Modelling of Close Proximity Manoeuvres in Shallow WaterChannels

Danilo Boulhosa Vizeu Lima

Thesis to obtain the Master of Science Degree in

Naval Architecture and Engineering

Examination Committee

Chairperson: Doctor Carlos António Pancada Guedes SoaresSupervisor: Doctor Sergey SutuloMembers of the Committee: Doctor João Alfredo Ferreira dos Santos

December 2014

ii

Acknowledgments

I would like to thank all my relatives that in moments of great doubts or weak always stand by me giving

all the support. I would like also to thank Centec staff for the continuous support on the development

of this thesis, specially Professor Guedes Soares that always gave me the needed advice and support

putting me in the right path and Professor Sutulo that was always there for providing me guidance on this

thesis. Special thanks to Machinery Chief Paulo Vitor Zigmantas, from CIABA, Brazil, for the possibility

to follow a lecture on a full mission simulator understanding the peculiarities of that system and general

guidance on this work.

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iv

Resumo

Um simulador de manobras offline e desenvolvido com o objetivo de analisar as situacoes mais tıpicas

a que os navios estao sujeitos em canais de acesso e zonas portuarias, o que, normalmente, configura

manobras proximas entre navios e aguas rasas. Estas manobras obrigam a uma mudanca do escoa-

mento ao redor do casco e, por conseguinte, forcas e momentos comecam a agir no corpo flutuante

(normalmente chamadas de forcas e momentos de interacao). O modelo, primeiramente, leva em conta

o comportamento em manobra do navio em aguas rasas pela introducao de fatores de correcao depen-

dentes da profundidade local dentro do caculo dos coeficientes de manobra. Posteriormente, o modelo

leva em consideracao os tipos mais comuns de manobras entre navios e entre navio e as fronteiras

do canal. A influencia do fenomeno de squat em interacao entre navios e estudada tendo em atencao,

em regressoes de interacao, termos relacionados a profundidade do local de operacao. Um estudo e

feito, utilizando formulas de squat, para analisar o caminho inverso, isto e, a influencia da interacao

entre navios no fenomeno de squat. As forcas e momentos de interacao calculados sao tomados entao

como dados de entrada das equacoes dinamicas de manobra nao-lineares acopladas do(s) navio(s) em

analise. Trajetorias podem ser analisadas e medidas de atuacao do leme sao estudadas para contrapor

as forcas e momentos de interacao e evitar colisoes.

Palavras-chave: Manobras proximas entre navios, aguas rasas, equacoes dinamicas de

manobra, modelos semi-empıricos.

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Abstract

An offline simulation code is developed to analyse the most typical situations that vessels must withstand

in entrance channels and harbour zones, which configure, normally, close-proximity manoeuvres and

shallow water situation. Close proximity manoeuvres and shallow water normally originates changes

in the flow past the hull and, thus, associated forces and moments appear acting on the floating body

(normally called interaction forces and moments). The model first modifies the manoeuvring behaviour

of ship in shallow water introducing water depth dependent factors in the manoeuvring coefficients.

These coefficients are used in Taylor expansions that calculate quasi-steady forces and moments. After,

the model takes into account the most common ship to ship and ship to waterway interactions. The

influence of squat effect on the interaction forces developed between ships is considered by taken into

account regression formulations from experiments that considers, among other parameters, water depth

to draught ratios. A study is made, by using simple squat formulas, to analyse the inverse path, which is

the influence of ship to ship interaction on squat. The calculated forces and moment come as an input

on the manoeuvring dynamic equations of free-surface vessels. Trajectories can be analysed and some

rudder actions are studied to overcome the interaction forces and avoid collisions.

Keywords: Close-proximity manoeuvre, shallow water, manoeuvring dynamic equations, semi-

empiric methods.

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Manoeuvrability in Shallow Waters . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.2 Bottom Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.3 Bank Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.4 Ship to Ship Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Description of the manoeuvring mathematical model 17

2.1 Reference frame coordinate systems and preliminary remarks . . . . . . . . . . . . . . . 17

2.2 Original Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Original Quasi-steady forces and moments on the hull . . . . . . . . . . . . . . . . 20

2.2.2 Propeller Force Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.3 Rudder Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Modified Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Modified Quasi-steady forces and moments on the hull . . . . . . . . . . . . . . . . . . . . 23

2.4.1 Modified Matsunaga [1993] model . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Regression Models of interaction forces and moments 27

3.1 Squat Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 Barrass Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.2 Tuck Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Ship passing near the Bank Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

ix

3.3 Encounter and Overtaking Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Overview of Varyani Generic Equations . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.2 Encounter Manoeuvre Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.3.3 Overtaking Manoeuvre Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Simulation Results 35

4.1 Bottom Interaction (Squat) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.1.1 Zero Static Trim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.2 Positive and negative trim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.1.3 Manoeuvre in Shallow Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Ship passing near bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Preliminary study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Ship passing near the bank with control . . . . . . . . . . . . . . . . . . . . . . . . 42

4.3 Encounter Manoeuvre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Overtaking Manoeuvre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Preliminary Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.2 Comparison between deep and shallow water . . . . . . . . . . . . . . . . . . . . . 54

5 Conclusions 57

Bibliography 62

A Matlab Code 63

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List of Tables

1.1 Shallow water general effects on ship behaviour. . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Mariner Class main dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.1 Encounter manoeuvre parametric variation. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 Values of filter parameters to be used in encounter yaw moment coef. regression . . . . 33

3.3 Overtaking manoeuvre parametric variation. . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4 Values of filter parameters to be used in overtaking yaw moment coef. regression Ship 1 34

3.5 Values of filter parameters to be used in overtaking yaw moment coef. regression Ship 2 34

4.1 Initial Conditions Bottom Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Initial conditions ship passing near bank. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Ship passing near bank interaction parameters. . . . . . . . . . . . . . . . . . . . . . . . 38

4.4 Initial conditions encounter manoeuvre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5 Encounter manoeuvre interaction parameters. . . . . . . . . . . . . . . . . . . . . . . . . 45

4.6 Initial conditions overtaking manoeuvre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.7 Overtaking manoeuvre interaction parameters. . . . . . . . . . . . . . . . . . . . . . . . . 49

4.8 Overtaking comparison study initial conditions. . . . . . . . . . . . . . . . . . . . . . . . . 54

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List of Figures

1.1 Common Interaction Situations: a) Vessel assisted by tug; b) Encounter manoeuvre; c)

Overtaking manoeuvre; d) Bottom and lateral channel boundaries interaction. (Fonfach

[2010]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Accidents due to interaction. Left: Collision in bridge pillar. Right: Collision due to over-

taking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Main Ship Carriage (left) and auxiliary carriage (right) arrangements (Vantorre et al. [2002]).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Interaction forces and moments. Left: forces and moments on the overtaken vessel.

Right: forces and moments on the overtaking vessel (Vantorre et al. [2002]). . . . . . . . 16

2.1 Ship passing bank sign convention and description of general simulation parameters. . . 18

2.2 Encounter manoeuvre sign convention and description of general simulation parameters. 19

2.3 Overtaking manoeuvre sign convention and description of general simulation parameters. 19

3.1 Squat versus manoeuvring convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Main geometrical parameters used in the bank force computation. . . . . . . . . . . . . . 31

4.1 Mean and Maximum Sinkage, Trim: Zero Initial Static Trim . . . . . . . . . . . . . . . . . . 36

4.2 Mean and Maximum Sinkage, Trim: Positive Initial Static Trim . . . . . . . . . . . . . . . . 37

4.3 Turning Manoeuvre in Deep (blue line) versus Shallow Water (red line) . . . . . . . . . . . 37

4.4 General evolution in time of kinematics of the ship. . . . . . . . . . . . . . . . . . . . . . . 39

4.5 General evolution in time of forces acting on the ship and heading, ship passing near the

bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.6 Interaction forces and moments acting on the ship passing bank manoeuvre. . . . . . . . 40

4.7 Vessel trajectory without control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.8 Maximum sinkage in ship passing near bank condition. . . . . . . . . . . . . . . . . . . . 42

4.9 General evolution in time of kinematics on the ship with control. . . . . . . . . . . . . . . . 42

4.10 General evolution in time of forces acting on the ship with control. . . . . . . . . . . . . . . 43

4.11 Interaction forces and moments acting on the ship passing bank manoeuvre with control. 43

4.12 Vessel trajectory with control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.13 General evolution in time of kinematics on the ship in Encounter manoeuvre. . . . . . . . 46

4.14 General evolution in time of forces acting on the ship and heading in Encounter manoeuvre. 46

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4.15 Interaction forces and moments acting on the ship in encounter manoeuvre. . . . . . . . . 47

4.16 Vessels trajectory in encounter manoeuvre without control. . . . . . . . . . . . . . . . . . 48

4.17 General evolution in time of kinematics on the ship 1 in overtaking manoeuvre. . . . . . . 49

4.18 General evolution in time of forces acting on the ship 1 and heading in overtaking ma-

noeuvre. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.19 General evolution in time of kinematics on the ship 2. . . . . . . . . . . . . . . . . . . . . . 50

4.20 General evolution in time of forces and heading acting on the ship 2. . . . . . . . . . . . . 51

4.21 Interaction forces and moments acting on ships 1 and 2 in overtaking manoeuvre. . . . . 52

4.22 Vessels trajectory in overtaking manoeuvre. . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.23 Maximum sinkage in ship 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.24 Interaction forces and moments acting on ships 1 and 2 in overtaking manoeuvre com-

parison study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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Nomenclature

δ∗ Desired rudder position

δ0 Rudder Dead-band Zone

δm Rudder angle saturation point

δR Actual Rudder Angle

δR Actual Rudder Turn Rate

r angular acceleration on z axis

u linear acceleration on x direction

v linear acceleration on y direction

η0 the ship to bank non dimensional separation distance

ηc y component of the centroid position vector relative to earth fixed coordinate system, transfer

κjiii Adjustment coefficient

µ11 added mass on x(surge) direction due to surge motion

µ22 added mass on y(sway) direction due to sway motion

µ26 added mass on y(sway) direction due to yaw motion

µ66 angular added mass on z(yaw) axis due to yaw motion

∇ Ship submerged volume

ψc yaw angle

ρ water density

Θ Mean dynamic trim

εm rudder turn rate saturation point

ξ′ stagger

ξc x component of the centroid position vector relative to earth fixed coordinate system, advance

xv

A(ξ′) Ship to Ship Interaction Adjustment Coefficient

Ad propeller disk area

Cb Block Coefficient

Dp Propeller diameter

fhcijj Water depth correction Coefficients

Fn Froude Number

g acceleration of gravity

H mean local water depth

heff effective mean water depth

Izz ship mass moment of inertia in z axis

k mean bank slope factor

L,B,T Lenght, Breadth, Draught of ship

m ship mass

n Propeller rotation frequency

Nq Quasi-steady Yaw Moment

Ns Interaction Yaw Moment

r angular velocity vector component on z axis, rate of yaw

S Blockage factor

SM Mean dynamic sinkage

Sp Separation distance between ships centreline

tp thrust deduction coefficient

TR Turning Gear Time Lag

u linear velocity vector component on x direction relative to ship coordinate system

V Speed

v linear velocity vector component on y direction relative to ship coordinate system

wp Wake fraction Coefficient

W0sChannel half width measured at the tool

X ′uu(∞) resistance coefficient due to forward motion in deep water

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X ′uu(heff ) resistance coefficient due to forward motion in shallow water

xg ship center of gravity longitudinal position, positive fwds midship

Xp Propeller Thrust Force

Xq Quasi-steady Surge Force

Xs Interaction Surge Force

Y0 Distance between ship Centreline and NSL

Yq Quasi-steady Sway Force

Ys Interaction Sway Force

zmax maximum sinkage

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Chapter 1

Introduction

A vessel spends most part of its lifetime navigating in deep waters and normally naval architecture

analyses are mainly devoted to investigate ship behaviour on that operational scenario. However, in the

past few decades, the increase both in world fleet and vessels size and speed added to the intensification

of use of marine resources (mainly with offshore platforms, wind farms, among others) result in an

increasing of interaction between floating structures. This factor, added to the fact that restricted waters

are becoming shallower compared to the vessels size, imposed a scenario in which safety in traffic flow

becomes a very important concern of the Maritime community (mainly Port Authorities, Pilots, Seafarers,

Regulation Organisms and Naval Architects).This situation is specially true in European waters, where

collisions and groundings represents 71% of accidents in 2011 according to European Maritime Safety

Agency (2011, apud ALLIANZ).

A huge variety of interaction configurations takes place during ship operational life meanwhile the main

ones are:

• Ship assisted by tugs (Figure 1.1 a).

• Encounters between ships (Figure 1.1 b).

• Overtaking, which can include as a subclass vessel passing moored ship (Figure 1.1 c).

• Navigation in narrow channels (Figure 1.1 d).

• A combination of the mentioned above.

1

Figure 1.1: Common Interaction Situations: a) Vessel assisted by tug; b) Encounter manoeuvre; c)Overtaking manoeuvre; d) Bottom and lateral channel boundaries interaction. (Fonfach [2010])

1.1 Motivation

With the accurate prediction of interaction forces and moments, Port Authorities, Pilots and Seafarers

could benefit of a more accurate basis for training enhancing the actual notion of the personal based

on “what I can or can’t do with that ship regarding her manoeuvring and coursekeeping abilities”, that is

normally based on experience.

Moreover, adding as a supplementary tool for the actual AIS and ECDIS systems, the insertion of inter-

action calculations inside offline simulators could help to take fast preventive and corrective actions on

board of ships, avoiding such serious accidents that can happen, sometimes with loss of people, ships

and nearby infrastructures (Figure 1.2).

In a last sense, naval architects could study the directional stability of ships when in presence of pertur-

bations evaluating new design solutions and keeping good operational profiles for ships.

Additionally to interaction, the passage from deep to shallow water originates a large number of phe-

nomena regarding the intrinsic ship dynamic behaviour. Some of the phenomena are listed in Table 1.1.

2

Figure 1.2: Accidents due to interaction. Left: Collision in bridge pillar. Right: Collision due to overtaking.

Ship passage from deep to shallow water effectsParameter Effect ObservationSpeed decrease considering same delivered power deep and

shallowResistance increase both viscous (due to squat) and wave compo-

nentsPropeller rotation rate decreaseAmplitudes of ship waves increase blockage effects due to confined waters and

increased FnhVibration level increaseIMO standard Manoeuvres pa-rameters:Tactical Diameter increaseAdvance and transfer increaseSinkage and trim increase squat phenomena

Table 1.1: Shallow water general effects on ship behaviour.

1.2 Objectives

The main objective of this work is to develop an offline simulation code to deal with interaction phe-

nomenon and the changes in intrinsic ship manoeuvrability in shallow waters. The author had previously

developed, using matlab, an offline simulator code on the basis of the solution of non-linear coupled

dynamic equations of motion given the parameters of a study vessel simulating some standard trial ma-

noeuvres, which is described in section 2.2 of this dissertation. The original manoeuvring code was

augmented to handle the harbour and channel offline simulation problem, where the interaction forces

and moment are modelled by means of regression formulas taking into account a large number of in-

teraction parameters, including water depth dependence coefficients. The intrinsic dynamic behaviour

of the ship was also modified by means of manoeuvring coefficients accounting for water depth depen-

dence factors.

3

1.3 Organization of the thesis

Chapter 1 presents a broad overview of the problem followed by the motivation, interested parts and

objectives regarding the close-proximity manoeuvres in shallow water. Then the chapter presents more

deeply the manoeuvring in shallow water and each type of interaction, starting with a more detailed

definition of each main parameters that influences them, its weights and behaviour when analysed its

influence isolated. It will be given an overall work review about Manoeuvring in Shallow Water and

Interaction calculations and the normal tools used namely:

• Empiric and semi-empiric methods.

• Potential flow methods disregarding free-surface effects i.e. accounting for only inertial hydrody-

namic loads assuming very low Fn or not. The main submethods involved are:

Slender Body Theory

Panel Method

• Theoretical and numerical methods accounting for free surface and, possibly, viscous effects.

As it will be seen on next section, each method of calculation has its inherent advantages and disad-

vantages. Modern approaches try to combine one or more of those methods searching a compromise

between them.

Since regression formulas originated from experimental tests will be used in modelling, the main fea-

tures of the tank tests and experimental design, for each component of the model (intrinsic manoeuvring

in shallow water, bottom, bank, ship to ship interactions), will be presented as well what are the main

constraints of the regression formulas that have the origin on the experiments.

Chapter 2 revisits the original non-linear coupled dynamic manoeuvring model that must be solved for

each ship and modifies them to account for the effect of interaction forces and moment and shallow

water effects on the intrinsic manoeuvring behaviour of the vessel.

Chapter 3 shows the scenarios of the solution of the dynamic equation of each vessel presenting the

coordinate systems used at each case study that will be performed on next chapter. The interaction

calculation models for each type of interaction are shown.

Chapter 4 presents the complete model applied to each interaction case study: ship bottom interaction

(Squat), ship passing near bank, encounter and overtaking of ships. Particular analyses are performed

with the addition of Proportional Derivative controller simulating the helmsman and comparisons of deep

and shallow water forces and moment models in overtaking.

Chapter 5 contains the conclusions and recommendations for further research.

4

1.4 State of the Art

1.4.1 Manoeuvrability in Shallow Waters

Empiric and Semi-Empiric Methods

Kobayashi [1995] investigate 3 types of vessels in various captive and free running model tests and ob-

tained correction factors for the MMG model (Kose [1982]) related to manoeuvring, rudder and advance

resistance coefficients. The shallow water simulation model demonstrated good adherence against ex-

periments for H/T values ranging from 1.2 until 11.4. The last value that is typically deep water situation

suggests that the method is even valid for deep water case and thus disposing the need for simulators

to have one model for shallow and other for deep water inside the programs routine.

Matsunaga [1993] proposes also correction factors to be used inside manoeuvring mathematical models

for a somewhat mixed manoeuvring coefficients terms that considers considers cubic model coefficients

as also also modulus terms. The model proposed by the author will be used inside the present offline

simulation code with a modification to consider modulus terms as cubic equivalent ones as suggested

by Sutulo and Guedes Soares [2011]. In addition, Matsunaga model did not consider corrections for

advance resistance and the model cited by Roteveel [2013] will additionally be implemented.

Vantorre et al. [2003b] analysed the manoeuvrability and coursekeeping abilities in small and negative

underkeel clearance. The situation of negative underkeel clearance was associated to the existence of

a ”mud” bottom that the vessel could navigate with keel submerged inside the mud until a certain level

of mud specific density and dynamic viscosity. It was verified in this preliminary study that even some

aspects of manoeuvrability and coursekeeping could improve towards a more dense mud level (lower

levels) as in contrast from what will be normally expected due to the proximity of the bottom.

Numerical Methods

Turnock and Molland [1998] studied the problem of manoeuvring performance in the presence of shallow

water and channel walls using a code called interaction velocity field that applies mainly panel method

theory. Wind tunnel tests were used to compare results obtained from the code. Due to the scale

used to the original model and wind test facility size, a somewhat truncated mariner hull was studied

considering stern sections from the original model. Nevertheless results for non-truncated model were

also compared by means of tests in Glasgow facility. Some problems were faced when very shallow

waters were studied and authors attributed the problem to keel panels arrangements used.

1.4.2 Bottom Interaction

Bottom Interaction originates the phenomenon known as squat. Squat is the decrease in underkeel

clearance caused by vessels forward motion or moored vessel facing ebb tide or river currents. The

physical consequence of squat on a vessel is the presence of sinkage and trim. It must be emphasized

that Ship Squat is not the difference between the draughts when stationary and the draughts when the

5

ship is moving ahead. For example, the difference in bow draught readings due to forward motion might

be 2 m, whilst the decrease in underkeel clearance might only be 0.40 m.

Ship Squat has always existed on smaller and slower vessels when underway. But it was not considered

because it was a matter of centimetres and thus came with no consequences. However, from the mid-

1960s to the present, ship size has steadily grown until we have Supertankers and Valemax vessels of

the order of 400000 dwt and above. These vessels have almost outgrown the ports they visit, resulting in

static underkeel clearances of the order of 1.0 to 1.5 metres. At the same time Service speed of vessels

has been increased namely containerships and passenger vessels, where speed has increased from 16

knots up to 25 knots.

As the static underkeel clearance have decreased and as the Service Speed increases, ship squat has

become more and more noticeable. For the modern vessels, squat values could be from 1.0 m to 1.5 m

as contrary to few centimetres from the past.

Some recent accidents and costs involved in repair and time out of service are:

• Costa Concordia

• Cruise Ship at Isle of Giglio, Italy

• January 2012

• Sea Empress

• Supertanker at Milford Haven, United Kingdom

• February 1996

• Repair Bill: $28million. Loss of earnings can be as high as 300000 pounds per day.

• Herald of Free Enterprise

• Ro-Ro vessel at Zeebrugge

• March 1987

• 193 lives were lost.

Some of the main factors that affect squat in shallow waters and canals are:

• Froude number based on depth (Fnh = V/√gH) mainly the squat varying with the square of this

value. Sinkage and trim can even change in sign if Fnh greater than one (semi-empirical formulas)

or even less, as proved by Tuck and Taylor [1970] which compared empirical formulas with test

results. Velocity must be considered relative to current/tide speed.

• Squat varies directly with the block coefficient (CB). This factor will additionally determine the trim

sign of a ship when moving if it is initially even keel when stationary. A full form vessel will trim by

the bow and finer form will trim by stern.

6

• The relationship between the depth of water (H) and the static mean draught of the ship (T ). H/T

ratio decreasing, increases squat phenomena.

• The presence of asymmetry due to propeller rotation, banks asymmetric configuration or another

vessel presence induces additional ship squat. Barrass [1979] stated that squat can double due to

interaction between passing or crossing vessels. This is confirmed also in Gronarz [2006].

• Blockage factor calculated as a function of the ratio of ship section dimensions to canal cross-

section dimensions. Even if vessel is in shallow “open water”, Barrass [1979] consider an artificial

cross-section width for calculation purposes defined according to the type of the vessel. The main

approaches in order to calculate ship squat can be classified as empirical and semi-empirical or

theoretical and numerical.

The typical first question regarding squat is how to account it. It is known that squat is a dynamic effect

but many formulations treats squat calculation outside the simulation loop (as “static” contribution to the

static draught and trim). This is mainly justified due to the fact that if inserted inside a loop (consider-

ing, for example, effective depth (heff ) inside the squat formula instead of static depth (H) as usual),

the transient state is apparent only in narrow initial region of changes between advance resistance and

sinkage. Of course, the squat dynamics will be more pronounced if bottom surface is irregular and/or

pilot changes velocity commanded, if there is the presence of encounters and overtaking manoeuvres

inside the channel or the presence of transients related to proximity to the bank.

The main approaches in order to calculate ship squat can be classified as empirical and semi-empirical

or theoretical and numerical.

Empirical Approach

In order to use regression formulas from experiments with confidence, it is very important to know the

experimental setup that the author uses to propose such regressions. Normally questions to be an-

swered regarding an experimental setup are:

• What models are used, regarding main forms and dimensions of the model.

• What are the parameters that are used and what are the variation interval (and values) of such

parameters in the tests.

• What are the degrees of freedom that are free and constrained. Which ones are measured and

which ones are reported by the author (sometimes author said it is recorded one value but didn’t

appear in the final report).

• The model was simulated with propeller (and rudder) or just bare hull.

• The model was simulated in the presence of currents and/or waves and/or wind.

7

• Is the model coupled with other effects (i.e. 1.Squat test with vessel trajectory not in the neutral

steering line (NSL) in the case of sinkage formulas that account blockage effects. 2. Ship to

ship test in which each ship is influenced by water depth coefficients and distance to the bank

coefficients).

• Regarding the overall experimental setup and the resulting regression formulas: are the tests with

statistical meaning regarding the population. The regression formulas fit well the data, what is the

R2 value of each expression.

To answer some of these questions about regressions from squat, it was found hard and that is jus-

tified due to the fact that they are very ancient tests and it was found hard to find the original data. The

main information about them is taken from Brix [1993]. In order to work with the regressions proposed,

we will make some assumptions about squat tests and try to answer the previous questions:

• Different models are used and the regressions are valid from fine forms to full form ships (as can be

concluded at least in Barrass [1979] research when telling about the effect of the block coefficient

on the final trim of ships due to squat).

• The main parameters tested, of course, will be the ones that appears on the regression formulas.

From Barrass [1979], it will be assumed that the parameters used are Cb (block coefficient of the

ship), S (ratio between ship dimensions and waterway dimensions), Vk (relative velocity, in knots,

between ship and the fluid, mainly to take into account the difference between vessel speed and

current speed).

• The model is free in heave and pitch only in order to take the sinkage measures fore and aft. The

other degrees of freedom are constrained.

• The model is equipped with propeller but it will be assumed that only one propeller rate will be

tested, the exception being Lataire and Vantorre [2008] which considered different propeller rates.

• The main assumption about the models will be that the vessel is in the (NSL) of the waterway if

blockage effects are considered in the formulas. The only exception will be Lataire and Vantorre

[2008] model which explicitly talks about sinkage measurements out of the NSL and so taking into

account bank coupling effect in the formula.

• The model was not simulated in the presence of wave or currents (calm water standard condition).

The main experimental regression formulas available on the literature are given by Barrass [1979],

and Millward [1990]. Lataire and Vantorre [2008] besides the normal parameters stated before incre-

ments the coupling with asymmetric bank configuration and propeller rotation developed for the study

of bank interaction of irregular configurations. Barrass [1979] proposed a method to calculate maximum

sinkage and that will be used further on the offline simulation model developed.

8

Theoretical and Numerical Approach

Tuck and Taylor [1970] used slender body theory to obtain mean sinkage and trim by means of simple

calculation formulas. The authors show the important dependence of squat and the depth based Froude

number. The formulas also works on the definition of Cz and Cθ parameters that according to the author

must be defined on a case by case basis by experiments. Higher values of Fnh dependencies were

presented suggesting the adequacy of the method to high speed crafts. Further studies developments

of the formulas also shown the adequacy to displacement hulls (PIANC [1997]) inclusively giving values

of Cz and Cθ. This method will be implemented inside the offline simulation model since it was found

important to know the distribution of sinkage along the hull besides the maximum value of sinkage.//

Doctors and Day [2000] analysed the squat of a vessel with transom stern (Lego model) by mean

of the near field solution to the flow using thin-ship theory. The authors describes that, after the forces

and moments are found, from the solution of the potential at the field panels, the vessel will not be in

equilibrium and iterating procedure must be used for finding the final sinkage and trim of the vessel,

still reporting that the use of traditional hydrostatic stiffness coefficients worked well for the iteration pro-

cedure. Comparison between theoretical and experimental results for a large range of Froude number

shows not good correlation. Some variations to the original mathematical model were performed but

still showing the presented model as the best correlation (near field plus free to sink and heave). The

differences are attributed to the not precise shape of transom-stern hollow for low speeds and non em-

ployment of proper form factor formula for the frictional resistance. Later the authors (Doctors and Day

[2002]) implemented form factor correction together with some attempt to better predict the shape of

transom in smaller Froude numbers. The first implementation shows improvement on the results but the

last one not with the linear and correction of sinkage and trim iteratively showing the best agreement

with the experiments.

More recent studies were developed by Gourlay [2014], who studied the problem using the complete set

of waterway geometries given by shallow water slender body theory and including arbitrary bathimetry

case. The study focused only on low Froude numbers neglecting terms on the shallow water equations

related to wave generation. Varyani [2006b] revisited formulations using Bernoulli equation, and division

of the hull in strips given by Dand and Ferguson [1973].

1.4.3 Bank Interaction

The preferable position for a ship moving through a narrow channel is the Neutral Steering Line (NSL) of

the waterway. This is normally the sailing line which no suction forces or turning moment (due to appli-

cation point aft of the suction force) are balanced and thus no rudder angle input is needed. Outside this

position the vessel will pass to the bank interaction situation where normally the modified distribution of

pressure around the hull develops suction force towards the nearest bank and bow away moments.

9

Vantorre et al. [2003a] calls the attention to the fact that this is the normally expected behaviour, but

some combination of parameters may cause an adverse situation. Some of the main factors that affects

ship-bank interaction are:

• Bank shape (surface piercing or submerged, vertical, like quay, or sloped, like normal banks in

channels).

• Water depth to the ship draught ratio.

• Ship-bank distance

• Blockage effect.

• Ship characteristics (e.g. L,B,T ,Cb or non-dimensional regarding geometry ratios)

• Ship speed (normally interaction forces and moment vary with square of these values).

• Propeller rate.

Mainly works found exclusively on this topic are of empiric and semi-empiric nature. One of the

ancient works regarding the study of ship-bank interaction was performed by Dand [1982]. The author

performed extensive tank tests and some preliminary behaviour of the influence parameters stated on

previous section could be observed. Norrbin [1985] carried out extensive experimental and analytical

investigation on manoeuvring in general and bank effects in particular. Based on model tests with a

propelled tanker model (L = 5.024m .B = 0.852m,T = 0.339m,Cb = 0.821), the author varied bank

configuration including vertical, sloped and submerged banks at different forward speeds. At the end,

the researcher proposed regressions formulas to deal with the forces and moments generated by ship

passing near bank interaction. The regression model developed by Norrbin [1985] will be detailed on

section 3.2 and it will be used in the offline simulation model.

More recent extensive tank tests were performed by Vantorre et al. [2003a] taking advantage of the al-

most automatic Flanders Hydraulics facilities. The authors used three different ship models and on each

ship large variations of ship bank distances, water depth ratios, forward speeds and propeller rates were

tested.

Regarding the ship speed it was stated that it could not be clearly approximated by the square power

dependency inside some combination regions of the other parameters, namely depth draught ratio and

proximity to the bank. Normally, the increase in bank proximity and decrease in depth do increase the

interaction with the bank proximity normally which is usually modelled by a linear relation.

The relation of interaction force and moment with depth to draught ratio is very complex. The sway

component the suction can even change in sign to a repulsion in the interval between 1.2 and 1.1. Phys-

ically, for a given ship speed a long bow wave is generated between the bank and the ship inducing a

high pressure region there and thus transforming the attraction transform in repulsion. The speed value

where the sign change happens also depends on water depth and bank proximity.

The rotating propeller appears to result in attraction between stern and the bank. As a result of that bow

away moments are increased. Repulsion forces at zero propeller rate in very shallow waters can even

turn into suction adding the propeller action. Outside critical shallow water the suction can even increase

than compared to the zero propeller rate as even duplicating. Regarding yaw moment, the application

10

point seems to be related with the ship-bank distance moving forward with decreasing underkeel clear-

ance and increasing speed and aft with decreasing bank clearance. Vantorre manipulate regression

expressions in order to disregard Froude number term and insert this influence only in separate squat

computations entered that was coupled in the formula by means of an effective depth (heff/T ) modified

term.

Vantorre et al. [2003a] study was augmented by Lataire and Vantorre [2008] who aimed to study irregu-

lar bank configuration. Lataire and Vantorre [2008] developed regression formulas on the basis of tank

tests of Flanders Hydraulics. The author report that 3 different ship models are used combined with 6

different bank configurations and different propeller rotation rates. In conclusion the study reported in

2008 builds on the previous experimental setup (from Vantorre et al. [2003a]) with different bank config-

urations. The author reported a total of 10000 runs in the tank tests with 25 runs per day in an almost

completely automatic facility that can run experiments and store the runs data 24 h/day!

The model also presents formulas to calculate sinkage not presented in Vantorre et al. [2003a] as ex-

pression to deal also with surge interaction besides, sway and yaw. Firstly, the author defines new

horizontal reach of interaction effects formula considering not only geometric vessel properties but also

Froude number depth dependent coefficient proving this need in a scatter plot.

Then, the author modifies somewhat the original Norrbin [1985] approach regarding ship-bank distance

calculation and inserting equivalent blockage effect in the following way: The author uses the centre of

gravity of the fluid domain portside and starboard of the ship giving an average ship bank distance and

blockage effect instead of a single value that is measured half draught considering only one side bank,

used in Norrbin [1985]. The author claims that this approach makes the model more robust regarding

changes of these parameters as is the case of irregular banks. The author calculate also the equivalent

blockage considered in a different manner than used in Norrbin [1985].

The final expressions then, are able to deal with the complete set of influence parameters that are mainly

found in the literature plus the modified terms that give the model the ability to be applied in every type

of bank configurations. The formulas, on the other hand, didn’t gave the hull geometric related parame-

ters meaning further consult to the author is required in order to be able to model using this regression

expressions. Of course the model has some limitations, which are pointed out by the author, namely

they do not deal with:

• Very high blockage coefficient such that found on locks.

• Ship at maximum aligned with the toe of the sloped bank.

• Ship just working on other than the first propeller quadrant

• Supercritical and transient water depth Froude number.

11

1.4.4 Ship to Ship Interaction

Theoretical and numerical methods accounting for free-surface and possibly viscous effects

Numerical methods accounting for free-surface and viscous effects seem to be very promising as they

may compute interaction forces without too strong assumptions concerning the absence of wave mak-

ing or viscous effects. These methods should result in an accurate prediction for any arbitrary mutual

position and motion of a large variety of ship hull shapes.

An earlier developed free-surface RANSE code was applied by Chen et al. [2003] to the ship-to-ship

interaction problem in shallow channel. The sway force and interaction yaw moment were computed

for two encountering ships with the depth based Froude number from 0.13 to 0.47. The results were

compared to the experimental data and to the double body solution obtained. A good agreement was

obtained with the experiment but it was found that neglecting free-surface effects underestimates the

interaction forces by a factor of 3-5. Fonfach [2010] modelled free-surface and viscous effects in the

interaction of a tug and a vessel. On that technique convergence analysis is done and validation is

performed against towing tank test results.

Despite of apparent advantage, online applications using CFD are not practical yet due to lack of com-

putational power and some uncertainty on which turbulence models to use. In fact, they are still adapted

in a case by case basis. Additionally, it is needed to perform a convergence analysis regarding the mesh

size and validation studies.

Potential flow methods

Potential flow methods depend on a rather strong assumption that the potential zero-Froude-number

interaction dominates over interaction caused by wave making or viscosity. There are strong reasons

to believe that in many situations the main part of the interaction is captured by inertial hydrodynamic

forces described by double-body potential flows. This statement constitutes in fact the so called Have-

lock hypothesis whose formulation can be found in Abkowitz et al. [1976]. It is also underlined that

accelerations, velocities and positions of the interacting bodies must be described accurately.

Earlier contributions dealt with simple geometries like spheres or ellipsoids while nowadays numeri-

cal methods were applied to real ship forms. For instance, a group of publications was based on the

slender-body theory and matched asymptotic expansions (Yeung and Tan [1978]; Tuck and Newman

[1974]). Solutions based on the slender body theory and matched asymptotic expansion can be faster

but are substantially less versatile and accurate.

A recent group of works applied the panel method (Hess and Smith [1964]; Hess and Smith [1967]) to

the hydrodynamic interaction problem (Sutulo and Guedes Soares [2008]). A convergence study was

performed in order to check the number of panels. It was discovered that a coarse mesh can produce

good estimates for the interaction forces and moments. The interaction forces and moments calculation

method was compared against results described in Brix [1993] showing good agreement.

The code was revisited and it was used coupled to an offline simulation code (Sutulo and Guedes Soares

12

[2009]) demonstrating its adequacy inside such a loop that other methods like RANSE or potential with

free surface are not able to attain. The model was compared against RANSE models (Fonfach et al.

[2011]) and validated against towing tank tests (Sutulo et al. [2012]) for the case of tug near vessel

operation. The model demonstrated good adherence in the sway and yaw moments and greater errors

occurs for the sway force at smaller horizontal clearances, which could be explained by free-surface and

viscous effects. It was also noticed greater discrepancies at the surge force.

The code was also able to deal with shallow water situation using mirror image techniques and with

inclined bottom (Zhou et al. [2010]).

Within the effort of interaction calculations Varyani et al. [2002] uses a discrete vortex distribution numer-

ical technique and slender body theory to perform parametric variations of water depth, ship size, speed

and separation distance to obtain new regression models for maximum peaks of sway forces and yaw

moments for encounter and overtaking. New generic models were developed using the numerical results

that are capable to estimate the transient behaviour of ship during the entire manoeuvre. Varyani et al.

[2003] studied and validated the model for ship passing by moored ship against tank test results show-

ing good agreement of results. The complete generic model was presented in Varyani [2006a], where

encounter and overtaking (including ship passing moored ship) were presented. The generic equations

model proposed by Varyani [2006a] were implemented inside the offline simulation code. Chapter 3 will

describe the model in more detail.

Empiric and Semi-empiric Methods

This is one of the most accurate methods since it is usually based on tests with larger models in order to

obtain both Fn and Rn flow similarity approximating the best as possible the flow on a full scale vessel.

Models studied are also equipped with propeller and rudder. The fact that the experiments normally

require a second carriage or some special tank arrangement poses some difficulties since there are

few facilities in the world that possesses such capabilities. Figure 1.3 shows typical arrangements of

experimental tests.

Figure 1.3: Main Ship Carriage (left) and auxiliary carriage (right) arrangements (Vantorre et al. [2002]).

13

Data recorded in the tests enables the construction of models that can run inside of online simulators

and predict interaction forces and moments. Normally, the data acquisition has also limited capabilities

and the consequences of that will be analysed later on. Additionally, all the numerical methods stated

before depends of tank tests to be validated.

In fact, this is the most expensive and time consuming method, due to the fact that inside just one inter-

action mode comes into play a large number of parameters that needs to be taken into account while

planning the tests. To overcome that, experiments focus their attention on just a limited number of inter-

action modes and parameters. The experiments are performed keeping certain fixed parameters while

varying a specific one around an interest range of values and recording interaction forces and moments

values. From that it is developed a set of regression formulas regarding the parameters considered

before. The parameters that are usually taken into account are:

• Lateral Separation: normal considered parameter in tests despite some divergence on the non-

dimensional technique between authors leading to differences on the regression coefficients found.

• Longitudinal positions of the vessels regarding to each other: normal considered parameters in

regression formulas when it is possible to analyse the transient approach and departing behaviour

of the interaction.

• Water depth under the vessels: despite this is a parameter normally considered in the tests, there

are still some discrepancy of regression coefficients and resulting peak interaction values accord-

ing different authors as analysed by De Decker [2006].

• Speed of the vessels

• Geometrical properties: most models consider just similar geometric vessels and normally regres-

sion formulas only account for ratio between ship lengths in a very narrow band.

• Relative headings of the vessels: not normally studied in towing tank tests.

• Influence of rudder and propeller: normally interaction tests provide the models with rudder and

propeller but only few studies try to vary those parameters and express it in regression formulas.

One well-known set of regressions was developed by Brix [1993]. He developed regression formu-

las for predicting forces in overtaking manoeuvres that were calibrated in terms of tank tests results.

The method accounts for all horizontal forces and moment components but it is applicable only to small

values of the ratio between ship sizes, only for overtaken vessel, in deep water. Vantorre et al. [2002]

developed a set of regressions formulas for the surge, sway force and yaw moment peak values in en-

counter and overtaking (including when one of the vessels has zero velocity). The regressions take into

account the under keel clearance, separation and speed ratios between ships as also uses different

size and ship types in the evaluation. Despite of the considerable number of tank tests performed it

was found not so clear what are the regression coefficients to be used inside the regression equations

proposed by Vantorre et al. [2002]. Additionally, the results don’t show non-dimensional evolutions of the

parameters turning it difficult to perform comparisons beyond the qualitative behaviour of curves against

values found in the present thesis.

14

Gronarz [2011] proposed new algorithms that could solve the problem of the maximum generated num-

ber of peaks that commonly appear when using regular regression equations with smaller vessel over-

taking a larger one. The same author studied problems regarding interaction between vessels in en-

counter and overtaking on inland waterways (Gronarz [2006]). Different vessel types are used and with

different main dimensions. Separation distances and velocities ratios are varied. Special attention is

devoted to the data acquisition of sinkage and trim during the interaction transient which normally is not

the main parameter recorded on interaction tests. It was found that the test with self-propelled models

shows significantly differences in the interaction behaviour when compared to the simple towed model

configurations. Additional analysis are found regarding the entrance in locks of larger vessel with smaller

lateral and under keel clearances (Vergote et al. [2013]). It was found that larger entrance times are en-

countered on such situation and a study of waves generated inside the lock is performed.

Main remarks for comparisons with Vantorre et al. [2002] experiments . In order to validate the

shallow water calculations developed presented latter on against tank test, data obtained from Vantorre

et al. [2002] is used. On Figure 1.4 appears plots for different vessels combinations (vessels combi-

nations in the legend, models C,D,E,H) for overtaken and overtaking model used by Vantorre (Left and

Right columns respectively).

For the yaw moment, one could misunderstood the apparent anti-phase of the yaw moment coefficient

and the fact that some another plots on the literature show both graphs in phase for both vessels (i.e.

(Sutulo and Guedes Soares [2008]). It is also known that regarding sign convention defined above and

physics of the phenomena, the graph bellow is incoherent. This is due to the test setup that is used to

produce the graphs bellow that is:

• Just the main carriage is acquiring the data.

• The auxiliary carriage is always positioned on the right side of the towing tank. Those configura-

tions imply that to gather data from:

• Overtaking, the main carriage model must have greater velocity than the auxiliary and the overtak-

ing is performed by portside.

• Overtaken, the main carriage must have lower velocity than the auxiliary and the overtaking is

performed by starboard.

And in conclusion, the yaw moment coefficient will be in anti-phase as opposing numerical methods

like panel method where both vessels have “sensors” producing the plots for both vessels in just one

overtaking manoeuvre.

The same apparent incoherence happens with sway forces when it is observed that on stagger equal to

zero both overtaken and overtaking vessels shows negative forces which physically would be expected

opposite signals revealing suction if just one manoeuvre and both vessels sensored were performed.

15

Figure 1.4: Interaction forces and moments. Left: forces and moments on the overtaken vessel. Right:forces and moments on the overtaking vessel (Vantorre et al. [2002]).

16

Chapter 2

Description of the manoeuvring

mathematical model

2.1 Reference frame coordinate systems and preliminary remarks

Section 2.2 will present the original offline simulation model implemented by the author for the analysis of

ship trial manoeuvres. The modifications from the original set will then be commented on the remaining

sections.

The modified simulation code developed in this study uses data for three different types of manoeuvres

in a channel or harbour area:

• Ship passing near the bank

• Encounter manoeuvre

• Overtaking manoeuvre

A detailed study of Squat will be performed considering different initial static trim and the effect on Squat.

Squat effects will be considered inside each manoeuvre and the effect of interactions cited above inside

squat will be analysed.

In Encounter and Overtaking manoeuvres, the simulation was performed putting in the scenario two

ships. Thus, the dynamic model, developed in next sections, must be solved twice (once for each ship)

and simultaneously taking in consideration the coupling between vessels given by the interaction forces

developed on each vessel due to the presence of the other.

The reference frame coordinate systems are one attached to each vessel and another common to all the

vessels (earth fixed reference system). Figure 2.1, Figure 2.2 and Figure 2.3 shows common situations

of ship fixed reference coordinate system regarding each analysed manoeuvre. Earth fixed coordinate

system will be positioned at the centreline of channel at its mouth.

For each simulation case study, it was assumed a vessel well known from the literature and whose ma-

noeuvring coefficients needed to input in the model are available, the mariner class vessel. Dimensions

of the Ship 1 are shown in Table 2.1 and the Ship 2 was chosen in a first analysis to have the same

17

dimensions of Ship 1. For encounter and overtaking, it is also assumed that the channel or harbour

area is very wide compared to ships greater breadth (Wchannel ≥ 10Bmax) so that there was no need to

account possible ship bank interactions on top of other manoeuvres. It was not computed any related

surge interaction forces inside the model. Therefore it will be assumed that the vessel is able to overtake

the additional resistance varying the propeller rotations.

Figure 2.1: Ship passing bank sign convention and description of general simulation parameters.

Parameter Value DimensionL 120.0 mB 26.0 mT 8.7 mCb 0.7 [-]

Table 2.1: Mariner Class main dimensions.

18

Figure 2.2: Encounter manoeuvre sign convention and description of general simulation parameters.

Figure 2.3: Overtaking manoeuvre sign convention and description of general simulation parameters.

2.2 Original Equations of Motion

The dynamic equations (2.1) are organized in such a way that acceleration derivative force coefficients

are presented on the left hand side as added masses (µ11, µ22, µ26 and µ66). Quasi-steady (Xq,Yq,Nq)

forces and moments and propeller thrust (Xp) are placed on the right hand side. Additionally, m will be

assumed the mass of the ship, xg and Izz the longitudinal position of the centre of gravity and moment

of inertia referred to the ship fixed coordinate system. u,v,r are the state variables surge, sway velocities

and yaw rate respectively, referred to the ship fixed reference coordinate system. The upper dot signs

19

on top of those variables refers to their related accelerations. Equations 2.2 present the kinematics of

the 3DOF model for each vessel, where ξc,ηc,ψc are the state variables advance, transfer and heading

of the ship related to earth fixed coordinate system. The rudder angle δR will complete the definition of

each ship state vector and equations regarding to it will be described on 2.2.3.

(m+ µ11)u−mvr −mxgr2 = Xq +Xp

(m+ µ22)v + (mxg + µ26)r = Yq (2.1)

(mxg + µ26)v + (Izz + µ66)r +mxgur = Nq

ξc = u cosψ − v sinψ,

ηc = u sinψ − v sinψ, (2.2)

ψc = r.

2.2.1 Original Quasi-steady forces and moments on the hull

The quasi-steady forces and moments are calculated as a function of its respective non-dimensional

coefficients:

Xq = X ′qρV 2

2LT, Yq = Y ′q

ρV 2

2LT, Nq = N ′q

ρV 2

2L2T (2.3)

where ρ is the water density, V =√u2 + v2 is the speed, L is the length of the ship, T is its draught

at the midship and the non-dimensional forces and moments coefficients (X ′q,Y ′q ,N ′q) are calculated by a

Taylor Multivariate Expansion approach regarding the non-dimensional kinematic parameters:

u′ =u

V, v′ =

v

V, r′ =

rL

V(2.4)

The non-dimensional terms calculation expressions are presented in next equations. It will be used

expansions until the cubic order. The manoeuvring coefficients considered and the computation of the

related quasi-steady forces and moments are given by:

Xq = X ′uuu′2 +X ′vrv

′r′ +X ′δδδ2r ,

Yq = Y ′0 + Y ′vv′ + Y ′rr

′ + Y ′vvvv′3 + Y ′vvrv

′2r′ + Y ′δ δR + Y ′vvδv′2δR + Y ′vδδv

′δ2R, (2.5)

20

Nq = N ′0 +N ′vv′ +N ′rr

′ +N ′vvvv′3 +N ′vvrv

′2r′ +N ′δδR +N ′vvδv′2δR +N ′vδδv

′δ2R +N ′δδδδ

3R.

where each hydrodynamic coefficient (X ′uu,. . . ,N ′δδδ) is calculated by:

X ′uu = −(2mCTL)/(L2T ), X ′vr = −(1.3µ22)/(L2T ), X ′δδ = KrX′δδ0;

Y ′0 = Y ′00, Y ′v = (1 + b1τ′)Y ′v0,

Y ′r = (1 + b2τ′)Y ′r0, Y ′vvv = Y ′vvv0, Y ′vvr = Y ′vvr0, (2.6)

Y ′δ = KrY′δ0, Y ′δvv = KrY

′δvv0, Y ′δδv = KrY

′δδv0;

N ′0 = N ′00, N ′v = (1 + b3τ′)N ′v0, N ′r = (1 + b4τ

′)(N ′r0 +m′x′gu′), N ′δ = KrN

′δ0,

N ′vvv = N ′vvv0, N ′vvr = N ′vvr0,

N ′δvv = KrN′δvv0, N ′δδv = KrN

′δδv0, N ′δδδ = KrN

′δδδ0.

where on each expression the final manoeuvring coefficient was found by the multiplication of the

original coefficients from the ship (terms with sub index 0) by the adjustment coefficients κjiii (i.e. κxuu

factor in X ′uu) and trim correction coefficients. Not all the adjustment coefficients are placed since they

are mostly equal to one. The exception are κnv = 0.7,κnr = 1.3,κyv = 1.2,κnδδv = 0.8. KR is the

rudder area coefficient and is assumed equal to one. Trim correction coefficients appears only in the

linear manoeuvring terms as in the original model from Inoue et al. [1981].Crane et al. [1989] shows

the constant base parameters appearing on equations for the Mariner model that can be found by the

expressions below:

CTL = 0.01, X ′δδ0 = −0.02,

Y ′00 = −0.0008, Y ′v0 = −0.244, Y ′r0 = 0.067, Y ′δ0 = −0.0586,

N ′00 = 0.00059, N ′v0 = −0.055, N ′r0 = −0.0349, N ′δ0 = 0.0293,

Y ′vvv0 = −1.702, Y ′rvv0 = 3.23, Y ′δvv0 = −0.25, Y ′δδv0 = −0.0008, (2.7)

N ′vvv0 = 0.345, N ′rvv0 = −1.158, N ′δvv0 = −0.1032, N ′δδv0 = 0.00264, N ′δδδ0 = −0.00482,

m′ = 2m/(ρL2T ), Izz = 0.0625mL2, x′g = xg/L, τ ′ = (Tstern − Tbow)/T ,

b1 = 0.667, b2 = 0.8, b3 = −(0.27Y ′v0)/(N ′v0), b4 = 0.3.

where B is the breadth of the ship. The added masses are calculated by:

µ11 = k11m, µ22 = k22m, µ66 = k66Izz, µ26 = µ22xg, (2.8)

k11 = 0.5T/L, k22 = 2T/B(1− 0.5B/L), k66 = 2T/B(1− 1.6B/L).

21

2.2.2 Propeller Force Model

For the propeller force computation it is used the propeller 4th quadrant approach instead of the normal

B series approach. The fourth quadrant formula is fitted on the present propeller by the coefficients CT0,

CCT , CST ,CCCT and CSST with values given bellow. The effective thrust calculation will be given by:

Xp = TE = (1− tp)ρ/2AdCTV 2B (2.9)

where tp is the thrust deduction coefficient, Ad is the propeller disk area and

CT =

CT0 + CCT cB + CsT sb at cB ≥ 0.9336,

CccT |cb|cb + CssT |sB |sB , at otherwise.

V 2B = u2

A + v2cp, ua = u(1− wp), vcp = 0.7πDpn, cB = vcp/VB , sB = uA/VB , (2.10)

CT0 = −0.833, CcT = 1.02, CsT = −0.332, CccT = 0.099, CssT = −0.671

where wp is the wake fraction coefficient, Dp is the diameter of the propeller, n is the propeller rotation

frequency (rps).

2.2.3 Rudder Model

Additionally to the hull dynamics, the rudder dynamics was modeled (equation 2.11). δR describes the

state variable actual rudder angle and δ∗ the rudder order. Non-linearities include dead-band zone δ0,

rudder angle saturation |δR| ≤ δm and rudder turn rate saturation |δR| ≤ εm .

˙δR =

min[ 1TR

(|δ∗∗ − δR| − δ0), εm]sign(δ∗∗ − δR) at L = false,

0, at L = true.

(2.11)

where L is an auxiliary variable given by

L = (|δ∗∗ − δR| < δ0) ∨ [(sign(δ∗∗ − δR) = signδR)]

TR is the time lag of the gear, and

δ∗∗ =

δ∗, se |δ∗| ≤ δm,

(δm + δ0)sign(δ∗), se |δ∗| > δm.

is an auxiliary variable necessary to exceute helms at ultimate angles and to prevent the winding up.

22

2.3 Modified Equations of Motion

The modified dynamic equations accounts for the presence of other ship (or bank) in her own near

field. This is accounted in terms of interaction forces and moment (Xs,Ys,Ns) in addition to the normal

quasi-steady forces and moments on the right hand side of equation 2.12. The dynamic and kinematic

equation (2.12 and 2.13 respectively) will need to be solved for each ship envolved.

(m+ µ11)u−mvr −mxgr2 = Xq +Xp +Xs

(m+ µ22)v + (mxg + µ26)r = Yq + Ys (2.12)

(mxg + µ26)v + (Izz + µ66)r +mxgur = Nq +Ns

ξc = u cosψ − v sinψ,

ηc = u sinψ − v sinψ, (2.13)

ψc = r.

For the equation solutions, the model is adjusted to achieve the Cauchy form and each equation

could be solved using discrete numerical integration methods for each time step. Euler method was

implemented in order to solve the equations due to its simplicity, provision of the required accuracy and

more computational speed for the properly chosen time step. Modified quasi-steady forces and moments

are adapted to account for shallow water situation and will be detailed on the next section. The propeller

model and rudder dynamics model will remain the same as in the original model. The terms Xs,Ys,Ns

related to the interaction forces and moments will be explained on the next separate chapter.

2.4 Modified Quasi-steady forces and moments on the hull

The modified quasi-steady forces and moments are calculated in the same manner as in the original

method, using the same Taylor Multivariate Expansion expression. The manoeuvring coefficients will

be again the multiplication of the originals from Mariner model, by trim corrections, adjustment factors,

but now with the additional multiplication by depth dependent factors fhcijj(equations 2.14). The terms

are the modified shallow water correction factors related to Matsunaga [1993] original model that will be

explained in more detail on section 2.4.1.

X ′uu = −(2mCTL)/(L2T ), X ′vr = −(1.3µ22)/(L2T ), X ′δδ = KrX′δδ0;

23

Y ′0 = Y ′00, Y ′v = (1 + b1τ′)fhcyvY

′v0,

Y ′r = (1 + b2τ′)fhcyrY

′r0, Y ′vvv = Y ′vvv0, Y ′vvr = fhcyvvrY

′vvr0, (2.14)

Y ′δ = KrY′δ0, Y ′δvv = KrY

′δvv0, Y ′δδv = KrY

′δδv0;

N ′0 = N ′00, N ′v = (1 + b3τ′)fhcnvN

′v0, N ′r = (1 + b4τ

′)fhcnr(N′r0 +m′x′gu

′), N ′δ = KrN′δ0,

N ′vvv = fhcnvvvN′vvv0, N ′vvr = fhcnvvrN

′vvr0,

N ′δvv = KrN′δvv0, N ′δδv = KrN

′δδv0, N ′δδδ = KrN

′δδδ0.

Propeller force model and rudder force models remains the same from the original model, but now it

is needed to be solved for each ship involved as also as the quasi-steady forces.

2.4.1 Modified Matsunaga [1993] model

The original model was given by Matsunaga [1993].This model was a mixed cubic and modular approach

containing terms like associated r′|r′| correction factors instead of r′3 ones.

In the present modified model, the modulus related coefficients are assumed as equivalent r′3 terms

enabling the modular coefficients to be used in the cubic expansion terms instead of modular terms.

This approach is not new and it was even suggested in Sutulo and Guedes Soares [2011].

The modified shallow water correction coefficients can be seen on the set of equations 2.15. Some of

this factors will not be used for the Mariner model since some manoeuvring coefficients for this model

are zero (equations 2.14).

fhcyv = 1/(1− Teff/H)0.4bdt − Teff/H

fhcyr =1.0 + (−5.5 ∗ bdt2 + 26.0 ∗ bdt− 31.5) ∗ Teff/H+

+ (37.0 ∗ bdt2 − 185.0 ∗ bdt+ 230.0) ∗ (Teff/H)2+

+ (38.0 ∗ bdt2 − 197.0 ∗ bdt− 250.0) ∗ (Teff/H)3

fhcyvvv = 1/(1− Teff/H)−0.26∗bdt+1.74 − Teff/H

fhcyrrr =1.0 + (−0.156e5 ∗ cb15) ∗ (Teff/H)+

+ (1.16e5 ∗ cb15) ∗ (Teff/H)2+

+ (−1.28e5 ∗ cb15) ∗ (Teff/H)3

24

fhcyvvr =1.0 + (2.15e4 ∗ cbt2 − 0.48e4 ∗ cbt+ 220.0) ∗ (Teff/H+

+ (−4.08e4 ∗ cbt2 + 0.75e4 ∗ cbt− 274.0) ∗ (Teff/H)2+

+ (−9.08e4 ∗ cbt2 + 2.55e4 ∗ cbt− 1400.0) ∗ (Teff/H)3

fhcyvrr = 1/(1− Teff/H)−0.213∗dbt+1.8 − Teff/H

fhcnv = 1/(1− Teff/H)0.425bdt − Teff/H (2.15)

fhcnr = 1/(1− Teff/H)−7.14ar+1.5 − Teff/H

fhcnvvv =1.0 + (−0.24e3 ∗ cbl + 57.0) ∗ (Teff/H)+

+ (1.77e3 ∗ cbl − 413.0) ∗ (Teff/H)2+

+ (−1.98e3 ∗ cbl + 467.0) ∗ (Teff/H)3

fhcnrrr =1.0 + (−0.196e4 ∗ cbt2 + 448.0 ∗ cbt− 25.0) ∗ (Teff/H)+

+ (1.222e4 ∗ cbt2 − 2720.0 ∗ cbt+ 146.0) ∗ (Teff/H)2+

+ (−1.216e4 ∗ cbt2 + 2650.0 ∗ cbt− 137.0) ∗ (Teff/H)3

fhcnvvr =1.0 + (91.0 ∗ dbt− 25.0) ∗ (Teff/H)+

+ (−515.0 ∗ dbt+ 144.0) ∗ (Teff/H)2+

+ (508.0 ∗ dbt− 143.0) ∗ (Teff/H)3

fhcnvrr =1.0 + (40.0 ∗ bdt− 88.0) ∗ (Teff/H)+

+ (−295.0 ∗ bdt+ 645.0) ∗ (Teff/H)2+

+ (312.0 ∗ bdt− 678.0) ∗ (Teff/H)3

where for the modified depth correction terms stated above the influence factors are calculated as

function of the squat and hull geometry parameters (expressions 2.16).

Teff = T + zm, cl = Cb ∗B/L, cb1 = 1.0− Cb, ar = 2.0 ∗ Teff/L,

cbt = cb1 ∗ Teff/B, dbt = Cb ∗ Teff/B, bdt = Cb ∗B/Teff (2.16)

The model was checked to see if the modifications of the original model produce reasonable results.

The results of the test performed are presented in section 4.1.3.

25

26

Chapter 3

Regression Models of interaction

forces and moments

For the Squat, it will be used regression formulas developed by Barrass [1979] and Tuck and Taylor

[1970] associated to PIANC [1997] . No attempt will be made to correct water depth dependent co-

efficients inside interaction formulas from bank, encounter and overtaking regarding to squat, namely

instead of using H as input use heff as a variable function of squat. This is justified since, when tested

in shallow water tank, models are free to heave and pitch and thus squat is already considered inside

regressions although not apparent due to the fact that formulas, derived from tests, normally asks for

static vessels H/T ratios as input for calculations.

On the other hand, it will be interesting to see the effect of encounter and overtaking inside squat. No

regression equations were found but from tank test results, the author attempted to use common squat

formula, considering an additional blockage effect parameter due to the presence of the other vessel.

Model test data for sinkage and trim was found only for the overtaking case and thus will be presented

only on this case study.

For the ship passing near bank case, it was not found in the literature any formula that could handle

both approximation and moving away transients to a bank (normally called short bank problem). Then

interaction forces and moment regressions implemented were obtained from Norrbin [1985].

The generic equations for calculating the transient interaction sway force and yaw moment obtained by

Varyani [2006a] will be used for encounter and overtaking in harbours and access channels. The generic

equations developed by Varyani are preferred since they provide a simple and fast estimate of interac-

tion forces and moment. The method takes into account important parameters such as water depth

dependence coefficients permitting its use in shallow water problems. The analysis was performed with

larger parametric variations that permit the model to be adapted to different simulation scenarios.

27

3.1 Squat Model

Barrass [1979] provides a set of formulas that can deal with maximum sinkage in open waters and

confined waters. The last one could represent vessel navigating in narrow channel in the NSL. In a

first analysis, it was attempted to analyse squat effect separately from other interactions (open water

situation). In order to do that, it was applied a water depth corrector to the surge force as presented

in Roteveel [2013]. Squat effects were also inserted inside the shallow water correctors for the ma-

noeuvring coefficients inserting effective depth (heff ) instead of the static value (H).It is mainly aimed

to verify the convergence of resistance and squat in straight path and curvilinear path.

Figure 3.1: Squat versus manoeuvring convergence

Barrass [1979] method will be inserted on the ship to ship and ship to bank interactions giving the

squat in each case considering the coupling effects between the interactions.

It was implemented Tuck and Taylor [1970] method in addition to PIANC [1997] to verify the separate

behaviour of squat and trim in different initial loading conditions. Although useful to see the sinkage and

trim separately, those formulas was not implemented together with others interactions since it doesn’t

considers blockage effects.

3.1.1 Barrass Method

Barrass [1979] proposes a set of equations to deal with squat situations both in open and confined

waters for a ship navigating in the last case on the Neutral Steering line.

zmax = Cb/30(S/(1− S))0.81(0.514V )2.08 (3.1)

where: zmax is the maximum sinkage with the maximum location dependent on the value of Cb:

• If Cb > 0.700 trim by the bow.

28

• If Cb < 0.700 trim by the stern.

• If Cb = 0.700 no trim occur.

V is the ship velocity in knots, Cbis the block coefficient and S is the blockage factor calculated by

the ratio between the sectional midship area of the ship (BT )and waterway sectional area (WH). It is

clear that this formula doesn’t report the change in sign of squat and trim due to ship velocity and related

Depth based Froude number reported by Tuck and Taylor [1970], but it will be preferred since it considers

blockage effects not considered in Tuck’s formulas. Tuck’s formulas change in sign normally occurs on

depth based Froude number around one and, for the present case study and normal other merchant

vessels case, maximum value achieved was 0.52, far below the critical value. For low blockages around

0.100 and 0.265, an even simpler formula can be used:

zmax = Cb/50(0.514V )2 (3.2)

This formula use can be justified most of the time since channel designs normally consider the blockage

around the values mentioned above for a target design vessel. The last formula will be used when

reporting squat from now on. Since squat exists even when the vessel is navigating in straight path, a

correction for surge was provided (Roteveel [2013]):

X ′uu(heff ) = [0.125 + 0.875(KWD0 + 0.4(B/T )KWD1)]X ′uu(∞)

KWD0 = 1.0 + 0.97 exp(−2.74CWD)

KWD1 = 0.75 exp(−4.875CWD) (3.3)

CWD = (T/(heff − T ))−1

(3.4)

where heff is the effective depth calculated by: heff = H−zmax ,X ′uu(∞) is the resistance coefficient

due to forward motion in deep water and X ′uu(heff ) is the corrected due to squat.

3.1.2 Tuck Method

Tuck and Taylor [1970] provides separate formulas giving mean sinkage(SM ) and trim(Θ) but not con-

sidering the effect of confined waters.

SM = CZ∇L2pp

F 2nh

1−√F 2nh

Θ = CΘ∇L2pp

F 2nh

1−√F 2nh

(3.5)

29

where ∇ is the ship submerged volume, Cz and CΘ are, respectively, sinkage and trim related coeffi-

cients that must be fitted on a case by case basis recurring to tests on Models. It will be used respectively

1.46 and 1.0 for those coefficients suggested by Hooft (1974, apud PIANC [1997]).

With some simple manipulation mean sinkage and trim can be converted in maximum sinkage and then

a comparison study between the methods can be performed.

Smax = Sm +1

2LppΘ (3.6)

where it is assumed as simplification that Longitudinal Centre of Flotation is located at midship.

3.2 Ship passing near the Bank Model

Norrbin [1985] developed the following relations to compute the forces considering a vertical bank (slope

factor (k) equal to zero):

Y ′B(k=0) = [0.0926 + 0.372(T/H)2]Fn2η0 (3.7)

N ′B(k=0) = −[0.0025 + 0.0755(T/H)2]Fn2η0

(3.8)

where: Fn is the common Froude number, η0 is the ship bank nondimensional separation distance given

by,

η0 = B/(W0s − Y0)

where W0sis the half width of the channel, Y0 is the distance from the ship centreline to the NSL and B

is the breadth of the ship. Additionaly Norbin gives formulas to account for a sloped bank:

Y ′B = Y ′Bk=0(1 + 0.377η0k + 19.53Fnk + 0.0673k3 − 0.0988(T/H)k3) (3.9)

N ′B = N ′Bk=0(1− 0.750η0k + 81.8Fnk + 0.0331k3 + 0.0195(T/H)k3)

(3.10)

where the bank slope is defined by 1 : k and so k = 0 reduces to the limit vertical bank case. Figure

3.2 presents a sketch with the main geometric parameters used in those formulas.

It is also important that the separation parameter is now limited to the configuration of the sloped

bank η0max = 2(1 + 2k/(B/T )). The final interaction force and moment will be somewhat different than

usual due to the type of non-dimensional operation performed:

30

Figure 3.2: Main geometrical parameters used in the bank force computation.

YB = Y ′Bρg∇

NB = N ′Bρg∇L

(3.11)

where g is the acceleration of gravity.

3.3 Encounter and Overtaking Model

The stagger (ξ′) is non-dimensionalized such that the values -1, 0, +1 correspond to bow-bow, midship-

midship and stern-stern situations on encounter and bow-stern, midship-midship, stern-bow in overtak-

ing. It will be calculated as:

ξ′ = 2(ξ1 − ξ2)/(L1 + L2) (3.12)

The sway force and yaw moment interactions are first expressed with regressions in terms of non-

dimensional coefficients:

CY i = Yi/(1/2ρV1V2BiTi); CNi = Ni/(1/2ρV1V2BiTiLi) (3.13)

where i can be 1 or 2 related each coefficient to each Ship. In order to perform internal calculations Y i

and Ni come in evidence knowing the values of he non-dimensional coefficients. Next section shows

the general form of Varyani generic equations to calculate interaction coefficients. Then the regressors

will be shown on the last subsections with the complete expression both for encounter and overtaking.

31

3.3.1 Overview of Varyani Generic Equations

Varyani (2002), as reported in chapter 1, studied the influence on the peak values of H/T , L1/L2, V2/V1

and Sp/L and construct regression for the peaks for these parameters. After that, Varyani constructed

regression expression for the sway and yaw coefficients for the evolution pattern of the interaction in-

corporating relative position parameter (ξ′) inside the expression. The standard pattern (i.e. with the

standard values of H/T , L1/L2, V2/V1 and Sp/L) (standard parameters values will appear for each type

of manoeuvre and will be shown on next sections) will be given by:

CY = k1 cos(k2πξ′)e−k3ξ

′2

(1 + k4ξ′) (3.14)

where:

• k1: adjusts the size of the profile to the size of the main peak

• k2: adjusts the width of the repulsion – attraction-repulsion pattern of peaks.

• k3: restricts the domain of influence so that only the correct numbers of peaks are reproduced.

• k4: adjusts the relative sizes of the repulsion peaks.

And for yaw interaction moment,

CN = k5 cos(k2πξ′)e−k3ξ

′2

(1 + k4ξ′)(ξ′ + ∆)A(ξ′) (3.15)

where:

• k5: adjusts the size of the profile to the size of the main peak

• ∆: allows the pivot point not being amidships.

• A(ξ′): adjustment coefficient that reduces moments generated by values of H/T , L1/L2, U2/U1

and Sp/L when given outside the standard parameters.

A(ξ′) is given by:

A(ξ′) = 1− ae−b(ξ′−ξ′0+∆)2 (3.16)

where a determines the severity of the reduction, ξ′0 is the nominal point of applications, ∆ is a shift along

the stagger axis. Each parameter of A(ξ′) will be defined for each manoeuvre on the next sections.

3.3.2 Encounter Manoeuvre Model

For the encounter manoeuvre, the parametric variation performed is shown in Table 3.1.

H/T L1/L2 V2/V1 Sp/L

1.2, 1.3, 1.5, 1.8, 2.0 1.0 1.0 1.01.5 0.8,0.9,1.0,1.2 1.0 1.01.5 1.0 0.5, 1.0, 1.5, 2.0 1.01.5 1.0 1.0 0.2, 0.25, 0.3, 0.4, 0.5, 0.7,1.0

Table 3.1: Encounter manoeuvre parametric variation.

32

where the bold numbers are the standard ones, for the encounter manoeuvre, commented in previous

section. The pattern coefficients of sway force and yaw moment for encounter manoeuvre are calculated

by multiplying the previous ξprime expression by the correction non standard parameters values:

CY = −0.47 cos−0.86πξ′e−0.95ξ′2

(1− 0.18ξ′)[H/T

1.5]−2.25[2

SpL

]−1.25[L1

L2]−2.5×

[1

2

V2

V1+

1

2]

(3.17)

CN = 0.15 cos (−0.86πξ′)e( − 0.95ξ′2

)(1− 0.18ξ′)(ξ′ + ∆)A(ξ′)[H/T

1.5]−2.25[2

SpL

]−1.25[L1

L2]−2.5×

[1

2

V2

V1+

1

2]

(3.18)

where the values to be used in A(ξ′) according to the parameter used outside the standard ones are

presented in Table 3.2.

Constant for: a b ξ′0 ∆

H/Tm 0.30 1.40 -0.50 -0.10Sp/Lm 0.10 5.00 -1.00 -0.10L1/L2 0.30 1.40 -0.50 -0.10V2/V1 0.50 6.00 -1.00 -0.20

Table 3.2: Values of filter parameters to be used in encounter yaw moment coef. regression

3.3.3 Overtaking Manoeuvre Model

For the overtaking manoeuvre, the parametric variation performed is presented in Table 3.3. The bold

numbers are the standard ones, for the overtaking manoeuvre, commented in previous section. The

maximum coefficients of sway force and yaw moment for overtaking manoeuvre are calculated sepa-

rately now for Ship 1 and Ship 2 due to natural absence of problem symmetry on that situation.

H/T L1/L2 V2/V1 Sp/L

1.2, 1.3, 1.5, 1.8, 2.0 1.0 2.0 0.51.5 0.7,0.8,0.9,1.0,1.2 2.0 0.51.5 1.0 1.5, 2.0, 2.5, 3.0 0.51.5 1.0 2.0 0.2, 0.25, 0.3, 0.4, 0.5, 0.7

Table 3.3: Overtaking manoeuvre parametric variation.

Faster Ship 1 generic equations The maximum coefficients of sway force and yaw moment are

calculated by:

CY1= −0.11 sin (−0.49π(ξ′ + 0.37))e−0.95ξ′

2

(1− 0.98ξ′)[H/T

1.5]−2.2[2

SpL

]−1.3[L1

L2]−0.35×

[1

2

V1

V2− 1

2]

(3.19)

33

CN1= −0.1 sin (−0.49π(ξ′ + 0.07))e−0.9ξ′

2

(1− 0.3ξ′)A(ξ′)[H/T

1.5]−1.8[2

SpL

]−1.0[L1

L2]−1.5×

[1

2

V1

V2− 1

2]

(3.20)

where the values to be used in A(ξ′) according to the parameter used outside the standard ones are

shown in Table 3.4.

Constant for: a b ξ′0 ∆

H/Tm 0.00 0.00 0.00 0.00Sp/Lm 0.30 0.01 -0.75 -0.50L1/L2 0.01 0.10 -0.10 -0.01V2/V1 0.40 0.02 -0.75 -0.50

Table 3.4: Values of filter parameters to be used in overtaking yaw moment coef. regression Ship 1

Slower Ship 2 generic equations The maximum coefficients of sway force and yaw moment are

calculated by:

CY2 = −0.23 cos (−0.49πξ′)e−0.8ξ′2

(1− 0.18ξ′)[H/T

1.5]−2.2[2

SpL

]−1.3[L1

L2]−0.35×

[1

2

V1

V2− 1]

(3.21)

CN2= 0.34 sin (−0.65π(ξ′ − 0.05))e−1.5ξ′

2

(1− 0.18ξ′)A(ξ′)[H/T

1.5]−2.2[2

SpL

]−1.3[L1

L2]−0.35×

[1

2

V1

V2− 1]

(3.22)

where the values to be used in A(ξ′) according to the parameter used outside the standard ones are

shown in Table 3.5.

Constant for: a b ξ′0 ∆

H/Tm 0.65 0.27 -0.50 -0.01Sp/Lm 0.65 0.20 -0.50 -0.01L1/L2 0.67 0.22 -0.50 -0.01V2/V1 0.70 0.15 -0.50 -0.01

Table 3.5: Values of filter parameters to be used in overtaking yaw moment coef. regression Ship 2

34

Chapter 4

Simulation Results

On the following sections, it will be commented configurations regarding simulation general parameters,

initial conditions of ship(s) and interaction parameters of each manoeuvre. After that, results concerning

each manoeuvre will be presented and commented. The results involved will be related to general

kinematic, quasi-steady forces in time of ship(s), interaction forces and moments acting on the ship(s)

versus non-dimensional stagger and trajectory of the ship(s). The case-studies was chosen in such a

way to be inside the validity interval of the regression formulas.

4.1 Bottom Interaction (Squat)

A comparative study was performed for the three different values of non-dimensional static trim τ ′ (0.00,

0.05, -0.05) for the mariner model navigating on the NSL unrestricted shallow water. The objectives of

the study are mainly verify the convergence of speed versus squat parameters inside the model and

trace comparisons between Barrass [1979] and Tuck and Taylor [1970] models. The study also aims

to study the behaviour of squat subroutine programmed for different values of initial static trim and the

values of the dynamic trim. The initial condition are presented in Table 4.1 common for all the three

cases.

Ship 1u (m/s) 2.0n(rps) 1.0v (m/s) 0.0r (rad/s) 0.0Xg (m) 0Yg (m) 0Φ (rad) 0δ (rad) 0.0

Table 4.1: Initial Conditions Bottom Interaction

35

4.1.1 Zero Static Trim

It can be seen in figure 4.1 that the values of mean sinkage are very low less than 0.25m as also the trim.

The trim is negative and that is according to the higher value of Cb for this vessel. Although small values

are found, the maximum sinkage can achieve around 1.00 meter at the bow for the Taylor Method. That

value is even greater than proposed by Barrass method. Regarding kinematics it was confirmed the

stabilized value of 2 m/s after the transient following by stabilization of the squat parameters. It was also

analyzed the situations of dynamic squat.

Figure 4.1: Mean and Maximum Sinkage, Trim: Zero Initial Static Trim

4.1.2 Positive and negative trim

For the positive trim, the program work as expected increasing the total trim with the addition of the

dynamic trim, despite this was not of greater amount than 0.1 o. The mean sinkage is of the same

amount as before as also the maximum sinkage. For negative trims, it was also confirmed the increase

in negative trim due to the squat.

36

Figure 4.2: Mean and Maximum Sinkage, Trim: Positive Initial Static Trim

4.1.3 Manoeuvre in Shallow Water

It was also interesting to see the comparison between deep and shallow water manoeuvre. For that it

was simulated the turning manoeuvre in deep and shallow water (H/T = 1.2) for a desired rudder angle

equal to 20o (figure 4.3). It could be noted the increase in the tactical diameter as expected.

Figure 4.3: Turning Manoeuvre in Deep (blue line) versus Shallow Water (red line)

37

4.2 Ship passing near bank

4.2.1 Preliminary study

A simulation time of 16.66 minutes was performed with integration step of 0.05s.Table 4.2 shows the

initial conditions for the simulation. Table 4.3 presents the main interaction parameters used.

Ship 1u (m/s) 2.0n (rps) 1.0v (m/s) 0.0r (rad/s) 0.0Xg (m) 0Yg (m) -100Φ (rad) πδ (rad) 0.0

Table 4.2: Initial conditions ship passing near bank.

Ship 1H/T 1.20Ws (m) 80.0y0 (m) 25.0

Table 4.3: Ship passing near bank interaction parameters.

Figure 4.4 and Figure 4.5 show kinematic and dynamic effect of a long bank in ship’s starboard posi-

tion without rudder action. It is interesting to compare the sway force and yaw moment from interaction

and the sway and yaw developed by the vessel when in presence of such perturbation due to interaction.

It can be seen that the forces developed by the vessel counter reacts the interaction forces developed

and tends to avoid collision with the closer bank even without rudder action.

Figure 4.6 shows the overall expected effect of a bank modelled using Norrbin regression equations.

A suction force and a bow out moment are noticeable. It was not noticed the non-linear phenomenon

when the suction passes to bank repulsion probably due to the combinations of H/T and velocity used.

38

Figure 4.4: General evolution in time of kinematics of the ship.

Figure 4.5: General evolution in time of forces acting on the ship and heading, ship passing near thebank

39

Figure 4.6: Interaction forces and moments acting on the ship passing bank manoeuvre.

40

It can be seen from the trajectory plot (Figure 4.7) that the vessels approximates from the closest

bank (continuous straight line at starboard) but at the end didn’t collide as explained above. On the

other hand, it can be seen that the vessel will not assume the previous trajectory and there is danger of

collision with the other channel margin. The bow out effect is even not noticeable due to the quasi-steady

yaw moment that counter reacts it.

Figure 4.7: Vessel trajectory without control.

Another possible cause of problems in manoeuvre is due to grounding associated with squat in

shallow waters. As commented before, the formula used here for squat doesn’t account for the influence

of the asymmetric flow due to bank and thus a conservative margin must be present. Figure 4.8 shows

the overall evolution of the maximum sinkage. The first seconds must be disregarded since it was

applied to the model suddenly the depth restriction and doesn’t represent the real dynamic reality. After

the transient, it can be seen that maximum sinkage calculated was around 0.4 m. This results in a value

around 1.5 m of underkeel clearance for the present study and thus giving margin for the added sinkage

due to the flow asymmetry.

41

Figure 4.8: Maximum sinkage in ship passing near bank condition.

4.2.2 Ship passing near the bank with control

The ship passing near bank case was simulated with the same initial conditions but now considering a

proportional derivative controller (PD) in order to simulate actions taken by an experienced helmsman.

Figure 4.9 and Figure 4.10 present the kinematic and dynamic vessel behaviour. It can be noticed now

the change in the rudder actual position and relation to developed yaw angle and general kinematics.

Figure 4.11 shows the interaction sway force and yaw moment with rudder control.

Figure 4.9: General evolution in time of kinematics on the ship with control.

Figure 4.12 shows the new vessel trajectory. Instead of the vessel escape from the bank in direction

to another channel margin now it oscillates the trajectory trying to maintain the reference heading.

42

Figure 4.10: General evolution in time of forces acting on the ship with control.

Figure 4.11: Interaction forces and moments acting on the ship passing bank manoeuvre with control.

43

Figure 4.12: Vessel trajectory with control.

44

4.3 Encounter Manoeuvre

The simulation time will be configured as 8.3 minutes with integration step of 0.5 seconds. The real

interaction time is even shorter than the chosen for the simulation. This option was preferred in order

to give the needed time to stabilize the ships in the straight path before interaction takes place don’t

mixing the transients. The initial conditions on Ship 1 and 2 are presented in Table 4.4. The interaction

parameters are presented in Table 4.5. They are stated in such a way that A(ξ′) parameters used on its

formula will be referred to H/T non-standard experimental value.

Ship 1 Ship 2u (m/s) 5.0 5.0n (rps) 1.0 1.0v (m/s) 0.0 0.0r (rad/s) 0.0 0.0Xg(m) 800 -800Yg (m) 120 0.0Φ (rad) π 0.0δ (rad) 0.0 0.0

Table 4.4: Initial conditions encounter manoeuvre.

Parameter ValueH/T 1.2L1/L2 1.0U2/U1 1.0Sp/L 1.0

Table 4.5: Encounter manoeuvre interaction parameters.

Figure 4.13 and Figure 4.14 show the evolution in time of the main ship variables. Due to symmetry

of the problem the evolution in time will be the same for both ships. There is no control acting in terms of

recovering the previous ship trajectory thus, after the perturbation, the ship remains with residual sway

force and yaw moment thus generating some sway velocity and heading despite they are very small.

45

Figure 4.13: General evolution in time of kinematics on the ship in Encounter manoeuvre.

Figure 4.14: General evolution in time of forces acting on the ship and heading in Encounter manoeuvre.

46

Figure 4.15 shows the evolution of interaction forces as a function of the stagger. Despite there isn’t

restriction on the sway motion imposed in the model without rudder control, as can be seen in Figure

4.16, the qualitative behaviour of the evolutions seems similar to Varyani (2009) as well as Vantorre

(2002) for the sway interaction forces with two peaks of repulsion in the bow-bow stern-stern posi-

tions and one large suction on midship alignments. Regarding yaw interaction moments, the qualitative

behaviour appears similar to Vantorre (2002) and almost similar to Varyani (2006) results just not com-

pletely similar due to the smaller hollow and peak on larger relative distances not shown in Varyani

graphs. Physically, the ship experiences a large bow-out moment followed by smaller bow in, again even

smaller bow out passing midship-midship position and finally another bow in. Due to the disturbance

forces, the vessel is positioned in such a way that develops quasi-steady hydrodynamic forces and mo-

ments that acts in anti-phase with the interaction forces and can be seen comparing plots about Ys and

Yqand Ns and Nq. When ceased the interaction forces and moments the quasi steady forces continue

acting but with very low values and thus an action of the helmsman must take place in order to put the

ship in previous trajectory.

Figure 4.15: Interaction forces and moments acting on the ship in encounter manoeuvre.

47

Figure 4.16: Vessels trajectory in encounter manoeuvre without control.

4.4 Overtaking Manoeuvre

4.4.1 Preliminary Study

The simulation time will be configured as 8.3 minutes with integration step of 0.5 seconds. The real

interaction time is even shorter than the chosen simulation time. This option was chosen in order to give

the needed time to stabilize the ships in the straight path before interaction takes place don’t mixing the

transients. The initial conditions on Ship 1 and 2 are presented in Table 4.6. The interaction parameters

are presented in Table 4.7. They are stated in such a way that A(ξ′) parameters used on its formula will

be referred to H/T non-standard experimental value.

Variable Ship 1 Ship 2u (m/s) 8.0 5.0n (rps) 2.0 1.2v (m/s) 0.0 0.0r (rad/s) 0.0 0.0Xg (m) -800 0.0Yg (m) 120 0.0Φ (rad) 0.0 0.0δ (rad) 0.0 0.0

Table 4.6: Initial conditions overtaking manoeuvre.

48

Parameter ValueH/T 1.2L1/L2 1.0U2/U1 2.0Sp/L 1.0

Table 4.7: Overtaking manoeuvre interaction parameters.

Figure 4.17 and Figure 4.18 shows the time domain response of Ship 1. It can be seen oscillation

in magnitudes of kinematics and forces mainly due to the changes in sign of interaction forces and mo-

ments. Figure 4.19 and Figure 4.20 shows the Ship 2 kinematic and forces acting on her. It can be seen

now that the responses are different from one to the other due to the absence of symmetry and that

the forces on ship 2 has more clear path probably due to the different magnitudes of interaction forces

imparted on the interaction regarding the overtaken and overtaking ships.

Figure 4.17: General evolution in time of kinematics on the ship 1 in overtaking manoeuvre.

49

Figure 4.18: General evolution in time of forces acting on the ship 1 and heading in overtaking manoeu-vre.

Figure 4.19: General evolution in time of kinematics on the ship 2.

50

Figure 4.20: General evolution in time of forces and heading acting on the ship 2.

51

Figure 4.21 and Figure 4.22 show the interaction forces and moments as function of stagger and

the trajectory of the ships respectively. For the sway forces, it can be observed that the overtaken ship

passes with three peaks and the overtaking passes in two. Also the overtaken ship experiments a larger

time feeling the interaction. First the overtaken vessel is repelled then attracted and finally repelled

again. The overtaking ship is firstly repelled and then attracted by the overtaken one. Regarding the

yaw moment, both vessels experiments first a bow out moment and after that a bow in moment. The

last phase of the manoeuvre is in fact the most pronounced for the occurrence of a collision as can be

seen in the trajectory plot.

Figure 4.21: Interaction forces and moments acting on ships 1 and 2 in overtaking manoeuvre.

52

Figure 4.22: Vessels trajectory in overtaking manoeuvre.

Beside the risk of collision, ships also suffer from grounding originated from squat phenomenon and

maximum sinkage increased due to the additional blockage effect (Figure 4.23). The first seconds must

be disregarded since it was applied to the model suddenly the depth restriction and doesn’t represent

the reality. After the transient, it can be seen that maximum sinkage calculated was around 1.1 m for the

vessel overtaking. This value alone can represent dangerous one since coupling with asymmetric con-

ditions due to near bank is not accounted. When starting the overtaking the situation even come worst

and achieves around 2.7m when the vessels are with midship sections aligned. This situation results in

grounding knowing that the original draught was around 8.7m and the considered depth is 10.4 m.

53

Figure 4.23: Maximum sinkage in ship 1.

4.4.2 Comparison between deep and shallow water

A comparison study was performed in order to compare deep and shallow water behaviour given by

regression formulas from Brix [1993] and Varyani [2006a] (from now on just called Brix and Varyani).

The comparison will not be complete since Brix only analyses the forces and moments acting on the

overtaken vessel. Also Varyani do not show surge interaction forces regression formulas despite that

Brix shows. The comparison also aims to verify a possible combination of surge formula in Brix with

Varyani if the magnitudes and qualitative behaviour seems coherent between both models, then one

could deal with a complete set of interaction formulas for the horizontal plane manoeuvre in shallow

restricted waters didn’t found on the literature. Table 4.8 shows the initial conditions tested for the com-

parison between interaction models. The same simulation time and time step for the previous overtaking

simulation applies here.

Variable Ship 1 Ship 2u (m/s) 8.0 5.0n (rps) 2.0 1.5v (m/s) 0.0 0.0r (rad/s) 0.0 0.0Xg (m) -400 0.0Yg (m) 120 0.0Φ (rad) 0.0 0.0δ (rad) 0.0 0.0

Table 4.8: Overtaking comparison study initial conditions.

54

Figure 4.24 shows Brix and Varyani interactions regression formulas results for sway forces and yaw

moments. It can be seen a similar qualitative behaviour but is noticeable the interaction in shallow water

is greater than in deep water. The same happens with the yaw moment being a few times greater when

analysing the peak magnitudes.

Figure 4.24: Interaction forces and moments acting on ships 1 and 2 in overtaking manoeuvre compari-son study.

55

56

Chapter 5

Conclusions

A review of the literature was performed and experimental results translated in regression formulas are

used to model close proximity manoeuvres in shallow waters. Intervals of use of those formulas and

comparisons of results between different sources were pointed out. Three most common situations of

access channels and harbour area manoeuvres were simulated using regression formulas inside a time

loop. Vessels involved behaviours, interaction forces and moments and trajectories are analysed.

First the equations of motion of the vessels were modified to account for the shallow water more limited

manoeuvrability. A separate study regarding squat was performed comparing different formulas found

on the literature for different initial loading conditions. Then, an attempt was made to analyse the cou-

pling effects of squat and the other interaction phenomena that were explained inside each of the other

interaction models.

Regarding ship passing near the bank, despite the interaction forces and moment are rather small if

compared with the other interactions the ship may overtake during its operational life, it is visible colli-

sion risk anyway requiring rudder action. A simple controller proportional derivative (that controls based

yaw angle differences and in the rate of yaw) was implemented and improvements on trajectory keeping

was observed. Regarding encounter, the results were compared against other shallow water simulation

sources showing good agreement. It was observed that route deviation was rather small and leading to

the application inside the boundaries of the interaction regression model. Of course this could not be

always true for all the combination parameters simulation and attention must be placed to the probably

need to use a controller to be inside the regression equations limits.

Regarding overtaking, comparisons between Deep versus Shallow water and Shallow water interaction

models from different sources are performed. The comparison between Shallow water models from dif-

ferent sources demonstrates some initial qualitative discrepancies that was clarified by further analysis

of the experimental setup and the way that was performed data acquisition. The comparison between

Deep and Shallow water revealed that Shallow water forces and moments are indeed greater than in

Deep water and in order to use surge interaction equation for deep water into shallow water model some

correction factor must be found. The influence of the Overtaking Manoeuvre inside Squat was found to

be significant even at moderate speeds of the vessels involved.

57

For further research, it will be desired to increase the data bank of regressions and see some more

measurable differences between results. It was found not clear the effects of coupling between shallow

water effects (squat, bank, ship to ship and intrinsic ship behaviour) despite some effort is currently

been done to understand those coupling effects better, that is found in recent researches. It could be

implemented rudder actions in encounter and overtaking manoeuvres in order to counter react interac-

tion forces and moments and try to approximate ships trajectory from the tank regression formulations

applicability boundaries without putting constraints in the degrees of freedom of the models. Also the

insertion of surge interaction forces computation by means or using correction terms from deep water

expressions would be interesting.

58

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62

Appendix A

Matlab Code

The code is programmed in Matlab R2013a and is composed by one main routine, subroutines and

some functions. The basic description of each item is given on the next table.

file name type task

main.m routine calls simulation 6DOF, and postprocessor

plots like variables, vessel trajectory and Dis-

play Ship

addedmasses.m function calculate added masses based on approach

similar to Munk Method and crossflow theory

quasi force calc shallow.m function Treats the initial manoeuvring coefficients

considering trim, shallow water and labora-

tory adjustment factors (k) and generate ma-

noeuvring coefficients to be used in the Taylor

expansion series for the hydrodynamic adi-

mensional forces and moments calculations.

Propeller force calc.m function Based on geometric and hydrodynamic pa-

rameters defined for the propeller as also the

rpm of the main engine, gives the Thrust force

generated by the propeller.

Display Ship.m function Based on information about center of gravity

posistion of the vessel as also the drift angle

on that location, plots an image of the vessel

sumperimposed on his trajectory.

SI6DOF1, SI6DOF2, SI6DOF3 subroutine Called by main routine, Enter the main pa-

rameters (Simulation, Ship(s) properties, In-

teraction parameters).Call functions. Con-

tains simulation loop. Save the main data to

be used by post processor or in tabular for-

mat.

Post-procesing 1,2, 3 and 4 subroutine Generate plots regarding ships kinematic dy-

namic, interaction force and moment and tra-

jectories for each close-proximity manoeuvre

scenario. Call function Display Ship.

63

64