On the flow field and the Hydrodynamic forces of an obliquing ship

1.

Introduction

Estimation of the flow field and the hvdrody-namic forces of an obliquing ship is important to grasp mainly the manoeuvrability of a ship

and many investigations were performed. As to

the estimation of lateral force, Fuwat) Mizoguchi2' and Matsumura et al showed

methods to use the separated vortex and the slender body theory, and these methods are

now in practical use in the still water surface. From the observation of ship-side wave profiles. however. Matsumoto et al41 reported that the local wave effect near the bow appears

clearly in che lateral force distribution. Then in

order to estimate accurately the hydrodynamic forces

acting on an obliquing ship,

it is necessary to take account of the free surface

effect in such calculation.

Nishiyama treated the lateral force of

ellipsoid from the standpoint of the linearized

wavemaking theory, and Hanaoka and

>onaka investigated the lateral force of thin

Kvushu tíniversitv

Kawasaki heavy Industries. Ltd.

103

wing under the wavemaking effect. Recently Hon et al8' presented a method based on the low speed wavemaking theory, and Song91

obtained the wavemaking lateral forces and wave patterns about an obliquing ship based on

the newly developed slender body theory, and

moreover Xia-Larsson10 presented a method

to use vortex distribution on the yacht hull and

Rankine source method..

This paper presents a method to calculate

the flow field and the hydrodvnarnic forces of an obliauing ship using vortex distribution on the hull

center plane and Rankine source

method on the assumption that the ship hull. is a thick wing of small aspect ratio.

2.

Basic Equation

We describe the basic equations to represent a ship hull advancing straight with a speed V

and a drift angle '. As shown in Fig.1. two kinds of coordinate systems are used. One is O-xyz coordinate system fixed to ship hull and the other is O-XYZ coordinate system fixed in

space with X axis in accordance with uniform

flow V Then we assume the potential flow

Deift University of Technology

Ship Hydromechanics Laboratory

Library

Mekelweg 2, 2628 CD Deift

The Netherlands

Phone: +31 15 2786873 - Fax: +31 15 2781836

s. On the Flow Field and the Hydrodynamic Forces of an Obliquing

Ship

Kuniharu NAKATAKE, Member Atsushi KOMURA* iVIember Jun ANDo , Member

Katsumi KATAOKA, Member

(From T. WestJapan. S.N..1. Japan. iVo.80, .4ugust 1990) Abstract

This paper presents a method to calculate the flow field and the hydrodynamic forces of an oblicuing ship under the free surface effect. We treat the ship hull as a thick wing of small aspect ratio. Ship hull is represented by the source distribution on the hull surface and the vortex distribution on the center plane of the hull. The strength of these singularities are obtained from the boundary condition on the hull arid the Kiittas condition. In order to obtain the wave flow, we use Kvushu University method. This is a kind of the Rankine source method and does not use th finite difference operator to satisfy the radiation condition. Since the flow field is calculated using these singularities, the wave pattern, the pressure distribution and the wavemaking lateral force and resistance acting on the hull are obtained. Using this method, we calculate the flow field around Wigley hull with drift angle. The wave pattern

and the hydrodynamic forces are compared with the experimental results. The agreements are

104

Kuniharu ìAKATAKE. Atsushi KOryIURA, Jun ANDO, Katsumi KATAOKA

z.z

V

_---;----..---.-.. -..--.

-.:--.

_..

_.-:.

Fp

Fig. 2 Bound vortex distribution

F.p.

A.P. Fia'. io Coordinate systems

around the obliquing ship hull and denote the

flow velocity vector around the hull by q and

the uniform velocity

_{vector by qu}

In order to express the flow field around the

obliquing _{hull, three kinds of singularity}

distributions are utilized. One is _source distribution m on the hull surface to express the hull shape, second one is bound vortex '7

distributed on the ship center plane Sv with

constant strength in the longitudinal _direction

and stepped strength in the draftwise direction,

which represents the lifting action originating from obliquing _{(vide. Fig.2) From bound} vortex distribution, there should exist trailing

vortex which, we assume, sheds from each_strip with angle _{to infinite rearward (vide Fig.3).} We call above vortex system the nonlinear vortex model in contrast to the linear _vortex model (i.e.

_{=0). The third}

_{one is source} distribution °pon the still _{water surface Spto} express the free surface wave.

Then we denote the velocity vectors due to singularity distributions m, 7 and F by q5, q and qp, _{respectively, whose mathematical}

expressions are given in Appendix. _{m and 7}

qn=0

₍₅₎

A.?.

Fig. 3 Vortex system on one strip

are assumed to be splitted into two parts, _i.e. m0 and 'fo which represent the double-model flow, and m and _{7 which do free surface} effects. That is,

mm4j±m,

77±7

₍₁₎

Then total velocity vector q is written as

q= q0 + ₍₂₎

where

q0q±q+q, q1q.q5q1

(3)

In Eq.(3), q,.o, q,.,o' qi and _{q1 express the} velocity vectors due to singularities _{mo, 'fo,}

mand _{y, respectively. Here, we represent}

the vector component in each direction by each suffix, for example, in case of q, (q,., qy, q) are

components wich respect to 0-zyz coordinate

systems and (qx, qy, qz.) are with. respectto O-XYZ coordinate systems. Then, among above components. there hold the following relations.

/ qx

qy

\qzl

/ cos

= sin!?

sin 9cos

o

0

_/q,.

o qy (4)

1/

_\q,.J

Next we deal with the boundary _conditions to determine the singularity _{distributions.}

Firstly, the boundary condition on the hull

surface S becomes

where a is the outward normal vector to 5. Secondly, as the Kutta's condition, _{we adopt the} equal-pressure condition on both face and back sides at the trailing edge. _namely

Lastly, on the still water surface, we satisfy the double-model linearized free surface condition given by DawsonLi). Expressing this condition

using gravitational acceleration _{g and O-XYZ} coordinate system, we havi

A.p

(six

( Six

ai

ax

)

qoxqoy a y a x 7

(aqiy

qoy aqoy

ix

2 _ox

ax)

ax)

Hqav(

/ aqoy

_qy±gqz

/ioY\

_a)

aqo

±

J 2

(

aq0 go xo y a Y 1qox

_ax)

ao

)oY

2

(

ao

a

ori ZO

(7)

After obtaining the singularity distributions

m, _{'7 and} Ffrom Eq.(5) through Eq.(7), we calculate the wave elevation ' the wave

resistance R, the wavemaking lateral force RL

and the wavemaking yaw moment M about the

OZ axis by the following equations.

= (V2qxqy2qoxqix

2qoyqiy)/(2g) (8)

R=

_-

ffSg (pk) nS

RLff(p)nydSs

Sg

M ff (pk)

(y±x)

d55 (9) where

ppo=0(V2

q2)/2

In Eqs. (8) and (9), we define water density

the force acting in the dYrection of the

uniform flow R and the force acting in the perpendicular direction to the uniform flow RL.

The word "wavemaking" indicates that the

forces include the free surface effects in

addition to the forces due to the doublernodel

flow.

Calculation Method

At first, assuming the free surface as the

rigid wall, i.e. qi = O, we obtain unknown singularity distributions rn0 and 'fo from the

hull surface boundary condition Eq. (5) and the Kutta's condition Eq. (6) - Though rn0 and Yo are included in Eqs. (5) and (6). we can not

solve both equations at a time because Eq.(6) is nonlinear equation. Then we obtain mo and

'7o iteratively using Eqs. (5) and (6) until

converged values are obtained. Yo is obtained

from the Kutta's condition making use of Newton-Raphson method.

Next we perform the calculations taking into account the free surface effect. Since rn0 and Yo

are known, unknown values are m, 7 and

0p which should be obtained by solving Eqs. (5) and (6) and the free surface condition Eq. (7). Assuming at first m and y as known values, we obtain F making use of KUM (Kyushu University Method) 12) Then using known p, we obtain m and '/ iteratively

as before. After repeating above procedures several times, we obtain converged values of

7 and

p.

Lastly, we calculate the wave elevation, the wave resistance. the wavemaking lateral force

and the wavemaking yaw moment from Eqs. (8)

and (9).

Calculation Results

We perform numerical calculations for

Wiglev model whose half breadth y is expre-ssed as

BI

72x211

(z21

Y'L)

1)

where L, B and d are ship length, maximum breadth and draft, respectively, and B/L=O.1

and d/B=1.O.

At first, in order to check the validity of thick wing model used in the present paper, we calculate the normal force of the wing of small

aspect ratio and thickness in the infinite fluid domain and compare the experimental results13) which were for the rectangular plate

wing of aspect ratio 0.2. Since our thick wing model cannot cope with complete plate wing, we

(lOI

r-106

Kuniharu NAKATAXE. Atsushi _{KOMURA, Jun .Ao, Katsumi}

KATAOKA

adopt the thin wing of thickness _{ratio B/L}

0.02 and the aspect ratio 2d/L=0.2. Fig.4

shows the comparison of the normal force

coefficient c.,(=N/ 0.5

p V2 (2 _L)}

_N:

normal force) of this wing against the drift angle ¡9 . In Fig.4 the experimental results'3 and the results calculated by the linear model

are also indicated. In case of small _aspect-ratio

wing, we know that the present nonlinear

model is more suitable for _{the normal force}

calculation of the wing. Next we calculate the

flow field and the

_{hydrodvnamic forces of}

Wigley hull taking into account the free surface

effects. The mesh division of the hull surface

are 44 divisions in the longitudinal _direction

and 10 divisions in the draftwise direction on

both sides as shown in Fig.5. The still water surface is also divided into 35 divisions in the main stream direction and 20 divisions in the

lateral direction, and Fig.6 _{illustrates the mesh}

Ø: preet ieUiod LLneac-ode! ci.'

°5r

00

F.P.

-: Expriiea

10.0 20.0 30.0

Fig. 4 _{Comparison of normal force coefficient}

Tz X

Fig. 5 _{Mesh division of hull surface}

z

A.P.

division in case of j9 10°.

Then we show the singularity distributions obtained by numerical calculations in case of

Froude number F,0.267.

_{Fig.7 shows the}

source distributions of the ship hull _advancing

straight and

_{we find that the wave effect}

appears near the bow. Fig.8 shows _{the source} distributions of the ship hull _{advancing with}

drift angle

9 = 10°. With

_{drift angle, the} source distributions differ between _{the face}

and the back sides and the free surface effect appears near the bow. Fig.9 shows the vortex

distributions on the ship center plane in case of /9 _{5°, 10°. We learn the vortex strength} in-creases with increase of drift angle and decreases a little in the _{calculation with free}

surface effect. Fig.10 _{shows the source} dis-tributions on the still water surface in case of

¡9 10° and we find the source strength on the face side becomes larger _{than that on the back}

side.

In Fig.1 1, the_{wave patterns are displayed at}

U*

,uuauu

uwuum

--e

Fig. 6 _{Mesh division of still water surface}

l-0 A.P. F.P. F?. -0.2 -0.4 -0.6 -05 -,.'... _-l. Exclu n Fr,. jrl.c. Ef _{c nclulri} Frs. Sczrfac, Effect

Fig. 7 _{Source distribution on hull surface (F=0.267,}

r

-1.0

ER

c.

On the Flow Field and the Hydrodynarnic Forces of an Obliquing Ship 107

4r 22

i, 2. 1.8 1.6 1.4 1.2 1_O 0.8 0.6 Face Side Exc ud i ng ER V _-20 t -22. Back Side Free Surface Effect

o

1 -1.0

Fig. 9 Comparison of bound vortex distribution

-0.6 -1.0 -1.2- -1.4- -1.6-rny -1.8- -2.0--22. z 0.0 Face Side Including Free

Fig. S Source distribution on

0.05 -4,c.y/V 0.1 hull Surface -z--d surface 0.0 Back Side Effect (F,=0.267, 3 =10) 0.05 - _{47t y/V} 0.1 ox ox cx cx oc ß=5. X Exctudin $ =10 cx cx cx

d

cx _{Free Surface Effect}

Oc

Oc Including cx

108

Kuniharu NAKATAKE, Atsushi KOML'RÂ. Jun ANDO, Katsumi KATA0KA 02

0.1

0.0

0.1

0.2

Faca Side

Fig. 11 _{Comparison of calculatedwave patterns (ß =}

0, 5°, 10°) 0.2 0.1 -0.0 -.

0.1

Fiz. 10 _{Source diszributon on still water surface ( =10°)}

FQ.267 in cases of

9 0°, 5°, 10°, where

the solid lines and the broken _{lines express}

elevation _{and depression of wave height.} respectively. With increase of drift angle, the wave height on the face side becomes _larger

and, ori the other hand. the wave height on the back side decreases. We _{think these}

phe-nomena agree with real ones. Fig.12 shows _a

comparison of calculated _{wave pattern in case} of _{9 =10° wich the experimental}

_{one. The}

positions of che crests and the troughs of

calculated wave pattern almost _{coincide wich} the measured ones.

FigJ3 shows the _{pressure distributions on} the ship hull surface, where the solid lines

c_l Cu ti

Fig. 12 _{Comparison of wave patterns =10e)}

L.

Sack Side

1.0-o Face Side Back Side cace Sice

T:TT;--;;

show the positive

_{pressure and the broken}

lines show the negative _{pressure. We learn that}

high pressure zones increase with increase of drift angle and the free surface _{effect appears} near the bow.

Lastly we describe about the _hydrodynamic

forces and moments which are obtained by integrating the pressure distribution on the ship hull

_{surface. Fig.14 shows the}

_wave resistance coefficient Cw(Rw/0.5 °SV2, S

wetted surface area) at ¡9 0 compared with

experimental vaiues1 . The agreements of both values are satisfactory. Fig.15 shows the wavemaking lateral force coefficient _{CL (} RL/0.5 °SV2, RL: wavemaking lateral force)

in case of 9 50, _{100 and F'ig.16 does the}

wavemaking yaw

_{moment coefficient C(=}

dO

ZO

-EdD.,,,..,.,A

Cw. ._CaIe.j0,

Fig. 14 _{Comparison of wavemaking resistance} coefficients

A.?.

Fig. 13 _{Calculated pressure distributions on hull surface}

s

2.0 x1O 4.0 CA. 30

EaeIudIr' Fe.. rf.ce t1.ct

InIiin F.. S..i,f.e. EU.ct

to

M/05p V2L2d, M: wavemaking _{yaw moment).}

In tfiose figures. _{we indicate the values} calculated for the double model flow. _{The free} surface effect gives rise to fluctuation of CL, C1 with change of Fraude number. In order to examine the calculated results, we indicate the

experimental values°

_{at F=0.181 and we}

know present calculation method gives the near

values _{o experimental ones on the whole.}

LSi(F,..O.1t.3.tO' )

3

-Eaa.rIeflt 1S)(,,,_o.11..ß-5 I

Fig. 15 Comparison of waveoì.aking lateral force

coefficients

On the Flow Field and the Elydrodyriamic Forces of an Obliquing Shtp ₁₀₉

Back Side A.?. Y.?.. Sack Side A.?.

Y.?.

Face Side A.?. Y,?.

Face Side A.?.

A.?. T.?. _{A .P.}

0.20 0.2 _0.20 Ft, _0.3e

0.20 025 o-30 . - F1, 0.35

Y.?.

Saca Side A.?. Y.?.

Sack Side A.?.

Ezc(uding Free Sijrface Effect

Including Free Surface Effect

ß S de A.?. Face Side A .P. Y.?. 3 =0 [7/(í Y r. Y.?. ß=5.

r

110

Kuniharu NAKATAKE. Atsushi K0MURA. Jun .ANDo. Katsumi KATAOKA

to

S-10.

S

-5-0.20 _{0-25 0.30 Fn 0.35}

Fig. 16 Comparison of '.vavemak-irig_{yaw moment}

coefficients

5.

_{Conclusion and Acknowledgement}

In order to obtain the flow field around the ob!iauing shiD

_{hull and the hydrodynamic}

forces, we proposed a calculation method to use a thick wing model for a ship hull and a kind of Rarikine source method to take into account the

free surface effect. Then we showed that

calculated results seemed to be reasonable and our method might be useful for such problem. As to the quantitative agreement with experi-ments, we must check for another ship form

and _{improve the calculation method, for} example, to treat with the _{trim, sinkage and}

healing.

We wish to _{express our sincere thanks to} Mr. Kokichi Oda and Ms. Keiko Matsuki for

their work to complete this _manuscript.

This research _{was partly supported by the} Grand-in-Aid for Research of the Ministry of

Education, Science and Culture. _{Further we are} deeply indebted to the staffs of the Large-Sized computer Center of Kvushu University _for their work using FACOM _{M780/20, VP-200,} 0S IV/F4.

R efere n ces

i) K.Fuwa; Hydrodynamic _{Forces. Acting on} a Ship in Oblique Towing, Journal of the Society of _{Naval Architects of Japan,}

.Vo.L134Dec. .1973...

... 2)' S.Mizoguchi;

_{Calculation of Flow with}

Three-Dimensional Separation Vorticities around _{Ships, Journal of the Kansai}

society of Naval _{Architects, Japan.} No.188, Mar. 1983

K.vÍatsumura; LTanaka, T.Oki, _S.Kishi; On the Nonlinear Lift Characteristics of

Slender Bodies at Incidence, Journal of the Society of Naval Architects _{of Japan,}

Vol.154. Dec. 1983

K.Matsumoto, K.Suemitsu; Hydrodynamic Force Acting on a Hull in _Maneuvering

Motion. Journal of the Kansai Society _of Naval _{Architects, Japan, No.190, Sep.}

1983

T.Nishiyama On the Lateral Resistance of a Prolate Spheroid and Ship, Journal of the

Society of Naval Architects of Japan. Vol.85, 1952

T.Hanaoka; Non-Uniform Theory of _Wave

- Making on Low Aspect

Ratio Lifting Surface. Proceedings of the 10th Japan NaionaI Congress for Applied Mechanics, 1960

K.Nonaka: Free Surface Effects _{on the}

Side Force and Moment Acting _{on a Ship} Hull with a Drift Angle, Journal of the

Society of _{Naval Architects of Japan.} Vol.138, Dec. 1975

T.Hori, K.Matsumura, I.Tanaka; On the Lateral Force Caused by Wave Generation Acting on the Ship Hull with Steady Drift Angle, Journal of the Society of _Naval

Architects of Japan, Vol.159, Jun. 1986

W.Song; _{Wave.making Hydrodynamic}

Forces Acting on a Ship with Drift _Angle

and Wave Pattern in Her _{Neighborhood,} Journal of the Society of Naval Architects of Japan, Vol.211, Mar. 1989

F.Xia, L.Larsson: A Calculation _Method

for the Lifting Potential Flow

_Around

Yawed Surface-Piercing 3-D Bodies,

Pro-ceedings of Sixteenth Symposium on Naval

Hydrodynamics. Berkeley,.1986

C.WiDawson; A Practical Computer Method for Solving Ship-Wave _Problems, Proceedings of Second International Con-ference on.Naval Hydrodynamics, Berkely, 1977

J.Ando, K.Nakaake; A Method to

Calcu-late

Wave Flow by Rankine

_Source.

xl 02 _F,..

F,..

If.t

20 -

Q'Jj

V ! 4 T JJ3 rn(x',J, 1) X G3(x, y, z; i, j, í)dS where '7k (L12

y,

qvx

_{4r J-U2 (xe)}

On the Flow Field and the Hydrodynamic Forces of an Obliquing Ship 111 Transactions of the West-Japan Society of where

Naval Architects, No.75. Mar. 1.988

3) W.Bollay; A Non-linear Wing Theory and O (x, y, z; z', y', z')

its ADplication to Rectangular Wings of

1

Small Asoect Ratio, ZAMM, 1939

-

2

14) 17th ITTC Resistance Committee Report:

-±

)± (zz)

Cooperative Experiments on Wigley Para-

1

bouc Models in Japan, 1983 _{/ 2 ' 2 ' 2}

is) TKashiwagi;

Study on Manuevering Hy-

(xx) ()

(zz)

drodynamic Force Acting on a Ship

Moving in Following Sea, Doctor Thesis.

(Expression of q',)

Osaka tjniversity, 1984

We divide the ship center plane into K

divisions in the draftwise direction and give

Appendix

the number k(=1--K to each strip and the

(Expression of q)

number

k'(=lK±l) to each dividing line

The velocity vector q5(q33, _{qsy. qsz) induced} () from the bottom to the still water surface.

by the source distribution m on the hull surface Then w assume the constant strength of bound is expressed as vortex y, on the k-th strip and the free vortex

7 a a a sheet shedding with the angle ( ¡9/2) from

(q5, q5. q)

_{= (} )

(A-1)

the k-th strip as in Fig.2. The velocity vector q

a

Sy 3z /

_{(q, q,q) induced by the vortex systems is}

is the disturbed velocity potential due to expressed as m and becomes

27

H

(A-2)

z

k'

(x)2±y2+ (z_)2

(x)cose±ysine

Slfl O , i d

4r

J-L12

(z) --- ycosO - (x)sin

-

/(x) H-y-e- (z)

('L12

'sine

, x

4ir

J-u2 (z-1)-ycose(x)sine)

(x)cos

±ysin C

(x)2-i-y2± (z'

7k 1u2

x

Zr

4

Ju2

(x)±y2 V(x_)2±y2+(z_)2

f(x_)2+y2+(z_

I

(x.)cosC±ysinO

-___'cose)

₍ sin O)

7(x) 2±y2± (z)

+1 4

Juz (z')2±)ycosO\xç)

q, q-,, =

q

(A-3)

z k'-- I

(x)2-f-y2+(z-1

)2

112

Kunjhu NAKATAKE, Atsushj _{KOMURA. Jun ANDo, Katsumj}

KATAOKA r112

+

cosE)j J112

_(ZThk'j

L

aF aF

ax' a' az

7

-ycosE) - (x)sinE) 2

-

'

r

VzR_J

_(x.)2±

ycosO - (x)sinE) 2

u

'

r ¿12

ycos E) - (x ) sin E)

J

J_U2

(zi)2

_{ycOsE)(x_)sin}

2 t

(Expression of qp)

The _{velocity vector q}

(qFx-,

qy,

_Fz) induced by the _{source disributjon ' ori the}

still water surface _{is expressed as} (qFx, FY, FZ)

(A-4)

p _{is the discurbed velocity pocential due}

to 0p _{and becomes}

y')

dS-x

- (A-5)

Y')2+Z2

dE

17

(x_)2±y2-

(xe) cos E)

+-y sin E)

(z

(x)cosE)

_±ysinE)

(t_)2y2

(z_)2T

1 _dE

(x ) cos E)

-+-y sin E)