Hydrodynamic analysis and computer simulation applied to ship interaction during maneuvering in shallow water

I

To my father... Tibie of contentS V Nomenclature Introduction 6 1.1 Problem Definition 10

Chapter-i: Methods and Assumptions 13

Chspter-3: Problem Formti lation and Analysts 20

3.1 Reference Systems 20

3.2 Potential Flow Description 23

Chapter-4: Added Masaeg 34

4.i Added Maas for Sway in Shallow Water 34

4.2 Added MaSses for Yaw in Shallow Watet 39

Chapter-5: Computatioñ of Forces and Momenti 40

5.1 Tine Derivative Operationa 43

5.2 Lift Force Estimation 44

5.3 Viscosity Effects 49

5.4 Berge Forms 51

Chapter-7: Numerical

Computations and RCsult 60

LIST OF ILLUSTIONS

7.1 Captive Model Test _Results

65 a)Passing Ships in a Canal

65

1)

2)

Co-ordinate systems

Locationa of lateral erray of imagea

107 108

b)Pessjq a MOoed Ship in _{Shallow Water}

68 3) Locations of vertical array of imagea 109

dRank Suction in a Canal

70 4e) Lateral Force: SOPA caca-1 110

d)Sank Suctlo In er. Aqymmetic Canal

72

4b) 4c)

Yaw Moment! SOPA case-1

Lateral Force; SOPA case-2

11 1

112 e)Paasing Barge-rows in n Canal

14 4d) Yaw Moments SOPA case-2 113

e)Overtakjg Chips in shaLlow water ₇₆ 4e) Lateral Force: SSPA case-3 114

7.2 SIoulatong of _{rsplenjsent Operations}

77

4f.)

4g)

Yaw Moment: SOPA case-3

Lateral Force: SOPA cece-4

115

il.'

7.3 Simulations of Passjng _{Ships in a Canal}

8] 4h) Yaw Moment: SOPA case-4 117

7.4 Collision Analysis

90 4i) Lateral Force: SOPA case-S 113

43) Yaw Moment: SOPA case-5 129

Conclusions Sa) Lateral Force: WOMB CaSe-1 120

99 5h) Yaw Moment: WOMB case-1 121

Sc) Lateral Force: NOMS case-2 122

List ot References Sd) Yaw Moment: WOMB case-2 123

103

Se) Lateral Force: WOMB ceae-3 124

5f) Yaw Moment: NOMS caca-3 125

AppendiX-A

194 6a) Lateral Force: Mariner SO' separation 126

AppCndi-8 6h) Yew Moment: Mariner - 50 separation 127

197

Appafldj-C 6c) Long. Force: Mariner - 50' separation 128

198 _{6d) Lateral Force: Mariner - 100' separation 129} AppendiS-D

199 6e) Yaw Moment: Mariner- 100' separation 130

6.1

Equations of Motjo

'( i 7d)

bank SUCtjo. moment.

SS??. - H.'T 1.25 8a)

Bann suction force:

7'ujino_H/T 1.3 Sb)

BaOk suction moment: _{FUj jno_,} 1.3 Ac)

Bank suction force: _?ujiaO..R/p 1.5 Ad)

Bank suctjo0 moment: Fuj3.no.1p _1.5

8e)

Bank Suction force:

Fuj1no_fl/ 1.9 8f> _{Bank suction}

moment: _{Fujjno_H/T} 1.9 9e)

Bank Suction force: _Moody

135 136 137 138 139 140 141 9bj _{Bank suction} moment; Moody 142 105) Lateral force:

Passing Barge.tows. _{VBD casej} ¡Ob)

Yw

moment: Passing Barge_tows; VBD Case-1 bc) Long. force: P5951mg 'Barge..tows. VBD CSse-1 boa) Lateral force:

Passing Bargetows. _{VBD Casp-.2}

10e) Yaw moment:

Passing 8arge...to5. VBD case-2 10f) Long. force:. Passing Barge_.t0w9 VBD casa-2 10g) Lateral force: Passing Barge..tows:. VBD Cane-3 10h Yaw moment: Pasjng Barge_tows: VBD case-3 101) Long. force:

Passing Barge_-tows; _{VBD Case-3} lOj> Lateral force:

Passing Barge_tows: _{VBD Case-4} 10k) Yaw moment:

Passing Barge_tows;, _{VBD Case4} 101) Long. force:

Passing Barge_tows: _{VBD cese_} lOm) Lateral force; _{Passing Serge-tow,;}

VBD case-5 iOni Yaw moment:

Passing Barge_tows:

VBD Case-5 100) LOng. force:

Passing Bargetow5: _{VBD casa-5}

lia) Lateral Position - Mariner tJN)tJp 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 159 lib) Lateral

Velocity - Mariner UNREP simuiatjo0_j lic) Heading

159 160 ng1e _{- Mariner (llORE? simulation_i}

lid) Lateral 161

PoSition - Merjfle _UNREp

simulation_2

11e) Lateral 162

Velocity - MarIner _UNREP

sL'sulatjon_2

11f) beading _Atgis 153

- Mariner (llORE? _aimuIation.2

11g) Lateral 164

Position - Mariner _UNR5P

_simu1aj03

11h) Long. 165

Posjtjo _{- Mariner (lURE? simulation_3}

11?!) Heading 166

Angle _{- Mariner (lURE? Simulation_3}

111) Rudder 157

Angle _{- MarIner mREp simulation_3}

12e) Lateral 168

ror.g _{- Seje50 ship in Canal} (Moody)

Yaw Moment _{- Series-60 ship in Canal (Moody)} 170 Lateral Position - Series-SO Pessing in Canal:A.B-O 171

Heading Angie _{- Serles-60 Passing in Canal:A=B=O 172}

Drift Angle - Sanas-60 Passing in Canal:A.BO 173 Lateral Position - Sanas-60 Passing in Canal:A-B-4 174 Heading Angie _{- Series-60 Pasaing in Canai:h.B.4 175}

Drift Angie - SerIes-So Passing in Canai:A-B-4 176 121) Rudder Angle _{. Seriee-60 Passing in Canal;A=B'.4 171}

Lateral Position _{Serins-60 Passing in Canai:A.B-6 178} beading Angla _{- Sanies-60 Passing in Canal:A-B-6 179}

121) Drift Angle _{- Series-60 Passing in Cal:A96 180}

12m) Rudder Angie _{- Sanies-60 Passing in Canel:MB-4 181} 13e) Trajectory 'NEW YOPJC' overtaking: Case-1 182 Trajectory - 'lOEN YORE' overtaking; Case-2 183

Trajectory - 'NEW YORE' overtaking: Case-3 J.84

i3d) Trajectory - lO59 YORK' overtaking: Case-4 lAS Trajectory - 'NEW YORE' passing: Case-5 186

Trajectory - 'NEW YORK' paasing: Case-6 181

Trajectory - 'NEW YORE' bank suction: Cana-7 188

Trajectory - 'MEW YORE' bank suction: CaSe-8 189 131) Trajectory - Barge-tow overtaking: Case-9 190 13j) Trajectory - Barge-toi overtaking; C se-10 191

Flow Chart of Main Program; 192

Flow Chart of Interaction Force Program: ₁₉₃

lx

i

2M

Hydrodynamic yawing moment. pL

2ML

NL NL Contribution to the total hydrodynamic yaw

PL3U2 moment from Legally terms. NOHENCLATtJ]E

t _22!i₃₂ Contribut!on to the total hydrodynamic yaw

L U moment from unsteady terms. A22

Latera L sectional added mass coefficient. B

Added inertia coefficient in yaw.

Displacement of ship.

N. N: Pirat order coefficient used in representing pL N as a function of r.

CB

Block coefficient. N

N-

2H' First order coefficient used in representing

L N as s function of y.

CD

Cross-flow drag Coefficient. CL

H1

Rate of CL with change in s.

Water depth in j15 chal. I. I

Second order coefficient used in representing

pL N as s function of V

F

F2=

-Hydrodynamjc force along X-direction.

N N' *

!4i.

_{Second in}

PL _{order coefficient used representing}

pL - N as a function of vr.

F Fy

Hydrodynamic force along Y- direction.

Pirat PLUUU

ID 21a _{Moment of inertia of Ship about X-axis.}

N' order coefficient usad in representing pi. u N as a function of r. PL5 N I I N I I

2Hj.I _{Second order coafficent usad In representing}

pi. N as a function of r

Moment of Inertia of ship about T-axis..

N. N. 2N First order coefficient used iii representing

L -N as s function of t.

¡ ¡ 212

Z D Moment of inertIa of chip about Z-axis. K2 _{Lateral added maas coefficient of}

Xc

666

N6

Third order coefficjnt used in

_repzesenting

QL u

N as e function of

-n

Propeller revolutions per miüts.

q

_{Source strength.}

r

_{Yaw rate (radians per second)}

S

_{Sectional Area}

sçuare feetj

SectIonal Draught (feet)

L'

Forward speed of- ship (feet per

sac.

Induced

velocity

in

the. X-direction

o

ship-i

Induced velocity

in

the

_Y-dIrection

_of

ship-i.

Xe o L Xc c . 2X QL22 3

X-Coordjnate of the c.o.g of ship.

Distance of c.o.g from mid-ship.

liydrodynamic torce along che X-direction.

L x

-

Contribution to the totA! hydrodynaiejc

'QL-u

_{X-Force from -L.agally terna.}

x

x'

-2!!-

_{ContributIon to the total hydrodynamic}

P. u

_{X-Force from unsteady terms.}

X X

.i_

_{First order coeffjcistt usad 'in}

representing

QL 5 . X' as -

function' of

u-X. X.

-_.

_{-!irst- order coefficient used in repreéenti-,g,}

u

PL3

_{X no a functot, of .c).}

X,1,1 X,

x x

Secàñd àtder coefficient used in, representing

pc.

X as a function of vr.

-x x

_{Second order coeff1cent used in representing}

l'i

'I

I _QL

X a's a function of r

X X66-

Second order coeffici1nt used in representing,

X as a functiio

of 6

Y-coordinate of the c.o.g. of ship.

distance of c.o.g. from canai centerline.

Hydrodynamic lateral force.

YO

Second order -coefficient used in representing

pL

X as a function of y

Yo L. y '

2!

L 2fL pi. 2Y Y Y - 22 L U 4

Contribution to the total hydrodynaisic

latéral force from -L.agal'ly terms.

Contribution to the total hydrodynamic

lateral force from unsteady terms:

'T' Y'

First order coefficient used in representing

QL U

Y as a function of

V.

Y Y' ,

-

1.L!.L

Second order coeffi'c'ent used in representing

pc.

Y as a function of' y

Y,1

I'

'

I

_pi

Second order coefficient used in representing

Y as a functIon of

T. Y.

- ---

First order coeffIcient used ir. representing

Pl.

H

_L order Coefftc4er.t used In _reprasenting Y as C function of r. T. Y8 -Pl.2jJ2 PL' 2Y5

22

PL U ç

Pirst order coeffjcLet usid in rePresenting

Y as s function of z-.

Z,aterel force when $ = I - O.

First order Coefficient _{used in representing} Y as a function of .

rirst order coefficient used _{in representing}

Y as a tunctjo _{of 6.}

2 Third order coefficient _{used in representing} pL /5 Y as s. function of, y

Third order coeffIci1t

used in representing

L U _{T as a function of I} Drift angle

Rudder deflection angle. Heading ang1e

VELocity potential Doublet strength

Stagger or longitudinal _separation

INTRODUCTION

Navigation of ships in conf med waters,

particularly in narrow channels hes received considerable attention in recent yeàrs. Sound knowledge of the behavior of ships opereting in close proximity In canale and harbors

is very important for ship owners, operators and _{everyone at}

large. It _{is well known that ships pásaing in canals are} extremely difficult to control because of strong hydrodynamic interaction forces and moments. There is _CiBo considerable interest in problems concerning the navigation

of barge flotillaa in rivers apd inland watere. These

barge-tows are

highly

susceptible to collision which in recent years have resulted in mil'lione of dollars lost in

damages. Another related meneuver is the procedure of replenishing ships while they move along parallel _{course. As} the ships nove close to each other for a considerable emount of time, the Interaction forces come to play which

3f shins under the ebo'e circumstances

Free rUflnig;-node1

tests with radio

_cont

lied models in shallow water are very

exper.aive to conduct

_{and the small models}

wbch are usuay

employed

jo

such

tests

are

susceptible

to

large

scale

effects. The number cf

parameters involved In the

_{studi' of}

maaeuver4nç of ships

i

reetrjced channels are too large

(dimeosjo09 of the canal,

_{vessel apeed, proxiity}

of second

shIp, vessel orientation

_{steering control}

...) to permit a

complete model

experiment Investigatico Computer situiat0

on _the

_other

_hand

can be éed

_very

_efficiently

for

simulation

of

_all

_types

_of

inaneuvero. _under

_different

conditions,

_Permitting

_atudy

of

the

_{Contributions}

_to

maneuvering forces made

_{by various changing}

parameters, once

the

basic

data

_for

_the

hydrodynamic

_coefficient9

_are

evailable.

Computer simulators

_{presently running at}

research

statjo0s

_like

_the

_Army

Corps

of

Engineers Waterways Experiment Static11,

_Vicksburg,

_{MS, and}

Computer Aided

Operations Research

_{Facility, Great Neck, Mew}

York, empjoy

empirical

modeling

of

_interaction

_and

_shallow

water

phenomena

_{based on}

model experiments

_{The present wòrk}

combines s generalized

hydrodynamjc force model with a ship

maneuvering model

_{to provdp a}

complete closed-loop ship

trajecto-y prediction

sYstem.

Thi

_{would have uritity}

_in

studying maneuvers

involving

ships

and barges jfl:

close

proximity in restricted waters.

_{Such a model can be used in}

training pilote who opérate in harbors,

_{canals and rivers,}

and also in studying the effects

_{of varIous control symtems}

in the early design stages

There have been a number of theoretical

_studies

and a few

experimental

ineetigationm

coúcerning

the

hydrodynamiic interactions between

ships in close proximity

in restricted waters. A brief

survey of these works can be

found

in

Sankmranarayanan

(39].

The

_{analyses range from}

simple

_{two_djmansjo0al}

_{approximations}

_to

_are

complex

analyses

involving

mathed

asymptotic

expansions and

integral equations (as in Ref.

2,10 and 45).

All of these modele and numerical results show that

respectiv.

methods

provide

reasOnable quatnitat ive

predictions of both lateral force and

yaw moment. However

all

of

them need

further

improvement

in

order

to

give

acceptable

_quantitative

_predictions.

_it

_is

known

that

two-dimensional

theories

are

especially

objectionable

because of

the

fact that

the

_{two-dimensional assumption}

breaks down even with a small clearLnce bàtween the shIp and

the canal bottom.

_{The flow beneath the ship reduces}

_the

9

forces and moments.

Neglecting this flow as in the case of

tWO-dimensioùal analysis is _{not considered as spprcpriate.}

All of these analyses have

considered the ships to

be moving parallel _{to each other, as well}

as parallel to the canal walls. The _{degree of agreement}

between theory and

model experimental

data varies, with better _{agreement shown} by lbs more complicated

analyses-computational atudies. The form of the results _{as well as the effort required}

for he CompUtations, precluded their _{utility for simulator studies.}

This was _{a major motivation in carrying}

out experimental studies of forces on passing

ships, as was carried out for the Panama Canal

eimuletion study recently _{carried out by}

CAOR! (Computar Aided

Oper5tioñs Aesearch Facility) (41)..

In a

recent, theoretical _{study carri-ed -out} at VPI&ST.J(39( (also published

n (22)), a theoretical enalyaja

was _{carried oiac for}

the case of _{passing chips in}

n

asynsuetric canal.. _{The ships were conátrained}

to be moving

perel,lel to eech other

and to the c'nel walls. _{The method} of analysis used the

taqailvthenrne_c...--.. s'

-l_o

computation, in contrast to earlier approache6 Compari-sons

of the calculated results for passing ship cases in the

Panama Canal tests at (Swedish ShIp Modal Teating Tank)

SSPA (411, as well as for paeeing e moored tanker in NSMB

(Natherlamds 9hip Model Basin) teats (38), ehowed good

agreement with the teat date. Some limited compensad wIth

test data on steady forces on a single ship in so- asymmetric

shallow canal (sa given in (291) also showed fairly good agreement, with some featüraC of the results indicating a

possible need for further refinement.

All of -the work referred to above did not consider arbitrary motions of the shipS, which would properly reflect their behaviOt in realistic cases, but only the constraint of- parallel motión at fixed lateral separation distances. The moat uoofui tool for a .aimulátiOfl study would be a mathematical model that wodid remove thät constraint and also include the interaction effects due to the ship dynamic

motiofls

11

rIver. The first part of _{this work descri,es the ieteractjon} force model, and the second part deals with motion

simulatIons. The canal' (or channel) can have asvmgflecrjc

depth conditions since

export-oriented' ports can consider

deeper dredged regions for

out-going cargo-laden ships. 'The

intent 'is _{tO calculate the' generajia}

hydrodynamic forces acting on each ship due to _{the effects of the other chip,} including the influence of the _{dynamic notions of that ship} (i.e. sway and yaw), se well

as the' influence of the canal

valle, when the ships!

operate in shaì'iow wster _{Thó' case of} a 'singlo _{chip in}

e canal or the ao-called bank' suction

problem im ajeo' covered in thé _{scope of thi'a work.}

The forces to 'be _{found represent the 'generalized} forces due to another ship _{and the physical boundaries, and}

are those not included hen modeling the convéntional

hydrodynainic forces acting oñ' a ship in either deep or shallow water,. _{Thé' inclusion of theae forces,}

whén also accounting for the effects of the _{dyùamic motions of}

the ships on the forces, allows full modeling of sh!ip

interactions which is not _{comoletãiy carried out at presént}

n _{ship simulation, studies.}

Aside from cases

of shji

passing, and/or overtakjg,, the inf'lueàce of

orientacjo, crcbig- motion, etc. fòr the case of two Ships as well as 'for a sLngl'e

sbip near a canai, we'll havé

12

not been' either fully analyzed or placed in a form readily

adaptable for use 'In a simulator. The application to

simulator stüdiec. which provide information appropriate to the problems of ship control aàd navigation, as well SS! to

harbor

end waterway development, are illustrationc of the

CHAPTER-2

NETHODS AJt1 ABSUMP'rIoHS USFD _{X THE AN.%LYSIS}

The method of approach wIll be similar to that in (24 J _{and. (39], where each, ship'}

was initially represented' by

a _{two-dimensional}

vertical center piane distr!.bution of sources, _{This was done since that}

procedure wee initially considered to be a way co reflect the Cffect of the _finite

draft of the ship relative

to the canai depth, which is _flot done when one-djcensjonaj

center line source distributions

ara _{used1 e.g.,}

'(1) end (31j. _{Associated vertical}

and' Lateral imagos are established

to account for the influe,ce

of the finita depth,

casal walls, and the free _{surface rigid}

wall boundary conditions.

While satisfying these _boundery Conditions, the bocedery

condition on the ship hulls _{is no}

longer satisfied so that

correcting dipole distributions are eStablished whose strengths

are proportional to the induced

velocitiea arising from

the image system and the.other shi.

14 functional representations.

The basic analysts assumes an Invicid incompressible fluid, for

which

ideal potential flow concepts are applicable. _{Since ships in close proximity in}

canals and othar restricted wetarways move st low forward

speedS, it is assumed that wave making effects are negligibie. The bounding free surface condition then

easumes the limit condition of a rigid free surface, so that double-body reflected models are used for the ships being considered (just sa in the work of (24J end J39, as well as

earlier atudies represented by (2,10 and 45J).

In the mathematical modeling in (24J and J39), an important procedure is the uso of the lateral added _mess term applicàble to shallow water in the establishment of the

correcting dipole' strength. This is a heuristic extension of Taylor's theorem (42J, relating the dipole strength and added mass, to the case of a bounded fluid region. Since the major hydrodynaxsic force in the cace of passing shipe is

15

slender body theory results, since auch terms

_{were found to}

be

_{dominant when}

considering bank

Suction

forces

for

a

single ship in a canal

_{(for the cases considered}

in (24)

and [39')).

On the basi'5 of

_{the procedures and resulta}

_in

(24) end (39), where the

_{major influence, of finite}

depth, was

manifested

In

the added

0558 dependence on

depth,

the

influence of the

_{vertical diettibution, of}

source strength

for the

ship

_{representation}

_{was found}

to

be

_relatively

unimporteflt

_{As a Consequence of}

this reault, it will be

agswsed that the

_{ïnitja'l source distribution,}

for the body

can be represented in terms; of a, one-dimensional

_{center line}

distribution,

_{This will reduce the}

integrations required

for the final force

_resulto,

_{thereby reducing Computation}

time COnsiderably

_{For the present problem}

it will alad be

necessary to Include dipole

distributions for both ships'

to

represent

their 'sway and yaw

_motions.

_The

formulation

allows an arbitrai-y

_{orientation of jach ship relative}

to the

canal veils (which will be assed to be parallel

_{to each}

other),. First all of

_{the necessary images}

to account for the

bottom,

walls

nd

free

_{surface, are}

_established

_for

he

source distribution for each

_ship.

_{The correcting dipóle}

distributions in each ship

_{are then determined, recogn'hjng}

the effecte of varying

_{orientation ad changing positions}

16

that are possible due to the motions of each ship.

Furthér imprdvements in the modeling of )24j and

(39)

are

also

applied

to

the

slender

body

lift

force

representation.

The procedure ued had followed a method

similar to that

of

,3acobs

(19), whih contained certain

semi- empirical features.

Och features are inherent in the

application

of

elender

body

theory

modeling

for

ship

stability derivatives, and generally represent the present

state-of-the-art in such estimation procedures. However the

application in (24) süd (39) treated the sectional added

mass and resulting lift force at the after end of the ship

separately for the ship hull', the effective skag addition,

and the rudder, with each considered as

a

vertical

flat

plate.

The same type of shallow water and free eurface

influence function vas applied for each element, which is

not appropriate for all ships.

A more refined treatment of

the slander 'body lift force, in terms of the proper model of

the added mass effects of their envisioned trailing edge or

stern end in a shailow water region, is provided here as an

improved modal.

In. addition another effect often used in slender

'body theory analysis of transverse forces on bödiea, viz.

17

analysis of the hydrodynamic forces. _{Phis effect, using :he}

distribution of induced lateral cross-flow Velocjtjes due to

images, is expected to hive its major influence _{on probleme} associated with single ship bank suction effects due _{to the} -lominance of inertial effects in the unsteady passing ship

case. This vLscou ames-flow dreg eff act is expected to have a primary influence on the yaw moment _{rather than the} side force, since the local _{cross-force representation will}

tend to cancel when integrated over the hull length.

Similar type results have been found _{1.n other cases where} croes-ulow drag is considered for determining lateral _forces

and momentfl, _{as an adjunct to the pure potentiel slender} body theory results (e.g. see (21j).

While these different improvements _{are useful} refinements, the major effort involves the _{treatment of: the} effects of varying orientation end the _{inclusion of}

_ship

dynamic motjot effects on the reeulelng genera'ized

hydrodynamic forces. _{The mathematical model includet all of}

these effects and will therefore reflect the

influence of the maneuvering of each ship. The _{geueraìjze force module}

18

responses, so that they will in turn influence the subsequent responses. Thus a coupling exists which is

considered to be a realistic modeling of ship interaction in a purely dynamic sense, which is not present

in

existing simulator models.

The present analyeis is intended co represent

these Interaction effects in the most complete manner poaeible. Results for a single ship are also expected to be

a more realistic representation of the forces acting on a ship near a wall, since the analysis represents the influence of ship beading orientation es well as the ship

sway and yaw motions. The model that is eetablished as the

end product of this work Le expected to be the moat complete possible for generalized interaction forces, and will be in a form suitable for direct utilisation in a simulator syetem.

The computer program for determining the forcam on both ships will also be established in this work, with the intent to have a rapid computation capability so thatit

i

19 need for aciditional special

experimental or empirical date. In this way the force module

will be similar to the major elements that are usad in _{predicting ship motions in}

waves

(e.g.., as in (37)j, with only similar type input data necessary to apply such a theortjcal _model.

This is a

definite improvement in the field of _{maneuverability} analysis that has _{reilad largely on experimental}

data to establish hydrodynanic force modale.

3. i) UFERENCE SYSTENS

In order to develop convenient analytical expressions, three coordinate eyntéma are used In the analysis. 'two of these x,y,z) and (x.y2,z2)J are body fixed coordinates. and the third one is fixed at the see or canal surface. The body fIxed reference System

20 CMAPTER-3

PRODLEX FORMULATION AND ANALYSIS

The problem considered here is that of two elender bodies moving through an ideal fluid; the method is general and je applicable to ships moving in shallow and

asymmetric canals. The ships may have arbitrary

orientations end velocities, both of which may be changing with time. All possible maneuvers euch as ships movIng close to each othar in shallow harbors, a single ship In a canal (or the bank suction problem), replenishment at sea,

ships passing a moored vaeeel, etc. are all special situations which can be investigated using the general procedura developed hare.

- -

I'}i e1,4_I..s..h

..

along the vertical step in the canal bottom. These

reference systems are illustrated _{in Fig. 1.}

The positions and heading angles of both _{ships are meaeured with respect}

to the earth fixed reference _system.

We derivatives directions

also have the

with respect

- cos (

tollowing relationships for the to the various coordinate

- P) (3.5)

When evaluating various quantities it is frequently required to transform variables between

systems. _{If we denote the origins of body axes as}

(x51,y)

and

(X05,10),

and the respective heading angles _es

and

any point (x1,y) or (x5,y3) _{can be expreseed in terms}

of the fixed coordinata system _{as below:}

X0 - + x, cos _- sin p, _(3.1) sin P, + _{coe p} (3.2 âx = -sin _(p1 coe (p sin (P1 P1) - P,) - P1) (3.6) (3.7) (3.8) ay, ay, L... 21 27

on each ahip has its respective _{origin at the intersection} - (X01 - x0) cas P1 + x cos(p - p,) of the mtd-ship plane, _{center.pjane,}

and the load

water_plane of the corresnondjng ahlp. _{The x-axis peints}

0

_{(y01 - y01)}

sin P, y, sin(p, - P,) (3.3)

forward, y-axle points starboard _{and the z-axis is positIva}

(X01 - X0)

sin p + x sin(gs - p)

downwards, _{in accordance with the right hand}

convention. 'oj

+ _, cos(P - (3.4)

23 3.2) _{POtentiai flow}

Description.

a

first approximation. _{the flow field}

around

the ships are _repreeented

by

_ontj05

Center_line

distrjbutl00

of sources end

sinks moving with the ships in a

fluid at, rest.

The' source

strengths for axi_syerj flow about a clender

'body are obtained _as:

dS-qI(x>

-475 dx (3.9)

where u- _{is the forward}

speed, S1 _{i's the local} Sectional area and _{the subscript}

'i' _{stands for the}

ship

Under consideration

This represents _{the flow}

around-the

double bodi5

obtained by

Considering the ships _{and their}

reflection at the _free

surface. _{Then the}

veloity

potentials for the longituj

flow about these _{doüble body}

models are given by

where the _Subscript

'i _{stands for}

the reference SYstem

Considered ând

'j' stands for the

ship on whIch the_'sources are lOc5td

(3.10)

24

The boundery condition on the body surface_requires that the normal component of the veloci'ty _{of the body be}

equal _{o the normal component of the velocity of the} fluid in contact with' the body at all points on the surfaCe. _This

condition will be Satisfied by the above _{potential only if} the Incident flow is longitudinal. _{In order to allow for} the lateral motions of the body, doublets are introdüced

along the body center-line. _{The strengths of these lateral}

doublets for the appropriate ship are given by, the fouowing

expression. (j + k57)

r1x1). 4jT

where k27 te the lateral added mass coefficient, S is the

local sectional erse, V1 is the lateral velocity and r7 is

the yew rete of the corresponding ship. _{The potential due}

-to the dipoles along the center line of éhip-i is then

j'

d7

(3.12)

where,

S (x 2 _{+ (3.13)}

like the Canal walls end the shsilou bottom. _{The wel'1 and} bottom conditions of zero normai velocity are satisfied by Introducing arrays of image bodice reflected on the canal

bottom and the wells.

ro determine the image locations, consider _{a source or} a _{doublet on the center line of ship-i,}

at a longitudinal

position . _{Then the image locations of this source} or

doublet in the body fixed reference frame _{are given by:}

The potential f unctione for the image system are to

be expressed in such a way thee they are applicable when the ships are in either of the two channels. The water depth h

depends on. the location of the ship with respect to the step in the canai bottom, (j-1 for y00; j-2 for y0<O. 'The.

potentiel. is expreeeed, allowing for the different water depth regions relative to the seep location, by making use

of the. Reòvieide unit step function, which has the velue 25

doublets epresencs the motion of a double body model in _the n O corresponde to the double body itself (i tekee horizontal

potential

plane, in an infinite fluid. However this

does not account for' the boundariee in the flow

on valnee i end 2 which

consideration).

corresponde to the ship under

unity when the argument le positive and zero when the

X1

-+ n(w+ w) sin V' _{(3.14) argument} is negative. The potential field due to the

y1 zs(w, W) cos V' _(3.115) sources and its imegee on ship-i can then be expressed as:

for n an even number, and

x1, cos 2fs + [n+1.w1 4 (n-l)w - 2yJ 5 V'i. (3.16)

Y1 for n -. ein 2V', *

[(Il+l)w

₊ (n-1.)w2 an odd number. - 2y013 cou V', (3.17) _V" Z 2mb (3.18)

OXE

- (x,-

fl(w.l. W)

ath 0X01 -

(x1-

¿coß

2

_{((flfl)W+(fl_1)w 2y}

Jein (y1-. n(w+ w)Jcos

OTO _{(y.14. Ç1si0}

2'jNn+1)w14(ii_l)w lY JCOS fr

t'i = 2m (hH(y ) + hH(-Y

)J (3.20) (3. 21) (3. 22) (3.23) (3.24) in these _{expressj00g we}

have _{neglected the} image

sources reflected _{on the relatively}

-saaL) _vertical face of the _{step in the canal}

bottom. Since the _{affecte of}

11 image flows

ere evaluated at the

center of the double body, and algo

because the _{difference in w5tar}

depths is

relatively small,

the influence of _{the images due} co the

vertical face of the

step can be assumed to be negligible. Combining the

source potentieje

corraapondjag to

both 0f the chipa-,

the total aouzce

potential is then given

by

'

ii

LnL

28 q1(DXE+DYE+DZ) "2d q, (DX0+DYO+DZ) t12d ( 3 25)

This syitem of self image potentials and the other-body potentials will induca lateral and longitudinal

velocitiec on the center linee of both the ships.

Considering also the- leteral and angular (yaw rate)

valocitiee of the shipe, we can obtain the net induced fluid velocity et the center line of both ships. In calculating

these induced velocities, contributions from the internal singularities of the correspoñding ship are not considered as their effect on the forces and moments identically_cancel

out as pointed out by L.endweber and Tih (25-J. Proper

consideration lo given to the potential functions in their respective regions of apolicability, as expreesed by the

ffeavjsjde- unit step functions,.

28

fine due to the totdj source _potential

field CSn be

expressed in _{ts loca! moving refernince}

system as

Substituting the _{expression for}

the _potential (!g. 3.25) we obtain n [1q1(D j(DXE+DYS+DZ2) (3.21)

A.nn.:

:í

DxE2. DTE

.i_n;; :2f

DXO

+DYO?.! ](DXO:+oYoa+D22)-'2d

30

may be. Appropriate coordinate transformations are used to

expreso theoe equations in either of the two ship-fixed reference systems.

By. a similar procedure we can obtain the expression or the induced velocity in the longitudinal

direction due to the source potential as given below.

:11xEiI (DXE+DYEl.DZ)312d1

+ (DXO+DTO+DZ) -+

1

b2 Ç q2 ax DICE

-e- +

28x .2 I DICO ax + ax1 (3.28) aya

DYE

-°ax

}DXE:+oYEoz./2d.

ay2

DYC

-2oxI _I

O+0y0+OZ)'2d2

Where ¡s O when n - 0. ThiS induced flow along

(3.26) ay1 y1. o

31

along the center lines of _{bàth. the chipa, the strengthè of} which are determined by use of the fo1lowing relations,

which also take _{proper account of the body motions.}

+ k)

S,(X1)(.V'(X) -

V1- r1xJ (3.39) 47T u,,' _{S (c ) u (X-,)} (1 k11) (3.'3O 4n

where ¿t is the strength of the doublets _{in, the}

X-direction u is _{the strength of}

the doubiet, in the

T'-direction, k2, is _{the local sectional lateral added} mass coeffIcient and k is, the local sectional iongttudin6j added sass coefficient of, the chip-ï.

The potential field due to these _{doublets te givén.by}

51

d,

where,

(x,1_ ¿)5

,2

Then the induced 'lateraj,. velocIties _{on chip-j due}

xu

.L.

d,

(3.31)

(3. 32 )

32

to the doublets on stip-i can be obtained sa:

CI

[,,

r

a1

(,3.33)

Similarly the expression for the induced longitudinal

velocity on' chip-j due to the doublets on ship-i can be

obtained from:

-- y1

'

,

i

d,

(3:. 34)

The total induced velocities are then obtained as

+ (3.36)

, d

U, - u + U,

The doublet 'strengths ere now recalculated using these values' for the induced velocities. However St this time the effects of body motions (per se) are to be excluded

as- we only need the lntarectïon affecta; which will directly include such 'body motion terme in 'the induced velocities. In other words, this procedüre Involves calculating' the Induced

33

and its motions. The effect of the motions of the 'other ship' is obtained this way as an induced fluid flow. This will give rise to djoole distribution along the center-lInes

of- both the ebbs. In order to calculate these dipole dietrtbuton, the self body motions are not considéred _as they will only give rise to a 'self' term in the Lagaily's

expressions. This is 50 because sil the 'self' terms are included in tbe maneuvering modal dlscribed in Chapter-6, end here we only seek the interaction forces and moments.

Chapter-4

ADDED MASSES

4.1.) Added K5s far Sway Accelerfttiofl

in Shelby Water

The added mees In sway motion (including the effect

of shallow water) iB needed for establishing the

-strength of

the doublet distribution required to correct for the induced velocities on each ship. The relation between the doublet strength and the lateral added maso is easentiallY- the

result of generalizing a relationship first established

by Taylor (423. That relation was applied to an infinite fluid domain, ànd in the present case the seme type of relation

is

assunêd to epply to a body in a shallow- bounded fluid region. Thus the added mass 1.5 assumed to be that for

a

body in shallow water, subject to the rigid free

surface

3h

water. Most full ship forms

such as those of tenkers _and bulk carriers can be _{approximated as}

rectangular farms. Other _{sectjos ¡re}

represented, by an _eouiva1snt rectangular Section where the

equivalent draft is _calculated

by dividing the _{sactjoflal area by the}

respective beam.

In order to evaluate

the added mass for very _shallow

water cases, _{a small parameter}

c _{is' introduced which is}

defined as

where T _{is the section draft}

and H is the water depth.

Then the _{two-dimeflsjoflSl}

lateral added mass rectanq,1lar cylinder, of breadth B and depth 2T,

by Newman (32] is

- -.log (4c')

+ O(c)

For full ships in _{very shallow water, this would}

give a good 'estimate

of the sectional lateral _{added mass'.} The resulting

expression for 'the sectjoaj _{lateral added}

mass _{Coeff'cjent used}

ic the oresent anelysis Is given below, where the _{division by two réducea}

to the resizl:s for the ship alone

Lthout the surface raflect5d_body.

of a as given B c*B 2c' k2 -36

(4.3)

log(4c')* - __ + + +

O(C*)}

Average value of k5 has been used for the whole

ship' in the present analysis, where,

k2(ava.)

[ e Cdx

Vol.

L

where Val. is the diepl'acad volume of the ship. The

longitudinal added mass coefficIent, k, is very siedI for

long slender bodies in an 'infinite fluid. It ja aaoumed in the present etudy that this added mass term is also

relatively small even in the case of shallow water. Hence

it will be ignored, i.e. asusned to be effectively zero in any, of the calculations carried out using the present theory.

For not so shallow situations, where TIR is lesa

than O. 0,, the asymptotic approximation as given in equation

(43) breaks down. In such casee,- a complete golution of Gurevich's (161 equations are needed. This Ls obtained by the computational -scheme outlined in Flagg 5 Newman (,131.

(4.4)

T

JI

See aooendjx-A for details.

This flatbed iflvojve _{the conforiai sapping of the}

fluid domaj, bounded by the rectançular section end the

canal betto, into a semi-infinite half-plane. _{Then the added}

mass coeffIcient for away acceleration _{can be expressed, as}

in Curevich (16) in terme of elliptic functions and integrals as below: A22-

P{_48T

--_+

03(a,q) O(a,q) 2.s

KB

M - -- log q K 37

The parameters in this equation_{are as defined below:}

.1i

(4.5)

T2(a)

q(%)

cos (2n+1)a dt and K

J

(1-t )(l-k t )

and

8,(a,q) -

2 gV4

(_j)nqn(nnußjfl

(2n+1)s where, O a a 5

nP

Oiga 1

K

38

This set of equatons is solved iteratively for

k See Appendix-Pt for detmils.

(4.11) (4.12) (4.14) r P 9(0) [ ( T2(0).T(0) 12 (4.13)

r2) :a

O(IT!/K)

T2(a).T3(s) j log fT P 8(0) R

l4i)/l

_k2P2 (4.6) - "' ein 2na

n.'

(4.15) 13(a) n

-i- +q"°

cos )2na)

dx

r

for each respective. Ship.

39

4.2) Added FIu±d

Inertia for Yaw Angular

Acceleration in

Shallow Watir:

The added fluid inertIa

coeffiejert for yaw is found for each _{ship by means}

using the

sectional values

coefficient determined above (i.

According to this procedure, _the Coefficient för yaw is given by

of strip theory _procedure,

of the sway added mass

e. _{as given in 2g.}

(4.3J).

value of the added inertia

(4.16)

Chapter-S

Còmputation of thr Forces and -Mdmnts

Having obtained all the expressions for the

strengths of the singularities, their potentials, induced velocities, etc., we can use the generalized Lagally's

theorem to determine the forces and moments on both the

bodies. For the present singularity distribution, Lsgally'í theorem for the total force acting on the double-body as

given by Landwaber and Fib (25 J is

-F - FI + Ft

yI .71 71

F1

_-

r

+

where the superscripts s and t stand for the

steady and unsteady components, and the subscripts x and

y represent the respective components of the total force,

respectively. The various components of the force and moment equations found by the method of (25 J, which also include the forcas due to self body motions as well as the

interaction effects are: 40

y' 41 - 4npJq,(v - V

_{- rx)d}

- 8V,

41Tpf de.

s at

4flPfg,(uu)

bi 8th

_4npJ

de

_B "i at bi 4mp _{Ji621(v,_}

_V,)d+

[ r,A]

whe B _{is the displacement}

of the respectjve

ship. The

cofltthutjonø from the self _{terms are present in;}

the ordinary

meneuvering ship motion

mathematical model (see

Chapter..6), and the, _parts

desired here are oñly the interaction effects due t

the second ship, canal _{walls and}

the _{shallow water.}

Ths

_{interaction force}

and moment (5 4 )

42

components are obtaiñed by eliminating the forces and

moments, thee wu1d act IL the ship were to move in an

where the quantity u in Eq. (-5. 15) ii the

atrength of the doublet distribution that would represent the body with unit yaw rare. Such a term is present regardless of whether the ve8sel has a yaw rate or not. By analogy with the expressions in Eqs. (3.29) end (3i3O), u62 la given by

unrestLcted ocean, from the eqüations 5.4 to 5.9 as below;

(5.5) F;, - 4flPfq,v,d, (5.10) (5.6) 5.7 Ft, . F'1 -

4ipJ

de,

- 4mp (5. 11-) (5. 12) (5.9) 4itp (5.15) bi

- 4nPJq,x(v_

V,-

f

F,,, - 4n0,,f,

-d,

(5.13)

+ 4nPJU

(u -(5.8) -bI u -U,)dC, (5.14)

4m- XS1 (X )

with

the

value

_of

_k'

_{found from}

q. (4.16)..

_In

_the

exprejo05 for the

_{steady Legally force above}

(Eq. 5.10 and

5.12), terms from 1.4v is not included as they are of higher

order in the Slenderness parameter r (where e

_Beam/Length,

compared to other terms considered For the same reason

j

V term

Is

not

included

in

the

steady

Ligally

moment

express io n-.

5.1) Time Derivatives

_-Operations

As can be seen from

_the

above expressions, the

tine derivatives

of

various

_quantities

_are

_needed

_for

Cai.çulating the unsteady

_{forces and moments.}

These are, Obtained

using

the

_following

_second

order

backward

differencing formula

df o

_{3fo -}

4f_-dt

_2at,'

(5. t7)

where At is the t-inc

_utep,

_f0

_{ts the valtie o}

the

function at time

t,,

'.d

f

_{is the value cf, the}

fuct-jo.

at time t - nat.

(-5. 6)

Force Contribution

Ail the interaction forces and moments treated

so

far

are

of

non-1ifting nature,

end

arise

from

t)e

potential flow representation in ternis of singularities.

In

addition to these forces, slender bodies, in steady motion In

a

iateral

flow

field

also

develop

a

lift

force.

-An

expression to evaluate this lifting force uoing slender body

theory ta- given below.

The forcé acting on, a cross-sectional plane of a

elender body moving- in an ideal fluid with forward velocity

U,

in an- arbitrary literal flow field v(x,t)-,

is given by

the time rate of change of fluid momentum.

This gives the

general expression for the etr.tp wise lateral force as

df

y-E

[ A(X) v(x,t) ]

(5.18)

where A

- ta the

sectional

added mesa. The

force

associated with unsteady acceleratiOn terms is an added mese.

sill tarn that is included in the- shi

equationsO1 motion,

so it is not considered here. -The

rernai.ninq part which is

44

quasi-steady

(because

y

_is

_changing

_with

time),

on

integrating over the

_{shin length, gives the}

Lollowing:

P

_{- U f}

____

.u2(.x) V(X,

t) J dx

- u { A(x5) Y())

_- F(X.)PJ.

_{(5. LS)}

As the bow is almost always pointed, the

added

mass at the bow can be

_{'taken to be 1ero.}

Thera will not be

any contribution from this

_{term if the trailing edge}

also

pointed.

_{For a body with finite.}

trailing edge span, as in

tho case of Bkeg In

_{a ship, the steady lifting force}

i

given by

F

_{U v(x)}

_A52(x)

(5.20)

The yaw momañt in

_{calculated in a similar}

.manner

from the lift Lorca and its assumed point

_{of appiicstionj}

with the general

expraesion for the moment given by

46

+ U j

V(X)A2dX (5. 23.)

The

last

terni

in tne above

eq.iation

is

the

moment acting on a body 'moving in an ideal fluid at an angle

of attack, which is present even for bodies with a pointed

traIling edge.

Thia is included in the Munk moment On a

non-lifting body in steady tranniatlon.

The contribution

due to this, term is already accounted for in the expression

for the steady Lagally moment,

eò the only additional yaw

moment is that due to the lift forca acting at the aft end,

which is given by the first term in the aboya equation.

rhie formulation has not been found to be

completely adequate to represent the lateral force and yaw

moment ou ships, although some of the basic' ideas have been

'adapted in

a. pragmatic. manner in some. uneul procedures

-'(e.g.

(19)).

In. such cases the separate contributions of

the bare hull,

the skeg, end the rudder ara determinad,

assuming no interference between each clament, and' the total

force and moment are found by swinsing the individual terms

47

the case of shallow water and the free surface

reflection represefltCtion of the appropriate end section. By extending

these idese to the present

case, where the ship hull and the

shag are considered as two different low aepect ratio alr±oi'ls, as _{interpreted by slender body}

theory, and the rudder ae

_{an effety5 wing of}

large aspect ratio, _the

lateral force is represented by

F _-

_{v(xb)A

'(X) +

V(x)As(xzj

where X5, x and x _{are the locations of the after} ende of the hull and skeg,

resPectively, and the mid-chord location of the rudder; s _{is the lateral projected}

area of

the rudder; and C _{is the rudder lift Coefficient}

rate given by

2

with , _{the rudder aspect ratio.}

Thé yaw moment due to lift

forces is given by

U [v(x

A22(xx5 +v(x»A22(x.)x.]

The added mase at the end of the skeg A(x5) is

taken as the two-dijeensional lateral áddad mase of a

vertical flat plate section in shallow water which is given

in (33) as,

where T is the ship draft, with this quantity considered to be appropriate for the end location of the

skeg. Since the ship generally has fullness, the use of a flat plate representation is not proper for the location of the after end of the hull. This requires a modification to

the added mase representation at the location of the

effective end of hull, which includes the effect of the ship

width, which is also, consistent with other models that apply

slander body theory to determine ship hydrodynamic force and

moment representations (e.g. (7)).

The ship section at the hull after end is, not

generally rectangular, but an equivalent rectangle method has been utilized' earlier in this work. That equivalent (5.24) p +

-r

_SUv(x)C

(5.25) i + 2/ CL

a.

(5.26) p + -T

SUV(X»XCLer

(5 27) 40M5 iTT - _{in (cos}

W

49

rectangle rapresentatio was based on using the affecive

draft (found by dividing the section area by the section bC8In) to determine sectional ödded meases as well as the total vessel added mase coefficient k2. Since more detail is required to esteblish these local sectional added mass than the overall total added mass for the vesgel, another possible approach is to use the concept of an equivalent

rectangle with an effective beam found by dividing the

section area by the actual section draft. The formujas given in Eq. 4.3 and 4.16 are used to evaluate the added mass in this wsyj uning the proper effective values. The

added mass values found by use of both of Zhese

effective-type reoresentations ara expected to bracket the

actual section added mass, so, that the valué to be used for the added mase at the hull end is then assumed to be ¿he arittusetic meen of the values foucd by use of the effective

draft and effective beam models discussed above.

5.3) Viscosity Etfect

All of the preceding analysis is beaed än

potential flow theory, although the presénca of a llft force implies the existence of vorticity and a trailing vortex

sheet. Since this trailing vortex sheet is only present in

the fLow field behind the ship, lt wIll not affect the

50

potential flow model-jag although such vorticity arisés on1y as e result of viscous effects in the fluid. One possible

effect of the trailing. vortex sheet may occur after passing

ships start traveLling in the wakes of each other. That may

possibly be the reason for large oscillatory forces and moments after the modela pass each other, which was noted in recent model taste involving passing chipe at SSPA (41-) as

well as similar type model tests at DTNSRDC. No analysis of such effects ja made here, se it je beyond the scope of the

present work.

A particular manifestation of viacou affecte that réSulta in hydrodynamic forces and moments on various bodies

(ehips, èircreft, missiles, etc.) is that in the concept of cross-flow drag (e.g. sea (21) aúd (431), with the cross-flow in the present casa arising au the induced

velocity due to the different image systeme and ciao the other ship. The total lateral force and yaw moment due to

crosS-f low are given by b

v1(X)

Iv1(x)I

dx (5.28)

5'

where the local draft at each sectioc Is

represented by 1(x). In the above exprwssions- the

cross-flow drea coefl'cient has. the suggested value Cn 2 for representatIve ful ship sections, and the absolute value on. the Induced velocity is used to produce the proper sign of the Local force regardless of the flow direction.

5.4) BARGE FORMS

Barges have simple box shape which can be, analyzed by a simplified'theoretjcel model. By the procedure established io the previous sections we obtaïn. taro source

strength along the length of thè berge -except at the two

extreme ends. The flow field around the barges are thus. represented by s source-sink pair along the center line,

movIng with- the barge. Theòe source strengthè for axi-synuzetric flow about berges can pe- expresSed as:

-q1(x) _i.!i {

a(x..Xb)

- ¿(X1-X) J

where U, is

the foard velocity

S1

is re

sectional, ares and 'the suDerscrlpts s and b stand 'io:

52

stern and bow of the barge. This form for the source strength simpl'ifïes all, the integrations needed to obtain potentials and velocities. As an example, the expression

for induced velocity Eq.3.27 becomes:

U1,S,(DYE1 (DXE+DYE+DZ)315 - 4n DYO1 J (DXO+DYO+DZVS n ndd 4!e e DXE + D5E2 J

.n

_:n:

4m(DXE+DYE+DZ2)3/2 ax ay5 nxo2 + DY01 ] -u ea 4n( DXO+DYO.DZ)'5 _a.x.a (-5.31)

This requires only e two term summation in the place of a lengthy integration whIch cuts down the

computation time, by a large factor.

Barge flotillas also- have a- tow boat pushIng

them from 'behind. These tow boats are much- smaller and l-igter -than the barge systems themselves. The- contribution

-

î

I

t

i

53

to the i.nteractjon forcea and moments from these tow boats Is similar to that of a skeg attached, tO the harge stern:. The Location of the tow-boat and its lateral sectional added: mass are the only input required for the compu'tat!on.

This can be taken as, the sectional edded mass at the aft shoulder of the tow-boat. There is aleo provision in tha program to calculate this value at any given aection of the

tow-boat. The aft end of barge hulls are nornally rectangular sections and the added masses at these nections ere caiculated.using the asyoptotic approximation given in

Eq. [4.3).

Chaptar-6

Maneuvering Model

Having established a procedure for obtaining

interaction forces and moments on ships while passing or meeting in shallow asymmetric canals it is now possible to

predict their trajectories nd study their responses to

control. The procedure uses standard mathematical model for vehicle dynamics together with the interaction forces added to all the other forces which would actduring an open sea

maneuver.

6.1) Equations of Notion

Motions: of e ship can be conveniently expressed when referred to a body fixed axes system. The reference system used here is the same right-handed orthogonal system referred in the previous sections, as illustrated in Fig.l.

the oertinent equations

_{of notion are:}

X m(uv,)

-Y

m(vu

₊ eX M - ₊

_mX(+ru)

T5 is zero

because

of

symmetry

_{with respect to}

the X-Z

piane. The forces x _and

_{and the moment Nhave}

aavsraj componens _which

are functions of the properties _of

the ship, motion of

_{the ship and prOpertjes}

of the

fluid.

In the case- of interaction

of ships in canals, x,

_{Y and N}

_are

also

functions of the orientation

_{of the shjp, proper-ties}

of the meeting, or passing shjp and the -properties

_{of the}

canal.

These

_{contributions}

_{are calculated separately}

and added

_{to the hydro4ynic forces on, a single ship in open}

sea _{naneuver, to get}

the-total forces

and

moments. This reduces-

che

hydrodynaec

interaction problem to that of

maneuvering

in

_unrestricted

_waters.

Here we chose to use the model equations -lu which

the -non-linearities are represented by a 'square absolute'

technique

_{wherein, all non- Linearitlea}

are -

reduced tq a

linear and quadratic

_f-it,

_where

_add

_fuoctiona

are

represented by Linear and _{square terms. This}

is _preferred

over a third order Taylor

expansion representation- for the

56

physical nature- of the interaction phenomenon., At larga drift angles, which is generally the- case for the- maneuvers considered here, the dominant part of- forces and moments arise due to croas flow drag affect which is proportional

_to

the square of velocity. This is the reason for adopting, a

'square absolute' representation

of

non-linearitjes

as

in

Goodman

et

al. (l5 and Miller (2BJ.

_(Góoda

foilowed. the

mathematical model of Gertler et al. (48) which is used -in

the U.S. 'successfully for- a number of years for submarina

simùlatjôns .-)

- Then we have

the

following as- or equations

of

motion:

m(ú-vr-xr2)

O.SPl'[XÙ +

X'vr]

+ O.5plX'v

+ O.Spl2uu

_;6

+ O.5p1' X'r (-6.4)

rn(+ur+xt)

o.5pi1[Y't +

'11-ririJ

+

o.5p13

+ O.5P1.[Y'ur

+ T'11v

_rIJ +

0.51n[:+

* ur)

o.spi3{ut ;,,r rl]

+ 0.5p11 H: 0.5pl[Nur + H,1,rJvl] + o.5p13u[N. _{H4 ]}

+ O.5P1.[N'uv + N'jjvlvI]

Here it is assumed that the chip

is at its self

propulsion point (il). These equations cre solved at every

instant for the accelerations and t) whIch are

integrated to obtain the velocities u.v and r. Than the

trajectories are obtained, by integrating the following

equations

x(t) U COG

- V

Sin * (6.7)

- U Bin * + V COB (6.8)

r (6,. 9)

Is

order

to

provide

realistic

simulations

similar to that of a helmsman steering a chip,

it is siso

5e

a ( _- +

b'+ c(y- y)

+

d'-a

(6.10)

where t is the time delay in rudder activation, 6

is the coiwsand rudder angle,

6 is the equilibrium rudder

angle

and 4 is the

actual rudder

angle. The coefficients in

the above equation (a, b', c' and d') are the non_dimensional gain constants which can be varied to

simulate different ersonalitiea in control of the vessel.

This equation is siso solved along with the aurge, sway and yaw equations to obtain the accelerations ù,v,t and 6. Por

practical considerations

maximum rudder angle is limited to

135

degrees

and

maximum

rudder

rate

is

limited

to

±2.7deg/BaC.

A

typical value of t

is about

Ô.t,

which is

the value used for the computations presented here.

Eda (12) bee performed an eigen

value anslyis of

this set of equations to determine the range of values for

59 r

Stability i _{that channel is maximum when a}

6 or the case

where _{b'c'd'o. These guidelines will be}

oIlowed while choosing gain constants for the simulations with the present

model.

Chapter-v

Numerical Computatione and Results

Based on the theoretical modal described iO the previous sections, FORTRAN programe were written to evaluate interaction forces and moments ou ships and barge-tows in shallow canals or rivers. The basic hydrodynamic analysis

described above includes various improvements and

refinements to the original analysis in (24) and t"). The main effort in the present generalized model involves the treatment of the effects o varying angular orientation of the ships relative to each other (and to the canal walls) as

wéll as the inclusion of ship horizontal plane dynamic motion effectG on tho resulting hydrodynamic forcee. The

detailed determination of such effects computationally is only to be considered às involved due to the great care

required to represent geometric changes in à pröper manner.

Other computational aepects applied to this work

include the effects Of us. of a one-dimensional center line source distribùtion in place of the earlier two-dimensiOnal canter plane source distributiOn in representing the basic

hLlls as well as the method of determining time derivative

terms in the force expressions. Th source distribution used has its strength proportional to the lonqitudLnal rate of change of crosa-sectioal aree,

w)jlé

the previous source strength was proportional to the iongicudna1 rate of change of the lateral sectton offset. The net result of thia change is to e1mjnate one of the integration operations (in the vertical direction for each term appearing in the final expreesions foc forces), thereby reducing the computation

time (by a large factor since 11 vertical points for

integration at every station were used in (24) and (39)).

The tIme dervative operation in the earlier effort ((24) and (39)) wee carried out analytically in terms oi the theoretical expreaaions developed there. However, for the generalized case, there would be too many quantities in the geometric. representations of distances between the hipe

that would be changing i'ith time; viz. longitudinal aûd

lateral separations, angular orientations, other motion

viriablas, etc. to consider an analytic evaimatton, from

both the analytic formulation effort jnt_gf view es well

application (See Eq. (5.l)).

For numerical computations, the center line of the ship is divided into 20 segmenta by 21 eguispaced stations along the length. Sectional areas add drafts at

these stations are input to the Program, which then

evaluates the source _{strengths and induced velocities at}

these stations. Computations only involve numerical

integrations and summations of algebraic expreBsons derived

in Chspter-3. All integrations are carried out using

impson's second rule.

-The Main Program calls the interaction force

module at every time step providing it with, velocities cOd positions of the ships. the force modula returns to the Hain Progress all the hydrodynamic interaction forces and momüt acting on both the ships at. that instant, these external forces are combined with the four equations of motion and

control, which are solved simultaneously to obtain the

accelerations. These are integrated to obtain the respective

62 61