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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.
iii
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.
v
vi
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.
vii
viii
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
x
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
xi
xii
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
xiii
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
xiv
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
xvi
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
xvii
xviii
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