Numerical Simulation of Maneuvering Motion

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Twenty-Second Symposium on Naval Hidrodynamics

ABSTRACT

A numerical simulation method has been developed for solving the manoeuvnng motion of

blunt ships by coupling the equations of motion with the NS equations. Since a coordinate system is fixed to an arbitrarily moving ship, inertia forces must be

incorporated into the NS equation as body forces

The hydrodynamic forces acting on the hullofa ship are obtained by solvmg the mcompressible and

time-dependent NS equations numerically. Forces and

moments by a rudder and a propeller and their

interactions

with a

hull are

computed by a

mathematical model. This method was applied to the simulation of the 10-degree Z manoeuvring motioû of

two. VLCC models

with the saine principal

dimensions and different aft-part frame lines, thai is,

one is so-called V-shaped and the other is U.. The

results of the simulations agreed well with those of

model tests and revealed the typical difference in

manoeuvrability between the two ships.

INTRODUCTION

Manoeuvzbiity is one of the most

important properties for all kinds of ships for safe

navigation. At present the prediction of the property for a newly designed ship is most likely to make use

of data obtained from the sea trial of similarly-shaped

ships or/and tank tests. However the former often lacks reliability and the latter always consumes

excessive time and costs. Therefore, there has long been an expectation för computational simulations to accurately predict the manoeuvr ing performance of

ships. Some mathematical models have been developed for this purpose, but the accuracy of such

simulations entirely depends on the modelling of hydrodynainic derivatives" and none of them has

been veiy successfùl to date. Since the forces and the moments acting on the hull of ships can be evaluated

with sufficient degree of accuracy by our NS solver'S 3.

the coupling of the equations of motion of ships

and the NS equations can be an answer to this

Numerical Simulation of Maneuvering Motion

T. Sato, K. Izumi, H. Miyata (University of Tokyo, Japan)

724

012111 UnIversity of Tecbnoloy

Ship Hy omeciaics

Laboratory

library

Mekelweg 2 - 2628 CD Deift The Netherlands

Phone: 3115 786873 - Fax 3115 781838

problem. Recent oil-flown-out accidents of Navotoka

and Diamond Grace near Japan's coasts have strongly motivated this study.

For solving the manoeuvring motion of ships

by CFD, there seem to be three steps in terms of the sophistication of methods from a CFD point of view.

The most sophisticated may be regarded that the flow

about a rudder and a propeller is also solved by CFD together with that about a hull. Davoudzadeh eraL6'

have introduced a rotating propeller in the flow

simulation about a submarine by using the technique of a rotating dynamic grid scheme. Their manoeuvring öonlröl is, however, not done by pure cFD, but either by fixed rudders with pre-set angles

or by an (jfrfJy.glven external yawing moment.

Although the similar treatment will

have few

theoretical difficulties for Ships like a VLCC, it may, unfortunately, take much time to be compared well

with tank tests.

The second step may be that a propeller is modelled as a body force someway at its position and

the flow about a rudder issolved

by D. In this case,

the proper grid systeth should be generated and fit to

a moving rudder whether it

is re1enced to ä

different computational space from that of the hull. This type of simulations has already been done for simply-shaped submerged vehicles with a body-force propeller model7". For ships like a VLCC, it can be forecasted that similar simulations will be realisedin the near future though a lot of efforts still have to be devoted to overcome technical difficulties, such as the accuracy of the modelling öl the propeller in the complicated waite of the hull and the generation of grid system satisfying the reqwrement from both a

hull and a rudder.

The present study has taken thethird step, i.e. making use of a mathematical model for estimating the forces and moments of a rudder and a propeller and their interactions with a hull, bi other words, the

hydìodynamic faites acting on a hull in such a

mathematical model are replaced by those obtaitied by CFD prediction. Considering the convenience to designers, this approach may be the most practical

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Wê focus on VLCCs in

this study, that means that double-model flow assumption may be acceptable when evalüating hydrodynanüc forces

acting on its hull and that only the planar motion in the horizontal plane, which are surge, sway and yaw, are considered. lt is well known that the

manoeùvrabiity of a VLCC is very sensitive to the shape of its frame Unes, especially in the aft part;

Motiön simulations extracting all the forces from the mathematical model have usually failed to represent

the differences in the performance between ships

with different frame lines in the aft part, because the modelling of the hydrodynamic.derivatìves is mostly based only on the principal dimensions. Therefore,

the main objective of this study ¡s to express such

differences by the present manoeuvrability simulation

method to the sufficient extent from a designers view

point.

EQUAflON OF MOTION OF SHIP

Two coordinate systems shown in Hg. I are considered; one is fixed to the globe (1g: Og-XgYg)

and the other to a ship (!s: Os.XsYs) making the

planar motion. The origin of Is, Os, is set onto the midship on the still-water plane level. Positive Xs

and Ys indicate the backward and the starboard of the ship, respectively. The position of Os coincides with that of 0g initially.

Yg

y8

o

Xg

Fig.1 Global and ShIp Coordinate Systems

When (F; , Fy1, O) and Mz are the total forces

acting on the ship and the consequent moment about the vertical (Zg) axis, respectively, the equations of the planar motion of the ship in !g are

m; =Fx, =Fx,cosa+Fvsina

rn;1

=Fy=Fx,sina+Fy,COSa

¡za = Mz

725

where a is the heading angle between the positive

Xg and the negative Xs axes, ¡n the mass and Iz

the moment of inertia of the ship. The

time-integration

of ;

_, and

a

gives the

translational and the angular velocities, V;(n x1.),

es y) and w (na), respectively. Märeover,

the aine holdS for the position (x1,y1, O) in 1g

and the beading angle a. The transiatiónal velocity

(Vx,Vv,, O) and the acccleration(; y , O)in s

are obtained by the transformation of coordinate by

using a, i.e. just a rotation n this case. The angle

in Fig. 1 is the drift angle formed by the negative

Xs and the

translational velocity vector V =

(Vx, ,Vy;,O).

NS SOLVER

OUr NS solver is ndled WISDAMS, which

has been successfully used for the flow simulation

about slñps3'4' dealing with wave-making, viscous boundary layer, turbulence, etc. The method adopts

the staggered allocation of pressure and velocity

components in Cartesian coordinates with MAC-type

algorithm for incompressibility, finite-volume

formulation in a cuÈvilinear body-fitted stthctured giid system,

a moving

free-surface-fitted grid

system2 or the m&ker density funelionn for the treatment of free surface and a combination of a Smagorinsky-type subgnd scale and a Boldwin

Lomax turbulence models4. The propeller action in the form of a body-force was incorporated into the method for the simulations of a self-propelled ship in

steady straight course. As mentioned above, we

focus only on VLCCs in this study, so that the Froude

number is low and double- model flow assumption may be acceptable. Accordingly, the free-surface algorithm is not activated. See the references about

each computational procedure for more details. In this study, we have provided the WISDAM5 with

some modifications for motion simulations, which

are described in the following sections.

INERTIA FORCES IN NS EQUATION

Since a coordinate system is fixed to a ship,

inertia forces are incorpórated into the NS equation as

bodyforcesin Is:

(I)

8u,

lp d(."

17

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Twenty-Second Symposium on Naval Hydrodynamics

transformation of coordinate and is given by

f =-2Ckx(t.cr)rI

(5)

where fl is the angular velocity vector (0,0, w) and

y the flow velocity vector (u. , v, O) and r the

position vector (x, y, 0) in s The terms in the

RHS of Equation (5) indicate the Coriölis force, the

centrifugal force and the unsteady forces due to

angular and translational accelerations. Considering

only plahar motiOn, Eqúation (5) is simplified as

follows:

BOUNDARY CONDITIONS

The boundary coñditions for the NS solver

are transient because the computationäl region is

fixed to an arbitrarily moving sbi. Al the inflow, let the position vector on the boundary r, the velocity

is given by

v(r0)=VfZxr0

(9)

In the case of planar motion, Equation 9 becomes as

follows:

u,(Xo,Yo,Zó)=.4'x +o,yo (IO)

v(x07y0,z0)=-5, Olz:0

(Il)

w,(x0,y0,;

)=O

(12)

For veIocitç the other conditions are the same as those of steady state simulations, ¡e. ño-sup on the

body and zero-nOrmal-gradient Neumann condition at

the outflow. Since doùble model flow is assumed,

symmetric condition is imposed on the still water

plane.

The Neumann-type pressure condition on

the body boundary has to be

aife

(13)

which differs from steady state simUlations. The

others are a Dinchiet condition (p = O) at the mflow and a zero-gradient Neumann at the outflOw ad. on

the still water plane.

MATHEMATICAL MODEL FOR RUDDER,

PROPELLER AND THEIR INTERACTION WITH

HULL

We have chosen the so-called MMG model

as a mathematical model, where the forces of a

rdder and a propeller are represented as point-wise loads at their positions, respectively, including their

interactions

with a

hUll. They arè schematically

drawn in Pig2..

X,q NH

YA NR

Fig.2 Schematic of Forces and Moments Actin9 on a Ship

The forces and moment of a rudder are

calculated by the following equations9t.

where ali

and x-

are the interaction coefficients

between the rudder and the hull, _XR nondimensionalpositionof the rudder, 6 the tudder angle and (I IR) the tithist reduction coefficient

due to the rudder. F,, is the hydrodynainie normal

force of the rudder usually modelled by

FN =ApUR2 A+Z25'C1l

(17)

were A is the area and A the aspect ratio of the

rudder. UR and a,, are the flow speed at the rudder

position andthe effective rudder angle:

U

=.,,Iu2+v2

aR

=ö+tan'-

(19)

"R

The MMG model proposed the fotiowing form of the effective velocity at the rudder position, «R 'and VR,

726

fr=2wv,+w2x,

+wy,Vx,

(6)

)5'=-2ai,

+wy,+wx,

Vv3

(7)

ft=0

(8)

XR =(ir,)isiaö

(14) Y

_{=(Iaff)FNcosô}

(15) NR = (x +a x,, )L,,,,FNcosô (16)

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r

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considering the effects of the propeller9.

lw

Uk UE X

r

(20)

£Lk

_{+f1&.L(2_..)1si}

He)

_L

Hc

_)j VR =YR('S+1RLPp) (21)

where D,, is the diameter of the propeller, H the

height of the rudder, ¡ç the propeller acceleration

ratio, 7, the proportinal coefficient for the lat&àl

component of the effective flow velocity at the rudder

and 'R the noñdìmensiônal lever for the rudder. The

experiments indicated that YR is almost constant

when the propeller is notteaction but, otherwise, VR

is nonlinear function of +IRLPPO4 9) In this study,

we assume that YR is constant just for simplicity. The propeller slip ratio, S, and the ratio between the

effective wake coefficients of the rudder

and the propeller, L, are given by

5=1

(iwji,

np

£ 1=W1

1w,

where n is the revolUtion per second and P the

pitch of the propeller. Séveral fUnctions for (i w)

based on (iw0). the wake coefficient in straight

course, are suggested in the MMG modél but none of

them was successful under the conditon of large

rudder ang1e For simplicit we use

(1w,) in

place of (iw,,) in this study as long as the rudder

aflgle is not very large.

In this study, thrust is a point-wise load at piopeIIer position sirnpy given by the simulated

negative resistance to a Ship in straight coUrse as is

given by

(24)

Therefore, the thrust reduction coefficient (i:,,),

which is not explicit in Equation (24),

is also

regarded as constant daring the ship motion without

excessive rudder angle.

lt should be notéd chat there are four kinds

of parameters in the MMG mödel in the present

method. i.e. the given dimensions ( r, 4.

727

Dr,,

H. P), the given manoeuvring parameters

(8, n), thé values empirically obtained in advance

(a,,,

x,, ,

(1tR), (iw,,),

. _Y _'R)

and the transient results of the NS and the motion

equations(u5, V5, W).

CONBINATION OF EQATION OFMOTION AND NS SOLVER body (OEce Grid

Generator P

NS Solver WISDAMS Flg.3 Flow Chat

Fig.3 shows the flow chart of the present

simultión method Firstly, a grid system is generated about a ship by a grid generator and is transportad co

our NS solver, WISDAM5. Then the steady-state

solution of a ship in straight course is solved as the minal state for manoeuvring motion simulations At

this stage, thrust which will be used invariably

throughout the simulation is obtained by Equation (24). Here one gives the steerage of desired ship motion. In the MMG model described above, the

forces and the moments of a flidder and a propeller are calculated considesing their interactions with a

hulL Since WISDAM5 solves the unsteady flow

about the ship time-evolutionarily at every time step

it provides the equations

of motion with the

hdrodynafluic forces acting on the hull. Then the equations of motion are solved and the resultant

translatiOnal and the angular velocities and

accelerations are used for the calculation of the inertie forces, which are incorporated into the NS

equation as body forces. This procedure except the

grid generation is iterated at evéxy time step. The

WISDAM5 adopted explicit cime-integraticm and,

therefore, time increment is thought to be small

enough to express the ship motion.

Cal. of Inertia Forces X,,Y11N EqUazin Mathemacai

L

of Motioñ Model (MMG Model) NRXp

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SIMULATION OF DRIFT FLOW

Before simulating the manoeuvring motion of ships, we solve drift flows in order to validate our

NS solver WISDAM5. It can be said that the

sufficient degree of accuracy of the

drift flow

simulation is a necessary condition for the successof

the following Z-manoeuvring ¡notion simulations.

Rßcently

cimiIr

simulations of chiftflow were done

by Ohmori et al. using WISDAM5 with a moving

grid system. in theirinethod, when a ship changes its

heading angle from suaight' to drift ¡n the fixed

computational domain, the grid System fitted to the ship has to move and alter its shape. This may limit the maximum drift angle because of the distortion of

grids. Here we alteaupt the similat sfrnii1eruns by

using the fixed grid system fitted to a ship.

Table I Princial Dimensions

The Reynolds number(Re) of thesimtilalion is 1.0 X

106, which is smaller than that of the model test (2.8

X 1O _{because the tUrbUlence model used in the}

WISDAMS has been tuned to match forces and 'ake velocity distribution with those of messi for

this Re and thegiven grid spacing. In terms of the forces related to the planar motlön of ships such as

lazerâl force and yawing moment are not thought to be very sensitive toReas long as the flowisturbulent

and as the Simulation Re has the same order of

Tnagninlde as that of the model test. Using the

steady-state flow result in straight course as an initial steady-state, a

ship starts drifting ai the nondiniUnzional time (t) of O

with keeping the trnklaiioI velocity constant and

reaches the target drift angle at t=0.5. Thesimuladon

lasts until the flOw reachSs the steady state (r>2).

We have picked two VLCCs (ModelA and -B) with the same principal dimensions and different

aftpart frame hues; one is so-called V-shaped and the other Is U. The principle dimensions of these

models are listed in Table I. The drift angle is set to 9

degree. The Speöd of the models is 0.807tn/sec

corresponding to I5knot for the 320m long ships. We have the model tests results Under these conditions,

where the motion of any degree of freedom is

prohibited.

728

Ag.4 GridSystem for Ship MotionSimulation

As shown in Fig.4.thecoinpulational region

is half-pipe-shaped of the longitudinal length of 2.5 (-1.0 to 1.5 from the midship) and the radius of 1.0

ñoridlmeusionalised by 14,p. The number of grid

points are 190,991 (101X31X61 in the longitudinal, the radial and the girth directions respectively) The

nnndimerjöJjs

niinimum gnd spaces in the

radial directiOn are 1.OX 10 at F.P, 0.IX 106 at

midship and 0.4X 106 at AP.

Fig.5(a) Longitudinal Distribution of Lateral Force

of Model-A OA 02 0.t .01 .02 .0.4 .02 0 z

Particulars Model-k I Mödel-B1

Lpp(m) 3.50 B (m) 0.634 D(m) 0.328 draft (m) 0.211 Cb 1 0.802. Op 0.806

Cpa.

0.7504 0.7557 Icb 2.45O -2.610 02 02

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The comparison of the longitudinal

distribution of lateral force between the simulations and the experiments of MOdel-A and -B is shown in Fig.5(a) and (b), respectively. Although the tendency

of the simulated (Orce profile agrees very well to those of the experiments, Ìnall discrepancies are

observed near FI' and AP in both figures, especially near AP of Model-A. where the recovery of laterál

fore to the positive is

larger than that of the

experiment The difference near FI' may be due to the

effect of bow wave which is excluded in

the simulations. Fig.6(a) and (b) indicate the contour map

of pressure nthe transverse plane at the position of 0.45. The presSure is nondimerLsiönàlisCd by pV0' where V0 is the ship speed in straight course Solid

and dashed lines indiOate the positive and the

negative pressure, respectively, the interval of which is 0.02. A negative pressure zone can be seen in each

figure, which corresponds to separated stróamwse

vortices- in the wake óf dtift flow. It is noted that the position and the intensity of these simulated vortices

affect the lateral force y AP to the significant

extent.

Flg.6(a) Contour map of Simulated Pressure at

x=0.45. Model-A

Fig.6(b) (continued) Model-B

Fig7(a) and (b) show e time history of the total lateral force and the yawing mo cnt of Model

A and -B,

respectively,

obtained both by the

729

simulations and the experiments. Henceforth, forces

and moment are nondimenSionalised by

..pt'dLpp

and -pVf)dLpp2, respectively. The

FrOude nuïnber of the experiments is 0.074 in order

to make wave effects as small as possible. The

sirTíulated lateral force and the yawing moment for

both models are in considerablygoodagreement with

those of the experiments. Therefore, ii can be said

thatthe prOsent NS solver petforms very well in the drift -flow and that the success in these simulations satisfies a necessary condition for ItS applicability to

motion simulations.

Flg.7(a) Time History of Total Lateral Forceand

Yawing Moment of Model-A

Fig.7(b) cçntInued) Midel.B

SiMULATION 0F Z MANOEUVRE

Using the same grid systems as those Used in

the drift simulationior Model-A and -B, respectively, we apply the presentmethodto the simulation of 10-degree Z-manoeuvring motion. Empirical parameters in the MMG model are set to those obtained by the model tests and rudder and propeller dimensions are

Siso those used- in the tests shown in Table 2.

o,. ala el -000 200 0ø ON a 4 0'Z

r.

-e o. N -o .001 L -e - o.o - - O 2 20 0

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Table 2 Dimensions ofRudderandPropeller

The position of the centre of ravity is 0.03 from the

midShip and. the inertia radius is 0.24 for both models as set in the model test. The speed of steering is given

by a constant value of 22.2 degreelsec. The Reynolds

number (Re) of the simulation is 1.0 X 10°. The

steady-stare flow result in straight course is again

used as the initial state of the Z-manoeuvnng motion

simulation. Firstly, the rudder Starts moving at the given constant speed until its angle becomes IO degree. Then the ship gradually starts yawing. As

soon as the beading of the ship becomes 10 degree,

the counter-steer is cmmaflded to set the nidder

angle to -10 degree. After the overshoot of the

heading the ship turns towards the extension of the

initial heading direction. The saine helm is made

when the heading becomes -lO degree and,

consequently, the ship is manoeuvred in a zigzag

curve.

-I

FIg.9(a)'Time History of' lciIate Heading."

Rudder and Drift Angles of Model-A

Fig.9(a) and (b) show the time history of the simulated beading, rudder and drift angles of Model-A and B, respectively; The horizontal axis indicates

the tine nondimensionaliSed

by LpI V0:

1.0

coiresponds io the real time of 4.34sec. These angles obtained by the expenments for Model-A and -B are

also shown in Fi.lO(a) and (b), respectively. The

simulatòd results clearly indicate the typical

characteristics of manoeuvrabiiity due to frame lines in the aft part as wéll as the measUrements, i.e. V shaped Model-A responds slowly to steering and, on

the contrary, U-shaped Model-B makes quick

response. Although the simulated timings when the

heading angle reaches the extrema are about 10%

earlier than those of the experiment for Model-A, the values of the. extrema are in good agreement with those of the measurement. In the case of Model-B,

such timings are much earlier, say 20%, and the

extrema are in moderate agreement with those of the

experiment. The simulated drift angles are a little larger than those of the experiments for the both

models.

- 2 4 8 I S '2 I ie

Fig.1 O(a) Tlme History of Measured Heading, Rudder and Drift Angles of Model-A

Fig. 10(b) (continued) Model-B Diménsions - Values Rudder Area(m2) 0.011455 Heiqht (m) 0.1372 Aspòct ratio 1.653 Propeller Dlameter(m) 0.1028 Pitch (m) 0.0645 S a a à G '2

Fig.9(b)(conthued)Model-B

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V 02 e .00

0/

o. o

--.--Fîg.1 1(b) (continued) Model-B

i

Fig. i(a) rime4Histcry of 6CalcatedShi'Spe1 and w Rate of ModekA

Flg.Il(a) and (b) indicate the time history of

the simulated speed and yaw rate of the models A and

B, respectively. These velocite obtained by

measurements for the two models are in Figi2(a)

and (b), respectively. IO these figures, the yaw rate is

stated in radian per noadimensional

me of 1.0.

Thnist in the simulations ¡s simply given by the

negative resistance of a ship in straight course

Nevertheless, the decrease tendency of ship speed in Figure 11 agrees well with that of the measurements,

except for the difference in time axs Similarly, the

simulated tendency of the yaw rate is in good

agreement, though its

values are larger

in the siniulatiöns thOñ those in the expeninents for the both

models. This large values, especially for Model-B,

correspond to the large heading angleS shown in

Fi&9(b). 731 'o o. 02 e .02 .04

Flg.12(a) Turne History of Measured Ship Speed and Yaw Rate of Model-A

'2 O. e. 0e 02 .00 .0i V 2 4 O I tO IO 4 lO '4 3 4 ts 12 ti 1* 0 2 4 0 2 Fig.12(b) (continued)viod°l-B

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Twenty-Second Symposium: on Naval Hydrodynamics

Figé 13(a) Time History of CalculatedLateral Force f Model-A

Fig.1 3(b) (contlrued) 'Mdeì.B

Fig. 13(a) and (b) indicate the time history of the simulated total latera] force and its components, i.e the one anting on the hulls and the one calculated by the MMC fliodel at the rudder position, of the two models, respectively. The imihr components of the simulated yawing moment and their tota] are given in Fig.14(a) and (b). It is observed that the forces of the hulls are dominant for the both models On che other hand che moments of the hulls are almost matched

with_{those of the MMG model, especially}_in_{the cuse}

of Model-A. This means that steer does not work so well for Model-A, hence, the worse manoeuvrability

compared with Model-B.

732

.0.511

Fig.14(a) Time Hitory of Calculated Yawing

Moment of Model-A

2 4 1 0 IO 12 - 14

Fig.14(b) (continued) Model-B

M(6

The simulated flow fields around both two models dunng Z-manoeuvrmg motion are shown in

Fig.15(a) and Flg.16(a), respectively. Both figures were output at the end of computation i e z=16 for Fzg.15(a) and 1=13 fOr Fig.16(a). Similar figures

obtained by the abovementioned drift flow simulations are also shown in Flg.15(b) and Fig..16(b),

respectively, for cmnparison. Velocity vectors and pressure contourswith intervals of 002 are drawn on the 10th and che 42nd pIane roughly normal to the

girth direction Because there are 30 planes in the

girth direction for the half side, they are loca ed about

onç third of a right angle from the still water pialle. The taxnarkable difference betvèen Fig.15(a) and (b)

is the values of pressure. This is because of the

reduction of ship speed shown in Flg.11(a), where the ship speed becomes about 0.8 at ton 16. However, the

contours of pressure have similatily in these two figures. This can be explained by the fact that the

drift angle of 9.85 degree at-i=16 showñ in Fig.9(a) is

close to the angle set in the steady drift flow

simulation and that the yaw rate is trivial at this instance as shown in Fig il(a) so this temporary

flow field resllmbles that of- the drift flow. It can be

also said that the unsteady force caused by w does

not have significant effects even though the yaw race

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--On the other hand, contours of pressure, in Fig.16(a) is quite different from those in Fig.16(b).

For nctinc

_{positive pressure near the bow in}

Fig.16(a) seems symmeiric unlike that in Fig. 16(b)

and the negative pressure region starting from the

stem is leant more than that in Fig. 16(b). The latter cannot be explained by the fact that the drift anile at

this instance of Fig16(a) is 6.80 degree, which is

smaller than that of F1gi6(b). This can be attributed

to the lEge yaw rate at r=13as shown in Flg.l 1(b),

for the centrifbgal force bends the flow direction throughout the ship length as shown by velocity

vectors in Fog.16(a).

733

::-.--- -

_____

::--.._---CONCLUSIONS

A numerical simulation method has been developed for solving the manoeuvring motion of

bLUnt thips by coupling the equations of motion of ships and the NS equations. Our target is VLCCs so that the only planar motion in the horizontal plane is considered and the effects of wave are thought to be

negligible because ita Froude number is very low.

The hydrodynarruc forces acting on the huil of a ship are obtained by our NS solver WISDAMS and the

forces and the inonts of a rudder and a propeller

and their interactions with the hull are computed by a

mathematical model called MMG. Since a coordinate

system is fl,ced to an arbitrarily moving ship inertia forces are incorporated into the NS equationas body forces.

In order to confirm the accuracy of our NS solver, we compared simulated results with those of measutemenls for the 9-degree drift flow about two VLCC modeis with the same principal dimensions

and different frame lines in the aft part: one is

so-called V-shaped and the other is U. It was elUcidated that the low pressure near AP due to the strearawse vortices separated from the hull plays a significant

rOle to affect the total lateral force and, especially, the yawing moment. The simulation performed vexy well

to predict the lateral force and the yawing moment. lt

--Fig.1 5(a) Calculated Velocity Vectors and Fig.1 6(a) Calculated Velocity Vectors.and Contour Map of Pressure at t=1 6 inZ- Contour Map of Pressure at t=1 3 in

Z-Manoeuvring Motion of Model-A Manoeuvrtng Motion of Model-B

-Fig.1 5(b) (contInued) in Drift.Flow Fig.1 6(b) (continued) in Drift Flow

(11)

is regarded that this fact sacisfiés a necessary

condition for applying the present method to motion 3.

simulations.

Then we moved to

the 10-degree

Z-nlanoeuvring motion of the two models. The results

of the simulations are in qualitatively good and

4.

quantitatively reasonable agreement with those of the

experimenc and represent sufficiently the typical

difference of manoeùvrability betweefl the models, i.e. Model-A with VShaped frame lines shows

directional instability with the large overshoots of

heading angle and, on the conary, Model-B with

li-shaped frame lines responds quickly to rudder

5.

movement. More test simulations for VLCCs will be

able to accumulate the reliability of this method in predicting the manoeuvrability of newly designed

ships. 6.

ACKNOWLEDGEMENT

This study is conducted as part of SR229 project sponsored by the Shipbuilding Research

Association of Japan. The authors thank the LINEC 7.

Research Group for their msUuctive comments Particularly, we would appreciate the offer of grid

systems and experimental data from Dr. T. Ohmori of

1H!, Mr. M. Takai of SF11 and the SR221 project

team.

8.

REFERENCES

Hearn, G., Clarke, D., Chan, H., Incecik, A. and

Varyani, K., "The Influence of Vorticity upoti 9.

Estimation of Manocuvring Derivatives",

Proceedings of- -the 2O SYP0T on Naval

HydrndynimicA, National Researth Council, Jun. 1994, pp.669-681.

Miyata. H.. Thu. M. and Watanabe5 O.-, 'Nunwrîcal

Study on a Vicous Flow with Free-Surface Waves

about a Ship in Steady Straight Course by a

Finite-Volume Method", Journal of Ship Research. VOI.

734

36, No. 4, 1992. pp. 332-345.

Kanal, A. and Miyata, H., 'Numerical Analysis of

Structure of Free-Surface Shock Wave About a

Wedge Model", Journtl. of Ship Rcearch. Vol. 40,

No. 4, 1 996, pp. 278-287.

Kawamura, T., Mashimo, K.. Masuda, S., Kimura. K.. Mitsutake, IL and Ando, .1., "Finite-Volume

SimUlation of self-propelled Tanker MOdels", Jthceedings of-the std:_{Korea-Japan Joint Work.chop}

nn Ship and Marinß Hydrodynamics, Society of Naval Architects of Korea and Society of Naval Architects of Japan, 1996, pp. I 05 114.

Ohmori, T., Fujino. M. and Miyata, H, "A Study on Flow Field around Full Ship Forms in

Maneuvering Motion', Jonrnal of Marine- Science nd TechnoJri,, to appear.

Davoudzadéh. F., Taylor, L. K., Z1erke W. C.,

Dreyer, J. i., McDonald, H and Whitfield, D. L.,

"Coupled NaviOr-Scokes and Eqúacions of Motion

Simulation of Submarine Maneuvers Including

Crashback", roceedingc of Fluids Engiqçering

Didsion Summer Meeting, ASME, 1997,

FEDSM97-3 129, pp. l-8.

Dreyer, J.. i., Taylor, L K., Zierke, W. C. and

DavöudvsdPb, F., "A First-Principal Approach to

the Numerical Prediction of the Maneuvering

CharacteriStics of Submerged Bodies", J1rnceedings

nf Rids

Engineering Division Summer- Meeting, ASME, 1997, FEDSM97-3 130,

pp. l-8.

-Takada, N., "3D Motion Simulation of Advancing

Bodies with Mobile Wings by CFD". Master

Thesis, University of Tokyo, Mar.

1998 (in

Japanese).

Kose, K., Yoshittiura, Y. and Flarnamoto, T.,

"Mathematical Model Used for Prediction of

Manoeuvrabthty and Model Test, ßulletm of the

society of Naval Architects nf Yapan. Society of Naval Architects of Jàpafl, Vol.668, 1985, pp.

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httpiw.nap.eduIapenbooWC3D9O65372fllml735.hsiiI, conyright, 2000 The Nalianel Academy of Sciences, et rights reser'ed

DISCUSSION

S. Yang

Korea Research hstitute of Ships and Ocean Engineerùig, Korea

I would first like to thank the authors for

presenting an interesting paper. This paper pro-vides a valuable step toward the direct simulation of ship maneuvering motions by CFD. Although

they use an empirical model for the rudder and

propeller, their results axe vexy enco ra

My first question is con erned *ith the treatment

of boundary conditions. The paper states that incoming flow condition is given at the inflOw

boundary and a zero gradient boundary condition

is applied at the outflow boundary. But its not

clear how you divide the outer boundary into the

inflow boundary and outflow boundary when incoming flow is changing continuously. You

seem not to be able

to take

in enough

computational domain. Do you think that your computational domain is sufficiently wide to

remove any reflections from the outflow

boundary?

When we predict maneuvering motions by a mrtthernatirajmodel, MMG model for example,

the inflow angle into the rudder (or VR in

Equation (21)) is as important as the forces

acting on the ship bull. So, it would be very

interesting to see how much CFI) can predict this property

wèll. Could you show me your

computed velocities at the rudder compared with experimental data?

AT.JTHORS' REPLY

We would like to thank Dr. Si. Yang for his

discussiön. His first question is about boundary conditions and the size of computational domain.

Figure 4 denotes the present computational

domain around a ship The half circle locatmg at

the tight-down in Fig. 4 is the inflow boundary on which a Diricblet condition for velocity is

imposed. The other half circle at the left-top and

the side wall of the half pipe are the outer

boundaries with the zero-gradient condition for

velocity. It might be notiöed that this is normally

recognized as a set of boundary conditions for

the flow in saight course. Here the same is

applied to drift flow or Z maneuvering motion tithulations. When drift angle is small, say IO degrees like the present simulations, we have experienced no problem for computation with

735

this domain and this set of conditions. If the

angle is large, it might be necessary to consider widening the domain and dividing the side into

t*o different kinds of boundaries. In this case, the inflow may be dern$je4 by the sign of the

mass flux through each cell-face of the side

boundary If free-surface waves are taken mio

account, the domain ought also to be enlarged for damping the waves and avoiding their unphysical reflections from the boundaries.

Secondly, the discusser asks for the comparton

of the velocity at rudder position between CFD

and measurements. As be mentioned, the value of VR is particularly important to demonstrate

the MMG model because it determines the attack

angle of the rudder as shown in Eq. (17). Here the method to obtain VR is the empirical way

shown by Eq. (21). Hence, the prediction of VR

totaly depeflds on the form of Eq. (21) and

hardly on the velocity at rudder position obtained

by CFD. Just

like experiments, CFI) only

provides Eq. (21) with balk values súch as and w,the lateral componeút of ship speed and

the angular velocity of a ship in the global

coordinates. Therefore, the comparison

of

velocity requested by the discusser may nØt

satisf' him in this context.

Howevet, such comparison is very important for the

validation of CFD. Figure

17 denotes

velocity vectors and the nominal wake contours

by CFI) and measurement, respectively, on the

section plane at A.P for Model A The condition of CFI) and measurement is for 9-degree steady drift flow without rudder and propeller.

Similarly, tbroe for Model B are showñ Fig.

18. It is fair to say that CFD predicts the velocity

(13)

...

'I....

\

a _Ç,,,

-:

Fig. 17 Velocity vectors and nominal wake

contours at x-03 in 9-degree drift flow for

Model-A. Contour interval is 0.1. Simulated

(upper) and measured (lower).

Fig.

18 Velocity vectors and nominal wake

contours at x-0.5 in 9-degree drift flow for

Model-B. Contour interval is 0.1. Simulated

(upper) and measured (lower).

DISCUSSION

D. Lint

Virginia Polytechnic Institute and State University, USA

What sort of solver method was used to solve for the differential equations of motion?

AUTHORS' REPLY

The equations of motion of a ship shown in Eqs. (1) to (3) are not solved by any special method. Once accelerations are obtained by solving them,

736

velocities and positions are calculated by

time-integration stated below.

X"

=Xg"+dtXg"

(25)

.

Xg =X"

+dtxg"++dt2 -xe"

(26)

DISCUSSION

G. Hearn

University of Newcastle Upon Tyne, United Kingdom

Thank you for your very interesting paper. Some

years ago in a discussion wìth van Hooft at

MARIN, in Holiand, van Hooft suggested that forces and momenta rather than hydrodynamic

derivatives should be used in analyzing the

maneuvering ship. In this paper you show in

Figure 6 the existence of the vortex for both

models A and B. In earlier commenta made by myself during the conference, I have indicated

the importance of the presence of the vortex

upon the hydrodynamic forces. Additionally, I wouJd expect that your methodology would be

capable of demonstrating a pair of vortices which

move relative to the bull as the drift angle

increases, upto the point where the drift angle is sufticiently large to produce vortices on one side

of the hull. In other words, demonstrate the

physical properties expected of a good mathe-matical modeL Have you undertaken such an

exercise? Could you provide some intermediate resulta?

AUTHORS' REPLY

We appreciate the discusser's instructive

commpnt and question. Figures 19(a) to

(C)

denote the distribution of heicity (the scalar

product of velocity and vorticity vectors) on the section plane of x-0.45 at the moment when the drift

angle becomes 0,

3

and 5

degrees, respectively, at the beginning of Z maneuvering

motion for Model B. They correspond to the

nondimensional time of O to 2, approximately, in

(14)

coyyright. 2000 The Nalionul Academy of Sciences. of rights reserved

Fig. 19 (b) (continued)

fi

3°

Fig. 19(c) (continued) fi ...50

Fig. 19 (a) Contour map of simulated heicity at x-0.45 during Z manoeuvring motion. Contour

interval is 10.0. Solid contours indicate

anticlockwise rotation. Model-B fi=0°

737

In Fig. 19(a), there are a pair of symmetric and

remarkable vortices in the starboard and the

portside of Model-B. Each one is probably the nnxfllre of a so-called bilge vortex and the one

generated in the boundary layerofupward flow

due to the longitudinal change of hull shape. Moreover, there is a very thin counter-rotating

vortex along the side wall near the water plane in

each side, which wedges between the wall and the upper part of the upward-boundary-layer

vortex. This may be resulted from the boundary

layer of downward. In Fig. 19(b) and (c), the windward mixed-bilge vortex shrinks. On the

contrary, the leeward one becomes larger and has two humps in Fig. 19(c). As is mentioned above,

the hump near the wall is

originated from

upward boundary layer and the other

corresponds to the bilge vortex separated far

from the hull by cross flow.

The transient behavior of sireamwise vortices

ought to be paid special attention to, because it is

thought that such dynamics must affect the

unsteadiness of forces acting on the hull. Such

vortex dynamics will possibly be a new target of CFI) motion sinmiations in the near firture.