Notes on Irregular Frequencies

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Notes on Irregular Frequencies

GP. Mlao and Y.Z. Llu

Abstract

The existence of the irregular frequencies is a kind of special phenomena in

solving the linear hydrodynamic problems for floating bodies in the

frequency domain by the source-sink distribution method. lt was pointed out in the past investigations that the phenomenon comes from the continuation of the outer fluid field to the imaginary inner fluid field.

Although the existence of irregular frequencies does not seriously influence the applicability of the source-sink distribution method to normal seakeeping calculations, it may have evident effects on the computational results of the

second-order steady wave drifting forces and bi-harmonic wave forces as

shown in the present paper.

Based on the previous investigations, this paper briefly describes the

method to remove the irregular frequencies and further point out the

necessity of the removing in the computation of the second-order wave

forces.

I.

Introduction

Nowadays, pulsating source-sink distribution methods are widely used in

solving the radiation problem and the diffraction problem, which relate to the harmonic oscillations of floating bodies on the water surface, still or wavy. lt

has been aware long before, however, that the methods are invalid for a series of discrete frequencies[11l[8]'l18H20]. No reliable numerical results could be obtained at these frequencies or in the narrow frequency bands

nearby them. These special frequencies are usually called irregular

frequencies.

lt was pointed out in the past investigations that the

phenomenon comes from the continuation of the outer fluid field to the inner

imaginary fluid

field. When we derive the source-sink distribution

expressions for the outer field potentials from the Green's theorem, it has

been implied that the inner region of the floating body is filled of fluid and the

motion of the inner fluid satisfies also the linearized free surface condition.

Hence, a series of discrete eigen-frequencies exists. Resonance will happen when the frequencies of external disturbances are near them. This effect will

reflect to the outer fluid field through the continuation condition on the body

surface , which regulates that the potentials of inner and outer fluid field are equal there, and lead to the source-sink distribution solutions of the outer

fluid field invalid.

Nevertheless, the source-sink distribution methods are still being widely

used in the linear hydrodynamic computations. The main reason may be that

the lowest irregular frequency is usually higher than the frequency range in

which we are interested and the influence of the irregular frequency is

limited to a narrow frequency band around the irregular frequency

concerned, which may be' avoided in the computations. The hydrodynamic

characteristics at the irregular frequencies may be obtained by the

interpolation of the computational values at the nearby frequencies.

Furthermore, it

is also one of the reasons that compared with the other

methods, such as the multipole expansion method etc., the source-sink

distribution method is more flexible and suitable for various body shapes.

However, the computational results of the present paper show that the

influence of the irregular frequencies is rather serious for the second-order steady wave drifting forces and the second-order bi-harmonic wave forces.

We need an effective means to remove the irregular frequencies rather than to avoid them in the computations.

li.

_{Generation of irregular Frequencies and Their Removal}

The linear radiation or diffraction problem for two-dimensional bodies is

considered here. As indicated in Fig.1, the fluid boundaries consist of the

body surface C, still water surface CF, C00 at the infinity on the left and the right sides and the horizontal fictitious plane CB far down the water surface. The disturbed velocity potential ( x,y,t) in the fluid field D may be written as

-kot

cb(x,y,t) =1Re{

(x,y) e

_}

(1)

where Re{ } denotes the real part of the variable,

is the oscillating

frequency. The spatial potential 4( x,y

₎

is a harmonic function, which

satisfies the linear wave type free surface condition on the still water surface,

i. e.

and the radiation condition

ik4 = O

ax

onC

±00

(3)

in which k= o)2/g is the wave number. Besides, _{is also required to satisfy}

the body surface condition and the condition on the infinite deep plane _CB, i.e. V = O there.

lt has been proved that if the Green's function G(P,Q) ¡s chosen to satisfy the

same boundary condition as 4(P) except the body surface condition, the

Green's third theorem gives

JE

(Q) aG(P,Q) -

G(P,Q)

ad(Q) _J dl

-2,t

(P) PE D

- it(P) PEC5

(4) an C O

PçDC

s

-where the integral is taken over the body surface C. We now assume that

3

the inner region D of the body is also filled of fluid with the velocity potential

( x,y,t ) = Re{ ( x,y)

e0t

}. The inner fluid field D1 is bounded by the body

surface Cs and the inner free surface CFI (the still water surface y O). By

using the same Green's function G(P,Q) as introduced above, the Green's

third theorem gives

f

aq

f [4(Q)

aG(PIQ)...Gi]dl

_an

G(P,Q) [km. ]d

C _CR

2ir.(P)

PED1 = 1<

q(P) P C5

CF. (5) O

PDj+Cs+CFI

Subtracting (4) from (5), we have

a.

J [(.-)

i an

G(

an an

)]dlfG(k.

_ai1 C _CR

2ir(P)

PED

PECE

2it.(P)

PED.

it4.,

_PECF.

If we let 4(Q) = 4(Q) on the boundary C, let

k.- L

a1 on CFI and wnte

(6)

(9)

a.

+

J

G(P,Q) _L dl

_an

=

f 4.(Q)

L aG(P,Q)

an

dl, PED.

When P approaches to the body surface along the normal of the body from

the inner fluid field D, the normal derivative of the above equation becomes

5

an

an

(8)

on Cs, the velocity potential in the outer fluid field can be written as

(P) = _{(Q) G(P,Q) dl} 2ir

s

where 4(P) is represented by the distribution of sources (and sinks) on the body surface, the source density c(Q) will be determined by the the body surface condition. Usually, the normal derivative f(P) of the potential on the body surface is given. Hence, the following boundary integral equation can

be formed, i.e.

.ta(P) -f-i-f

c(Q) aG(P,Q) d!(Q) = f(P) , PeCe (10)

2

Once y(Q) is obtained from the integral equation (10), we may determine the velocity potential everywhere in the fluid through the equation (9).

lt has been proved by John[fl that the solution of the outer fluid field should uniquely exist. So we can only find reasons why the irregular frequencies would appear in the integral equation (10) from the source-sink distribution expression. In

fact, no irregular frequency exists when the multipole

expansion method etC-E16] [17]

are used to solve the boundary value

problem for the outer fluid field.

On the other hand, if 4 satisfies the free surface condition (7), the equation

4(P)

+ __

a(Q)

aG(P,Q) _{dl =}

f

G(P,Q) dl

anan

2 an

25

nP 21tC PECS

Its left hand side has the same form as the left hand side of the equation

(10). Obviously, the irregular frequencies of the equation (10) will also

appear in equation (11). In fact, eigen-values exist for the problem of fluid oscillating

in an open vessel of finite volume. The eigen-functions

correspond to the non-trivial solution of the homogeneous form of the

equation (11), which is obtained by using 4 = O on C. These eigen-values

are a series of discrete frequencies, known as the irregular frequencies.

Several methods have been put forward in order to remove the irregular

frequencies, in which can be listed the multipole expansion methodl16], _the

eigenfunction expansion method[171, the null-field method[l, the method by locating source or dipole on the origin of the inner fluid field[1 8] _{and the} method by adding a fictitious lid on the still water surface of the inner _fluid

field[6]. For revising the existing computer program based on the source-sink distribution method, the last treatment is the simplest means, which does not

require the inner potential _{to satisfy the free surface condition (7). In this}

case, we get

(P) =

-i--f

y(Q) G(P,Q) dl +

f

(Q) G(P,Q) dl (12)

a.

from equation (6), where

on CF

has been let equal to k4

-Compared with expression (9), we may find that the main difference is the appearance of the source-sink distribution on the inner still water surface _CFI A new boundary integral equation similar to equation (10) can be obtained with the normal derivative f(P) on CFI equal to v(x). Ohmatsu[6] pointed _out that the boundary condition on CFI may be written as

onC.

_Fir.

(13)

where v(x) can be defined quite arbitrarily, the only demand for v(x) is that an even function should be regulated for symmetrical fluid motion and an odd

III.

_{The Effects of the Irregular Frequencies on the}

Second-Order Wave Forces

The interaction between non-linear water waves and offshore structures has

drawn much interests recently. An effective way to treat the non-linear

problem with small wave steepness or small oscillation of body is by using

the perturbation method. If we define the wave steepness as a small

parameter e, the velocity potential _{may be expanded as a power series of}

e, i.e.

=

cci) +e22 +

and the original non-linear boundary value problem for

reduces to a series of linear boundary value problems of j() with different order of _{e. The} hydrodynamic forces up to the second-order are considered here, which already exhibit some interesting features observed in offshore engineering.

For example, the wave forces exerting on bodies in regular waves include

the second-order steady wave drifting forces and the

_second-order

bi-harmonic wave forces besides the first-order

wave forces

( the linear

solutions ) with zero mean and the oscillating frequency as that of the

incident waves. The oscillating frequency of the bi-harmonic wave forces equals to the twice of the incident _{wave frequency. These non-linear wave}

forces are of importance for the analyses of the maximum forces, fatigue and springing vibration of offshore structures.

In the diffraction problem of regular waves on a fixed two-dimensional body, it can be proved that the velocity potential c1 _{in the fluid field and the wave} forces F may be expressed as

-2iût

1( x,y,t ) = Re{

e)t+

_{c242 e}

_}

-ROt ₂ -2icot

F(t) = Re{ eF.1(t) e

+ e { F2 + F2 e

_{J }}

(15)

respectively, where i (=1,2,3 ) denotes the direction of the wave force, i.e. F1 represents the swaying force, F2 represents the heaving force and F3 the

rolling moment. The superscripts (1) and (2) denote the first- and the

second-order parts of the variable respectively and the subscript s denotes the steady component of the wave forces. Each force component may be written in non-dimensional form as follows, i.e.

f(l)=

F1 ¡ _pgaB ko gaB Cs (1) _{n. dl} F2

-(2) S E (1)X (1)X + (1)y (1)_y

] n dl

i '

-pga

2ga2 c k

- E

(1) (1) + (1) (1) L 2) 2 = 2ko 5 (2) dl

5

[q)(1)2

_{I n. dl}

pga ga _{4ga2 c} k

_[(1)2n

I

'+

(1)2n I _L 4ga2

in which the first-order potential (1)_{contains both the regular incident wave} potential and the first-order diffraction potential Jl) , a is the wave

amplitude of the incident waves, B is the breadth of the 2-D body, n1 is the projected component of the unit outward normal of the body surface Cs in the i-th direction, Subscripts W and L denote that the attached variable is

taken the value at the intersection of the body with the still water surface on

the weather side and lee side respectively, and (1) denotes the complex

conjugate of1).

is the solution of the linear diffraction problem, which may be solved by the source-sink distribution method as outlined above. The right hand side of

equation (10) now becomes to -

a4'/an

, in which is known. The first-order wave forces (i.e. the linear part of the wave forces) exerted on

the body can be obtained by equation (16) onceq(1) is completely solved.

2ga

9

Fig.2 shows an example of the first-order wave forces on a two-dimensional cylinder with Lewis section. The non-dimensional half breadth of the section is 1.0, draft is 0.8 and the sectional area coefficient is 0.95. lt is obvious that

the heaving force has one of the irregular frequencies around k = w2/g = 1.85.

The second-order steady wave forces can be determined by integrating the first-order potential (1)

and its coordinate derivatives on the body surface according to the expression (17). The computational results of the three

components of the second-order steady wave force on the above mentioned

Lewis section are shown in Fig.3-Fig.5 respectively. The existence of

irregular frequencies can be observed in these figures.

lt is obvious that the second-order diffraction potential q(2) is needed for the

computation of the second-order bi-harmonic wave forces, as well as the first-order potential. By the perturbation procedure, it is easy to know that

q(2) _{must satisfy the non-homogeneous linear free surface condition}

)(2)

4k

(2)

= P(x)

ay

ony=0

(19)

besides the Laplace equation, radiation condition and the condition at the infinitely deep boundary. P(x) is a known function in (19), which includes the

second-order coupling terms of the first-order potential and its

partial derivatives to the coordinate variables. Furthermore, 4(2) needs to satisfy the

body surface condition as a2 ia

n = 0.

If the solution of (2) can be found, which satisfies the condition (19) without

imposing the body surface condition, then we may seek another _{solution of} (2) _{, which satisfies the corresponding homogeneous form of (19) and the}

body surface condition a/an

_{= - a2/an ,}

so that the sum of

q(2) and 42) forms 'the required second-order _{diffraction potential q(2). it is}

the first-order diffraction potential 41) from the mathematical points of view, but with wave number K (=4k) instead of k.

The solution of

can also be obtained by means of the source-sink

distribution method. One can expect that when the incident wave number k

tends to 0.463, the wave number K (=4k) for

cj(2 tends to one of the

irregular frequencies 1 .85. As mentioned above, the irregular frequencies

are a series of eigenfunctions, so far when the incident wave number k

varies, the irregular frequencies would appear more often. Fig.6 to Fig.8

show the computational results of the three components of the second-order bi-harmonic wave forces, from which we can obviously see the existence of the irregular frequencies and how often they would appear.

Now we may clearly observe the serious influence of the irregular

frequencies on the computational results of the second-order wave forces. For example, according to the theoretical derivation of Maruo[13l, the

non-dimensional value of horizontal steady drifting force tends to 1.0 as

frequency increasing. However, the computational result without removing

the irregular frequencies shows a complete different tendency, as shown in Fig.3. lt

is also shown in Fig.3 that the computational result is largely

improved when the method of introducing a fictitious lid on the inner free surface is adopted to remove the irregular frequencies, which proofs the

effectiveness of the method. Other examples are shown from Fig.4 to Fig.8,

in which the solid lines denote the computational results after removing the

irregular frequencies. We may notice from the comparison of two

computational results shown in Fig.2 that the frequency bands influenced by

the irregular frequencies are rather narrow for the linear problem. In real

computations, we may simply avoid these frequency bands, and the values

in these bands may be obtained without unacceptable errors by

interpolating the computational results nearby. In fact, such a treatment has been used in practice.

However, the serious influence of irregular frequencies to the second-order

wave forces excludes us to adopt such a treatment. For example, as shown

in Fig.3, when the irregular frequencies exist, the computational results

exhibit obvious errors at almost all frequencies. After removing these

irregular frequencies, the computational results coincide with the Maru«s

conclusions and the experimental results. Furthermore, for the computation

of the second-order bi-harmonic wave forces, the appearance of the

irregular frequencies will be more often, and the influence bands of which

may approach closely and even overlap each other. Not only would the

obvious numerical errors be led in the whole frequency range but also the

tendency of the numerical results be deformed completely, as shown in Fig.7 and Fig.8.

We may conclude from the above examples that it should be considered to remove the irregular frequencies in the computation of the second-order wave forces by using the source-sink distribution method, otherwise no

IV.

Concluding Remarks

lt is revealed by the present computational results that the effects of irregular frequencies are limited only at these frequencies or within the narrow bands

around these frequencies for linear hydrodynamic problems, which agrees

with the previous work of others. However, the effects of the irregular

frequencies are rather serious for the non-linear interaction of water waves

and the bodies, as shown by the present paper. Special treatment should be introduced to remove the irregular frequencies.

The computational tests of the authors confirm that the fictitious solid lid at

the free surface of the inner fluid field can effectively suppress the resonance of the inner fluid, hence, remove the irregular frequencies. This treatment is

easy to be done by modifying the original computer program of the

source-sink distribution methods.

References

John, F., "On the motion of floating bodies, Il, Simple harmonic _motions ", Comm. Pure and Applied Math. 3, 1950, pp.45-101.

Frank, W., " Oscillation of cylinders in or below the free surface of deep

fluids ", Report No.2375, Naval Ship Res. & Dey. Center, _Bethesda,

MD.,1 967.

Potash, R.L., " Second-order theory of oscillating _{cylinders", J. of Ship} Research, Vol.15, No.4, Dec. 1971, pp.295-324.

Kan, M., " A program for the hydrodynamic pressure acting on a

cylinder with an arbitrary section oscillating in still water ", Rep. Ship

Res. Inst., Vol.11, No.1, 1974.

Report of the Seakeeping Committee, 17th hIC Conference.

Ohmatsu, S., " On the irregular frequencies in the theory of oscillating bodies in a free surface ", Papers Ship Res. Inst., No.48, 1975.

Ogilvie, T.F. and Shin, Y.S., " Integral-equation solutions for

time-dependent free surface problems ",J. of the Society of Naval Architects of Japan, Vol.143, 1978.

Ursell, F., " Irregular frequencies and the motion _{of floating bodies ",J.} of Fluid Mech., Vol.105,1981, pp.143-1 56.

Liu,Y.Z. and Miao,G.P., " An introduction of _{hydrodynamics for naval}

architecture and offshore engineering _{(in Chinese) ", Lecture Notes,} Shanghai Jiao Tong Univ. Press,1 983.

Wehausen, J.V. and Laitone, E.V., " Surface waves ", Handbuch der

Physik, Band 9, Springer-Verlag, Berlin, 1960.

Martin, P.A., " Multiple scattering of surface water waves and the

null-field method ", 15th Symp. on Naval Hydrodynamics,

_{Session III,} pp.1 4-27,1984.

Miao,G.P. and Liu,Y.Z., "A theoretical study on the second-order wave

forces for two-dimensional bodies ", Proc. of the

5th International Symposium on Offshore Mechanics and Arctic Engineering, _Vol.1,

pp.330-336, Tokyo,1 986.

Maruo, H., "The drift of a body floating on waves ", J. of Ship Research, Vol.4, No.3, Dec. 1960, pp.1-10.

Kyozuka, Y., " Experimental study on second-order forces acting on a

cylindrical body in waves ",

14th Symp. on Naval Hydrodynamics,

Session Ill and IV, Michigan, U.S.A., Aug.23-27, 1982, preprints

pp.73-136.

Fang, Z.S.,

"

Source and dipole distribution method for the

hydrodynamic forces on large offshore structures (in Chinese) ", Ocean Engineering of China, to be published.

Miao,G.P., " On the computation of ship motions in regular waves ",

Division of Ship Hydrodynamics, Rep. No.58, Chalmers Univ. of

Technology, Sweden, 1980.

Miao,G.P. and Liu,Y.Z., " Hydrodynamical coefficients of a column with

footing in

finite-depth waters ",

Proc. of the 3th International

Symposium on Offshore Mechanics and Arctic Engineering, Vol.1,

pp.199-205, New Orleans, U.S.A., 1984.

Sayer, P. and Ursell, F.," Integral-equation methods for calculation the

virtual mass in water of finite depth ", Proc. 2nd mt. Symp. on

Numerical Hydrodynamics, Berkeley, California, 1977.

Dai, Y.S. and Wang, T.S., "The prediction theory and methods for ship

seakeeping performance in waves (in Chinese) ", Lecture Notes,

Harbin Institute of Ship Engineering, 1980.

Inglis, R.B. and Price, W.G.," Irregular frequencies in three dimensional source distribution techniques ", I.S.P., 1981.

CF w-o

_CFj

L ¡ D1 C

s

1 If(1)' \ \ \

I1)I

/

_/

/

_/

/

_/

/

o

N

\ N

theoretical results (without irregular

frequencies)

theoretical results (with irregular

frequencies)

Fig.2

The 1st-order wave forces on the Lewis section

0.5

lb

20

2.5

1.0 f(2) si 0.5 o 2.s

//ò

-I

/.

f

theoretical results results results without I.F) _{(with I.F)} 114J

- - theoretical

experimental 05 10

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Fig.3 The steady horizontal wave drifting forces

on

f(2) s2

I.,

f(2) s3

o

0.5 0.5

theoretical results (without I.F) theoretical results (with I.F) experimental results {14J

s 1.0 s s p 20 kb

Fig.4 The steady vertical

wave forces on

the Lewis section

10

t.5

20

kb

Fig.5 The steady heeling

wave moments on

the Lewis section

2.5 2.5

theoretical experimental results results

results

(without (with I.F)

(143 I I.F)

____ - theoretical

s

-I f(2)

12

(2) f1 i

I

;

II

Fig.6

The 2nd-order biharmonic vertical wave forces on the Lewis section

I

Fig.7 The 2nd-order biharmonic horizontal wave forces on the Lewis section

-I. I

theoretical results (without

I.F)

- theoretical results (with

I.F)

experimentl results [14)

2 4

theoretical results (without

I.F)

theoretical results (with

I.F)

experimental result [14)

o 0.5

13

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0.2

(2)

f3

0.1

Fig.8 The 2nd-order biharmonic wave moments on

the Lewis section

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