with constant speed below the free-surface. As a result, it is shown that steep waves appear in the

region where a parameter ƒÑ0(= U0ƒÖ/g ; U0 is local velocity of the double body flow, ƒÖ the encounter

frequency and g the acceleration of gravity) is close to 0.25. The wave pattern shows the similar

tendency to the diffraction waves in front of a full hull form observed by Ohkusu 11).

1. Introduction

In the classical linearized wave-making theory, it is

understood that the diffraction waves do not propagate

upstream in the region where ƒÑ( = UƒÖ/g ; U is the ship's

velocity, ƒÖ the encounter frequency and g the accelera-

tion of gravity) is larger than 0.25. However, Ohkusu

observed that diffraction waves in front of bow of a full

hull form propagate obviously upstream even for ƒÑ

much greater than 0.25. Those waves propagate some

distance forward getting steep and break 11). The compli-

cated behavior of the diffraction waves can not be

evaluated in the framework of the classical linearized

theory, because the effect is not taken into account of

the deformed flow due to the presence of the ship hull

(we shall call hereafter the deformed flow as the steady

perturbation flow) . It seems that a rational considera-

tion of the steady perturbation flow is indispensable for

the improvement of the theory.

Several attempts have been made to take the steady

perturbation flow into account. Faltinsen et al. gave an

asymptotic formula of added resistance in short

waves 3). Their formula considers even approximately

the interaction of diffraction waves with steady pertur-

bation flow. Naito et al. calculated ray patterns of

incident waves and discussed the relation with added

resistance in waves 9). Hermans calculated the ray pat-

terns for several angles of incidence and the added

resistance in short waves 4). In these studies, however,

simple hull forms such as a rankine ovoid form, a

vertical cylinder and a sphere are dealt with.

Recently numerical approaches have been made to

solve directly the basic equations including the steady

perturbation flow. Method by using the Rankine sources is a typical one. The most advanced one is a hybrid

method presented by Kashiwagi and Ohkusu for a two-

dimensional half-immersed cylinder 7). At present, how-

ever, it is difficult to extend the method to the three-

dimensional problems, since the calculation scheme for

solving the basic equations is too complicated. Accord-

ingly, an effort has been made to develop a simply

treated method. Methods proposed by Takagi 15),

Yasukawa 17) and Sclavounos et al.14) are the extension

of Rankine panel method in the steady free-surface flow

problems 2). The Rankine panel method may be useful

for ship motion problems since first order hydrodynamic

forces such as added mass, damping and exciting forces

can be predicted with sufficient accuracy. However, the

flow field far away from the ship hull can not be evaluat-

ed exactly, since a numercal damping is employed to

satisfy the radiation condition. Then, for prediction of

second order steady hydrodynamic forces such as added

resistance in waves and wave drifting forces, a pressure

integration on the ship hull surface is the only way. At

present, however, it may be difficult to predict accurate-ly the second order forces by using the pressure

integration 6).

In 1986, Sakamoto and Baba derived a new free-

surface condition13) which makes a theoretical pair with

the steady free-surface condition in the low speed

wave-making theory", by taking the steady perturba-

tion flow into account. In addition, an asymptotic for-

* Nagasaki Experimental Tank, Mitsubishi Heavy

Industries, Ltd.

144 Journal of The Society of Naval Architects of Japan, Vol. 170

mula was introduced of added resistance in waves at the

low speed limit 13). However, no attempt has been made

to obtain a general solution of the boundary value

problem presented by Sakamoto and Baba.

In this paper, a theory is presented on free-surface

flow around slowly moving full hull forms in short

waves. Under the assumption of small perturbation over

the double body flow which is regard as the steady

perturbation flow, a new function is derived which satisfies

the free-surface condition presented by Sakamoto and

Baba 13). By using the function, velocity potentials re-

presenting radiation due to ship motion, diffraction and

deformation of incident waves are expressed. In such a

way, the effect of the steady perturbation on unsteady

free-surface flow is rationally considered. Further, in

order to know the basic feature of the present theory,

calculations are made of wave pattern generated by a

pulsating source moving with constant speed below the

free-surface. As a result, it is shown that in the region

where a derived parameter ƒÑ0( U0ƒÖ/g ; U0 is local

velocity of the double body flow) is close to 0.25, steep

waves appear and the wave pattern is quite different

from that by the classical linearized theory. The present

wave pattern shows the similar tendency to the

diffraction waves in front of a full hull form observed

by Ohkusu 11).

2. Problem Formulation

2. 1 Basic Assumptions

Let us consider a ship slowly moving at constant

forward velocity U into a plane progressive wave of

amplitude a, circular frequency ƒÖ0 and wave number K.

Here, it is assumed that the wave amplitude a is small

and the circular frequency ƒÖ0 is large. By using the

ship's velocity U which is assumed to be small, the

orders of magnitude of a, K and coo are assumed to be

The angle of wave incidence is denoted by x and

measured as in Fig. 1; ƒÔ=0 corresponds to the follow-

ing wave. Due to the effect of this incident wave, the

ship performs sinusoidal oscillations about its mean

position with the circular frequency of encounter co,

which is related to coo by ƒÖ= ƒÖ0- KU cos ƒÔ.

As shown in Fig. 1, we take a right-hand Cartesian

coordinate system o-xyz, translating with the same

velocity as that of the ship. We take z=0 as the plane

of the undisturbed free-surface, the x-axis positive

forward, and the z-axis positive upward.

Supposing the fluid is inviscid, irrotational and incom-

pressible, the velocity potential ƒÓ is introduced, which

represents flow around the ship and satisfies Laplace's

equation •Þ2ƒÓ=0. The potential ƒÓ is assumed to be

expressed as the sum of three components as :

( 1 ) ( 2 )

where

φ0: velocity potential representing steady double

body flow, which is regarded as the steady

perturbation flow.

φ1:velocity potential representing steady wavy

How.

φU:velocity potential representing unsteady wavy

flow.

In ( 2 ) Re denotes the real part of a complex number

and t the time.

The following assumptions are made about the order;

of magnitude for the•@ƒÓ0, 01 and ƒÓ1)10)U:

when operated on 00.

when operated

on ƒÓ1 and ƒÓU.

Next, consider about the order of magnitude of the

ship motion. The vector of local oscillatory displace-

ment of ship's surface ƒ¿ is expressed as :

( 3 )

where

Here ƒÌ and ƒ¶ denote the oscillatory translation and

rotation vector of the ship respectively. Subscript in E

denotes the mode of ship motion, where 1, 2, 3, 4, 5 and

6 mean surge, sway, heave, roll, pitch and yaw respec-

tively. r denotes the coordinate vector of the ship hull

surface. a is assumed to be the same order of magnitude

as the amplitude of incident wave a as follows :

when operated on

α.

2.2 Boundary Conditions

Exact free-surface condition is written as :

( 4 )where ƒÄ denotes the wave elevation. Taking the lowest

Fig. 1 Coordinate system and notations

A Theoretical Study on Free-Surface Flow around Slowly Moving Full Hull Forms in Short Waves 145

terms under the assumptions of the orders of magnitude,

two independent linearized free-surface conditions, one

for the steady wavy flow and the other for the time

dependent wave motion are obtained") as follows.

steady component :

( 5 )

where

( 6 )

(7)

unsteady component

( 8 )

Exact ship hull surface condition is written as :

(9)

where S denotes the submerged position of ship's sur-

face and n the unit normal vector which is defined to

point out of the fluid domain. Here, total velocity •ÞƒÓ on

S is expanded in Taylor series as :

(10)

where Sm denotes the mean submerged position of ship's surface. Substituting ( 10 ) into ( 9 ), two independent ship hull surface conditions are obtained as follows. steady component :

(11)

unsteady component :

(12)

The free-surface condition ( 5 ) and ( 8 ) should be

satisfied on z= ƒÄ0. Here, for the simplicity of treatment

a following nonconformal transformation of coordi-

nates is introduced :

(13)Then

when operated on ƒÓ0,

when operated on ƒÓI

and ƒÓU.

When taking the lowest order terms, ƒÓI and ƒÓU satisfy

the usual Laplace's equation in terms of the new vari-

ables x', y', z', and the primes can be dropped for

simplicity. The position where ship hull surface condi-

tions should be satisfied is transformed from Sm into S;n,

where S'm denotes the transformed ship hull surface.

Then, the hull surface conditions are rewritten by using

S'm instead of Sm in ( 11 ) and ( 12 ) . Here, however,

the hull surface conditions are approximately dealt

with not on S'm but on Sm, since the solution on Sm is

considered as the first step of an iteration cycle for

obtaining the exact solution.

Substituting ( 2 ) and ( 3 ) into ( 8 ) and ( 12 ) and

dropping the term of eiwt , the boundary conditions for ƒÓ

are obtained. In summary, the boundary conditions for

steady and unsteady flow components are written as

follows.

steady component :

(14)(15)

unsteady component :

(16)(17)

Here uo= ƒÓ0x(x, y, 0) and ƒË0= ƒÓ0y(x, y, 0). Eqs. (14) and

(15) coincide with the boundary conditions in the low

speed wave-making theory°. Further, eq. (16) coincides

with the free-surface condition presented by Sakamoto

and Baba 13). In this paper, the unsteady flow problems

with the boundary conditions (16) and (17) are dealt

with.

2.3 Components of Velocity Potential

The potential ƒÓ(x, y, z), which is to be solved, is

expressed in the following form :

(18)

Here ƒÓI is the velocity potential representing the inci-

dent wave, and can be expressed as :

(19)

φI does not satisfy the present free-surface condition

(16). In order to satisfy the free-surface condition, the

potential ƒÓA is added to the ƒÓI. ƒÓA represents the defor-

mation of the incident wave due to the double body

flow. ƒÓD is the potential representing the diffraction

flow, and ƒÓRj the potential representing radiation flow

due to the ship motion of j-th mode.

Substituting (18) into (16) and (17), the boundary

conditions are obtained for the following problems.

( a ) Radiation problem for j-th motion :

(20) (21)

where

(22)

( b ) Diffraction problem :

(23) (24)

( c ) Deformation problem of Incident wave :

(25)

where


(26)

Here, it is noted that I(x, y) stands for the residual term where 0/ do not satisfy the present free-surface condi-tion (16). The free-surface conditions in the problems ( a ) and ( b ) are the same form. By introducing a new function which satisfies the above free-surface condi. tion, the problems ( a ) and ( b ) can be dealt with. In the problem ( c ), singularity distribution representing

φA will be derived.

2. 4 A Function which Satisfies the Present Free-Surface Condition

A function, which satisfies the present free-surface condition for radiation and diffraction problems, is introduced. And by using the function, an expression of the velocity potential is obtained. The function G(x, y, z; x1, y1, z1) is assumed to be a following form :

(27)where

Here (x, y, z) is the field point, (x1, y1, z1) the singular

point. The term of mirror image of 1/r with respect to z=0, 1/r1, can be written as :

(28)

where

(29)Then G1(x, y, z; x1, y1, z1) is assumed to be expressed as follows :

(30)Here, F(x, y, z, k, ƒÆ) is determined in such a way that

G(x, y, z; x1, y1, z1) satisfies Laplace's equation and the

free-surface condition. It can be considered that F(x, y,

z, k, ƒÆ) contains the effect of the double body potential.

Therefore, the differential operation on F does not

change the order of magnitude. Then,

(31)

The second and third terms in the above bracket are of

higher order than the first term. The same rule is

applied to Gyy and G. When taking the lowest order

terms, G satisfies Laplace's equation. Substituting G

instead of ƒÊRj(or ƒÓD) into the free-surface condition (

20) (or ( 23 ) ), F(x, y, z, k, ƒÆ) is determined as fol-

lows :

(32)

Here ƒÊ is the Rayleigh's viscosity which is introduced to

satisfy the radiation condition, and u0= ƒÓ0x(x, y , 4 and

ν0=φ0y(x, y, z). It can be seen that F(x, y, z, k,θ)is

expressed in terms of solution of the double body flow . From (30) and (32) G(x, y, z; x1, y1, z1) is obtained

as :

(33)Here we put,

(34)where

U0 means magnitude of velocity of the double body flow and 00 the direction of the double body flow. Then (33) can be rewritten as :

(35)

After some reduction of (35), the following expres-sion is obtained as :

(36)

where

(37)

In the classical linearized theory, ƒÑ(= UƒÖ/g) is the

parameter for determining the characteristics of wave

propagation and the radiation or the diffraction waves

can not propagate upstream in the region where r is

larger than 0. 25. In the present theory, however, To

becomes a new parameter of the wave propagation. To

varies with the position of the field point. In the region

where To is smaller than 0. 25 the waves can propagate

upstream even if r is much greater than 0. 25.

In the field far away from the ship hull, it follows

that

Then, ( 36 ) can be written as :


(38)

Eq. (38) coincides with the Green's Function in the

classical linearized theory 16).

2. 5 Expression of Velocity Potentials

2. 5. 1 Radiation and Diffraction Potentials

By using the function G, an expression is introduced

of the radiation or the diffraction potential (ƒÓRj, or ƒÓD).

Here, it should be noted that in this section ƒÓRj and ƒÓD

are represented by ƒÓ.

By applying the Green's theorem, the velocity poten-

tial ƒÓ(P) is expressed as follows :

(39)

(40)

(41)

where P means the field point (x, y, 4 and Q the singu-lar point (x1, y1, z1). Sm denotes ship hull surface in steady condition and SF the still water surface (z=0). Superscript + means the values on the outer surface of Sm and superscript -the value on the inner surface of Sm.

Here, we put(42)

and the velocity potential is assumed to be continuous

through the ship hull surface, then ƒÓH(P) can be written

as :

(43)

Next, let us consider ƒÓF(P) which is the potential

represented by the singularity distribution on the free-

surface. When Q is on the free-surface (z1 =0), ƒÓ(Q) has

to satisfy the following free-surface condition :

(44)Then, G(P, Q) satisfies the following equation :

(45)

Here, u0 and ƒË0 are defined at P. Therefore, it should be

noted that eq. (45) is not the free-surface condition with

respect to G(P, Q) at z1 =0. Substituting (44) and (45)

into (41), ƒÓF(P) can be written as :

(46)

(47)

(48)

where

(49)

Here c means the cross contour of free-surface and ship hull surface. nx, and ny, are components of the outward normal vector on c with respect to x1-and y1-direction respectively. G*(P, Q) is a function constructed by G(P, Q) on z1=0, and satisfies the present free-surface condition. Due to the hull surface condition of the double body flow,

(50)

therefore ƒÓF1(P)=0. On the other hand, ƒÓF2(13) does not

become 0 except the case of P= Q in the present formu-

lation.

For reference, consider the classical linearized theory.

Then it follows that

and ƒÓF2(P) becomes 0 from G*(P, Q)=0. ƒÓF1(P) can be

expressed as follows :

(51)

Eq. (51) coincides with the expression of line integral

term in Neumann-Kelvin problem. Thus, ƒÓF1(P)

remains in the classical theory.

In summary, the potential ƒÓ(x, y, 4 representing the

radiation or the diffraction flow is expressed as fol-

lows :

(52)σis determined together with the velocity potentialφ

on z=0 from the ship hull surface condition.

2. 5. 2 Deformaion Potential of Incident Waves due

to Steady Perturbation Flow

A velocity potential reprensenting the deformation of

incident wave (ƒÓA) is derived. First we introduce a

potential ƒµ(x, y, z) which equals to I(x, y) at z=0 as :

(53)

Then ƒÓA(x, y, z) is assumed to be expressed as fol-

lows :

(54)

Here FA(x, y, k, ƒÆ) can be determined by the same

manner as the function G is obtained as :


(55)

Substituting ( 55 ) into ( 54 ), OA is obtained as :

(56)

By using the wavy term of the function G (the third

term of eq. (35)) as denoted by G.(x, y, z; x1, y1, 0), ƒÓA

is rewritten as :

(57)

φA is represented by singularity distributions on z1=0

with the strength of I(x1, y1).

3. Calculations of Wave Pattern generated

by a Pulsating Source

Calculations were made of wave pattern generated by

a pulsating source moving with constant speed below

the free-surface as shown in Fig. 2. Here, in order to

know the basic feature of the present theory, the equa-

tion which corresponds to the second term of (51) is not

dealt with.

3. 1 Equation of wave pattern

The potential 0 for unsteady free-surface flow prob-

lem can be written as :

(58)

(59)

(60)

where

the immersion of a source is defined by f( •„0; its

coordinate of z-axis is z=-f). co is source strength

for double body flow and c the strength of the pulsating

source. Then wave elevation -(x, y) is expressed as :

(61)

For the comparison, calculations were made of wave

pattern in the classical linearized theory as follows :

( a ) Calculation by the classical theory including

double body flow :

φ0:double body flow is used.

φ:Green's function in the classical theory is used

instead of the present function G.

( b ) Calculation by the classical theory :

φ0:a uniform flow of velocity U is used.

φ:Green's function in the classical theory is used.

3.2 Calculation Method of Function G

For the calculation of wave pattern, we have to

calculate the function G and its derivatives with respect

to x and y. Here, Hoff's method 5) was applied to evalu-

ate the present function. An outline of Hoff's method is

as follows. The present function may generally be for-

mulated in terms of a set of double Fourier integrals

involving contour integration. The double integrals can

be reduced to single integrals using the complex

exponential integral. Formulations in terms of the sin-

gle integrals are given both as function of the ordinary integration variable as well as for the transformations

required to handle the singularities in the integration

range. The ordinary integration is numerically perfor-

med by using the Gauss-Legendre method. In such a

way, Hoff evaluated the Green's function in the classical

theory and its derivatives with good accuracy.

3.3 Results and Discussion

Calculations were made for 1=1, ƒÐ0= 10r, ƒÐ=4ƒÎ and

K0(=g/U2)=4.0. ƒÑ was chosen to be 0.224 and 0.274. The

point source is located at x=0, y=0, z= and the

calculation of the wave pattern was made in the region

with -0 •ƒx/f•ƒ 10 and 0 •ƒy/f•ƒ 10. Fig. 3 shows the

stream lines of the double body flow on z=0. It is well

Fig. 2 Problem of the wave pattern generated by a

pulsating source moving with constant speed

below the free-surface Fig. 3 Stream lines of double dody flow on z=0


known that the point source in the uniform flow repre-

sents the flow field around a semi-infinite submerged

body.

For verification of the program code, we carried out

the calculations of wave pattern by the classical linear-

ized theory. The linear calculation has been made by

Kobayashi8) and Ohkusu et al 12). Here, the comparisons

with the wave patterns presented by Ohkusu et al. are

shown in Figs. 4 and 5. The agreement with their results

is fairly good.

Figs. 6 and 7 show a contour and wave patterns at t =0 by the present theory and the classical linearized

theory including the double body flow. In the field far

away from the source point, the present wave pattern

agrees with that by the classical theory. However, the

present wave pattens near the source point is quite different from that by the classical theory. It is found

that in the present calculations, half-circular shaped

steep waves appear along a contour line of to =O. 25

which is limit of the wave propagation. When r is

smaller than 0.25 the steep waves appear behind the

source point, and when r larger than 0.25 the steep

waves appear in front of the source point.

It seems that the present wave patterns resemble the

tendency of the waves observed by Ohkusu : the

diffraction waves in front of bow of a full hull form

propagate obviously upstream even for r much greater than 0.25, and those waves propagate some distance

forward getting steep and break". Here, we try to

explain Ohkusu's observation using the present theory.

For the full hull forms, the region where a is smaller

than 0.25 always exists near the stagnation point. In the

region the diffraction waves can propagate upstream.

Then, as shown in the present wave pattern of Fig. 7,

steep waves appear near the line of ro =0.25 in front of

the bow of the ship hull, which may get too steep and

break. Thus, the present theory can explain the ten-

dency of the diffraction waves observed by Ohkusu.

4. Concluding Remarks

Under the assumption of small perturbation over the

double body flow, a theory was developed on free-

surf ace flow around slowly moving full hull forms in

short waves. A new function, which satisfies the free-

surface condition including the effect of deformed

flow due to the presence of the ship hull, was introduced.

And by using the function, an expression of the velocity

potential was obtained as the form of the surface dis-tributions. The present function coincides with the

Green's function in the classical linearized theory in the

field far away from the ship hull. Therefore, the present

theory can be considered as an improvement of the

classical theory in the expression of flow field around

the ship hull.

In order to know the basic feature of the present

theory, calculations were made of wave pattern gener-

Fig. 4 Comparison of wave patterns in the classical

linearized theory (ƒÑ=0.224, K0=4.0, f=1.0)

upper : present calculation

lower : Ohkusu and Iwashita

Fig. 5 Comparison of wave patterns in the classical

linearized theory (ƒÑ=0.274, K0-4.0, f=1.0)

upper : present calculation

lower : Ohkusu and Iwashita

150 Journal of The Society of Naval Architects of Japan. Vol. 170

ated by a pulsating source moving with constant speed

below the free-surface. As a result, it was shown that in

the region where a newly derived parameter ƒÑ0( =

U0ƒÖlg ; Uo is local velocity of the double body flow) is

close to 0.25, steep waves appear and the wave pattern

is quite different from that by the classical linearized

theory. The present wave patterns show the similar

tendency to the diffraction waves in front of a full hull

form observed by Ohkusun). Therefore, it can be said

that the present theory is useful for better understand-

ing of the free-surace flow in waves.

In this study it was found that a newly derived

parameter ro has an important role for determining the

characteristifcs of the waves. Needless to say, an exper-

imental verification of the parameter ƒÑ0 is necessary.

Further, to solve the boundaryvalue problems present-

ed in this paper is a future work together with the

improvement of the theory.

Acknowledgements

The authors would like to express their sincere grati-

tude to Professor Emeritus R. Yamazaki of Kyushu

University for his valuable comment and encourage-

ment. Thanks are also due to Dr. E. Baba and Dr. Y.

Kayo of Nagasaki Research and Development Center,

MHI, for their guidance and valuable discussions.

Thanks are also extended to all the members of the

Nagasaki Experimental Tank for their cooperation.

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Fig.6　 re　contour　 and　 wave　 patterns(τ=0.224,　 K0=4.0,

f =1.0)upper : ƒÑ0 contour on z=0

middle : present theory

lower : classical linearized theory including

double body flow

Fig.7　 re　contour　 and　 wave　 patterns(τ=O.274,　 K0=4.0,

1 =1 .0)upper : ƒÑ0 contour on z=0

middle : present theory

lower : classical linearized theory including

double body flow


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