Wave generation and absorption by means of completely submerged horizontal circular cylinder moving in a circular orbit - Fundamental study on wave energy extraction -

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PAPERS

OF

SHIP RESEARCH INSTITUTE

Wave Generation and Absorption by Means of Completely Submerged Horizontal Circular Cylinder

Moving in a Circular Orbit

-

Fundaniental Study on Wave Energy Extraction

-By

Takeshi FUWA

October 1978

Ship Reseach Institute Tokyo, Japan

Lab... v. Sheepsbouwkunde

Technische Hogeschool

WAVE GENERATION AND ABSORPTION BY MEANS OF COMPLETELY SUBMERGED HORIZONTAL CIRCULAR CYLINDER MOVING IN A CIRCULAR

ORBIT

FUNDAMENTAL STUDY ON WAVE ENERGY EXTRACTION -By

Takeshi FUWA**

* Received on May 8, 1978. ** Ship Dynaiñics Division.

CONTENTS

SYNOPSIS 2

INTRODUCTION 2

THEORETICAL INVESTIGATION 4

2.1. Formulation of the PrOblem 4

2.2. SolutiOn Of the Problem and Theoretical Investigations 6

2.3. Submerged Circular Cylinder 8

EXPERIMENT 11

3.1; Experimental Apparatus - 12

3 2 Wave Generation Experiment 16

3.3. Wave Diffraction Experiment ...17

3.4. Wave Absorption Eperiment 26

4 COMPARISON OF THEORY AND EXPERIMENT 28

4.1. Numerical Results, . 28

4.2. Comparison of Theory and Experiment . ₃₂

4.3. Feasibility of the Rotatory Circular Cylinder System as a Device for

Wave Energy Extra tion ₃₈

5. CONCLUSION ₄₁

ACKNOWLEDGEMENT ₄₂

SYNOPSIS

In order to investigate the feasibility of the completely submerged circular cylinder rotating about a circular orbit as a device of wave ab-sorption and wave energy conversion theoretical and expenmental studies were carried out.

The application of the water wave theory to the problem showed the possibility of perfect wave cancellation and energy extraction for the regular incident wave. The conditions for the perfect energy con-version from the incident wave and some remarkable hydrodynamic

characteristics of the device wete presented.

Though the wave absorption experiment was not successful, ex-perimental results confirmed the characteristics of the device and new

knowledge about the circular cylinder were obtained.

The feasibility of the device at this stage and the further direction of research are discussed.

1. INTRODUCTION

It was not long ago that sea waves were generally thought of as only troublesome and ocassionally terribly powerful disturbances for ocean

going vessels and coastlines. On the other hand, the goal of utilising ocean

wave energy has been a challenge to ocean engineers for many years.

Various methods and mechanisms have been tried to capture the wave energy including the use of spring-mass-damper systems, pneumatic and

hydraulic devices. l)

As the shortage of conventional energy sources becomes evident and the demand for alternative energy sources increases, the projects on wave

energy extractiOn become more and more practical on a large scale.

From the hydrodynamical point of view, one of the most interesting

challenges is Salter's nodding duck. 2, 3) It makes use of its special charac-teristics for wave making and diffraction. The duck generates an outgoing

wave in one direction only, when it nods in calm water. And the duck does not reflect the incident waves because of its special shape. These features of the duck enable the Wave absorption and energy extraction.

A completely submerged horizontal circular cylinder rotating about a circular orbits has similar wave making characteristics, and moreover

it has special diffraction characteristics both for reflection and trans.

mission. 4) Therefore the submerged circular cylinder has better

possi-bilities as a device for Wave absorption and energy conversion than Salter's duck.

In order to investigate the feasibility of this type of mechanism,

theo-retical and experimental studies have been done. The investigation consists

of three parts They are wave generation, wave diffraction and wave

ab-sorption/Wave energy extraction

character-istics of the cylinder are discussed. It is shown that the completely sub-merged circular cylinder generates outgoing waves in one direction only

when it rotates along a circular orbit. It is also shown that a circular

cylin-der fixed beneath a free surface in infinite depth does not reflect the incident wave but the diffraction effect can only be seen in the phase shift of transmitted wave. Energy consideration tells that the rotating

circular cylinder absorbs the regular incident wave perfectly if the phase

and amplitude of motion are suitably chosen. The conditions for

ab-sorption are shown.

A small tank with wave generators was used for the experiments.

The wave generator composed of a horizontal circular cylinder which moved along a circular orbit. Two wave generators of same design were used for the wave absorption. One is for generating incident waves and the other for absorbing the incident wave by means of cancellation

Ex-perimental results proved the special characteristics of the. circular cylinder

for wave generation as well as wave diffraction. The wave absorption experiment, however, was not successful mainly because of difficulties

on the mechanical side.

Comments on further research and practical feasibility of the system

are discussed.

Although there is no actual design in detail, the following draft will be considered as a practical model of the wave energy extraction system.

The system is composed of: Monitor and Controller

Circular Cylinder(s)* and Driving Deyicés Regulator

Electric Generator

Measuring and analysing the incident waves, the moiitor detects

the dominant cornpoijnt of the incident wave with its wave length, wave amplitude and phase. The controller produces the control signal to the driving device in order to satisfy the cancellation conditions.. The driving device rotates the cylinder according to the control signal, adapting the

radius of rotation and the submergence of the cylinder. The regulator

controls the rotation of the cylinder with reference to the control signal. Once the operation of optimum condition is perfOrmed, it is. thought

that less power is necessary to maintain the rotation. Therefore the regula-tor is preferable in addition to the main driving and control device. So long

as the rotation is maintained by a small power supplied to the regulator,

the wave power will rotate the cylinder and electric generator will convert thefl wave energy into electric energy. In fig. 1.1 the. concept of the system is shown.

PLATFORM.

/1

WAVE MONITOR -ROTATORY CYLDDER. ELECTRIC GENERATOR. REGULATOR. -MOORING CABLES.

Fig. 1.1. Practical Model of the Wave Energy Extraction System.

2 THEORETICAL INVESTIGATION

Water wave theory gives us .a lot of information about wave proper-ties, wave generation and diffraction characteristics of bodies in water. In order to apply the water wave theory to the wave energy extraction

problem, the theoretical formulation and solution of the problem is

presented bnefly The numerical results are compared with experimental

results and discussed later.

2,1. Formulation of the Problem

It is customary to call the generation of waves by forced oscillatory motiOn of a body in an otherwise undisturbed fluid a "radiation problem" and the interaction of incident waves with the fixed body a "diffraction

problem". The combination of the two covers the problems of the response

of the body floating or moormg to the incident waves and the forced

oscillatory motion in the incident waves. The surface wave problem either

radiation or diffraction is formulated as a linear boundary value problem

of Laplace's equation with regard to the velocity potential after some

assumptions. 5,6)

Assume that a long horizontal circular cylinder is situated beneath

[F] : Free surface condition

--+Kc5=0

_ay

K w2/g

[co I: For field condition

at y=O (4)

x

IB] BODY BOUNDARY COND.

IL] LAPLACE'S EQ.

INFINITY COND.

Fig. 2.1. Governing Equation and Boundary Cônditibns of, the Problem..

(RI

axis and harmOnic with respect to time.

The velocity potential 0 (, .y, t) is governed by Laplace's equatiOn

throughout the intenor of the fluid domain with some boundary

con-ditions. (Fig. 2.1)

O(x, y, t) =Re[çb(x, y)eiu] (1)

[LI: Laplace's equation

0 in fluid domain

()

[B]: Body surface condition

(3) where Vn is the normal velocity of the body surface fOr a radiation problem and zero for a diffraction problem.

'- 0

.y +

(5)

(RADIATION PROBLEM]

A+

0

RADIATION WAVE RADIATION WAVE

FORCED OSCILLATION

y

Fig. 2.2. Radiation Ptoblem

]DIFFRACTION PROBLEM] x INCIOENT WAVE AT TRANSMITTED \WVE FIXED BODY

Fig.. 2.3. Diffraction Problem.

[RI: Radiation condition

0 A±e11

AR

REFLECTED V/AVE

z* ±

for a radiation problem

x-+ +qo

x.-,- -'

(6)

I

A(e+Re)

1

Te'

for a diffraction pmblem (7)

where A+ are complex coefficients, representing the wave arnplitude and phase at infinity R and T are reflection and transmission coefficient respectively. A is the amplitude of the incident wave. (Fig. 2.2, Fig. 2.3)

2.2. Solution of the Problem and Theoretical Investigations

For the solution of the problem any method of Solving general bOunda-ry value problem is available, but there are three different methods known

to be applied surface wave problems They are field equation techniques

e-quation methods, 8,9) 'and series solution methOds. The tter two are

popular for naval hydrodynamists Ursell's method 10) and Gnm's methodii) both belong to the series solution.

Let G(x, y; x', y') be the Green's function which satisfies EL], [F], [oo] arid [RI, then the genral solutior is given by

q5(x, ,i) ----ja(x'. ii') G(x, y; x', y' )dc (8)

or

a1G(

of)

c(x,y) ' ' (9)

The solution (8) is obtained by integral equation method 'and (9) by

series solution. ,(x' y') represents the strength of source distributed on the body surface and Cu is the strength of higher order singularities

at the centre of the body.

The Green's function of the two-dimensional surface Wave problem

for harmonic oscillation is given by

1

(x-x')2+(yY')2 21ec0skx)dk

G(x,y;x,y)_1n(x_x)2+(y+y)2+ J

kK

-_2ire''C0SK(XX')

(10)

For the wave problem of the completely submerged circular cylinder, Ogilvie (1963) obtained the complete numencal solution according to Ursell's method. Frank (1967.) and Maeda (1974) presented numerical

examples by means of the integral equation method for the radiation

problem.

Though the radiation problem and the diffraction problem are

differ-ent boundary value problems, there are some interesting and practically

important relations such as the famous Haskind, Hanaoka and Newman's relations for the linearized problem.'2.13) The relation has no benefit for the calculation of pressure distribution of the diffraction problem. How-ever, hydrodynamic forces and moments, reflected waves and transmitted

waves for the diffraction problem are calculated from the solution of

the radiation problem.

The objective of this paper is wave cancellation and energy extraction

from waves Therefore relations between incident wave, reflected wave and transmitted wave and cylinder motion are mainly investigated and hydrodynamic forces and moments are not treated.

General expression for the relation between wave amplitude ratio

in 'the radiation problem and transmitted coefficient and reflected

A±AR+AT= 0

where an asterisk denotes a complex conjugate and A± are complex coef-ficients, representing the wave amplitude and phase at infinity m a radi-ation problem, and R, T are complex coefficients representmg the

ampli-tude and phase of reflected wave and transmitted wave in diffraction

problem.

When wave absorption or wave energy extraction by means of an

oscillatory body motion is considered, the efficiency of the absorption is measured by the balance of wave energy transported by incident waye, reflected wave and transmitted wave. Efficiency of wave absorption is

E=1R1R TT

(12)

R1, and T1 are complex coefficients of reflection and transmission for the condition the body is moving in the incident waves. Assumed linear

superposition principle, one finds

N

R1 =R+A+

_{i= 1}

T1 =

T+EA-I?, T are reflection and transmission coefficients for the fixed body, and A± are wave amplitude ratio fOr i-th mode of motion when there are N possible modes of motion for the body. are complex variables which expresses the amplitude and phase of z th mode of body motion the phase is measured with respect to the incident wave.

When the configuration Of the wave absorbing mechanism is given,

its characteristics for wave generation and scattering, are determined. The

solutions of the radiation and diffraction problem show those

character-istics Now the mode of the motion is the only freedom left, and the

motion which gives optimal Wave absorption is considered. It is given by the condition

(15)

i=1,2,

N

a2E

_=0}

2.3. Submerged Circular Cylinder

For the completely submerged circular cylinder, Evans (1976) 14) and Mei (1976) 15) _{treated the spring-damper system and showed that perfect}

wave absorption is possible in which the centre of the cylmder moves a circular orbit Here the system which consists of submerged cylinder

Further Ogilvie's results and other nUmerical results fOr the submerged circular cylinder show

A1+ = iA2+ ₍₂₀₎

Substituting (18), (19) and (20) in (16) and (17) gives

AR 0 ₍₂₁₎

AR+

= 2e'Ai- = 2e+A2_

₍₂₂₎

and (21) show that the progressive wave exists on only one side of

the cylinder and that the wave amplitude is tWice that of heaving Or sway-ing motion.

The Other indirect proof is giVen here after which is based on the

rotating along a circular orbit is examined. As the resu1, it is shown that perfect absorption of incident waves by the rotating cylinder is possible.

It depends on the special charactenstics of the cylinder for both wave generation and wave diffraction. The completely submerged circular

cylinder generates progressive wave in one direction only, if the cylinder

centre follows a circular orbit Diffraction of incident waves by submerged circular cylinders fixed in the water of infinite depth occurs in such a way

that there is no reflection wave but transmitted wave has a phase shift

after passing the cylinder.

These two special properties allow perfect cancellation of transmitted

wave by radiated wave in downstream and no generation f outgoing

wave in upstream when suitable phase difference between the incident

wave and the motion of the cylinder is imposed Ogilvie (1963) investigated both of the properties Here different ways of investigation are undertaken

Because rotating motion along a circular orbit c is expressed by the

combination of vertical oscillation and honzontal oscillation E2 with

phase difference of r/2, the motion and wave amplitudes AR± are

expressed by (when anti-clockwise rotation)

= (I+E2)/(l+E2'Efl,

El =iE2 (16)

AR±EC = A1±E1 +A2±E2 (17)

Symmetry of circular cylinder shows that

A i+ = A1-A2+ =

(18) (19)

property of the special Green's function for travelling doublet along the

circular orbit. 16) The Green's function is given by

GR(x,y;p,f) =

_4En

X:

ijj: ±2pICOS(K2-0)t)

2j

e' cos kx

dk

In the far field urn GR(x,y) =O

x-+

1imGR(x, y) =

4,tpKe'COS(KXWt)

Surface elevation is given by

1 dq5

at yO

(26)

Provided the flow field around the circular cylinder is expressed by a

doublet, (24), (25) and (26) show that progressive waves exist in one side

of the cylinder only. Similar investigation into heaving and swaying motion

indicates that progressive wave amplitude by rotation is twice as much as

that of heaving or swaying.16)

The remarkable feature of the diffraction by the submerged circular c'linder was discovered by Dean (1948) 17) through theoretical

investi-gation. Ursell (1950) 18) gave more rational basis and Ogilvie(1963) ob-tamed complete numerical solution for the phase shift of the transmitted

waves. Here, the relation with the radiation problem is applied to the

investigation of the diffractiOn problem. 13) Applying the relation (11) to the heaving and swaying with the relation (18) and (19), one finds

(29) and (30) show that there is no reflection wave and that phase shift

of the transmitted wave is twice as much as phase difference between

progressive wave and heaving motion in radiation problem. Both coincide

R+T= A1+/A= exp(2iarg4i+) (27)

RT A2+/A+

exp(2iargA2+) further (20) leads to (28) R=O ₍₂₉₎ T= exp(2iargAi+) exp(i,9) (30)

with the results of Ogilvie's investigation.

For the Wave absorption, the efficiency E defined by (12) is investi-gated When the circular cylinder rotate along a circular orbit, (13), (14),

(21), (22), (29) and (30) shoW

R1=O ₍₃₁₎

T1 = ep(iI9)+AR_Ec = exp(i8)+AREcexp{i(a'+7)}

Where i7

çccce

-. A - A

-I-IRe then

E iT1T

A-2ARCcbs(/3---a-7)

(35) (15) leads = 0 = 0 (36) as the results =1/AR (37) y=a+(2k±1)r, k:integer (38) substitute (37) and (38)in (35) Emax = 1 (39)

(39) shows that perfect absorption of the incident wave is possIble, and

(37) and (38) give the condition of it

3. EXPERIMENT

In order to investigate the feasibility of the system for the wave ab-sorption, experiments were carried out using a small tank in the Hydro-dynamics Laboratory of Glasgow University. The experimental study

consists of three different kinds of expenments They were wave generation,

submerged circular cylinder.

In the first section the expenmental apparatus is descnbed After

that description of each experiment is shown with its results. 3.1. Experimental Apparattis

A small tank (3m x O.76m x O.23m) was equipped with wave

generat-mg devices and wave probes for the expenments There were a beach or

wave absorber made of porous plastics at both ends of the tank. On the side

wall of the tank there is a glass window through which the wave profiles were observed and their photographs were taken Three wave probes of resistance type were used for the measurement The accuracy and lineanty of the probes were satisfactory. The arrangement of the tank is shown in

fig. 3.1 and photos 1 &2.

2w

WWW

>zn-<Wx

zI-oz

iw

wuJ

>ucL z

wow

BEACH IAMERA FIXED

CAMERA _CYLINDER WAVE.

-GENERATOR

N° 1 N° 2

WAVE WAVE CAMERA GENERATOR GENERATOR

Fig. 3.1. Arrangement of Experimental Apparatus.

2m - im 75 cm 15cm 50 cm N°3 WAVE PROBE -NO.1 WAVE PROBE N°2 WAVE PROBE /v. -

I.

50cm 50cm 50 cm-N°1 W.P. N°2W.P. W.P.N°3 - 125 m m 75cm WAVE WAVE GENERATOR ABSORBER 1:25.m .lm 75cm 75 cm 50 cm 50 cm 50 cm N° 1 N° 2 N° 3 W.P. _w.P. W.P.

I

ci

27C IY1

Photo 1. Experimental Apparatus.

Photo 2. Tafik and Wave Probes.

The wave generating device is shown in flg. 3.2 and photo 3. It is

composed of a rotating circular cylinder [1], discs at both side [2], and belt drive [3] Along the slits in the discs, the rotating cylinder can be

set in suitable position, which gives the radius of the cylinder motion

As shown in photo 4, circular cylinders of five different size were prepared

The radu of the cylinders are given in Table 3 1 The driving wheel [4]

was rotated by a D C motor [5] via a gear box [6] and flexible drive [7]

BELT DRIVE TANK WALL TAN W SLIT. CIRCULAR CYLINDER. DRIVING WHEEL. Disc. SUPPORT.

Fig 3.2. Rotating Circular Cylinder and Driving Systen:.

FLEX!BLE DRIVE.

GEAR BOX. _D.C.,MOTOR.

Photo 3. Rotating Circular Cylinder and Driving Wheels.

change the position or rotatory direction Of the device. The revolution was controlled by the slidac [8] and displayed in a digital counter They

are shown in photo 5 A diagram of the control of the device and measure-ment is shown in fig 3.3.

The tank has rather a large width and shallow depth. Both properties are not desirable for the purpose of the wave expenments The small scale

of the tank and the wave generatmg devices restncted the accuracy of

the experiment and measurement. The shallow water effect on the wave propagation and the shape of the Wave profile is less than 4% in almost SLIDAC.

- .

Wave length: X - -

-

-Wave

Wave Amplitude: a photograph Probes

-Radius 0f Cylinder:

Input

.Output

Fig. 3.3. Diagram of Experimental Apparatus.

all range of interest where wave making characteristics are remarkable,

though it is not small in long wave range (X>3Ocm) The effect on the wave

making characteristics is seen to be negligible after the investigation for mirror images. Comparison of wave length obtained from wave probe

records on the deep water assumption with that read from photograph

shows good agreement. This means that shallow water effect is negligible. Photographic records were helpful for the analysis in the case when the

wave probe measurements show beat records.

Waves in the Tank Rotatory Cylinder W.G. Radius of Rotation: p Flexible Drive Amplifier Revolution Counter Gear Box Recorder Digital Display Motor

Photo 4. Circular Cylinders used m Experiments.

SLIDAC

revolution: N

Wave Amplitude: a

Wave Period: T

Photo 5. COntrol and Measuring Devices. Table 3.1. Dimension of Circular Cylinder.

3.2. Wave Generation Experiment

One Wave generating device was operated in calm water, and wave records measured by three wave probes and photographs were analysed.

Arrangement of the expenment is shown m fig 3 1 Parameters of the

experiment were the radius of the circular cylinder R, the submergence of the centre of rotation f, the radius of circular orbit of the motion p, and

the fre4uency of the motion o, Which are shown in Table 3.2.

As already stated the remarkable property of the honzontal circular cylinder rotating along a circular orbit for wave generation is that it pro-duces outgoing waves in one direction on1y This wave making property

was proved experimentally.

Photo 6 shows the wave generating experiment The circular cylmder

rotates in the anti-clockwise direction and progressive waves are seen only in the left side of the cylinder.

The wave records of No 3 wave probe put m "no wave side", showed

No. RADIUS cm LENGTH cm 1 0.75 67.7 2 1.10 677 3 1.75 67.7 4 2.10 67.7 5 2.71 67.7

K 10 80; Anticlockwise Rotation

Photo 6. Wave Generating Experiment.

the amplitude less than 5% of those of No. 1 and No. 2 wave probes in

"wave side", providing the cylinder was completely submerged and except

the case of short waves. The high frequency motion produces a relatively

small amplitude wave and, the disturbance by the side discs becomes

large. Therefore SN ratio of measurement in the short wave region increases.

The results of measurements are shown in figs. 3.4 - 3.11 in the form

of wave amplitude ratio A A /p. The frequency of the motion is

ex-pressed by the dimensionless wave number R.mw2IR/g. The submergence

and amplitude of motion are divided by R and expressed as f/R and mp/R

respectively. In photos 7 - 9 the profiles of the radiation waves in the wave side are presented.

Table 3.2. Experimental Condition (Wave Generation)

No. _{1cm Rem} f/R E=p/R

1 4.50 2.71 1.66 0.124, 0.174, 0.224, 0.274, 0.407, 0.474 2 4.50 2.10 2.14 0.195, 0.295 3 4.50 1.75 2.57 0.214,0.314 4 4.50 1.10 4.09 1.182, 2.182 5 3.50 2.10 1.67 0.145, 0.195, 1.095 0.295, 0.428, 0.495, 0.695 6 2.90 2.10 1.38 0.195, 0.295 7 2.90 1.75 .1.66 0.2 14, 0.3 14, 0.447 8 1.80 1.10 1.64 0.282, 0.382, 0.482

05 15

f 45cn,

R 2'71an

166

Fig. 3.4. Measured Wave Amplitude Ratio of Radiation Wave in Wave Side caused by Rotatory Motion.

(f 4.5 cm, Rs = 2.71 cm)

Fig. 35. Measured Wave Amplitude RatiO of Radiation Wave inWare Side caused by Rotatory Motion.

(1=4.5 cm, R4 cm) svsx- £ V 0-124 A 0174 o 0274 o O-07 V 0474 SYMBOL C 4 0-195 R 21cm 211. 0-295 1-0 a 15 A

Fig. 3.6 Measured Wävë Amplitude Ratio of Radiation Wave in Wave Side caüsëd by Rotatory Motion.

(f 3.5 cm, R4 2.1 cm) SYMBOL f 29cm 0195 R 21cm IR 18 0296 f = 3.5 cm R4= 2.1cm 167 SYMBOL C V 0.145 4 0.195 O 0.295 o 0.425 V 0.495 A 0.695 o 1.095

Fig. 3.7. Measured Wave Amplitude Ratio of Radiation Wave in Wave Side caused by Rotatory MotiOn.

1-0 05 8 15 -1-0 0-5 On 0-5 - 1-0 1-5 2Ô

Fig. 3.8. Measured Wave Amplitude Ratio of Radiation Wave In Wave Side caused by Rotatory Motion.

(f4.-5 cm, R3 = L7-5 cm) O 0 05 1-0 f 45cm R3 1-75cm 2-57 f 29cm R3 1-75cm 166

Fig: 3.9. Méasure'd Wave Amplitude Ratio Of Radiation Wave m Wave Side caused by Rotatory Motion

(f=2.9 cm, R3 =1.75 cm) SYMBOL C-0-214 0 0-314 SYMBOL & 0-214 0 0-314 0 0 447 1-5 2-0

A

Fig. 3.10. Measuied Wave Amplitude Ratio of Radiation Wa'ie in Wave Side caued by Rotatory

Motion.-(f 1.8 cr,. R2 Li cth)

1.5

1.0

0.5

00 0.5 1.0 1.5 -. 2.0

Fig. 3.11. Measured Wave Amplitude Ratio of Radiation Wave in Wave Side caused by Rotatory MotiOn.

.(f4.5 cñi, R2 1.1 cm) f 4.5cm P1 1.1 cm f,= _4.09 f '16cm R, 11cm 164 SYMBOL E/R o 1.182 2.182 SYMBOL £ 0282 0 0362 O 01.82

30cm

/Ay /977

77C O9 a). _R= 0.488

/A'y '977

e). _R 1.509

Photo 7. Wave Profiles of Radiation Waves in Wave Generating Experiment. (1 = 4.5 cm, R5 = 2.71 cm, p = 0.47 cm)

28June77

R4?-ICfl

28June 77

R4=

21 cr1

28Jtrne 77

R42 en

a). _R= 0.588 R=0756

28June 77

iT1II

4E2IJt

LLI_L-R= 0.924

R1197

Photo 8. Wave Profiles of Radiation Waves in Wave Generating Experiment. (f= 3.5 cm, R4 = 2.1 cm, p 0.90 cm)

e9T. .Jwe = 1-1 cm. a).

_R0198

29June '77

Rz=l.1c'

f=1scn b). _R= 0.396

29

irne Rz = 1.1 c't R= 0.484

29Jur,e '77

1.1 R= 0.627

Photo 9. Wave Profiles of Radiation Waves in Wave Generating Experiment. (f = 1.8 cm, R2 = 1.1 cm, p = 0.42 cm)

7No77

r

Photo 10. Wave Diffraction Experiment.

Photo 11. Circular Cylinder and Support for Wave Diffractioh Experiment.

3.3. Wave Diffraction Experiment

One wave generating device and one cylinder fixed beneath the water surface are used for the diffraction expenment Arrangement of the wave

generating device, the fixed cylinder and wave probes are shown ir fig. 3.1.

Photo 10 shows the wave diffraction expenment, and photo 11 shows

the circular cylinder and the support used in the expenment The phase shift of the transmitted wave were read from the photograph of the wave profile. The amplitude of the incident wave and transmitted wave were

6O

Table 3.3. Experimental Condition (Diffraction)

Wave height is chosen under conSideration of clearance between water surface and top of the cylinder

111

0 0 fi

Fig. 3:12 Measured and Calculated Phase Lag of Transmitted Wave due to Diffraction by Submerged Circular Cylinder.

measured by wave probes. Parameters of the experiment are the size and the submergence of the fixed circular cylinder and the frequency of the

mcident waves, which are shown in Table 3 3

The measurements were quite difficult, especially in the case of small submergence, because phase lag measurement was difficult in itself and small amplitude of the incident waves spoiled the accuracy of measure-ments. The measured wave amplitude of transmitted waves, however, show almost the same value as those of the incident waves This result is consistent with that of the water wave theory The water wave theory

tells that there is no reflected wave due to the completely submerged

circular cylinder in infinitely deep water and the diffraction effect can be seen only in the phase lag of the transmitted waves. The results for the

phase lag are presented in fig. 3.12.

No. f _Rcm f/R

fR

K 1 4.50 2.71 1.66 1.79 11, 18, 25,37, 45, 74 2 3 4 3.50 3.00 250 2.71 2.71 21 1.29 1.11 119 O.19 0.29 040 11, 25, 18 18, 25,37, 37, 45, 50, 25_37 45 45, 74 50 74 74 LIHEAR THEORY (R 271cm.)

-

EXPERIMENT O f1.5cm.. R165 (3 _f3Scm.. f3Ocm.. R=13O 18 O

I 'I

R= 0.298

R°03

,'4=__=

ai.t/ r

_'

R219

Photo 12. Wave Profiles of Incident Waves and Transmitted Waves in Wave Diffraction Experiment. (f = 3.0 cm, R5 = 2.71 cm)

.-

__I.

--R= 0.678

______

-

-R 1.003

Photo 13. Wave Profiles of Incident Waves and TransmittedWaves in Wave Diffraction Experiment. (f 4.5 cm, R5 = 2.71 cm)

In photos 12 & 13 the profiles of the incident waves and the

trans-mitted waves in the diffraction experiment are presented. 30cm

3.4 Wave Absorption Experiment

Two wave generating devices were used for the wave absorption ex-periment. Arrangement of the experiment is shown in fig. 3.1. The wave absorption or wave energy extraction is performed by the cancellation

of the incident wave by the radiation wave of the device.. No. 2 wave

generating device produces the incident wave tO No. 1, device. Provided the amplitude and phase of the No 1 device's motion are properly chosen,

the radiation wave cancels the transmitted wave. Because two wave generat-ing devices of the same design were used and the amplitude and frequency

were set equal, the only freedom left was the phase difference between

the two devices. The condition for the cancellation is given by (phase difference) = 180° - a- 13

where a is phase lag due to the distance between No. 1 and No. 2 device,

and 13 is the phase lag due to the diffraction by No. 1 device. a =361Y(Ei/An)

O 36O, n : integer

where E is the distance between No. 1 and No. 2 wave making devices, and X is the wave length.

Though this wave absorption experiment was performed, it was not successful after all. Practical difficulties prevented the cancellation of the waves. First difficulty existed in the inaccuracy of phase difference

setting. The other is the shortage of tank length. The tank nay be too

small to observe the cancellation of the propressive waves a long way from

the wave generator. The last and most important difficulties existed in the synchronization of the cylinder motions. Because of the slip of the belt drive, synchronous rotation was not achieved. Therefore the

cancel-lation conditions were not satisfied in the experiment.

4. COMPARISON OF THEORY AND EXPERIMENT

The water wave theory and its results concerning the submerged

circu-lar cylinder are presented m Chapter 2 Numencal results of theoretical

estimation and the comparison between theoretical nd experimental

results are shown in this chapter.

41.

Numerical Results

method.9) The velocity potentials, surface elevation, pressure distribution

on the cylinder, hydrodynamic forces and moments are calculated from

the density of source distributed on the cylinder surface given as the

solution of the integral equation. The problem was solved only for the

radiation problem, but some quantities of diffraction problem are

calculat-ed using the relations between the two problems. So long as the linear

theory is concerned, the only parameter governing the wave problem

around the submerged circular cylinder is the ratio of Submergence f and

the radius of the cylinder R when non-dimensional quantities are regarded

In figs. 4.1 & 42. the wave amplitude ratio of the progressive wave A and the added mass coefficient K for the forced heaving or swaying motion of the submerged circular cylinder are shown respectively. These results

are consistent with the previous numerical results. 4, 8,9) The wave

amplitude ratio of progressive wave in wave side for rotatory motion is

twice as much as those for the heaving or swaying motion, therefore

the scale for the rotatory motion AR is also presented in fig 4 1 In fig 4 3

the phase lag of the transmitted Wave due to the diffraction by the sub-merged circular cylinder is shown. From these results, one can see the effects of submergence and frequency. For the estimation of the wave amplitude ratio, the results by means of the point doublet method were obtained. The dotted lines in fig. 4.4 show the results of the point doublet method. It can be seen that the point doublet method is not only simple and convenient for calculation, but also gives good estimation when the

submergence is large and the wave number is small. Big discrepancies with

the results of integral equation method appear in the region f/R < 2.0,

ER< 0.75.

Fig. 4.1 shows that the wave amplitude ratio increases as the wave number increases when the wave number is small. The shallower the

sub-mergence is the larger the amplitude ratio is In large wave number region, A decreases as ER increases. The wave number, which gives the maximum

A shifts to the higher frequency region as the relative submergence f/R becomes smaller in the deeply submerged conditon. In the case where f/R is smaller than 1.5, that wave number decreases as the submergence becomes small f/R = 1 0 means the condition when the top of the cylmder touches to the free water surface, so it is the limit condition of the com-pletely submerged cylinder. There exists the "wave free frequency" in the condition with very small submergence At the wave free frequency there is neither radiation wave nor exciting forces by the incident wave. Another interesting result for very small submergency is seen in fig. 4.2. A negative value of the added mass coefficient is shown. For the rotating

circular cylinder negative added mass has the physical meaning that inward

centrifugal force instead of outward centrifugal force with rotatory drag

ñega-ASAHR

11020

1.5 3.0

1.5 2.0

Fig. 4.1. Calculated Wave Amplitude Ratio of Radiation Wave.

Fig. 4.2. Calculated Added Mass Coefficient for Heävin Or Swaying Circular Cylinder.

05 10 20

Fig. 4.3. Calculated Phase Lag of TranSmitted Wave due to Diffraction by Submerged Circular Cyllhder

f/

___-__1.

INTEGRAL EQ. METHOD. POINT DOUBLET

1.5 20

DEG, 100

I

go 80 70 60 50' 40 20 05 _10---20

Fig. 4.5. Calculated Phase Lag of Rotatory Motion to Incident Wave for Perfect Cancellation.

tive added mass means that the cylinder has thrust instead of drag as

the component of hydrodynamic force proportional to the acceleration.

When submergence is very large, the circular cylinder does not make

a wave The added mass coefficient approaches 1 0 for every wave number

which coincides with the well, known result for infinite fluid without free

Water surface.

The phase shift due to the diffraction is calculated from the phase difference between the hydrodynamic forces and the cylinder motion

in the radiation problem The numencal results are shown in fig 4 3

It is seen that the phase lag increases as' f/R decreases and it is large in short, wave condition when submergence is small When the submergence is large,

it decreases as wave number increases after it reaches the peak The wave

ntiniber, Which' gives a peak value, in the phase lag, becomes large as f/R becomes small.

In fig 4 5 the phase lag of the cylinder motion to the incident wave when the cancellation is performed is shown The condition for the ampli-tude of the motion is to be seen in fig. 4.1, because the ampliampli-tude of the' motion 'c is given simply by the quotient of the incident wave amplitude by the wave amplitude ratio of the cylinder f/R is the only parameter for

4.2. Comparison of Theory and Exprirnent

Experimental results 'are presented in Chapter 3. Here some

investi-gations and discussions are made by comparison with' the theoretical

results.

Figs. 4.10 -- 4.12 show that the wave making characteristics of the

rotatory circular cylinder obtained by the experiment have similar

tenden-cy as those of the linear water wave theory for the change of the

sub-mergence and frequency Relatively big quantitative disagreement, how-ever, can be seen in the whole region except for very small wave number.

The dependence of the wave making characteristics on the amplitude' of

the motion and the scale effect, which are not explained by the linear

theory, are obvious in the results. For some cases, especially small size

cylinder with big amplitude motion, there is very poor agreement, even for the tendency of the curve, or the wave number which gives maximum wave

amplitude ratio.

In fig. 3.4 the curves for e = 0.124 and = 0.174, and in fig. 3.5

the curves for 6 = 0.195 and 6 = 0.295 can be considered as the same

curve. They 'are the only exception of the experimental results for the

dependence of wave making characteristics on the amplitude of the motion.

Similar results were presented for the heaving motions of partially

sub-merged cylinder. 19, 20) Some of the previous results are explained that

the wave breaking suppresses the amplitude of progressive waves. 2i) The

amplitude of progressive wave produced by completely submerged body,

however, are relatively smaller than those by surface piercing bodies

The wave slope was not steep enough according to the wave breaking ériterion, and obvious wave breaking was not observed at all in the

ex-periment.

Therefore different reasoning from wave breaking is necessary The

non-linear effects of body boundary condition as well as those of free

surface condition, or viscous effects were examined by several researchers.19' 20)

Any concrete reason to explain the phenomena quantitatively cannot

be' discussed here, but the reduction of the wave amplitude seems to come

from wave making mechanism rather than the phenomenon occurring in propagation, becaUse the dependence is also seen in the small wave number condition in which the experiments were performed satisfactorily

and pure sinusoidal w.aves were observed.

The experimental results show that the dependence is small when the amplitude ratio of rotatory motion e is small. Some results of small

cylinder in shallàw submerged condition show different conclusion, but the difficulty of measurements in such conditions should be remembered

In order to investigate the effects of other parameters figs. 4.6 - 4.8 are presented Fig 4 6 shows the results for the same cylinder size R and different submergence f As the submergence was adjusted by the change

15 A 10 05 00 CONST. R21 cm CONST. fI.-5 cm 05 10

Fig. 4.6. Mean Line of Measured Wave Amplitude Ratio (R = 2.1 cm = const.)

15 20

15 20

ER

Fig. 4.7. Mean Line of Measured Wave Amplitude Ratio (f = 4.5 cm = const.)

of the water level in the tank, the depth of water h was also changed. Provided that the effect of water depth on the results are negligible as discussed later, the differences in the results came from the change of

relative submergence f/R. The effect of f/R is not so clear as the theory shows. The results may be suffered from the inaccuracy of the

measure-ment. The clearance between the top of the cylinder and the free surface is

cm /R .50 2.11. 350 167 290 1.36 Rcm !R 271 166 210 214 175 257

10 Fig. 4.8. I - CONSI. 166

/

//

/

//

1/ II / / /

//

/

/7.

I/I

/

Ii,

/

II

/

II

_/

0124 " N NN N'\ N N N N. N\ N_N

N

N N N N N N_5'

.1 -I .

10 20

Mean Line of Measured Wave Amplitude Ratio (f/R 1.66 const.)

C0 124 174 0314 0214 0382 0282 W#WE jO. 0145 0195 0274 0295 I I .-. I -. -I 20 60 _K 80

Fig. 4.9. Mean Line of MeasUred Wave Amplitude Ratio against Wave Ntuñber K (f/R = 1.66 = cojst.)

very small in the condition f= 2.9 cm, and c 0.295.

In fig 4 7 the results for the same submergence f and different cylinder

size R are shown The effect of the relative submerge?ice f/R can be seen in the scale of wave amplitude ratio and the wave number which gives

Rem 271 ASO 1 66 210. 350 167 175 290 166 10 180 1.64 Rc crn. 'R 271 450 166 210 350 167 175 ?90 166 110 180 161. A 10 P.S CONST. 166

3-0 2-0 1-0 138 EXP. f2-9cn R 2-1 cm. C 0-195 10 E EXPERIMENT.

INTEGRAL EQ. METHOD.

POINT DOUBLET METHOD.

15 2-0

Fig. 4.10. Compànsion between Measured and Calculated Wave Amplitude Ratib

(f/R= 1.38) 3-0 1-65 EXP f4-5cm. R = 2-71cm. C 0 -121. EXPERIMENT. INTEGRAL EE METHOD.

POINT DOUBLET METHOD.

05

1-01-5

2

Fig. 4.11. Cornpañsibn betWeen Measured and Calôulated Wave Amplitude Ratio (f/R 1.66)

20

1-0

£ / A

0 05 1-0 15 20

Fig. 4.12. ComparisiOn between Measured and Calculated Wave Amplitude Ratio (f/R= 2.14)

maximum wave amplitude, though anp1itude ratio of the motion are slightly different from each other

In fig. 4.8 the results for different cyliider size but approximately

same relative submergence f/R 1.66 are shown. COmparison of the curves between similar . shows the scae effect on the results. For the bigger

scale experiment, the wave amplitude ratio has larger value than for the small scale The wave number which gives the maximum wave amplitude

ratio shifts to the higher frequency side as the scale becomes bigger.

Though the linear deep water wave theory tells nothing about this nature, some reasonings may be possible The plots against the wave number K

instead of dimensionless wave number shown in fig. 4.9 shows the peak

at almost the same value of wave number K, and it can be seen in fig. 4.8 that the tendency of wave amplitudes increase as the increase of ER, is

approximately same at rather small wave number. Therefore-the

discrepen-cy of theory and experiment may originate from the existenc of a certain limit in maximum wave amplitude or wave slope for the experiments The configuration of the tank and cylinder may give a certam limit As mentioned before, the tank has wider breadth and shallower depth. The shallow water effects should be marked m the small wave number

con-dition From the theoretical and expenmental results for heaving

two-dimensional bodies it is noted that the shallow water effects on added mass

coefficint and damping coefficient; Le. wave amplitude ratio of the

progressive wave, are negligible for hIT> 5 0 22 23) Where h_{is water depth} 30

A Rr 214 £ EXPERIMENT.

EXP f45cm. _{- INTEGRAL EQMETHOD.}

R2-1 cm.

and T is depth of the body. If the equivalent depth of body is taken by

the lowest position of the circular cylinder during the rotation, every

experimental condition fulfils the criterion and shallow water effects can be considered negligible. Next the effect of breadth is examined. In some conditions of experiment, standing waves in the direction of breadth and three-dimensional wave patterns were observed. Though only the

investi-gation into the stability of two-dimensional plane wave will tell the effects

of the breadth, it can be said that the wider the breadth is, the easier

the transverse wave is induced. Clearance between the water surface and the top of the cylinder, the radius of the circular cylinder, wave length or wave slope in longitudinal direction may have some relations with the existence of the transverse wave, because the three-dimensional wave

patterns were observed mainly in the case of high frequency motion and

smaller size of the cylinder.

In fig. 3.12 theoretical curves of phase shift by the diffraction calculat-ed from radiation problem are shown with experimental results. In the case

f = 4.5 cm and R = 2.71 cm, measured results show good agreement with the theory. On the other hand for the condition f 3.5 cm and f 3.0 cm

with R = 2.71 cm have big discrepancies with theoretical results though their quaiitative agreement may be acceptable. Difficulty in the

measure-ment of accurate phase lag prevent the agreemeasure-ment between the experimeasure-ment

and theory especially when the submergence is small and the clearance between the water surface and the top of the cylinder is small. Precise measurement in big scale experiment will offer much more useful data

for the feasibility study.

4.3. Feasibility of the Rotatory CircUlar Cylinder System as a Device for Wave Energy Extraction

For the practical application of the submerged rotatory circular

cylin-der system to the full scale device, there are many problems to be solved.

Among them mooring and anchoring problem, energy storage and/or

transportation, mechanical and structural problems and irregularity and

three-dimensionality of sea waves are more or less common to any device

for wave energy extraction and ocean structures general. Here some

problems specific for the submerged circular cylinder system are examined.

The inconsistency of two-dimensional cylinder to three-dimensional sea

condition is less important, because a large part of the usable wave energy seems to be transported as the fOrm of swell. The rotatory cylinder system shown in fig. 1.1 will work well for one particular component of the wave

spectrum and is better with regular incident waves. To achieve good

per-formance in irregular wave it is necessary that the system be provided with

the capability to predict the waves in both amplitude and phase and to control the motion of the cylinder quite precisely. This is very difficult in

practice. This difficulty steths from the fact that the system is originally

an active type wave generator and the monitoring and control system makes the whole system respond to the incident waves This may be

called the narrow band property of the system to the waves, and

multi-cylinder system may improve the capability.

The system of one circular cylinder operating for one part of the

spectrum is considered here Consideration of the balance of wave energy

shows that some energy will be extracted from irregular waves. The energy

comes from the wave component whose wave penod comcides with the penod of cylinder rotation when the cancellation conditions are satisfied

However, more precise discussion is necessary to know how much energy is

required by the regulator to maintain the rotation Work done W on the

cylinder by the hydrodynamic forces and moments of the incident waves is given by

w = +XT1e{1t

e*(wi)f°f(w)eiwrdw}dr (40)

where T is a certain time period for averaging and w' is circular frequency

of the cylinder rotation The cylinder motion

(r, w) and the

hydro-dynäthic force on the cylinder F (r) are given by

e(r, w) = = (wi)ei01ei0)1T (41)

F(r) = f f(w)e"dw =

fj(w)e1eit)Tdw

=

ff(w).S(w)eiTdw

(42)

where S (w) is spectrum of the Waves and f '(w)is frequency response function of hydrodynamic force to the wave Substitutmg (41) and (42) in

(40), one finds

w.=

1)cos (a-3)cos (ww) rdwdr

(43)

f and a are amplitude and phase of hydrodynamic forces acting on the cylinder and they are to be calculated and j3 are amplitude and phase of the cylinder motion which are chosen to cancel the wave component of wi.

Therefore the Work done by the component WI is positive. Only the

numerical results for a suitable size and submergence of the cylinder with sea spectrum will give the answer Though no attempt for numencal m-vestigation is made, it seems difficult to extract wave energy from wide

banded irregular waves from the practical point of view.

Next point is the stability of the motion. The submerged circular

Table 4.1. Comparision of Wave Energy Conversion System

necessary. In Table 4.1 some comparison between the rotatory cylinder

system, Salter's duck and spnng-damper-mass system are shown The

essential difference between the first and the latter two is that the later two have restoring force and the systems are oscillatory by thernselves Practically it is quite important, because the resonance of the oscillatory

system is useful for the energy extraction Absorption of waves by

cancel-lation is the problem of freedom as discussed by Evans (1976) and Mei

(1976). Comparison from the point of view of the freedom are also shown in Table 4.1.

Txri SALTER'S_DUCK

SUBMERGED C. CYLINDER SPRING-DAMPER

-MASS SYSTEM

ROTAI ION IN CIRCULAR ORBIT

BODY SHAPE ASYMMETRY SYMMETRY

MOTION ROLL HEAVE & SWAY ROtATION RESTORING FORCE BUOYANCY SPRING NIL (REGULATOR) USUAL COND. SPECIAL COND. "N. WORKING MEDIUM NARROW -WIDE :

0

0

Another interesting difference between them is the means of wave cancellation. Salter's dblck cancels reflected waves and does not produce radiation waves in downstream by means of special body shape The

sub-merged circular cylinder produces no reflected wave by means of its special feature and cancels the transmitted waves.

The Salter's duck seems to have reasonable band width of the_response for the incident wave. The spring-damper-mass system works for full _range

of incident wave spectrum, though optimal efficiency _{is restncted On} the other hand the rotatory circular cylinder system works well only for a certain wave component The sea condition is not necessanly suitable for the optimal operation of this fmely tuned system.

According to the discussion above, the feasibility of the rotatory

circular cylinder System as a wave extraction device does not seem

promis-ing.

As a wave making device for the experimental tank, however, the

rotatory cylinder system has some advantages (e g good efficiency for wave

making, special feature of making progressive waves in one direction, usage

of the water surface over the wave marker). Even with constant radius of

rotatory mOtion for constant radius of the cylinder, wave amplitude can be

adjusted by the change of the submetgence. Therefore, the submerged

circular cylinder system seems to have many advantages and, possibilities as a wave maker of an experimental tank.

5. CONCLUSION

Fundamental feasibility studies of wave energy conversion by means of

rotatory circular cylinder system were carried out. Though the feasibility does not seem promising, some interesting results are obtained through

theoretical and experimental investigations.

It is known that the perfect absorption of the regular incident wave and

wave energy extraction is possible by means of the submerged circular cylinder rotating in a circular orbit Special property of the circular cylin der for the wave generation and wave diffraction makes the cancellation

of the transmitted waves by the radiated waves possible. The conditions of

the cancellation are obtained. Green's function of travelling doublet along a circular orbit is introduced to prove the special wave making character-istics of the rotatory cylinder. Numerical results obtained by means of the

integral equation method are presented

It is confirmed by the wave making experiment that the progressive waves are generated by the rotatory motion of the circular cylinder m one

direction only Measured results of progressive wave amplitude show similar

The discrepancy is large, and the dependence of wave making

character-istics upon the radius of rotatory motion and cylinder Size are seen. Small

motion of large cylinder has better efficiency than large motion of small

cylinder for the wave making.

The wave diffraction experiment Of the submerged circular cylinder shows that there is almost no reflection wave and that the effect of the

diffraction can be seen in the phase shift of the transmitted wave solong as

the cylinder is fully submerged. It is also coincide with the linear theory.

The phase shift of the transmitted waves are measured.

Unsuccessful experiment of the wave absorption tells that synchroni-zation of the cylinder rotation to the incident wave period is essential for

wave cancellation.

As a practical application, the submerged circular cylinder of rotatory

motion can be a good wave making device for experimental tanks, provided inechanical problems are solved. As a device of wave absorption and wave

energy extraction, however, this type of mechanism has great difficulties

inspite of remarkable characteristics in the laboratory. They are caused

from the narrow band spectrum for wave making characteristics and

response to the wave, and the instability of the motion. ACKNOWLEDGEMENT

The author is indebted to Professor D. Faulkner and Mr. N. S. Miller

who gave him the opportunity to do this research at the University of

Glasgow and their continuous encouragement, The author would like to

express his thanks to Dr. A. M. Ferguson and Dr. R. C. McGregor for their

encouragement and valuable suggestions. He also thanks all the members

of the Hydrodynamics Laboratory, Glasgow University for their assistance

combined with kindness and hospitality. He expresses his thanks to Mr.

R. Christison for his skill in the experimental devices and to Mr. M. R. C. Sharp in measurement and analysis assistance.

Professor M. Bessho gave the author valuable suggestions for the theo-retical investigation and Professor H. Maeda offered his computer program and valuable advice. The author's colleagues in the Ship Research Institute

enabled him to stay in Glasgow and gave him tremendous support. The author would like to express his thanks to these people. He also expresses

his deep gratitude to his wife for her devotion.

Work presented here was performed at the Department of Naval Archi-tecture and Ocean Engineering, the University of Glasgow, while the author

was working there as the Talbot-Crosbie Fellow of Engineering 1976-77.

It was also presented in the Glasgow University Report No NAOE-HL-09, in October, 1977.

REFERENCES

Leishman, J. M. and Scobie C. T.: The development of wave power - A techno-economic study, Department of Industry, National Engineering Laboratory, 1977.

Salter, S. H.: Wave Power, Nature, VoL 249, 1974.

Salter, S. H., Jeffrey, D. C. and Taylor, J. R. M.: The Architecture of Nodding Duck Wave Power Generators, The Naval Architect, January 1976.

Ogilvie, T. F.: First- and second- order forces on a cylinder submerged under a free surface, Journal of Fluid Mechanics, Vol. 16, 1963.

Stoker, J. J.: Water Waves, Interscience Publisher, 1957.

Wehausen, J. V. and Laitone, E. V.: Surface Waves, Handbuch der Physik, VoL 9, Springer, 1960.

Gallagher, R. H. et al (edt): Finite elements in Fluids, Wiley, 1975.

Frank, W.: Oscillation of Cylinders in or below the Free Surface of Deep Fluids, Naval Ship Research and Development Center, Report 2375, 1967.

Maeda, H.: Hydrodynamical Forces on a Cross-Section of a Stationary Structure, International Symposium on Dynamics of Marine Vehicles and Structures in Waves, 1974.

Ursell, F.: On the heaving motion of a circular cylinder on the surface of a fluid, Quart. J. Mech. AppL Vol. 2, 1949.

Grim, 0.: A Method for a More Precise Computation of Heaving and Pitching Motions both in smooth Water and in Waves, Proceedings Third Symp. on Naval Hydrodynamics, 1960. Hanaoka, T.: On the Reverse Flow Theorem concerning Wave-making Theory, Proc. of Ninth Japan National Congress for AppL Mech. 1959.

Newman, J. N.: Interaction of waves with two-dimensional obstacles: A relation between the radiation and scattering problems, J. of Fluid Mech. VoL 71, part 2, 1975.

Evans, D. V.: A theory for wave power absorption by oscillating bodies, 11th Symposium on Naval Hydrodynamics, 1976.

Mci, C. C.: Power extraction from water waves, Journal of Ship Research, Vol. 20, No. 2, 1976.

Fuwa, T.: Green's functions for two-dimensional surface wave problem with oscillating body of finite amplitude motion. Glasgow University Report No. NAOE-HL-12, 1977.

Dean, W. R.: On the reflection of surface waves by a submerged cylinder, Proc. Camb. Phil. Soc. Vol 44, 1948.

Ursell, F.: Surface waves on deep water in the presence of a submerged circular cylinder, Proc. Camb. Phil. Soc. VoL 46, 1950.

Tasai, F. and Koterayama, W.: Nonlinear hydrodynamic forces acting on cylinders heaving on the surface of a fluid, Rep. Res. Inst. AppL Mech. Vol. 24, No. 77, 1976.

Yamashita, S.: Calculations of the hydrodynamic forces acting upon thin cylinders oscillating vertically with large amplitude, J. of Society of Naval Architects of Japan, Vol 141, 1977.

Fuwa, T.: Investigation into characteristics of wave maker of Glasgow University Tank, Glasgow University Report No. NAOE-HL-11, 1977.

Kim, C. H.: Hydrodynamic forces and moments for heaving, swaying and rolling cylinders on water of finite depth, Journal of Ship Research, Vol. 13, 1969.

Takaki, M.: Ship Motions in Shallow Water. (Part I), Transactions of the West-Japan Society of Naval Architects. No. 50, 1975.

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March 1969.

No. 31 Some Aspects of the Correlations between the Wire Type Penetrameter Sensi-tivity, by Akira Kanno, July 1969.

No. 32 Experimental Studies on and Considerations of the Supercharged Once-through Marine Boiler by Naotsugu Isshiki and Hiroya Tamaki January 1970

Supplement No. 2

Statistical Diagrams on the Wind and Waves on the North Pacific Ocean by Yasufumi Yamanouchi and Akihiro Ogawa, March 1970.

No 33 Collected Papers Contributed to the 12th International Towing Tank Conference

March 1970.

No. 34 Heat Transfer through a Horizontal Water Layer, by Shinobu Tokuda, February

1971.

No. 35 A New Method of C.O.D. Measurement Brittle Fracture Initiation Character. istics of Deep Notch Test by Means of Electrostatic Capacitance Method, by Kazoo Ikeda Shigeru Kitamura and Hiroshi Maenaka March 1971

No. 36 ElastoPlastic Stress Analysis of Discs (The 1st Report in Steady State of

Thermal and Centrifugal Loadings); by Shigeyasu Amada, July 1971.

No. 37 Multigroup Neutron Transport with Anisotropic Scattering, by Tomio Yoshimura,

August 1971.

No. 38 Primary Neutron Damage State in Ferritic Steels and Correlation of V-Notch Transition Temperature Increase with Frenkel Defect Density with Neutron Ir-radiation, by Michiyoshi Nomaguchi, March 1972.

No 39 Further Studies of Cracking Behavior in Multipass Fillet Weld by Takuya Kobayashi Kazumi Nishikawa and Hiroshi Tamura March 1972

No 40 A Magnetic Method for the DeterminatiOn Of Residual Stress, by Seiichi Abuku,

May 1972.

No 41 An Investigation of Effect of Surface Roughness on Forced Convection Surface Boiling Heat Transfer, by Masanobu Nomura and Herman Merte, Jr., December

1972.

No. 42 PALLAS-PL, SP A One Dimensional Transport Code, by Kiyoshi Takeuchi,

February 1973.

No. 43 Unsteady Heat Transfer from a Cylinder, by Shinobu TokUda, March 1973.

No. 44 On Propeller Vibratory Forces of the Container Ship --Correlation between Ship and Model and the Effect of Flow Control Fin on Vibratory Foces by Hajime Takãhashi, March 1973.

No 45 Life Distribution and Dealgn Curve in Low Cycle Fatigue, by Kunihiro Iidà and Hajime Inoue, July 1973.

No 46 Elasto Plastic Stress Analysis of Rotating Discs (2nd Report Discs subjected to Transierrt Thermal and Constant Centrifugal Loading), by Shigeyasu Amada and

Akimasa Machida3 July 1973.

-No 47 PALLAS 2DCY A Two Dimensional Transport Code by Kiyoshi Takeuchi

November 1973..

No. 48 On the Irregular Frequencies in the Theory of Oscillating Bodies in a Free Surface, by Shigeo Ohrnatsu, January 1975.

No 49 Fast Neutron Streaming through a Cylindrical Air Duct in Water by Toshimasa Miura, Akio Yarnaji, Kiyoshi Takeuchi and Takayoshi Fuse, September 1976. No 50 A Consideration on the Extraordinary Response of the Automatic Steering Sys

tern for Ship Model in Quartering Seas, by Takeshi Fuwa, November 1976 No 51 On the Effect of the Forward Velocity on the Roll Damping Moment by Iwao

Watanabe, February 197.7.

No 52 _{The Added Mass. Coefficient of a Cylinder Oscillating in Shallow Water in the Lithit}

K -.0 and K oo, byMakoto Kan, May 1977.

In addition to the above-mentioned reports, the Ship Research Institute has another series

of reports, entitled Report of Ship Research Institute The Report is published in Japanese with Enghsh abstracts and issued six times a year