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Simulation of Ship-Wave-Ice Interactions in the Arctic

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Global climate change is presenting opportunities for new networks of maritime transportation through the Arctic. However, these sea routes are often infested by floating sea ice, which brings uncertainties to shipping operators, designers and builders. This work aimed to develop reliable simulation approaches for shipping scenarios in the presence of sea ice and investigate the associated changes to ship calm water resistance. For this purpose, computational fluid dynamics and ice solid mechanics were combined to model the potential ship-wave-ice interactions. Specifically, models were developed to simulate the two primary scenarios of a cargo ship operating in the Arctic, respectively a waterway with floating ice floes and an open-water channel created by icebreakers. Additionally, to build understanding of the Arctic sea condition, two other models were developed simulating the interaction of ocean waves with a rigid ice floe and then an elastic ice sheet, which provided a new solver capable of modelling hydroelastic fluid-structure interactions. Based on validation against experiments, these models provided the ability to accurately predict the ship-wave-ice interactions and the ice-induced resistance changes. Through conducting a systematic series of simulations, it was found that ice floes can increase the ship resistance by the same order of magnitude as the open water resistance, but this is strongly dictated by the ship beam, ice concentration, ice thickness and floe diameter. An open-water ice channel was found to increase the ship resistance by up to 15% compared to the situation without ice, particularly when the channel width is less than 2.5 times the ship beam and the ice thickness is greater than 5% of the ship draught. Moreover, this work developed a procedure to derive simple ice-resistance equations from the simulation results, enabling fast prediction of ship fuel consumption in sea ice fields and incorporation into a new Arctic Voyage Planning Tool.
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Simulation of Ship-Wave-Ice Interactions in
the Arctic
By: Luofeng Huang
Principal supervisor: Professor Giles Thomas
Co-supervisor: Professor Guoxiong Wu
A thesis submitted for the degree
of
Doctor of Philosophy
Department of Mechanical Engineering
University College London (UCL)
2021
Declaration
I, Luofeng Huang, confirm that the work presented in this thesis is my own. Where
information has been derived from other sources, I confirm that this has been indi-
cated in the work.
Signature:
I
Abstract
Global climate change is presenting opportunities for new networks of maritime
transportation through the Arctic. However, these sea routes are often infested by
floating sea ice, which brings uncertainties to shipping operators, designers and
builders.
This work aimed to develop reliable simulation approaches for shipping scenarios
in the presence of sea ice and investigate the associated changes to ship calm water
resistance. For this purpose, computational fluid dynamics and ice solid mechan-
ics were combined to model the potential ship-wave-ice interactions. Specifically,
models were developed to simulate the two primary scenarios of a cargo ship op-
erating in the Arctic, respectively a waterway with floating ice floes and an open-
water channel created by icebreakers. Additionally, to build understanding of the
Arctic sea condition, two other models were developed simulating the interaction
of ocean waves with a rigid ice floe and then an elastic ice sheet, which provided
a new solver capable of modelling hydroelastic fluid-structure interactions. Based
on validation against experiments, these models provided the ability to accurately
predict the ship-wave-ice interactions and the ice-induced resistance changes.
Through conducting a systematic series of simulations, it was found that ice floes
can increase the ship resistance by the same order of magnitude as the open water
resistance, but this is strongly dictated by the ship beam, ice concentration, ice
thickness and floe diameter. An open-water ice channel was found to increase the
ship resistance by up to 15% compared to the situation without ice, particularly
when the channel width is less than 2.5 times the ship beam and the ice thickness is
II
greater than 5% of the ship draught.
Moreover, this work developed a procedure to derive simple ice-resistance equa-
tions from the simulation results, enabling fast prediction of ship fuel consumption
in sea ice fields and incorporation into a new Arctic Voyage Planning Tool.
III
Impact statement
The emerging Arctic sea routes offer shorter voyage distances compared to their tra-
ditional counterparts and provide access to rich reserves of oil, gas, mines, fishing
grounds and tourism. Attracted by the unexploited resources and potential financial,
time and emission savings, the shipping industry is keen on maximising the oppor-
tunities afforded by Arctic shipping. Yet, the floating ice present in the Arctic sea
routes is holding back the stakeholders with navigational concerns. Therefore, reli-
able simulation models are required to correctly predict the ice resistance on ships,
so that the industry can prepare accordingly.
Addressing this challenge, this work provided new models that can accurately sim-
ulate ship operation in floating ice floes and in open-water channels between ice
sheets. The models can be applied to predict the ice resistance on a given ship. This
will in turn allow naval architects to optimise ice-going vessels by comparing po-
tential hull forms and comparing retrofits of a hull form; marine engineers to equip
the vessels with adequate propulsion systems; and structural engineers to assess ice
loads on a hull and plan scantlings to strengthen key areas.
The work also developed a procedure to derive a simple ice-resistance equation
based on systematic simulations. During the European Union’s Horizon 2020
project – SEDNA (Safe maritime operations under extreme conditions, the Arctic
case), the ice resistance question was combined with open-water resistance equa-
tions to provide a quick prediction of a ship’s total resistance and fuel consumption
for a given ice-infested route. This facilitated a voyage planning tool that links with
real-time metocean and ice data to calculate a ship’s fuel consumption along poten-
IV
tial routes, allowing ship operators to select routes with lower energy costs. This
software is currently being commercialised by project partner GreenSteam.
The output of this work can also contribute to future international guidelines and
regulations for polar maritime transportation. For example, based on the developed
simulation approaches, associations such as the International Towing Tank Confer-
ence could develop a fuller guideline on how to model ship-wave-ice interactions.
The findings regarding ice resistance on ships could help the International Maritime
Organisation to extend their Polar Code to formulate advice on ship operations in
floating ice floes and in open-water ice channels, and the Finnish-Swedish Ice Class
Rules may do the same to expand their scenarios.
Moreover, this work provided higher education and research institutions with tools
and knowledge to account for the transforming Arctic environment. As the Arctic
used to be covered by continuous level ice all year round, new knowledge for the
emerging ice-floe environment has been urgently required, not only for shipping,
but also to gain a better understanding of global warming and how to deal with it.
For example, contemporary climate models still cannot accurately predict Arctic
ice evolution and global temperature change. One of the main reasons is that they
need to improve the parametrisation that represents wave-ice interactions, because
ocean waves propagating in the ice fields dictate the ice layout and the associated
ice-reflected solar radiation. The provided computational models for wave-ice in-
teractions could fill this gap and potentially help remedy the inaccuracies in current
analytical methods.
V
Acknowledgements
Whilst I unexpectedly enjoyed this whole process of carrying out and presenting my
PhD, this project wouldn’t have existed if Professor Giles Thomas didn’t possess
such a keen eye and ear for what might prove valuable. Such a perspicacious and
enthusiastic supervisor is what I am most grateful for. He is a remarkable coach, a
terrific teacher and a wonderful editor. I learn something new in every one of our
conversations and always look forward to his calls.
I have eternal gratitude to Professor Guoxiong Wu who endorsed my research po-
tential and recommended me to retain at UCL as a PhD student. He taught me to
scrutinise the mathematical and physical links that are easily overlooked behind a
good-looking simulation. His rigorous and challenging attitude remains inspiring
as much as humbling.
Numerous scholars have demonstrated trust in my supposed skills and provided
the present work with constructive advice. They include but not limited to Jukka
Tuhkuri, Hrvoje Jasak, Jonas W. Ringsberg, Philip Cardiff, ˇ
Zeljko Tukovi´
c, H˚
akan
Nilsson, Luke Bennetts, Mike Meylan and Yiannis Ventikos.
Several important elements should be factored into the completion of this work:
the use of the UCL High Performance Computing Facility and associated support
services, without which this extensive computational project wouldn’t have been
possible; the technical input from Minghao Li and Bojan Igrec, who didn’t leave
me alone with my ambitious ideas and helped me break through; the writing and
presenting skills taught by Sunny Bains that not only help me interpret better but
also give me confidence; the countless hours I spent in gym together with Anna
VI
Chiara Faralli that provided much-needed fuel; and the unfailing companionship of
Daniela Benites Munoz and Katherine Wang over these years, during which I got
myriad helps as well as happy memories.
I am indebted to many colleagues who meanwhile are truly valued friends, espe-
cially Blanca Pena, Enrico Anderlini, Chris Ryan, Yuanchang Liu, Kang Ren, Tuo-
mas Ruma, Chris Cassar, Rui Song, Andrea Grech La Rosa, Abbas Dashtimanesh,
Faheem Rehman, Gabriela Lam, Sasan Tavakoli, Azam Dolatshah, Dimitris Stago-
nas, Zhiyuan Li and Yuzhu Pearl Li.
This work is fortunate to be part of a European Union’s Horizon 2020 project -
SEDNA: Safe maritime operations under extreme conditions, the Arctic case. I
hence benefited enormously from the project experience and expert opinions; spe-
cial thanks go to the SEDNA partners.
Last but not least, I am particularly grateful for the scholarships received from
Lloyds Register Foundation, UCL Faculty of Engineering Sciences and China
Scholarship Council.
VII
List of publications
Journal articles as part of this work
Huang, L., and Thomas, G., 2019. Simulation of wave interaction with a circular
ice floe. Journal of Offshore Mechanics and Arctic Engineering, 141(4), 041302.
Huang, L., Ren, K., Li, M., Tukovi´
c, ˇ
Z., Cardiff, P. and Thomas, G., 2019. Fluid-
structure interaction of a large ice sheet in waves. Ocean Engineering, 182, pp.102-
111.
Huang, L., Tuhkuri, J., Igrec, B., Li, M., Stagonas, D., Toffoli, A., Cardiff, P. and
Thomas, G., 2020. Ship resistance when operating in floating ice floes: a combined
CFD&DEM approach. Marine Structures, 74, 102817.
Huang, L., Li, Z., Ryan C., Ringsberg, J., Pena, B., Li, M., Ding, L. and Thomas,
G., 2021. Ship resistance when operating in floating ice floes: derivation, validation,
and application of an empirical equation. Marine Structures, 79, 103057.
Huang, L., Li, M., Romu, T., Dolatshah, A. and Thomas, G. 2021. Simulation of a
ship operating in an open-water ice channel. Ships and Offshore Structures, 16(4),
pp.353-362.
Huang, L., Igrec, B., and Thomas, G., 2022. New tools to generate realistic ice
floe fields for computational models. Journal of Offshore Mechanics and Arctic
Engineering, 144(4), 044503.
VIII
Conference proceedings as part of this work
Huang, L., 2018. An opensource solver for wave-induced FSI problems. CFD with
Opensource Software, at: Gothenburg, Sweden.
Huang, L., Li, M., Tukovic, Z. and Thomas, G., 2018. Simulation of the hydroelas-
tic response of a floating ice sheet. The 13th OpenFOAM Workshop, at: Shanghai,
China.
Huang, L., Dolatshah, A., Cardiff, P., Bennetts, L., Toffoli, A., Tukovic, Z. and
Thomas, G., 2019. Numerical simulation of hydroelastic waves along a semi-
infinite ice floe. The 34th International Workshop on Water Waves and Floating
Bodies (IWWWFB), at: Newcastle, Australia.
Huang, L., Li, M., Igrec, B., Cardiff, P., Stagonas, D. and Thomas, G., 2019. Sim-
ulation of a ship advancing in floating ice floes. the 25th International Conference
on Port and Ocean Engineering under Arctic Conditions (POAC), at: Delft, Nether-
land.
Huang, L., Li, Z., Ryan, C., Li, M., Ringsberg, J., Igrec, B., Grech La Rosa, A.,
Stagonas, D. and Thomas, G., 2020. Ship resistance when operating in floating ice
floes: a derivation of empirical equations. The 39th International Conference on
Ocean, Offshore & Arctic Engineering (OMAE), Virtual Conference.
Huang, L., Bennetts, L., Cardiff, P., Jasak, H., Tukovic, Z. and Thomas, G., 2020.
The implication of elastic deformation in wave-ice interaction. The 15th Open-
FOAM Workshop, Virtual Conference.
IX
Selected publications as extra research
Benites-Munoz, D., Huang, L., Anderlini, E., Mar´
ın-Lopez, J. and Thomas, G.,
2020. Hydrodynamic modelling of an oscillating wave surge converter including
power take-off. Journal of Marine Science and Engineering, 8(10), 771.
Huang, L., Tavakoli, S., Li, M., Dolatshah, A., Pena, B., Ding, B. and Dashti-
manesh, A., 2021. CFD analyses on the water entry process of a freefall lifeboat.
Ocean Engineering, 232, 109115.
Pena, B. and Huang, L., 2021. Wave-GAN: A deep learning approach for the pre-
diction of nonlinear regular wave loads and run-up on a fixed cylinder. Coastal
Engineering, 167, 103902.
Pena, B. and Huang, L., 2021. A review on the turbulence modelling strategy for
ship hydrodynamic simulations. Ocean Engineering, 241, 110082.
Ryan, C., Huang, L., Li, Z., Ringsberg, J. and Thomas, G., 2021. An Arctic ship
performance model for sea routes in ice-infested waters. Applied Ocean Research,
117, 102950.
Li, F. and Huang, L., 2022. A review of computational simulation methods for a
ship advancing in broken ice. Journal of Marine Science and Engineering, 10(2),
165.
Huang, L. and Li, Y., 2022. Design of the submerged horizontal plate breakwater
using a fully coupled hydroelastic approach. Computer-Aided Civil and Infrastruc-
ture Engineering, 37, pp.915–932.
X
Contents
1 Introduction 1
1.1 Background.............................. 1
1.2 Problemdenition .......................... 3
1.3 Literaturereview ........................... 4
1.3.1 Arctic ice and shipping activities . . . . . . . . . . . . . . . 4
1.3.1.1 Historical observation . . . . . . . . . . . . . . . 4
1.3.1.2 Future prediction . . . . . . . . . . . . . . . . . 5
1.3.2 Wave-ice interactions . . . . . . . . . . . . . . . . . . . . . 7
1.3.2.1 Ocean waves with small ice floes . . . . . . . . . 8
1.3.2.2 Ocean waves with a large ice sheet . . . . . . . . 10
1.3.3 Ship operation in various ice conditions . . . . . . . . . . . 13
1.3.3.1 Icebreaker in level ice . . . . . . . . . . . . . . . 14
1.3.3.2 Ship in a brash ice channel . . . . . . . . . . . . 15
1.3.3.3 Ship in an open-water ice channel . . . . . . . . . 16
1.3.3.4 Ship in floating ice floes . . . . . . . . . . . . . . 17
1.4 Researchgaps............................. 21
1.5 Researchvision............................ 21
XI
1.6 Researchapproach .......................... 22
1.7 Researchquestions .......................... 23
1.8 Thesisoutline............................. 24
2 Numerical methods 25
2.1 Fluidsolutions ............................ 25
2.1.1 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . 25
2.1.2 Turbulence modelling . . . . . . . . . . . . . . . . . . . . 26
2.1.3 Free surface modelling . . . . . . . . . . . . . . . . . . . . 27
2.1.4 Wavemodelling ....................... 28
2.2 Solidsolutions ............................ 29
2.2.1 Rigid-body motion . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Elastic deformation . . . . . . . . . . . . . . . . . . . . . . 30
2.2.3 Collisions........................... 31
2.3 Computational procedure . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.1 Time discretisation . . . . . . . . . . . . . . . . . . . . . . 32
2.3.2 Spatial discretisation . . . . . . . . . . . . . . . . . . . . . 33
2.3.3 Pressure-velocity coupling . . . . . . . . . . . . . . . . . . 36
2.3.4 Choice of software . . . . . . . . . . . . . . . . . . . . . . 36
3 Wave interaction with a rigid ice floe 37
3.1 Computational modelling . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Discretisation......................... 39
3.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 41
XII
3.2.1 Mesh sensitivity tests . . . . . . . . . . . . . . . . . . . . . 42
3.2.2 Validation........................... 43
3.2.3 Overwash........................... 45
3.2.4 Scattering........................... 49
3.3 Conclusions.............................. 50
4 Wave interaction with an elastic ice sheet 52
4.1 Development of a new hydroelastic solver . . . . . . . . . . . . . . 53
4.2 Computational modelling . . . . . . . . . . . . . . . . . . . . . . . 54
4.2.1 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 54
4.2.2 Discretisation......................... 55
4.2.3 Fluid-structure interaction . . . . . . . . . . . . . . . . . . 56
4.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Mesh sensitivity tests . . . . . . . . . . . . . . . . . . . . . 59
4.3.2 Validation........................... 60
4.3.3 Overwash........................... 65
4.3.4 Icedeformation........................ 67
4.4 Conclusions.............................. 69
5 Ship operating in an open-water ice channel 71
5.1 Computational modelling . . . . . . . . . . . . . . . . . . . . . . . 71
5.1.1 Hullmodel .......................... 71
5.1.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . 72
5.1.3 Discretisation......................... 73
5.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 76
XIII
5.2.1 Verication.......................... 78
5.2.2 Analyses on the ship resistance with ice . . . . . . . . . . . 79
5.3 Conclusions.............................. 84
6 Ship operating in floating ice floes 85
6.1 Computational modelling . . . . . . . . . . . . . . . . . . . . . . . 85
6.1.1 Shipow ........................... 86
6.1.2 Discrete ice floes . . . . . . . . . . . . . . . . . . . . . . . 87
6.1.3 Floe-distribution algorithms . . . . . . . . . . . . . . . . . 91
6.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 96
6.2.1 Validation........................... 97
6.2.2 Analyses on the ship resistance with ice . . . . . . . . . . . 101
6.3 Conclusions..............................106
7 Incorporation of simulation results with real-time applications 107
7.1 Derivation of an equation for ice-floe resistance . . . . . . . . . . . 108
7.1.1 Non-dimensional analysis . . . . . . . . . . . . . . . . . . 108
7.1.2 Unification for multiple hull forms . . . . . . . . . . . . . . 109
7.1.3 Validation...........................114
7.1.4 Full-scale extrapolation . . . . . . . . . . . . . . . . . . . . 117
7.2 Arctic ship performance model and voyage planning tool . . . . . . 118
7.3 Comparison with full-scale measurements . . . . . . . . . . . . . . 120
7.4 Fuel and cost comparison between the Arctic and traditional routes . 123
7.5 Conclusions..............................124
8 Concluding remarks 125
XIV
8.1 Summary ...............................125
8.2 Recommendations for future research . . . . . . . . . . . . . . . . 129
References 131
XV
Nomenclature
αWaterline angle
δWave period
ηFree surface elevation
γButtock angle
λWavelength
FForce vector
uDisplacement vector
vVelocity vector
µDynamic viscosity
νPoisson’s ratio
Torque vector
ωWave angular frequency
ΨWave energy
ρDensity
aWave amplitude
BShip beam at waterline
XVI
CIce concentration
Co Courant number
DEquivalent diameter of an ice floe’s upper surface
dWater depth
EYoung’s modulus
Fr Froude number
gGravitational acceleration
HWave height
hIce thickness
kWave number
LLength of an ice sheet
Lpp Ship length between perpendiculars
mMass
pPressure
RWave reflection coefficient against an ice sheet
Rchannel Ship resistance in an ice channel
Rice Ship resistance induced by ice
Row Ship resistance in open water without ice
Rwater Ship resistance induced by water
TWave transmission coefficient against an ice sheet
TMShip draught at midship
XVII
UShip speed
WWidth of an ice channel
6-DOF Six degrees of freedom
AIV Arctic In-service Vessel
ASPM Arctic Ship Performance Model
CFD Computational Fluid Dynamics
CSM Computational Solid Mechanics
DEM Discrete Element Method
FSD Floe Size Distribution
FSI Fluid-Structural Interaction
FSICR Finnish Swedish Ice Class Rules
FVM Finite Volume Method
ITTC International Towing Tank Conference
JBC Japan Bulk Carrier
KCS KRISO Container Ship
MIZ Marginal Ice Zone
NSR Northern Sea Route
NWP Northwest Passage
RANS Reynolds-Average Navier-Stokes
RAO Response Amplitude Operator
VOF Volume of Fluid
VPT Voyage Planning Tool
XVIII
List of Figures
1-1 Average monthly sea ice extent in March 2016 (left) and September
2016 (right): illustrates the respective winter maximum and summer
minimum extents. The coloured line indicates the median ice extent
during the period 1981-2010, from which a distinct ice reduction is
present[1]............................... 2
1-2 Comparison between the Arctic shipping routes (red dashed line)
and their current counterparts (black solid line) [2] . . . . . . . . . 2
1-3 A ship surrounded by floating ice floes (Credit: Alessandro Toffoli) 3
1-4 Satellite observation of the sea ice extent till the March of 2018 [3] . 5
1-5 Arctic ice extent in September of the year 1978 and 2010 [4] . . . . 5
1-6 The decreasing trend of the Arctic ice thickness [5] . . . . . . . . . 6
1-7 Pancake ice condition [6] . . . . . . . . . . . . . . . . . . . . . . . 6
1-8 A comparison among the Marginal Ice Zone (MIZ), continuous ice
cover and open ocean [7] . . . . . . . . . . . . . . . . . . . . . . . 6
1-9 The number of transit per year through the Northwest Passage
(NWP) and the Northern Sea Route (NSR) [8] . . . . . . . . . . . . 7
1-10 The decreasing trend of the Arctic sea ice: historical observations
and future predictions [9] . . . . . . . . . . . . . . . . . . . . . . . 7
1-11 Projected duration of the navigable period of the Northern Sea
Route and the Northwest Passage [10] . . . . . . . . . . . . . . . . 8
XIX
1-12 Experimental view of an icebreaker advancing in level ice [11] . . . 15
1-13 Commercial ships travelling in a brash ice channel [12] . . . . . . . 15
1-14 Open water channel created by an icebreaker (Credit: Aker Arctic) . 17
1-15 Experimental view of a ship advancing in floating paraffin slices [13] 17
1-16 Comparison of ship operating in three different scenarios: open
ocean, canal and open-water ice channel; Wis the width of the
channel and his the ice thickness . . . . . . . . . . . . . . . . . . . 18
1-17 Ice conditions along shipping routes of Western Arctic, measured
inAutumn2015[14] ......................... 18
1-18 Regular pancake-ice field used by Sun and Shen [15] . . . . . . . . 20
1-19 Illustration of ship-wave-ice interactions as broken down into sub-
processes ............................... 23
2-1 Illustration of free surface construction and the corresponding α
values[16]............................... 28
2-2 Illustration of two neighbouring cells in FVM [17] . . . . . . . . . 33
2-3 Illustration of the Gauss’ Divergence Theorem [17] . . . . . . . . . 34
2-4 Illustration of the first-order upwind scheme [18]: if v>0, ui1/2=
ui1,ui+1/2=ui; if v<0, ui1/2=ui,ui+1/2=ui+1........ 35
2-5 Illustration of the second-order upwind scheme [18]: if v>0,
ui1/2= (3ui1ui2)/2, ui+1/2= (3uiui1)/2; if v<0,
ui1/2= (3uiui+1)/2, ui+1/2= (3ui+1ui+2)/2 ......... 35
3-1 Schematic of the case: a circular ice floe is freely floating on the
water surface and subjected to incoming regular waves generated by
a numerical wavemaker, where the surge, heave and pitch motions
are its main hydrodynamic responses . . . . . . . . . . . . . . . . . 37
XX
3-2 Sketch of the computational domain and the applied boundary con-
ditions................................. 38
3-3 The geometry of the two disks. Left panel: the disk with an edge
barrier (Disk B); right panel: the disk without edge barrier (Disk NB) 39
3-4 Mesh layout of the model. High resolution was applied to the free
surface area and where the disk is expected to move . . . . . . . . . 40
3-5 Generated waves with different cell numbers per wave height. The
target waves are of λ= 2.38 m and a=40mm............ 42
3-6 RAOs obtained with different total cell numbers. The applied wave
condition was λ= 2.38 m and a=40mm .............. 42
3-7 Computational and experimental [19] RAOs, as a function of non-
dimensional wavelength . . . . . . . . . . . . . . . . . . . . . . . 45
3-8 CFD illustration of partial overwash and full overwash . . . . . . . 47
3-9 RAO comparisons between Disk B (dashed line) and Disk NB (solid
line), as a function of non-dimensional wavelength, alongside the
typeofoverwash ........................... 48
3-10 Heave RAO of Disk B (dashed line) and Disk NB (solid line), as a
function of wave amplitude, alongside the type of overwash. Trend
lines (blue) are also included . . . . . . . . . . . . . . . . . . . . . 48
3-11 The interaction of Disk B with waves, at different wavelengths . . . 50
4-1 Schematic of the case: a thin ice sheet is floating on the water sur-
face and subjected to incoming waves, with its elastic deformation
enabled ................................ 52
4-2 Development route of the hydroelastic solver . . . . . . . . . . . . 54
4-3 Mesh layout of the model: the fluid mesh is graded towards the free
surface area, while the solid mesh is uniform . . . . . . . . . . . . . 56
XXI
4-4 Flowchart of the FSI scheme . . . . . . . . . . . . . . . . . . . . . 56
4-5 Generated waves with different cell numbers per wave height. The
target waves are of δ= 0.7 s and H=0.017m............ 59
4-6 Vibration amplitude at different locations of the ice sheet, obtained
with a range of solid cell number (CN). Experimental data of Sree
et al. [20] are also included. The plate length is 1 m in total and
the x-axis shows the measured distance from its left edge. Applied
wave and ice conditions are shown in Table 4-1: Case 19. . . . . . . 61
4-7 Simulation examples of the wave interaction with a large floating
icesheet................................ 61
4-8 Computational and experimental R & T [21], obtained when over-
washwasavoided........................... 66
4-9 Computational and experimental [22] mean overwash depth . . . . . 66
4-10 R & T with and without overwash; dashed lines show the analytical
solutions of Nelli et al. [21] . . . . . . . . . . . . . . . . . . . . . . 67
4-11 Comparison of simulations between wave interactions with semi-
rigidandelasticice........................... 68
5-1 Sketch of the computational domain with dimensions (only one ice
sheetisshown) ............................ 73
5-2 Mesh layout of the model . . . . . . . . . . . . . . . . . . . . . . . 74
5-3 Wave patterns in ice channels of different channel widths, obtained
when Fr = 0.12 and h/TM=0.1.................... 75
5-4 Computational resistance of the model-scale KCS in an ice channel
of W/B>2.5, compared with experimental resistance of the model-
scale KCS in open water [23] . . . . . . . . . . . . . . . . . . . . . 76
5-5 CFD illustration of the KCS hydrodyanmcis (Fr = 0.18) . . . . . . 77
XXII
5-6 Free surface obtained with different discretisation resolutions . . . . 80
5-7 Ship resistance in ice channels of different widths, in comparison
with the simulation without ice, obtained when h/TM= 0.5 . . . . . 81
5-8 Ship resistance in ice channels of different ice thicknesses, in com-
parison with the simulation without ice, obtained when Fr = 0.03 . . 83
5-9 Wave patterns in ice channels of different ice thicknesses, obtained
when Fr = 0.12 and W/B=1.8 .................... 83
6-1 Illustration of the computational domain with dimensions . . . . . . 86
6-2 Mesh layout of the model, in which local refinements are applied at
the Kelvin-wave region, the free surface region and around the hull
geometry ............................... 88
6-3 Illustration of how ice floes are imported . . . . . . . . . . . . . . . 89
6-4 Simulation view of a ship advancing in a regular ice floe field . . . . 89
6-5 FSD measured by Colbourne [24]: ice floes in a sufficiently large
region are of a mixture of different sizes and the size against possi-
bility function follows a log-normal form . . . . . . . . . . . . . . 92
6-6 Flowchart of Algorithm (I) . . . . . . . . . . . . . . . . . . . . . . 93
6-7 Flowchart of Algorithm (II) . . . . . . . . . . . . . . . . . . . . . . 94
6-8 Ice-floe fields obtained by Algorithm (I), C= 40% (left side); and
by Algorithm (II), C= 70% (right side) . . . . . . . . . . . . . . . 95
6-9 Simulation view of a ship advancing in floating ice floes, with nat-
ural floe fields implemented . . . . . . . . . . . . . . . . . . . . . . 96
6-10 Heatmap of the ice-induced collisions on the KCS hull . . . . . . . 97
6-11 Time series of ice-induced impulse against the direction of ship ad-
vancement (obtained when Fr = 0.12, C= 60%, h= 0.02 m and
FSDfollows[24])........................... 98
XXIII
6-12 Time series of ice resistance: an oscillation is shown when the ship
just enters the floe field, and then Rice approaches a steady value
overtime ............................... 98
6-13 Experimental [23] and computational total resistance of the model-
scale KCS operating in ice concentration 60% and 70%, alongside
thewatercomponent ......................... 99
6-14 Ship advancement to make contacts with ice in different floe-shape
scenarios ...............................101
6-15 Ice-floe resistance for different ice concentrations and ship speeds . 101
6-16 Comparison between ship-ice interactions with and without waves . 103
6-17 Ship advancing in different-sized floes (left side) and corresponding
time-series of resistance impulse (right side); obtained when Fr =
0.15, h= 0.02 m and C=60% ....................104
6-18 Ice-floe resistance in different-sized floes (floe diameters of [23]
globally scaled by a factor), obtained when Fr = 0.15 and h= 0.02 m 105
6-19 Ice-floe resistance in varying ice thickness, obtained when Fr = 0.15 105
6-20 Ice-floe resistance as a function of ship beam (normalised by the
design beam), obtained when Fr = 0.18 and h= 0.02 m . . . . . . . 106
7-1 Geometries of the KCS, JBC and AIV ships, in the order of KCS,
JBC and AIV (their relative sizes in the images correspond to the
actualshipsizes) ...........................110
7-2 Simulations of different ships advancing in ice floe fields . . . . . . 110
7-3 Graphic expression of buttock angle and waterline angle, defined
following the International Association of Classification Societies
[25] ..................................111
7-4 Waterline velocity fields around different bows (Fr = 0.12), colour
contours show the velocity magnitude in the transverse direction . . 112
XXIV
7-5 Ice-floe resistance calculated by simulations (dots) and Equation 7.4
(lines), for KCS hull at model scale (1:52.667), when h= 0.02 m . . 113
7-6 Ice-floe resistance calculated by simulations (dots) and Equation 7.4
(lines), for JBC hull at model scale (1:52.667), when h= 0.0066 m . 113
7-7 Ice-floe resistance calculated by simulations (dots) and Equation 7.4
(lines), for AIV hull at model scale (1:52.667), when h= 0.0066 m . 113
7-8 Rice data from simulations nondimensionalised according to Equa-
tion7.4 ................................114
7-9 Comparison between ice-floe resistance of 1:52.667 KCS hull
model measured by experiments [23] (dots) and calculated by Equa-
tion7.4(lines).............................115
7-10 Ice model tests of 1:18.667 Araon hull model reported by Kim et
al.[26].................................116
7-11 Comparison among ice-floe resistance of 1:18.667 Araon hull
model given by Equation 7.4 (lines), by experiments (circles), and
by the finite-element method (crosses) of Kim et al. [26] . . . . . . 116
7-12 Calculation procedure of the Arctic Ship Performance Model . . . . 119
7-13 Two VPT-suggested routes for a ship travelling through the North-
ern Sea Route, obtained using the metocean and ice data of early &
late summer 2018. Colour bars show ice concentration in the left
panels and ice thickness (meter) in the right panels, and the xand y
axes denote the distance (meter) with respect to the North Pole origin 120
7-14 Full-scale measurement of engine RPM (solid line) and sea ice con-
centration (dashed line) . . . . . . . . . . . . . . . . . . . . . . . . 121
7-15 Predicted attained ship speed against full-scale measurements . . . . 122
XXV
List of Tables
4-1 Parameters of the Simulation Cases . . . . . . . . . . . . . . . . . 62
4-2 Relative Deviation of Ice Deformation with Different Solid Mesh
Densities ............................... 62
4-3 Relative Deviations of R & T and Mean Overwash Depth . . . . . . 63
4-4 Variation of R, T and Mean Overwash Depth with Different E . . . 69
5-1 Main Dimensions of the KCS Hull . . . . . . . . . . . . . . . . . . 72
5-2 Ship Resistance with Different Discretisation Resolutions . . . . . . 79
5-3 Ship Resistance in Ice Channels of Different Widths (W/B), along-
side Breakdown into Pressure and Shear Components and Increased
Percentage Compared with in Open-water, Obtained when Fr =
0.03 and h/TM=0.5.......................... 82
6-1 Comparison of Ice Resistance between with & without Ship-
generated Waves, when C=60%...................103
7-1 Main Particulars of the Studied Hull Forms . . . . . . . . . . . . . 109
7-2 Bow Angles of the Candidate Ships and the Corresponding Rice Co-
efficients in Equation 7.3 . . . . . . . . . . . . . . . . . . . . . . . 112
7-3 Particulars of the Propulsion System of AIV . . . . . . . . . . . . . 122
XXVI
7-4 Comparison of Journey Time, Fuel and Cost between Voyages
through the NSR and the Suez Route. VPT Simulations were Made
from Shanghai to Rotterdam based on the Historical Weather Data
of 2018 [27]. The Consumed Fuel is Assumed to be the IFO380
Type and the Price is £298 per Tonnes . . . . . . . . . . . . . . . . 123
XXVII
Chapter 1
Introduction
1.1 Background
With global warming, the sea ice extent in the Arctic is reducing quickly. As shown
in Figure 1-1, the Arctic ice extent usually hits a maximum in March and a minimum
in September [1], and satellite images have observed its summer minimum to have
decreased by approximately 12% per decade [28]. According to this trend, an ice-
free Arctic could appear by the middle of this century [29].
The ice reduction creates open water and leads to the notion that commercial ship-
ping through the Arctic will be viable [30], with numerous waterways opening for
travelling between continents and the Arctic, which are used to access oil, gas,
mines, fishing grounds and tourism. In addition, there are two major shipping
routes becoming navigable, the Northwest Passage (NWP) and the North Sea Route
(NSR), which can be used as alternatives to the Panama and Suez canals to con-
nect Europe, Asia and America [2], as illustrated in Figure 1-2. Compared to their
current counterparts, both new routes can reduce the travel distance by up to 40%,
signifying substantial time, cost, fuel and emissions savings [31].
There are formidable challenges coming hand-in-hand with the benefits of Arctic
shipping. One of the most obvious is to understand the potential navigation envi-
ronment for ships. The effects of ice reduction on the navigability of the Arctic
can be more complex than anticipated. Rather than providing a pure open-ocean
1
environment, the melting ice cover also evolves into numerous ice floes floating on
the sea surface, as shown in Figure 1-3. Such ice-floe fields have been predicted to
be the most ubiquitous ice condition of future Arctic [14], but its influence on ships
has yet to be fully understood.
Figure 1-1: Average monthly sea ice extent in March 2016 (left) and September 2016
(right): illustrates the respective winter maximum and summer minimum extents. The
coloured line indicates the median ice extent during the period 1981-2010, from which a
distinct ice reduction is present [1]
Figure 1-2: Comparison between the Arctic shipping routes (red dashed line) and their
current counterparts (black solid line) [2]
2
Figure 1-3: A ship surrounded by floating ice floes (Credit: Alessandro Toffoli)
1.2 Problem definition
To clarify the uncertainty brought by floating ice to Arctic shipping and facilitate
corresponding hull design, power estimates and route planning, this work aims to
develop reliable models that can represent shipping scenarios in the presence of sea
ice and investigate the associated changes to ship resistance in open water. There
are two primary scenarios for the proposed problem. The first scenario is a cargo
ship operating in a waterway infested by floating ice floes, as identified to be the
dominant ice condition for the emerging Arctic shipping routes. Moreover, certain
segments of the shipping routes can still be seasonally covered by consolidated ice,
which are unnavigable for cargo ships and icebreaker assistance would be required
in this case. Therefore, the second scenario represents a cargo ship operating in a
channel created by icebreakers in continuous ice, a standard solution to maintain
the navigability.
Additionally, the ice-floe environment allows ocean surface waves to propagate
through it. In such a event, the waves can dictate the ice distribution, and mean-
while the ice can affect the wave propagation. For example, waves can cause large
floes to break up into smaller floes, until the floes are sufficiently small to survive
intact in the waves [32]; meanwhile, waves attenuate when passing through ice
floe fields [33]. Therefore, following wave propagation and attenuation from open
3
ocean to permanent ice sheets, the size of ice floes generally increase [34]. Smaller
floes are more susceptible to melting [35] and waves also introduce warm water
to further increase the melting rate [36], significant for atmosphere-ice thermal
exchange. These processes are currently accounted for in climate models through
simple empirical equations that are not satisfactorily accurate [37]. To better model
wave-ice interactions is very important for predicting the environmental conditions
for Arctic shipping and achieving voyage planning. Thus, this work also intends to
build advanced models for the wave-ice interactions that can be useful to understand
the Arctic sea states and ice conditions.
1.3 Literature review
1.3.1 Arctic ice and shipping activities
1.3.1.1 Historical observation
Since the late 1970s when satellite observations began, the Arctic ice extent has
decreased rapidly, as shown in Figure 1-4. Such reduction is most significant in
September, at a rate of -12.9 ±1.47% per decade [38], evidenced by an obvious
transition from ice-covered areas to open water, as shown in Figure 1-5. The melt
of sea ice is actually a process of volume loss, which includes not only the reduction
of ice extent but also a decrease of thickness [39]. A combination of submarine and
satellite measurements indicates the Arctic sea ice has thinned by more than 50%
in the past 50 years [5], as shown in Figure 1-6.
The ice melting has created more complex conditions than level ice coverage and
open water. Level ice is also evolving into broken ice floes floating on the sea
surface, pronounced as a decrease of ice compactness [40]. Meanwhile, those floes
tend to be circular under the effect of wave wash and floe-floe collisions, thus known
as pancake ice, as shown in Figure 1-7. Such a status is known as the Marginal Ice
Zone (MIZ) [41], as illustrated in Figure 1-8; with the process of global warming,
MIZ is taking an increasing proportion in polar regions and is predicted to be a
4
primary environment of future Arctic [14, 42].
The reduction in ice extent, thickness and compactness leads to an increased impe-
tus for Arctic maritime activities. On one hand, the decrease of the ice-covered area
opens a larger navigable area and results in that vast quantities of natural resources
such as oil, gas and minerals are becoming exploitable; on the other hand, the ice
thinning increases the ability of ice-breakers to create paths, thus further improving
the accessibility of commercial ships to the Arctic.
There has been a significant growth of complete transits of the NWP and the NSR,
especially for the NSR that hosted more than 60 transits in 2019, as shown in Figure
1-9. Nevertheless, this amount is very small compared approximately 1700 transits
through the traditional routes [43]. Thus, there is still huge potential for the Arctic
shipping routes to share the carrying capacity.
Figure 1-4: Satellite observation of the sea
ice extent till the March of 2018 [3]
Figure 1-5: Arctic ice extent in September
of the year 1978 and 2010 [4]
1.3.1.2 Future prediction
Multiple prediction models have indicated that the retreat of the Arctic ice will
not slow down, and it has become a consensus that an almost ice-free Arctic will
appear by mid-century [29, 31], as shown in Figure 1-10. This will be accompa-
nied by a stable increase of navigable days per year, which can be seen in Figure
1-11. There seems to be bright promise for LNG carriers to bring Arctic gas to
market; oil tankers doing the same with oil; cargo shipping serving Siberian com-
munities; and various specialised ships, for instance, Japanese freezer ships could
5
Figure 1-6: The decreasing trend of the
Arctic ice thickness [5]
Figure 1-7: Pancake ice condition [6]
Figure 1-8: A comparison among the Marginal Ice Zone (MIZ), continuous ice cover and
open ocean [7]
purchase seafood from US fishermen and deliver it straight to Europe via the NSR
[29]. Moreover, the Arctic also contains 13% of the world’s undiscovered oil, 30%
of the gas and abundant deposits of valuable minerals [44]. With projects com-
mencing such as the Yamal LNG facility, exploration activity will only increase in
the next few years [43].
In practice, a navigable Arctic may come earlier than expectations. Existing predic-
tion models have not performed accurately against historical observations, which
have shown a larger loss of sea ice than predicted [45]. This suggests that large-
scale activities of Arctic shipping may occur even sooner, attracting special research
interest and investment from stakeholders.
6
Figure 1-9: The number of transit per year through the Northwest Passage (NWP) and the
Northern Sea Route (NSR) [8]
Figure 1-10: The decreasing trend of the Arctic sea ice: historical observations and future
predictions [9]
1.3.2 Wave-ice interactions
The changing ice conditions increases new wave-ice dynamics in the Arctic. As
shown in Figure 1-8, incoming waves from open ocean can interact with ice floes
in the MIZ and then the continuous ice cover. Based on the relative size of ice to
waves, relevant research has been broken down into wave interactions with small
ice floes and with a large ice sheet, which are reviewed respectively in this section.
7
Figure 1-11: Projected duration of the navigable period of the Northern Sea Route and the
Northwest Passage [10]
1.3.2.1 Ocean waves with small ice floes
In the MIZ, ocean waves can induce the movement of a solitary ice floe and lead to
the distinctive behaviour of multiple ice floes, such as collision [46], herding [47]
and rafting [48]. Meanwhile, the incoming waves can be scattered by an ice floe
to cause directional spreading thus attenuated by a group of ice floes [33]. Based
on the long wavelength and small amplitude waves that are common in the MIZ,
previous studies usually consider the ocean waves to compose of regular waves,
and the ice floe is assumed to be rigid according to its relatively small dimension.
Meylan et al. [49] conducted wave tank experiments to investigate the wave-
induced motions of a solitary floating disk. They recorded the movement trajec-
tory of the disk and decomposed it into six-degrees-of-freedom (6-DOF) motions.
Alongside drifting with the wave flow, the disk was found to undergo oscillatory
8
motions at the same frequency as the incident waves. They also observed wave
scattering when the dominating wavelength is less than twice of the disk diameter
and found the scattering can reduce the motion amplitudes of the disk. However,
they used a barrier around the edge of the disk to prevent overwash, i.e. wave run-
ning over the top of the disk surface. Based on the findings of Meylan et al. [49],
Yiew et al. [19] carried out further experiments to study the influence of overwash
on the disk motions, by comparing the motions of two disks, with and without the
edge barrier. They found overwash tends to happen with short wavelengths or large
wave amplitudes and has a suppression effect on the disk motions.
Considering the prohibitive cost of experimental testing, theoretical models have the
potential to provide more efficient and economical solutions. Grotmaack and Mey-
lan [50] applied a slope-sliding model to predict the wave-induced surge motion of
an ice floe. The model assumes the floe does not affect the incoming wave field, so it
is not applicable in a short-wave condition where scattering is expected to happen.
Meylan and Squire [51] and Montiel [52] developed a linear potential-flow/thin-
plate model that includes the wave field surrounding the floe due to scattering. It
can predict the wave-induced surge, heave and pitch motions of an ice floe; after
validation against the experiments of Yiew et al. [19], this model was shown to be
accurate for the case without overwash but lack accuracy when notable overwash
occurs.
Theoretical models are built upon certain ideal assumptions, by which exact analyt-
ical solutions may be obtained but are limited by corresponding applicabilities. For
example, they usually assume the nonlinearity, viscosity and turbulence of the fluid
are negligible and thus the wave amplitude should be sufficiently small. These as-
sumptions exclude some influential phenomena in wave-ice interactions, e.g. over-
wash. Although Skene et al. [22] recently incorporated the nonlinear shallow water
equations with the linear potential-flow/thin-plate model to predict the depth of the
overwash water, their method was only one-way coupling, i.e. no back coupling
from overwash to predict the surrounding fluid domain or the ice floe motions. Due
to the very small freeboard of sea ice, overwash is a highly frequent phenomenon
9
thus of great importance, while a gap remains on accurately modelling it and study-
ing its role in wave-ice interactions.
A remedy to this can be using the Computational Fluid Dynamics (CFD) technique
to numerically solve the nonlinear Navier-Stokes equations, by which it is possi-
ble to obtain a fully-coupled solution between the fluid domain and floating-body
motions. The wave-ice interaction including overwash can be considered as a com-
bination of two typical CFD applications: (a) wave response of a floating body (b)
green water load. For both of topics, related CFD methods have been applied suc-
cessfully [53], so technically it is possible to build a CFD model to simulate the
interaction of waves with a floating ice floe including overwash. Another reason to
apply the CFD method is that it would be easy to import complex geometry files
to further study the coupling wave-ice effect on a structure, important for Arctic
engineering.
CFD has barely been applied to the ice-floe problem. Only one publication has been
found, in which Bai et al. [54] used an opensource CFD code, OpenFOAM [55], to
predict the wave-induced movement of a solitary ice floe. Their computational re-
sults show good agreement with experimental results when overwash occurs, which
suggests CFD could be a suitable method. Addtional relevant work needs to be
conducted with CFD; for example, Bai et al. [54] modelled the ice floe in square
shape rather than the more common pancake shape, and the role of overwash and
scattering remains to be analysed. Furthermore, the CFD model of waves with a
solitary floe will need to be expanded to incorporate multiple ice floes.
1.3.2.2 Ocean waves with a large ice sheet
Different from the ice-floe scenario discussed above, ice sheets in the Arctic can
be kilometres long, having a very small thickness-to-length ratio. In this situation,
the wave response of an ice sheet is dominated by the elastic deformations rather
than rigid body motions, known as the hydroelasticity of sea ice. A review of this
phenomenon has been given by Squire [56], where the author notes its modelling is
a key challenge of polar science and engineering. Noting that the Arctic ice sheets
10
are getting increasingly thin [39], the effects of sea ice hydroelasticity are becoming
more significant.
Studies on sea ice hydroelasticity have mainly modelled an ice sheet as a thin elas-
tic plate subjected to regular ocean waves. Experimentally, Meylan et al. [57]
conducted wave basin tests to measure the wave response of a plastic plate. They
found that the plate deforms due to wave propagation. Performing similar experi-
ments, Sree et al. [58] observed that a high aspect-ratio plates tended to follow the
wave-shape, and they reported wave attenuation due to the presence of the plate.
Sree et al. [20] also found that the wavelength and wave celerity inside the plate are
larger than those of an open water situation. Dolatshah et al. [59] conducted real
ice tests to study the wave-induced ice vibration and breakup. When encountering
an ice sheet, waves partially pass through and are partially reflected, for which Nelli
et al. [21] measured the proportion of the transmitted & reflected waves, expressed
as transmission & reflection coefficients. With increasing wavelength, waves were
found to be transmitted more and reflected less. Their experiments also indicated
the reflection coefficient is insensitive to wave amplitude, while the transmission co-
efficient can be reduced by the energy dissipative “overwash” phenomenon, which
is strongly dictated by the incoming wave amplitude. The depths of overwash water
at different wave conditions were reported by Skene et al. [22].
Theoretical work started by obtaining the transmission and reflection coefficients
of surface waves propagating against a semi-infinite ice sheet. These models were
based on linearised theories that can also be applied to Very Large Floating Struc-
tures (VLFS) [60]. In this approach, potential flow theory is employed in the fluid
domain and the ice sheet is treated as a linear elastic thin plate. Fox and Squire
[61] considered the problem of wave transmission & reflection from open water
into an ice sheet. The Eigenfunction Expansion Method (EEM) was adopted for
the velocity potentials underneath the open water surface and ice sheet, and an it-
erative conjugate gradient method was used to impose continuity between these
two parts. The transmission and reflection coefficients of waves were obtained,
alongside their relationship with the incident wavelength, ice thickness and water
11
depth. Other methods were also applied to the same case, e.g. Chung and Fox
[62] employed the Wiener-Hopf method; Hermans [63] used the Green’s function
method. Although the above studies ignored the submergence of the ice sheet, the
ice draught was afterwards included by Bennetts et al. [64], William and Squire [65]
and William and Porter [66]. Apart from a semi-infinite ice sheet, relevant linear
models were also applied to the case of a finite ice sheet. Meylan and Squire in-
vestigated the hydroelasticity of a solitary ice sheet [67, 51] and a pair of ice sheets
[68]. Wang and Meylan [69] used the Green’s function method to solve the fluid
domain surrounding an ice sheet and calculated the wave-induced ice deformation
by the finite-element method. In addition, Smith and Meylan [70] investigated the
influence of ice thickness on wave transmission.
Works based on theoretical models have provided great insights into sea ice hy-
droelasticity; however, similar to the small ice-floe scenario, theoretical methods
could not include overwash in the large ice-sheet case. Although Meylan et al. [57]
provided a linear theoretical model that proves valid to predict wave-induced ice
flexure even for high-amplitude wave conditions, Toffoli et al. [71] and Nelli et al.
[21] demonstrated the theoretical approach cannot accurately predict the transmis-
sion and reflection coefficients when significant overwash occurs, which is because
of the exclusion of energy dissipation associated with overwash. To obtain more
realistic solutions, CFD may be considered as an alternative, but in this case the ice
can no longer be assumed as rigid. When the size of an ice sheet is sufficiently large,
a structural solution is required to model the ice deformation in waves, and then the
influences of ice deformation on the surrounding waves need to be accounted for. In
such a situation, a Fluid-Structure Interaction (FSI) approach is required to obtain
both structural and fluid solutions and couple them together.
Tukovic et al. [72, 73] developed an FSI code based on OpenFOAM (fsiFoam
solver). It employed a partitioned FSI scheme to include the two-way coupling
between fluid and structure, where the fluid and solid solutions are solved separately
and coupled via the fluid-solid interface. An advantage of this approach is that it
employs the Finite-Volume Method for both fluid and solid domains [74]. Most
12
current FSI works involve a combination of solvers, usually with a finite-volume
(FV) solver for the fluid flow and a finite-element (FE) solver for the structural
analysis, which requires a third code for coupling, data interpolation and simulation
management. Thus, the combined FV+FE approach for the fluid and solid domains
will tend to increase computational costs and impose limitations on the coupling
method. In contrast, the entirely FV approach of Tukovic et al. [73] makes an
all-in-one solver under the framework of OpenFOAM. Furthermore, a benefit of
its open-source nature is the flexibility to add extended models, e.g. viscoelastic,
thermoelastic, and poroelastic solids [75, 76].
One gap of this FSI approach is that it is only applicable to single-phase fluid mod-
elling [77]. In other words, it currently cannot be applied to maritime applications
containing both air and water. Therefore, in order to simulate sea ice hydroelas-
ticity, the approach needs to be extended to be capable of modelling multi-phase
flows. For this, one potential solution is to expand the approach Tukovic et al.
[73] to include the Volume of Fluid (VOF) method [78], a method well validated in
modelling ocean free surface.
1.3.3 Ship operation in various ice conditions
This section in order reviews four common scenarios of ship operation in ice con-
ditions:
The Arctic region used to be covered by consolidated ice all year round and was only
accessible by icebreakers. The ice coverage mostly has a flat surface, thus referred
to as level ice, which is the most classical ice condition for polar ship operation.
This is the first scenario to be reviewed - icebreaker in level ice.
To enable commercial, non-icebreaking, ships to navigate through regions covered
by level ice, a standard solution is to use icebreakers to create channels. There are
two types of ice channels identified for this purpose - brash ice channel and open-
water ice channel, which are respectively the second and third scenarios.
Moreover, global warming has induced a widespread transformation of Arctic level
13
ice into broken ice floe fields, presenting shipping routes that are navigable for
commercial ships but infested by floating ice floes. This gives the fourth scenario -
a commercial ship operating in floating ice floes.
1.3.3.1 Icebreaker in level ice
When an icebreaker is advancing in level ice, the ship-ice contact causes ice crush-
ing and produces broken ice pieces that can slide along the hull and provide friction
[79], as shown in Figure 1-12. The icebreaking process induces the majority of
the total ship resistance in level ice, alongside a smaller proportion induced by the
water underneath the ice [80].
The ship resistance in level ice has been widely studied since it is essential for
icebreaker performance. Model tests [81, 82, 11] have reported that the level-ice
resistance is dictated by the ice thickness and hull form. Based on experimental
and sea-trial data, empirical equations [83, 84, 85] were derived to express the ship
resistance as a function of influential hull particulars, ship speed, ice thickness, ice
strength and friction.
Modelling methods has also been widely developed for this classical scenario. Hu
and Zhou [86] applied a numerical method to predict the level ice resistance, which
is more time-consuming than empirical equations but can be incorporated with a
realistic hull geometry. They compared the proposed numerical method and previ-
ous empirical equations to calculate the resistance of an icebreaker in level ice of
a range of thickness and speed. Following validation against model tests, the nu-
merical prediction performs more stably than empirical formulae, with deviations
of less than 10% for all the tested conditions, while each of the empirical equations
is less accurate in a certain thickness/speed range. Li et al. [87, 88, 89] developed
a finite-element-based model for icebreaking vessels and validated the model pre-
diction against full-scale measurements. More modelling work for this case can be
found in the review of Xue et al. [90]. Generally speaking, various methods have
reported good accuracy in predicting the ice resistance in this scenario.
14
Figure 1-12: Experimental view of an
icebreaker advancing in level ice [11]
Figure 1-13: Commercial ships travelling in
a brash ice channel [12]
1.3.3.2 Ship in a brash ice channel
Brash ice refers to an accumulation of ice fragments. A brash ice channel, as shown
in Figure 1-13, is usually created by an icebreaker in level ice so that commercial
ships are able to operate as needed. Kitazawa and Ettema [91] compared model
tests of ships operating in brash-ice channels with in open water, which shows that
that the ice-added resistance can be several times greater than the water resistance.
They also concluded that the brash-ice resistance is almost linearly proportional
to the thickness of the brash-ice layer, and the resistance is little affected by the
channel width when the channel is wider than twice of the ship beam.
The Finnish Swedish Ice Class Rules (FSICR) [92] provide empirical formulae to
predict the resistance of commercial vessels in a brash ice channel, which has been
widely used. However, recent model-test work [93] reported that the FSICR formu-
lae overestimate the brash ice resistance, showing there is still space for FSICR to
improve the accuracy; for example, the size and shape of ice pieces have not been
taken into consideration [94].
On the other hand, numerical models have been developed and appeared to be more
comprehensive, typically using the Discrete Element Method (DEM) to model the
ice fragments [95, 96]. The DEM simulation of Knono et al. [97] shown good
agreement with experiments in predicting the ship resistance in a brash ice channel;
Luo et al. [98] also reported good accuracy of DEM in predicting the brish-ice
15
resistance. More examples can be found in the review of Tuhkuri and Polojarvi
[96], presenting promising capabilities of DEM in modelling the interactions of a
ship/structure with small ice pieces.
1.3.3.3 Ship in an open-water ice channel
Since the resistance increment induced by brash ice is a considerable burden for
commercial ships operating in ice channels, new technology has been developed
for modern icebreakers to clean the broken ice fragments produced during the ice-
breaking process. One such method is turning the azimuth-propulsion units inwards
by 15 30 degrees, called ”toe-in” mode, with which the flushing effect can push
the broken ice pieces under the ice sheets, thus cleaning the channel and making it
slightly wider. This technique has shown effective for negating the brash ice resis-
tance component, as well as reducing the chance of blockages in the new channel
due to freezing of the broken ice pieces [99]. This approach therefore results in an
open-water channel between two large ice sheets, as shown in Figure 1-14, an al-
ternative for brash-ice channels that will increase in likelihood. Since the shipping
season will remain variable and unreliable in the first half of this century, commer-
cial ships will continue to require ice-breaker escort assistance [8]. Therefore, the
importance of studying this scenario is rising.
To date, limited research into ship performance in open-water ice channels has been
conducted, with few experimental investigations found in literature. Leiviska et
al. [100] conducted model tests of an oil tanker with a range of open-water ice
channels of different ice thicknesses and channel widths. In their results, an open-
water ice channel was seen to induce a markedly higher resistance than that of a
pure open-water situation without any ice, and the resistance increment was found
to be influenced by channel width and ship speed. In particular, the resistance in-
crements are evident when the channel width is less than three times of the ship
beam. Heinonen [101] reported on model tests for an icebreaker in open-water ice
channels and also observed that the close proximity of the ice edge to the ship can
increase the resistance. However, both Leiviska et al. [100] and Heinonen [101] did
16
Figure 1-14: Open water channel created by
an icebreaker (Credit: Aker Arctic)
Figure 1-15: Experimental view of a ship
advancing in floating paraffin slices [13]
not give collective conclusions and claimed that more studies are required to clarify
the underlying mechanism.
There is no modelling work found on the case of open-water ice channel, but a
similar and well-studied scenario is the operation of a ship in a canal. Existing
studies found that the ship resistance in a canal relates to the wave reflection and
flow change induced by the canal walls [102]. However, the ship hydrodynamics
discovered in a canal scenario may not be directly applied to open-water ice chan-
nels, as there are certain important differences between these two cases. Figure 1-16
provides a schematic comparison of a ship operating in open water, a canal and an
open-water ice channel. The proposed problem is different from a canal as the ice
has a limited thickness and is floating on the water surface. In addition, the water
depth in a canal is usually restricted, thus limiting underkeel clearance, while an ice
channel tends to be in deep water. Nevertheless, in term of method, as CFD has been
validated in the canal case [103], it could be practical to employ similar numerical
theories while applying the ice-channel geometry to confine the ship flow.
1.3.3.4 Ship in floating ice floes
Satellite and field observations in recent years have reported very different Arctic
ice conditions from the traditional level ice. During the navigable summer sea-
son, numerous floating ice floes are observed along the emerging Arctic shipping
routes, usually in pancake shape. Figure 1-17 demonstrates the dominance of the
17
(a) Open ocean (b) Canal (c) Ice channel
Figure 1-16: Comparison of ship operating in three different scenarios: open ocean, canal
and open-water ice channel; Wis the width of the channel and his the ice thickness
ice-floe condition (green and blue) along the measured routes, reported by field
observations in the autumn of 2015, while level ice (red) only occupies a small por-
tion. The emerging ice-floe conditions are navigable for commercial ships without
icebreaking capabilities, despite requiring icebreaker assistance when encountering
level ice. Thomson et al. [14] predicted that the proportion of pancake ice will keep
increasing with the climate change effect and become the dominant ice condition of
future Arctic shipping.
Figure 1-17: Ice conditions along shipping routes of Western Arctic, measured in Autumn
2015 [14]
As an emerging scenario, the study of ship performance in ice floes has only started
in recent years. Guo et al. [23] measured the resistance of an advancing ship in a
towing tank with numerous pieces of floating paraffin wax, mimicking ice floes, as
18
shown in Figure 1-15. They reported that such floating floes can induce significant
resistance increments on the ship, indicating the importance of accurately predicting
the ice resistance. Subsequently, Luo et al. [13] conducted similar experiments but
in head sea conditions to assess the seakeeping performance of a ship surrounded by
floes. They reported the presence of floes can increase the heave and pitch response
of a ship.
Considering the complexity of such experiments and the shortage of field-
measurement data, developing a reliable computational model can be a cost-
effective way to provide insights into the ship performance in floating ice floes.
Successful modelling of broken ice-floe fields has been achieved using the Discrete
Element Method (DEM), since this method allows the calculation of the contact
force of ice-ice and ice-structure, which is essential for modelling ship operation in
ice floes.
Currently, one gap in related DEM simulations is how to accurately account for the
force of the surrounding fluid on ice, which is usually implemented by empirical
equations [96]. Due to this deficiency, previous simulations of a ship advancing in
ice floes ignored the effect of fluid flow [104, 105, 106, 26], which can make the
modelling insufficiently realistic. The process of a ship advancing in floating ice
floes can be summarised as the following ship-wave-ice interaction: ship advance-
ment generates waves; waves interact with ice floes; ice floes make contacts with
each other and with the ship. The ship-generated waves can play a key role within
the process; for example, it can change the velocity (magnitude and direction) of
floes, especially when the floes are small. Therefore, ignoring the wave effect may
considerably influence the ice load on a ship.
One solution could be using CFD to provide fluid forces for DEM ice floes, since
CFD has proved to be a mature method to obtain the wave generated by an advanc-
ing ship [107], as well as accurate in predicting the motions of ice floe in waves
[54]. Other potential approaches could be combining the Lattice Boltzmann method
(LBM) [108] or the Smooth Particle Hydrodynamics (SPH) [109] with structural so-
19
lutions to model the ship-wave-ice interaction, but the applications and validation
of SPH and LBM in ocean engineering are still rare when compared with CFD.
One disadvantage of CFD comparing with empirical/theoretical predictions is that
it needs much higher computational power. However, as DEM requests very high
computational power itself, compared with using DEM solitarily, using CFD to
provide fluid solutions for DEM will not significantly increase the required compu-
tational cost. Based on the above reasons, it is recommended to combine CFD with
DEM to achieve the ship-wave-ice coupling.
Another challenge in the modelling of ice-floe fields is how to import natural ice
distributions into computational models. In polar regions, ice floes are randomly
distributed and of a range of sizes [110]. Even though computational models are
capable of simulating the structure-wave-ice interaction, the initial size and location
of each floe need to be prescribed. For example, in the computational models of
Janssen et al. [108] amd Sun and Shen [15], ice floes are set to have a uniform
size and a uniform initial distance between each other, as shown in Figure 1-18;
this is not a natural condition and the result can be subjected to the initial setup,
e.g. the relative positions between ship and floes can considerably influence the ice
load. Therefore, there is the need to import natural ice distributions to realistically
simulate the physical processes associated with Arctic ice-floe fields.
Figure 1-18: Regular pancake-ice field used by Sun and Shen [15]
20
1.4 Research gaps
Following the literature review, there are clear gaps in the modelling of potential
wave-ice interactions and shipping scenarios in the Arctic.
For wave-ice interactions, analytical methods have been used as the standard in
previous studies, but their ideal assumptions have been found to induce inaccuracies
due to the exclusion of some nonlinear features, especially the overwash behaviour.
The CFD method, with the potential to remedy this issue, has however barely been
applied in this field. In addition, a new hydroelastic method is required to model
the interaction of waves with large ice sheets, as the wave-induced ice deformation
needs to be accounted for.
For ship operations in various ice conditions, numerous studies have been conducted
for traditional level-ice and brash-ice-channel scenarios, with corresponding mod-
elling methods well established. Yet there has been a lack of work investigating the
scenarios of ship operations in an open-water ice channel and in floating ice floes.
Both these scenarios appeared in recent years and so far there is no valid modelling
approach for either of them.
1.5 Research vision
Based on the identified research gaps, this work aims to develop valid computa-
tional models to simulate four specific cases and perform systematic simulations to
investigate them: (I) the interaction of ocean waves with a rigid ice floe; (II) the
interaction of ocean waves with an elastic ice sheet; (III) the operation of a cargo
ship in an open-water ice channel; (IV) the operation of a cargo ship in floating ice
floes.
Cases (I) and (II) aim to provide a set of modelling methods for polar wave-ice
interactions, which can cover the scenarios where sea ice needs to be modelled as
either rigid or elastic, depending on its characteristics. Case (III) and (IV) are pri-
mary ice conditions that a commercial ship can encounter in an ice-infested Arctic
21
sea route; specifically, an open-water vessel is able to operate by itself in ice floe
fields, while consolidated ice can exist in certain segments of its route, in which
case icebreaker assistance would be required and the ship would be operating in an
ice channel. Therefore, Case (III) and (IV) can combine into a modelling strategy
for the design and power estimates of a typical ice-going commercial vessel.
1.6 Research approach
To meet the vision of this work, valid modelling approaches are required to be es-
tablished for the four identified cases. Based on the literature review, it is proposed
to use CFD as the foundational technique for all of the four cases, since CFD has
revealed a superior capability in the modelling of ship and wave hydrodynamics,
which is essential in the identified cases. Additionally, since the research cases are
interactive problems between ship, wave, and sea ice, as illustrated in Figure 1-19,
CFD is required to be coupled with various solid solutions of sea ice, specifically,
in Case (I) with rigid ice motions, in Case (II) with elastic ice deformations, in Case
(III) with the restriction of ice sheets, and in Case (IV) with the collisions of ice
floes with each other and with a ship. It is found that rigid ice motions may be
modelled by the six-degree-of-freedom equations; elastic ice deformations may be
modelled by computational solid mechanics; the restriction of ice sheets may be
modelled using the wall boundary condition validated in a similar canal case; and
ice-floe collisions may be modelled by the discrete element method. In particular, it
is crucial to apply/develop appropriate coupling algorithms between CFD and these
solid solutions.
The work will start with the cases of wave-ice interactions and then move on to
the ship-wave-ice interactions, since the investigations for Cases (I) and (II) are
expected to provide constructive insights into modelling the ice sheets and floes in
Cases (III) and (IV).
22
Figure 1-19: Illustration of ship-wave-ice interactions as broken down into subprocesses
1.7 Research questions
Regarding wave-ice interactions:
What is the valid simulation approach to model the interaction of ocean waves
with a small ice floe?
What is the valid simulation approach to model the interaction of ocean waves
with a large ice sheet? This approach should account for the wave-induced ice
deformation.
What are the interactive behaviours between ocean regular waves and a floating
ice floe/sheet? This is a two fold problem, including the response of the ice in
waves, as well as the influence of the ice presence on the wave propagation. Also,
how do the interactions change with the relative wave-ice dimensions? What is
the influence of overwash on the wave-ice interactions?
Regarding ship-wave-ice interactions:
What is the valid simulation approach to model the operation of a ship in an open-
water ice channel?
23
When a ship is operating in an open-water ice channel, how does the resistance
change compared with a pure open-water condition and what is the underlying
reason? How is the ice-induced resistance change affected by relevant environ-
mental/operational variables, e.g. ship speed, ice thickness and channel width?
What is the valid simulation approach to model the operation of a ship in a water-
way with floating ice floes?
When a ship is operating in floating ice floes, how does the resistance change com-
pared with a pure open-water condition and what is the underlying reason? How is
the ice-induced resistance change affected by relevant environmental/operational
variables, e.g. ship speed, ice thickness, ice concentration and floe diameter?
1.8 Thesis outline
Following this introduction chapter, Chapter 2 will introduce the numerical theories
to be employed in the remaining chapters. Chapter 3, 4, 5 and 6 will in turn report
work on each of four cases, which includes practicalities on building the computa-
tional model, the model’s verification and validation, systematic simulations with
respect to influential variables, and analyses into the phenomena. Chapter 7 will
develop an approach to incorporate the results from relatively slow computational
simulations into applications that require rapid computing, e.g. a real-time Arctic
voyage planning tool. Chapter 8 will summarise the project and discuss its implica-
tions and limitations, and then provide suggestions for future work.
24
Chapter 2
Numerical methods
This chapter presents the numerical theories applied in this project and their prin-
cipal equations. In the first part, CFD solutions for fluid flows is introduced, e.g.
the pressure and velocity fields in a computational domain. Particularly, the fluid
solutions include the methods for modelling free surface and ocean waves, since
these are involved in this project. The second part of this chapter presents the solid
solutions for modelling ice rigid motions, ice elastic deformations and ship-ice/ice-
ice collisions. These solid solutions are incorporable with CFD solutions to model
relevant interactions. The final part introduces the computational procedure to yield
the solutions.
2.1 Fluid solutions
2.1.1 Navier-Stokes equations
Based on the continuity of mass and momentum, the velocity and pressure of a
fluid domain can be solved using the Navier-Stokes equations for incompressible,
isothermal and Newtonian flow, as expressed in Equation 2.1 and 2.2.
·v=0 (2.1)
(ρv)
t+·(ρvv) = p+·τ+ρg(2.2)
25
where vis velocity vector, pis pressure, ρis the density, τ=µv+vTis the
viscous stress, µis the dynamic viscosity and gis gravitational acceleration.
2.1.2 Turbulence modelling
Ship operations and wave-structure interactions can induce turbulent fluid be-
haviours. The turbulent flow contains unsteady vortices, which leads to fluctuat-
ing changes of pressure/velocity and certain dissipation of kinetic energy. Thus it
is of great importance to account for the turbulent effects in computational mod-
elling so as to secure accuracies. One approach for modelling the turbulent effects
is to extend the Navier-Stokes equations into the Reynolds-Average Navier-Stokes
equations (RANS) where instantaneous turbulent velocity is decomposed into its
time-averaged and fluctuating components, as expressed in Equation 2.3 and 2.4.
·v=0 (2.3)
(ρv)
t+·(ρvv) = p+·(τρv0v0) + ρg(2.4)
where vis the time-average velocity, v0is the fluctuating component. Since the
fluctuating one has increased the number of unknowns, an extra turbulence model
is required to close the equations. For ocean engineering simulations that have a
similar scale as the proposed cases in this project, the most common turbulence
models are the kεfamily and the kωfamily, both have a number of branches
specialising in different problems [111].
The solution to close the equations is given below using the standard kεmodel
as an example, since other kεand kωvariants are similar. It can be seen in
Equation 2.4 that the RANS equations introduced an additional term ρv0v0. This
additional term can be expressed through the Boussinesq hypothesis:
ρv0v0=µt(v+vT)2
3ρkI(2.5)
26
where µtis the eddy viscosity. The way the kεmodel works is by expressing
µtas the turbulent kinetic energy (k) and the rate of dissipation of turbulent kinetic
energy (ε):
µt=Cµ
ρk2
ε(2.6)
Then, kand εcan be solved through their transport equations:
(ρk)
t+·(ρvk) = ·[(µ+µt
τk
)k] + P
k+P
bρε (2.7)
(ρε)
t+·(ρvε) = ·[(µ+µt
τε
)ε] + C1
ε
k(P
k+P
b)C2ρε2
k(2.8)
where P
kis the production of mean velocity shear, P
bis the production of buoyancy;
Cµ=0.09, τk=1.0, τε=1.3, C1=1.44 and C2=1.92 are empirical values [112].
2.1.3 Free surface modelling
Since ocean-related problems involve both water and air, there is a need to split
the fluid domain into two phases and capture the free surface in between. For this
purpose, the Volume of Fluid (VOF) method [78] is applied. The VOF method
introduces a passive scalar α, denoting the fractional volume of a cell occupied by
a specific phase. In this case, a value of α=1 corresponds to a cell full of water and
a value of α= 0 indicates a cell full of air. The local density and viscosity of each
cell is determined according to Equation 2.9 and 2.10, and a linear approximation
is applied to satisfy the continuity of the free surface, as expressed in Equation
2.11 [113]. Furthermore, the evolution of free surface with time is solved by the
advective equation of α, as expressed in Equation 2.12 [114].
ρ=αρwater + (1α)ρair (2.9)
µ=αµwater + (1α)µair (2.10)
27
v=αvwater + (1α)vair (2.11)
∂ α
t+·(vα) + ·[vrα(1α)] = 0 (2.12)
where vwater and vair are the velocities of the nearest water cell and air cell respec-
tively and vr=vwater vair is the relative velocity between them [115]. In this
study, ρwater =998.8kg/m3,µwater =8.90 ×104N·s/m2;ρair =1kg/m3,µair =
1.48 ×105N·s/m2and gis set as 9.81 m/s2.
The free surface between air and water is constructed through all cells with 0 <α<
1, as illustrated in Figure 2-1.
Figure 2-1: Illustration of free surface construction and the corresponding αvalues [16]
2.1.4 Wave modelling
The modelling of waves includes generation of desired waves at the inlet bound-
ary and a wave absorption in front of the outlet boundary to avoid waves being
reflected back and interfering the desirable wave field. For the Arctic wave-ice in-
teractions investigated in this project, the incident waves are commonly of a large
wavelength and a small wave height, which may be modelled as regular waves
[57, 21]. By prescribing the fluid solutions in the upstream, regular waves can be
generated and propagate towards the downstream, according to the linear Stokes
28
wave theory [116]:
η=d+H
2cos(kx ωt)(2.13)
vi=πH
δ
coshk(ζ+d)
sinhkd cos(kx ωt)(2.14)
vj=πH
δ
sinhk(ζ+d)
sinhkd sin(kx ωt)(2.15)
in which, ηis the free surface elevation, viand vjare respectively velocity com-
ponents that parallel and perpendicular to the propagation direction, His the wave
height (double of the wave amplitude a), δis the wave period, kis the wave num-
ber, dis the still water depth, ζis the vertical location relative to the still water level
and ωis the angular frequency. In the computational models of this work, Hand
δare given in advance, and the wavelength (λ=2π/k) is solved by the dispersion
relation: ktanhkh =κ,κ=ω2/g.
Inside the wave absorption zone, waves are dissipated by an artificial damping force
so that a still water surface can be achieved [117]. Specifically, in the wave absorp-
tion zone the momentum Equation 2.4 is modified into:
(ρv)
t+·(ρvv) = p·(τρv0v0) + ρgρχ(vvstr)(2.16)
The last term is the artificial damping force that dissipates the wave motion, where
χis the damping coefficient in units of s1, and it increases smoothly in the wave
propagation direction. vstr is the background stream velocity that is exempted from
damping, which equals zero when there is no current.
2.2 Solid solutions
2.2.1 Rigid-body motion
With the fluid solutions, the fluid force (Fh) on a solid body can be calculated by
integrating the solutions on the solid surface, as expressed in Equation 2.17. If the
solid body is not fixed, the fluid force can induce its movement, known as rigid-body
motions. The motions can be decomposed into six-degrees-of-freedom (6-DOF), as
29
a combination of translation and rotation. This can be solved using the translational
and rotational equations based on the mass centre of the structure (G), as expressed
in Equation 2.18 and 2.19.
Fh=Z(Pn+τ·n)dS (2.17)
F=md
VG
dt(2.18)
= [J]·d
ωG
dt+
ωG×([J]·
ωG)(2.19)
where Fand are the total force and torque on the structure, induced by its gravity
and the fluid force; mand [J]are the mass and inertia moment tensor, and
VGand
ωGare respectively the translational and rotational velocity vectors of the floating
structure.
2.2.2 Elastic deformation
When the elastic response of a solid body is evident, the deformation on the solid
shape needs to be taken into consideration. To model a deformable solid body, the
solid body can also be descretised into a set of cells, and the solid deformation
can be described by integrating the displacements of all the cells, known as Com-
putational Solid Mechancis (CSM). The displacement may be solved according to
conservation of momentum. In order to account for nonlinear ice deformations, this
work applies the nonlinear St. Venant Kirchhoff hyperelastic law to solve the stress
inside a solid body, as implemented by Tukovic et al. [118] and Cardiff et al. [76].
The mathematical model in total Lagrangian form (reference configuration) may be
written as:
I
V0
ρsolid
t(u
t)dV =I
S0
n·(Σ·ΓT)dS +I
V0
ρsolid g dV (2.20)
30
where uis the displacement vector, Γ=I+ (u)Tis the deformation gradient ten-
sor, and Iis the second-order identity tensor and Σis the second Piola-Kirchhoff
stress tensor, which is related to the Cauchy stress tensor σthrough Equation 2.21.
The Cauchy stress may be obtained based on the fluid force (Fh=n·σ), and the
stress-strain relationship is dictated by Equation 2.22.
σ=1
detΓΓ·ε·ΓT(2.21)
ε=2Gϒ+ΛTr(ϒ)I(2.22)
where the Green-Lagrange strain tensor ϒis defined in Equation 2.23, and Gand Λ
are the Lam´
es coefficients, related to the material properties of Young’s modulus E
and Poisson’s ratio ν, following Equation 2.24 and 2.25.
Ψ=1
2[u+ (u)T+u·(u)T](2.23)
G=E
2(1+ν)(2.24)
Λ=νE
(1+ν)(12ν)(2.25)
2.2.3 Collisions
The Discrete Element Method (DEM) is suitable for modelling floating ice floes that
tend to have collisional behaviour. In DEM, ice floes are modelled as elements in the
Lagrangian framework and they are allowed to move in the Eulerian CFD domain.
The movement of each element is governed by the rigid-body-motion equations
(Equation 2.18 and 2.19). However, the total force on an element here also includes
a solid-solid contact force, Fc, i.e. F=mg +Fh+Fc.
31
The contact force Fcis calculated by a penalty method [95], in which two col-
liding solid bodies are allowed to have a small overlap, according to the motion
solutions over a timestep. Where a contact occurs, the overlap is modelled as a
linear spring-dashpot system where the spring (K) accounts for the elastic response
and the dashpot (b) reflects the energy dissipation during the contact, by which the
normal and tangential components of Fcare calculated according to Equation 2.26
and 2.27 respectively. Subsequently, the contact force pushes the overlapped bodies
apart so that the overlap is minimised in the solution of the current timestep.
Fn=Kdnbvn0(2.26)
Ft=
Kdtbvt0,if |dt|<|dn|Cf
|Kdn|Cf·n,if |dt|≥| dn|Cf
(2.27)
where dnand dtare overlap distances in the normal and tangential directions respec-
tively, vn,rand vt,rare the normal and tangential components of the relative velocity
between the two contact bodies, Cfis the friction coefficient. In this study, Cfis set
at 0.35 for ice-ice contact and 0.05 for ship-ice contact; Kis set at 6 ×104N/mand
b=2Cdamp pKMeq, in which Cdam p is set at 0.067 and Meq is the equivalent mass
of two contact bodies, calculated as Meq =MAMB/(MA+MB). These parameter
values are selected based on corresponding tests and guidelines [119, 120].
2.3 Computational procedure
2.3.1 Time discretisation
To obtain the computational solutions, within a certain computational domain and
over a certain time duration, the procedure in this work involves two types of dis-
cretisation, in space and time respectively. In space, the computational domain is
divided into a set of non-overlapping cells, known as a mesh; in time, the temporal
dimension is split into a finite number of timesteps; the interval between two adja-
32
cent timesteps is known as the timestep size (t). For a single timestep, the solution
of all the cells can be obtained integrally by solving assigned governing equations.
Then, the solution over a certain time duration is the connection of the solution at
each timestep.
The temporal discretisation is easier than spatial; in this study, it is realised through
the first-order temporal discretisation scheme that approximates a designated tran-
sient parameter (denoted as u) using the solution at the current timestep and the one
from the previous timestep:
d(ρuv)
dt=(ρuv)t=n+1(ρuv)t=n
t(2.28)
The second-order temporal discretisation scheme, which uses the solution at the
current timestep and two previous timesteps, is also tested for this study. Nonethe-
less, the results do not show a notable difference from those yielded through the
first-order temporal discretisation.
2.3.2 Spatial discretisation
The spatial discretisation of this work is realised through the Finite Volume Method
(FVM) [121]. To solve the governing equations over numerous cells, the procedure
can be explained through the example of two neighbouring cells, marked as the
owner cell and the neighbour cell as in Figure 2-2. The required solution includes
desired variables at the centroid of each cell, as well as the face flux between the
cells.
Figure 2-2: Illustration of two neighbouring cells in FVM [17]
33
Considering the owner cell P, the Navier Stokes equations in the spatial domain can
be integrated into the following form:
I
V
[·(vv) + 1
ρP·τg]dV =0 (2.29)
Then, each term can be integrated separately:
I
V
[·(vv)]dV =I
V
[1
ρP]dV +I
V
[·τ]dV +I
V
[g]dV (2.30)
The integration of a vector term is solved through the Gauss’ Divergence Theorem,
as given in Equation 2.31 and Figure 2-3, in which Fdenotes an arbitrary vector.
I
V
[·F]dV =I
S
[F·ˆ
n]dS (2.31)
Figure 2-3: Illustration of the Gauss’ Divergence Theorem [17]
The above process calculates the solutions at each cell centroid, but to obtain the
face flux between cells, the solutions obtained at cell centroids need to be interpo-
lated to the cell surfaces. In this study, the second-order upwind differentiation is
used to interpolate the results.
The first and second order upwind schemes are both introduced below for compar-
ison. The first-order upwind scheme determines the transported quantity of a face
depending on flow direction and only one neighbouring cell on each side, as shown
34
in Figure 2-4. The second-order upwind scheme determines the transported quan-
tity of a face depending on flow direction and two neighbouring cells on each side,
as shown in Figure 2-5.
Figure 2-4: Illustration of the first-order upwind scheme [18]: if v>0, ui1/2=ui1,
ui+1/2=ui; if v<0, ui1/2=ui,ui+1/2=ui+1
Figure 2-5: Illustration of the second-order upwind scheme [18]: if v>0,
ui1/2= (3ui1ui2)/2, ui+1/2= (3uiui1)/2; if v<0, ui1/2= (3uiui+1)/2,
ui+1/2= (3ui+1ui+2)/2
Both first and second order upwind schemes are tested for this study, showing that
the first-order one causes significant inaccuracies. Therefore, it is necessary to use
the second-order spatial interpolation.
35
2.3.3 Pressure-velocity coupling
Directly solving the Navier Stokes equations is currently one of the most challeng-
ing mathematical problems. However, the pressure and velocity solutions can be
obtained through numerical algorithms. In this study, the PISO algorithm is applied
[122]. The algorithm can be summed up as follows:
According to the initial boundary conditions (for the first timestep) or the solution
from the previous timestep (for an intermediate or final timestep), the discretised
momentum equation is solved to compute a velocity field, noting that this velocity
field does not satisfy the continuity equation.
This velocity field is used to calculate a pressure field, but this pressure requires
correction as it is from a velocity field that does not satisfy the continuity equation.
Then this pressure field is substituted into the continuous equation to get a new
velocity field, which is now satisfying the continuity equation.
The updated velocity field that satisfies the continuity equation is used to correct
the pressure field.
The corrected pressure field is substituted into the continuous equation to get the
final velocity field of the current timestep.
2.3.4 Choice of software
In this project, there were a total of three software packages used for the different
cases, i.e. Flow-3D, OpenFOAM, STAR-CCM+. It should be noted that, all three
packages have identical CFD solutions, but the choice was made based upon which
software provided the best possibility to couple the additional ice solid solution
required for each case. Specifically, OpenFOAM gives the required flexibility to
fully achieve CFD+CSM coupling, and STAR-CCM+ has a robust DEM package
that can be coupled with CFD.
36
Chapter 3
Wave interaction with a rigid ice floe
A floating ice floe can move with incoming ocean waves and meanwhile affect the
wave pattern. In this chapter, a computational model is built to simulate the inter-
action of regular waves with a circular ice floe, as illustrated in Figure 3-1. A series
of simulations are presented to categorise the motions of the floe in different wave-
length and wave amplitude conditions, and the simulations are validated against
experiments. Furthermore, overwash and scattering behaviours are demonstrated,
and their roles in the wave-ice interaction are analysed in detail.
Figure 3-1: Schematic of the case: a circular ice floe is freely floating on the water surface
and subjected to incoming regular waves generated by a numerical wavemaker, where the
surge, heave and pitch motions are its main hydrodynamic responses
A version of this chapter has been published as: Huang, L., and Thomas, G., 2019. Simulation
of wave interaction with a circular ice floe. Journal of Offshore Mechanics and Arctic Engineering,
141(4), 041302.
37
3.1 Computational modelling
3.1.1 Boundary conditions
A three-dimensional cuboid computational domain was established, defined by the
earth-fixed Cartesian coordinate system Oxyz, as shown in Figure 3-2. The (x,y)
plane parallels the undistributed water surface, and the z-axis is positive upwards.
The computational domain is 2 m wide and 1 m high, and its length is five times
of the target wavelength (λ). A no-slip wall condition was applied to the bottom
boundary to model the presence of the seabed, and a static pressure condition was
defined to the top boundary to account for the atmosphere. The domain was filled
with water to a depth of d=0.83 m, and the water surface was initialised as still.
This water depth in is set as per the experiments of Yiew et al. [19] for the purpose
of validation. The inlet boundary was set at the left, where regular waves are con-
tinuously generated and propagating towards the positive x-direction, and a wave
absorption zone was placed at the outlet to eliminate the reflection of waves from
the outlet boundary. The wave generation and absorption are both one-wavelength
long and mathematically realised by prescribing the free surface elevation and ve-
locity components, as introduced in Section 2.1.4.
Figure 3-2: Sketch of the computational domain and the applied boundary conditions
For the purpose of validation, the ice and wave parameters set in this study follow
an accordant manner with the experiments of Yiew et al. [19]. As shown in Figure
3-3, two rigid disks, with and without an edge barrier, were employed to model ice
floes, named as Disk B and Disk NB respectively. The edge barrier attached on
Disk B can prevent waves from flowing onto the upper surface of the disk, so that
38
the influences of overwash can be investigated by comparing the wave response of
the two disks. The density of the disks was set at 636 kg/m3.
Each disk was initialised as floating on the undistributed water surface, with its
mass centre locating at two wavelengths away from the inlet in the x-direction, the
middle section of the y-direction and its buoyancy-gravity equilibrium position in
the z-direction. The tested wave conditions cover a wavelength from λ= 0.69
4.91 m combined with a wave amplitude of a= 10 40 mm. For each single wave
condition, the two disks were put into the CFD model separately.
Figure 3-3: The geometry of the two disks. Left panel: the disk with an edge barrier (Disk
B); right panel: the disk without edge barrier (Disk NB)
With the generated waves continuously propagating from the inlet to the outlet, the
positioned disk is expected to leave its initial equilibrium position and move with
the incoming waves. No artificial restraint was applied to the movement, so the
disk was allowed to move freely. The wave-induced movement of a disk can be
considered as the combination of translation and rotation, which was solved with
the rigid-body motion equations in the body-fixed system based on the mass centre
of the disk Gx0y0z0, as expressed in Equation (2.18) and (2.19).
3.1.2 Discretisation
The computational domain was divided into a hexahedral mesh, as shown in Figure
3-4. Local mesh refinements were applied at the free-surface area and the area
where the disk is expected to move. The cells around the floating disk are partially
blocked by the disk volume, and the blockage is described by fractional cell volumes
and areas on cell sides [123]. The blockage updates after each timestep to satisfy a
39
two-way coupling between the fluid solutions and floe movement.
The size of each timestep was determined by a prescribed value, Courant number
(Co):
Co =vt
x(3.1)
where tis the timestep size, v/xis its normal velocity divided by the distance be-
tween the cell centre and the neighbour cell centre. For every timestep, there exists
a maximal v/xvalue in the domain, and tcan be calculated by the product of that
value and Co. This allows an optimal tbeing selected according to the transient
fluid state. The value of Co was prescribed at 1 in this study, which is a rule-
of-thumb value for similar problems. The computation was performed using the
Flow-3D solver [124], where the RANS equations coupling with VOF were solved
through FVM, alongside the renormalisation-group kεmodel [125] to account
for turbulent effects, as introduced in Section 2.1.2 and 2.1.3. The renormalisation-
group kεmodel has shown great performance in modelling curving flows that are
expected in this case [124].
(a) Plan view
(b) Profile view
Figure 3-4: Mesh layout of the model. High resolution was applied to the free surface area
and where the disk is expected to move
40
3.2 Results and discussion
With the waves continuously being generated and propagating, a regular wave field
gradually forms in the domain, and after that, the disk is expected to conduct peri-
odic motions with stable incoming waves. This indicates the simulation has entered
its steady state, and valid data were taken only after the periodic motions begin.
The wave-induced disk movement was decomposed into 6-DOF motions for anal-
ysis. As the incident waves are unidirectional, the wave-induced movement of a
disk is basically within the xzplane, composed by translational motions along
the x-axis and z-axis, alongside a rotational motion along the y0-axis, i.e. surge,
heave and pitch. The surge is a combination of a harmonic oscillation with a drift,
while the heave and pitch are just harmonic oscillations. These harmonic oscilla-
tions were induced by the elliptical wave motion, while the drift was forced by the
wave celerity.
To investigate the relationship between the disk motions and incident waves. The
surge, heave, pitch amplitudes were calculated and compared with the correspond-
ing wave motion amplitudes through the Response Amplitude Operators (RAOs).
After the drift was eliminated, the surge amplitude (as) was calculated as half the
difference between the average peak and trough values of its oscillation, and the
heave amplitude (aH) and pitch amplitude (aP) were calculated through the same
procedure. Subsequently, the motion amplitudes were used to obtain the RAOs,
following:
RAOsurge =aS
acothkd (3.2)
RAOheave =aH
a(3.3)
RAOpitch =aP
ka (3.4)
41
Figure 3-5: Generated waves with different cell numbers per wave height. The target
waves are of λ= 2.38 m and a= 40 mm
Figure 3-6: RAOs obtained with different total cell numbers. The applied wave condition
was λ= 2.38 m and a= 40 mm
3.2.1 Mesh sensitivity tests
As the computational cost increases with the cell number, mesh sensitivity tests
aim to get an accurate solution with as few cells as possible. In this work, two
tests were conducted. The first test was to verify the quality of wave generation
and propagation. For the large wavelength and small amplitude waves that are of
interest, at the free surface area the mesh density is sensitive in the vertical direction.
42
Therefore, the cell number per wave height (M) was varied to see the influence
on generated waves, from M = 5 15. For the mesh density in the wavelength
direction, 100 cells per wavelength were always used, which is the minimal cell
number matching with M = 5 15 to secure an error of less than 1% [126]. In
the wave tests, the floating disk was taken away and replaced by a probe to record
the free surface elevation. Figure 3-5 presents the recorded free surface elevation
at different M values, alongside the ideal value. Based on the test, M = 15 was
chosen to generate the mesh around the free surface, as the wave field obtained at
this density was very close to the target.
The second test was conducted to secure proper solving of the disk motions. The
mesh density was globally scaled, and four sets of mesh were produced, consisting
of 570k, 770k, 970k and 1.25 million cells respectively. The RAOs of the disk were
calculated with the four meshes respectively, and the results are shown in Figure
3-6. It shows the RAOs converge to stable values with the cell number increased,
and the convergent RAO values are close to the experimental data of Yiew et al.
[19]. The cell number of 970k was selected, as further increasing the cell number
did not effectively improve the results. This set of mesh corresponds to around 40
cells per disk diameter (D) and 6 cells per disk thickness.
3.2.2 Validation
The CFD model was validated against the experiments conducted at the University
of Tasmania [19]. For the simulations at different wavelength and amplitude con-
ditions, computational results include the surge, heave, pitch RAOs of both Disk
B and Disk NB. The comparison between the computational results (CFD) and ex-
perimental data (Exp.) is presented in Figure 3-7, as a function of non-dimensional
incident wavelength (λ/D). For error analysis, the deviation between the two re-
sults was calculated as:
Deviation =|CF D Exp.|
Exp.×100% (3.5)
43
Overall, the CFD model demonstrates a good agreement with the experiments in
predicting the disk motions. For surge, heave and pitch, the mean deviations are
2.4%, 2.5% and 2.8% respectively. The deviation does not vary significantly at
any specific wavelength or amplitude regime. This is because the mesh density
of each simulation was set according to the incident wave condition, rather than
a constant size. As the deviations are slight and there is no obvious over/under-
estimate trend identified, the deviations can be considered as the uncertainty of
numerical calculation [127] or experimental measurement, which means the applied
CFD approach is reasonable and accurate.
The average deviations of Disk B and Disk NB are similar, at 2.7% and 2.4% re-
spectively. Previous studies found it difficult to predict the motions of Disk NB. For
example, for the RAOs predicted by the potential-flow/thin-plate model, the average
deviations for the three motions are 1% 4% for Disk B but increase to 4% 7%
for Disk NB [19]. The reason for this difference is the existence of overwash with
Disk NB. The study of Bennetts and Williams [128] also indicates that the presence
of overwash causes considerable inaccuracy to their prediction models.
Overwash has been found to be a highly nonlinear process [22], so it is hard to be
included in a linear model. Even for a fully nonlinear CFD model, the method to
model the free surface should be carefully chosen. For example, the free surface
can also be considered as single-phase by enforcing the free surface boundary con-
ditions on it [129]. This method is not applicable in an overwash situation, where
the overwash water can be discontinuous on the floating body [128], so it can no
longer be treated as a continuous boundary. Therefore, a two-phase modelling via
the VOF method is recommended to handle the overwash problem. Using an ap-
propriate turbulent model is also important, since turbulent flow has been observed
when strong overwash happening [22].
44
(a) Surge RAO of Disk B (b) Surge RAO of Disk NB
(c) Heave RAO of Disk B (d) Heave RAO of Disk NB
(e) Pitch RAO of Disk B (f) Pitch RAO of Disk NB
Figure 3-7: Computational and experimental [19] RAOs, as a function of non-dimensional
wavelength
3.2.3 Overwash
Overwash was observed in the simulations of Disk NB. The extent of overwash
generally gets stronger with a larger wave amplitude or a smaller wavelength. From
45
weak to strong, two types of overwash were identified, named partial overwash and
full overwash. Both are presented in Figure 3-8. The process of partial overwash is
shown in (a) and (c): water only washes over the head and tail areas of the disk; full
overwash is shown in (b) and (d): water flows through the upper surface of the disk.
Since overwash was prevented by the edge barrier of Disk B, the influence of over-
wash was investigated by comparing the wave response of Disk B and Disk NB.
Figure 3-9 shows the RAOs of both the disks, together with the observed overwash
type. Data were selected from the largest wave amplitude group (a=40mm) and
in short-wave conditions, where overwash was most obvious. Overwash was incon-
spicuous in a long-wave condition (when λ/D>3). It can be seen that Disk NB
generally has smaller RAOs than Disk B. RAO difference between the two disks is
most significant for the heave and least for the surge. For surge and heave, the RAO
difference increases with a stronger overwash. This results from the load of the
overwash water on top of the disk, which suppresses the transitional movement of
the disk. Pitch seems to be unaffected by partial overwash, which means the water
at head and tail areas of the disk may provide a rotational torque, acting as an offset
to the overwash suppression. In contrast, full overwash can notably suppress pitch.
The influence of wave amplitude on overwash is presented in Figure 3-10, where
only heave RAO is presented since surge and pitch reveal similar trends. It shows
the RAO of Disk B is insensitive to wave amplitude. For Disk NB, its RAO is
insensitive to wave amplitude when partial overwash occurs. However, when full
overwash appears, the RAO decreases obviously as wave amplitude increases. This
is because a larger wave amplitude increases the water depth above the disk, which
can enhance the suppressing effect. The fact that the RAOs vary with wave ampli-
tude signifies the nonlinearity of overwash. In the linear theory, the RAOs should
be constant values for a single wavelength, but the present results indicate such a
theory is not applicable when notable overwash occurs.
46
(a) Partial overwash overview (b) Full overwash overview
(c) Partial overwash close-up
animation
(d) Full overwash close-up
animation
Figure 3-8: CFD illustration of partial overwash and full overwash
47
(a) Surge (b) Heave
(c) Pitch
Figure 3-9: RAO comparisons between Disk B (dashed line) and Disk NB (solid line), as
a function of non-dimensional wavelength, alongside the type of overwash
(a) λ/D=2.5(b) λ/D=1.725
Figure 3-10: Heave RAO of Disk B (dashed line) and Disk NB (solid line), as a function
of wave amplitude, alongside the type of overwash. Trend lines (blue) are also included
48
3.2.4 Scattering
The disks were observed to reveal varying wave-ice interactions with the incident
wavelength changed. When a disk is subjected to long waves, it hardly influences
the wave transmission. By contrast, in short waves, a disk can scatter the incoming
waves, which generates directional-spreading waves surrounding the disk and influ-
ences the transmitted waves. Examples are shown in Figure 3-11, where Disk B is
presented, in order to eliminate the distraction of overwash. In a long-wave condi-
tion, wave scattering is not obvious, and the waves are nearly intact after passing
through the disk, as shown in Figure 3-11 (a). At a shorter wavelength, wave scat-
tering becomes visible, while the waves can still transit with the incident amplitude,
except the part right behind the disk is distorted, as shown in Figure 3-11 (b). For
even shorter waves, a stronger scattering can be observed, and the transited waves
are significantly attenuated, as shown in Figure 3-11 (c).
The disk motions are related to the scattering phenomenon. As shown in Figure 3-7,
in the long-wave regime (λ/D>3), wave scattering is negligible and the RAOs are
close to one, which means the disk approximately moves at the same amplitude of
the incident waves, agreeing with the fact that the wave transmission is unaffected.
In the short-wave regime (λ/D<3), the RAOs start to decrease rapidly with the
scattering becomes stronger, since the scattering can attenuate the incident waves
so that the disk motions cannot be induced at the original wave amplitude.
Although both scattering and overwash have been found as sources to disturb the
incident wave and can reduce the disk RAOs, they reveal different relationships
with the incident wave amplitude. As shown in Figure 3-10, for Disk B, where only
scattering occurs, the RAO is insensitive with a changed incident wave amplitude,
which suggests a linear effect of scattering on the disk motions. However, for Disk
NB, overwash has shown obvious nonlinearity on the disk motions. This explains
why the linear potential-flow/thin-plate model can predict the RAOs accurately in a
scattering condition as long as there is no overwash.
49
(a) λ/D=3.525
(b) λ/D=2
(c) λ/D=1.525
Figure 3-11: The interaction of Disk B with waves, at different wavelengths
3.3 Conclusions
A CFD model has been presented to model the interaction of regular waves with an
ice floe. The ice floe was treated as a floating rigid disk to represent the pancake-ice
scenario, and it was subjected to waves of large wavelength and small amplitude,
according to the common environment in the MIZ. A series of simulations were con-
ducted to investigate the behaviour of the ice floe in different wave conditions, and
the wave-induced surge, heave and pitch motions of an ice floe were presented as
RAOs to analyse their relationship with incident waves. The computational results
were compared with the corresponding experimental data, showing the proposed
50
approach is capable of simulating the wave-ice interaction including overwash, to
which previous analytical models have been reported to contain inaccuracy.
Two specific behaviours with the wave-ice interaction were displayed within the
simulations, namely overwash and scattering. Both tend to happen in a short-wave
condition and were found to be significant sources to cause wave dissipation and
reduce the ice floe motions. The scattering and ice floe motions were found to
follow a linear relationship that is not affected by a changed wave amplitude, but
such a relationship becomes nonlinear when overwash occurs, where a larger wave
amplitude can increase the water depth on top of an ice floe, resulting in extra load
to suppress the motions.
In this chapter, a valid CFD approach has been developed that can provide high-
fidelity simulations for wave interactions with a floating ice floe, where the floe is
assumed as rigid based on its relatively small dimension. The next step is to develop
a computational approach that can account for the elastic deformation of sea ice in
waves, thus covering the modelling of relatively large ice floes/sheets - this will be
presented in the next chapter.
51