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-Norwegian University of

Science and Technology

Doctoral theses at NTNU, 2009:268

Xiangjun Kong

A Numericat Study of a Damaged

Ship in Beam Sea Waves

Deift University of Technology

Ship Hydromechanics laboratory

Library

Mekelweg 2

26282 CD Delft

Phone: +31 (0)15 2786873

Xiangjun Kong

A Numerical Study of a Damaged

Ship in Beam Sea Waves

Thesis for the degree of philosophiae doctor

Trondheim, December 2009

Norwegian University of

Science and Technology

Faculty of Engineering Science and Technology

Department of Marine Technology

ONTNU

Norwegian University of

Science and Technology

NTN U

Norwegian University of Science and Technology Thesis for the degree of phifosophiae doctor Eacufty of Engineering Science and Technology Department of Manne TechnoLogy

©Xiangjun Kong

ISBN 978-82-471 -1958-7 (printed ver.) ISBN 978-82-471 -1959-4 (electronic ver.)

ISSN 1503-8181

Doctoral Theses at NTNU, 2009:268 Printed by Tapir Uttrykk

Abstract

An important objective of survivability criteria for passenger vessels is to ensure that there is sufficient time called Time To Capsizing (TTC) to evacuate passengers and crew members in case of emergency. This requires a realistical modeling of the phases following an accident. The ship motions can interact with the ingress/egress flooding.

water-on/off-deck and/or sloshing of the floodwater in compartments or on vehicle decks. Each of these subproblems is challenging to handle. There is a large variety of phenomena

that are neglected in the models proposed so far to predict passenger ship stability. In the

sequence of events leading a passenger ship to capsize, numerous questions can be raised on the importance of nonlinear effects. I'he present thesis is based on the following three considerations. The first one is that the ship motion equations used in the simulations have six Degrees-of-Freedom (6D0F) in order to represent realistic ship capsizing dynamics. The second is to correctly model the flooding flow through the openings. The final aspect is the three dimensional simulation of the floodwater flow on the deck or in the compartment and the prediction of induced loads on the damaged ship, since the flooding flow can not generally be described adequately by a two-dimensional model.

Potential flow theory is used to study the damaged ship motions in waves with ingress/egress flooding through the damaged opening considered. 1'he whole problem involves the exterior and interior flow domains which must be properly described. Here a floodwater solver (FWS) is used for the interior problem. A shallow water solver (SWS) or a multimodal solver (MMS) is used according to different filling ratios. The FWS can also take into account the interior problem with ingree/egress flooding flow conditions. For the exterior flow, the 6DoF ship motion equation system is solved with the flooding water interaction accounted for. The communication between the exterior and interior is enforced by means of corrections of the exterior and interior boundary conditions, except when the Hull Reshaped Method (HRM) is used because it models the exterior and the interior

domains simultaneously so that the communication is automatically handled.

The flooding mechanism has been physically analyzed, categorized and mathematically modeled accordingly. Three different approaches, i.e., shallow water equations, multimodal method and hull-reshaped method, are formulated to model the different flooding

scenarios. The former two are of a nonlinear nature and the HRM is a linear approach.

Additionally, the adaptive nonlinear roll (viscous) damping is determined from the free decay tests. Sorne necessary verifications are performed. Validations against experiments show that the present formulation is a promising way to simulate the complex damaged

ship and flooding flow system. The important parameter, TTC, for a damaged ship can be

predicted.

Finally, suggestions are proposed on how to minimize the commercial risk and maximize

the safety at an initial design stage, as well as to provide a rational analysis or

Acknowledgements

My deepest gratitude goes lirst and foremost to my supervisor, Professor Odd Faltinsen, who led me into the world of Hydrodynamics, not only for his great contributions to this

thesis but for his patient guidance during these more than five years. It is his constant encouragement and unwavering support that sustained me through frustration and

depression esp. during the last stages of my PhD work. Without his pushing me ahead, the completion of this thesis would be impossible. His brilliant insight into physics, enthusiasm for research and conquering difficulties are worth 'learning in a lifelong time.

I own Prof. Marilena Greco a great debt of thanks for her generous help and friendship. Her encouragement and patient proofreading of this thesis finally led me into a right track. High tribute shall be paid to Prof. Torgeir Moan, who has harmonically organized and directed the Centre for Ships and Ocean Structures (CeSOS) where I have been doing my PhD study. Especially, I whole-heartedly appreciate his kind help and caring for my study.

I am so grateful to Marrianne KjøMs for her thoughtful assistance and help to students, and me in particular.

Thanks to all my fellow PhD students and the staff in CeSOS and the Department of

Marine Technology for constructing an enjoyable environment. Their friendship and

discussion meant a great deal to me. Particularly, I wish to thank Dr. Sun Hui, Dr.

Renato Skejic, Dr. Trygve Kristiansen, Dr. Zhen Gao, Yanlin Shao, David Kristiansen for their help during my thesis writing.

Many other friends (too long to list) have contributed in various ways to my study life and given me tremendous help. I thank all of you infinitely.

I cannot express in words how I should appreciate the love, encouragement, and support from my wife Wei Zhu and our parents in China. I will say thanks to my ten-month-old daughter Yangyang, who has been very healthy and enjoying her toys everyday.

Nomenclature

General Rules

e All symbols are at least defined where they appear for the first time in the thesis

Vectors are indicated by vector arrows over the boldface letters

Matrices are donated by the square-bracketed vector's notations, except the Jacobian matrices in math-bold font

Over-dots and over-stars signify differentiation with respect to time, but an over-star

of a vector means that the vector is regarded as a scalar in the time differentiation

The same symbol donates the same thing in the context, unless explicitly explained

Abbreviations

2D two dimensional

3D three dimensional

BEM Boundary Element Method BVP Boundary Value Problem FWS Floodwater Solver DoF Degree-of-Freedom

CFD Computational Fluid Dynamics

FVM Finite Volume Method WAF Weighted Average Flux

FDS Flux Difference Splitting

SWS Shallow Water Solver MMS MultiModal Solver TTC Time To Capsizing RCM HLL HLLC CFL COG RORO TRRS S RRS RSRS TSRS HRM SWE TTF

Random Choice Method Harten-Lax-van Lee

Harten-Lax-van Lee & Contact Wave Courant-Friedrichs-Lewy

Center OF Gravity

Roll-On/Roll-Off

Two-Rarefaction Riemann Solver Shock & R,arefaction Riemann Solver Rarefaction & Shock Riemann Solver Two Shock Riemann Solver

Hull Reshaped Method Shallow Water Equation Time-To-Flood

Subscripts and Superscripts

s ship associated variable

os origin of the ship-fixed system

fd flooding associated variables

w incident wave associated variable

Roman Letters

L load vector including forces and moments

Ñ normal vector with six components

gravitational acceleration vector

Û state variables

P

flux variable in xaxis or force vector

M moment vector

P

momentum vector

Ô flux variable in yaxis

Greek Letters

xdisplacement

OmXmymZm system or damping ratio

r ymdisplacem1t in OmXmYmZm system

displacement in OrnïrnYmZm system

» roll Euler angle from OmXmYmZn to oX8y5z

(1 pitch Euler angle from QmXmYmZm to O3XsYsZ8

yaw Euler angle from OmXmYmZm to OXSySZS

homogenous velocity potential angular velocity vector

S tokes-Joukowski potential vector

3 generalized coordinate

Table of Contents

Abstract

Acknowledgements

Nomenclature

Table of Contents

List of Tables

List of Figures

i

Introduction

i

1.1 Background and Motivation

1.2 Problem Description 4

1.3 Previous Studies 8

1.4 Present Study 12

1.4.1 Structure of the Thesis 13

1.4.2 Major Contributions 14

2

Ship and Floodwater Dynamics and the Equations of Motion

17

2.1 Coordinate Systems and Their Relationships 18

2.2 Nonlinear Equations of Motion in the Ship-fixed Coordinate System 21

3 Hydrodynarnic and Hydrostatic Loads Acting on a Damaged Ship

25

3.1 Loads due to the Exterior Flow 26

3.1.1 Nonlinear Froude-Krylov Loads 27

3.1.2 Nonlinear Restoring Loads 27

3.1.3 Linear Scattering and Radiation Loads 28 3.1.4 Nonlinear Viscous Damping Moment and Free Decay Test 28

3.2 Loads due to Interior Flow 31

3.2.1 Shallow Floodwater Loads 31

3.2.2 Non-shallow Floodwater Loads 32

CONTENTS

4

Ingress-, Egress-, Cross- and Down-Flooding in Damaged Compartments 37

4.1 Flooding Calculation Formulation 39

4.1.1 Flooding through a Small Opening 40

4.1.2 Flooding through a Large Opening 42

4.1.3 Other Effects 45

4.2 Time To Capsizing Calculation 47

5

Shallow Water Solver for Shallow Floodwater on a Damaged Ship

49

5.1 Shallow Water Flow and its Coupling with Ship Motions 52

5.1.1 Basic Shallow Water Assumptions 52

5.1.2 I'he Local Shallow Water Flow in the Body-fixed Coordinate System 53

.2 Conservative Numerical Scheme and Riemann Solver 56

5.2.1 Elementary Wave Solutions of the Riemann Problem 58

5.2.2 HLLC Riemann Solver 61

5.3 The Weighted Average Flux Method as the Shallow Water Solver 66

5.4 Practical Numerical Problems 69

6 Multimodal Solver for the Non-shallow Floodwater on a Damaged Ship

73

6.1 Multimodal Analysis of Nonlinear Ship-Floodwater Problem 75

6.2 Formulation of the Free Surface BVP in the Ship-fixed Reference Frame 77

6.3 Fourier's Expansion of the Modal Functions and Bateman-Luke Formulation 80

6.4 Forces and Moments 86

6.5 Advantages and Limitations 88

7 Numerical Studies: Verification, Validation and Physical Investigations

89

7.1 SWS Application and Verification 90

7.1.1 Two Dimensional Dam-breaking Simulation 90

7.1.2 Two Dimensional Sloshing Simulation 96

7.1.3 Three Dimensional Down-flooding Simulation 98

7.1.4 Space Discretizationi Convergence 98

7.2 Numerical Solution of Ship Motion Equations and its Verification 99

7.2.1 Analytical and Numerical Solution of the Roll Motion in Waves 102

7.2.2 Analytical and Numerical Solution to Coupled Sway-roll Motions in

Waves 104

7.3 Numerical Simulation and Comparison with Experiments 106

7.3.1 Equilibrium Sorting Process 107

7.3.2 Hydrostatics and Righting Arm GZ Curves 108 7.3.3 From Model rfts to Full Scale Numerical Simulation 110 7.3.4 Free Decay Tests for Intact and Damaged Ships: only opening or only

free surface tank 113

7.3.5 Free Decay Tests for Intact and Damaged Ships: both opening and

free surface tank 125

CONTENTS

7.4 Parametric Analysis on the Time To Capsizing (TTC) 152

7.4. 1TheTTC Dependence on DoFs, Wave Amplitude/Period and

Inter-polation of Hydrodynamic Coefficients 155

7.4.2 The TTC Calculations Based on the Rigid-lump Mass Model 157 7.4.3 The TTC Calculations Based on the Hydrodynamic Model-SWS/MMS 157

8

Conclusive Remarks and Suggestions for Future Work

161

8.1 Conclusive Remarks 161

8.2 Suggestions for Future Work 164

A Riernann Problem

167

B Rankine-Hugoniot Jump Condition

169

C CFL Condition

171

D The Shallow Water Equations arid the FDS Solver

173

D.1 Governing Equations 173

D.2 Boundary Conditions 173

D.3 Shallow Water Equations and the FDS Solver 175

D.3. i The Tensor Form of the Shallow Water-on-deck Equations 175

D.3.2 Eigenvector Space Mapping Scheme 176

D.3.3 The Flux Difference Splitting Scheme 180

D.3.4 '1' he Approximation to the Conserved Variables 186

D.3.5 Modified Roe Riemann Solver 187

D.3.6 The Compact Formulation of Flux Difference Splitting Method 188

D.3.7 The Second-order Corrections with Flux Limiter 191 D.3.8 Conserved Variables Approximation and Entropy Condition 192

D.4 Sorne Confusions and Challenging Points Concerning the Existing Theories 193

E The Strang Splitting Method

195

E.i Fractional Dimension-splitting Method 195

E.2 Fractional Source-splitting Method 196

F Analytical Linear Sloshing Loads

197

F.1 Formulation of the Sloshing Boundary Value Problem and the Analytical

So-lutions 197

F.2 Mutual Verifications between the Analytical Solutions and WAMIT

Calcu-lated Results 202

G Method of Equivalent Linear Damping

205

H Equations of Motion in a Body-fixed Coordinate System

207

List of Tables

1.1 The RORO vessel accidents from 1953 to 2006. Most of the RORO vessel

tragedies are caused by water-on-deck or sloshing loads of floodwater through the loading doors or vehicle decks. The FSE is the acronym for Free Surface

Effect. 2

1.2 Relevant flooding parameters that influence damaged ship motions and

cap-sizing. The studied terms by the present work are in italic and the regimes of

water depth (*) refer to Faltinsen and Timokha (2009) 6

1.3 Relevant opening parameters that influence damaged ship motions and

cap-sizing. The studied terms by the present work are in italic 7

2.1 The notations used for the coordinate axes and the kinematic variables in

different coordinate systems 20

4.1 The flooding models generally applied in a capsizing analysis. The models in

italic have been built up and used in this study. The integral hydraulic model

for an intermediate opening was given in Kong and Faltinsen (2008). . . . 39

7.1 Geometric data of the tanker ship (model 'TNK') used in the ITTC benchmark

test C (cf. Papanikolaou and Spanos, 2004b) (also from http://www.naval.

List of Tables

7.2 The ITTC experimental tests Cl, C2, C3 and C4 for the tanker ship (model 'TNK' from Papanikolaou and Spanos, 2004b). The corresponding water depth values, in the sloshing compartment (length L=82.5Om and breadth B=31.76m), are 000m, 100m, 400m and 1600m, respectively. The natural periods T8 and are the ship's roll and transverse sloshing natural period /(gir/B) tanh(7rh/B), respectively. The underlined numbers are the ratios between the water depth and breadth of the compartment and the critical

List of Figures

1.1 The flooding phenomena are distinguished into: (a) water run up the edge

and freeboard exceedance; (b) water shipping on-/off-deck; (c) water sloshing within deck area. Note these phenomena are analogous to the cases of flooding

through an opening in Figures 1.2 and 1.3 for 2D and 3D problems respectively. 5

1.2 A Flooding Model is applied to the flow near an opening boundary and a Floodwater Model is applied to the interior flow (interior domain is marked by a circled I and exterior by a circled O). A Rigid-lump Mass Model is the

floodwater model which assumes that the interior water surface is horizontal (dashed) versus the curved free surface (solid) described by a hydrodynamic

model

1.3 Three Dimensional Flooding Model and Floodwater Model. The Flooding Model deals with different opening conditions through which the fluid can

ingress or egress the compartment. 15

2.1 The Earth-fixed (inertial) coordinate system OXYZ, ship-fixed coordinate

system and tank-fixed coordinate system oxyz. The wave heading

angle ¡3 is defined as the angle between the positive x8axis and the direction in which the wave propagates. In this convention, ¡3 = 1800 is for head sea,

13 = 0° is for following sea, and ¡3 = 270° is for beam sea. 18

3.1 The improvement through a modification of the underwater geometry with an

intermediate opening, S is the intact underwater surface. S0 is the opening

List of Figures

4.1 Five cases of flooding scenarios in a damaged compartment with the waterlevel

configuration following the convention of inside of the compartment marked

by a subscript 'j' and outside by 'o' The pressure differences at upper and

lower edge are = p

- p

and = p _{- p'j»} respectively. 40

4.2 Small opening flooding scenario, h1 and h0 are the inside and outside water levels, respectively, and (a is the incident wave amplitude, h0 is the instan-taneous wave elevation relative to the opening and it can also include the

diffraction and radiation effects through the sum of the radiation and

diffrac-tion velocity potential near the opening. 41

4.3 Body plan of the 'PrrOl' model and the ship particulars from the ITTC

bench-mark tests in Papanikolaou (2001); ITTC (2003). The natural roll period (Ta) is 14.55s in the damaged condition. Note the roll radius of gyration values for the intact and damaged conditions are 909m and 9.56m, respectively, which

are estimated from experimental period by eq. (7.19), not the value 653m

from Papanikolaou (2001), since the latter one is not consistent with the given

natural period. 43

4.4 Drawings of midship damage geometry of the ITTC passenger RORO Ship

('PrrOl' model from Papanikolaou, 2001) or (ITTC, 2003)) with the dark area

being floodable 44

4.5 The five floodable compartments of the ITTC RORO ship ('PrrOl' model from

Papanikolaou, 2001) are combined into a large conipartment where the colors

indicate the different original compartments (see Figure 4.6a for their names). 46

4.6 (a) Large scale midship floodable compartments of the ITTC RORO ship

('PrrOl' model from Papanikolaou, 2001); (b) the reshaped underwater

geom-etry with the heel angle minus 100. 46

4.7 Reshaping procedure for complex damaged geometry. The five floodable

com-partments of the ITTC RORO ship ('PrrOl' model from Papanikolaou, 2001) are combined into a large compartment (left) and then put inside the exterior hull (middle) with the original opening. The sign '=' indicates the final model

(right) used in the hydrodynamic calculation. 47

List of Figures

5.2 Shallow water flow over a bottom defined by z = b(x, y) with z = ((z, y, t)

donating the free surface profile. For a flat bottom, the coordinate system

ozyz is built such that b(z,y) 0. 53 5.3 Discretization of the domain (left) and local Riemann problem to be solved

in a local cell coordinate system (right) with its origin o at the grid interface

i+l/2.

57

5.4 '1he lines for a left rarefaction wave. The first line L1

: x/t =

- a

and

the third line L3 : x/t = u - a connect the left state ÛL and right state respectively, with the state Û inside the rarefaction fari (line L2 : z/t = ua).

The fourth line L4 : z/t = u + a indicates the path along which the right

Riemann invariant u + 2a is kept constant across the rarefaction wave. . . . 63

5.5 IntegraI average of the exact solution of the Riemann problem in the HLL

Riemann solver. Srrijn, Smax and S are the slowest, fastest and contact wave

speeds corresponding to the minimum, maximum and intermediate slopes of

the characteristics, respectively. 63

5.6 The Weighted Average Flux method and the half time-step wave structure of the solutions of the Riemann problem with the three speedsSmin (or A1),

S (or A2) and

Smax (or A) from the slowest to the fastest. The intervals

Wk, k = 1. .. 4 indicate the fractions of the cell size x 70

5.7 Boundary condition at the opening area. v

and v, are the x and yaxis

projected components of the total incident velocity vector vfd, respectively. . 70

6.1 Coordinate system for a 2D sloshing tank. 78

7.1 The SWS and analytical solution for the water depth (in the dam-breaking

problem where the initial water depths at z < O and z > O are 10m and

00m, respectively. The water has zero velocity before the dam breaks at

time t = 0.

Analytical solution: U, A and

at t = 0.5s, t = lOs, and

t = 2.Os, respectively. SWS solution: dashed, dash-dot, and dash-dot-dot lines at t = O.5s, t = lOs, and t = 2.Os, respectively. 92

7.2 The SWS and analytical solution for the water particle velocity u in the

dam-breaking problem with the same conditions and legend definitions as Figure

List of Figures

7.3 The initial conditions for the bore propagation problem proposed by Stoker

(1948) where the analytical solution is provided. The dam is placed at x = O

and the water depth and velocity at both sides of the dam are shown in the figure where the subscripts L and R represent the left and right side of the

darn, respectively 93

7.4 The numerical and analytical water depth for the bore propagation problem

with the conditions given in Figure 7.3. The solid, dash-dot and dash-dot-dot

lines are the results from the SWS against the analytical ones, E, V, and

_O

at the time instants t = O.6s, t = lOs and t = 2.Os, respectively 94

7.5 The numerical and analytical water velocities for the bore propagation

prob-lem with the conditions given in Figure 7.3, (see Figure 7.4 for legend

defini-tions). 95

7.6 Dimensionless wave elevations at O.001270m from the left wall of the tank studied by Chester and Bones (1968)'s experiments, Faltinsen and Timokha (2002)'s multimodal theory, and the present shallow water solver, marked

by , y, and

, respectively. Steady-state wave height H = max - Ç_min

noridirnensionalized by the water depth h = 0050800m versus nondirnensional excitation frequency o7ci. The water depth-to-length and forcing

amplitude-to-length ratios are h/La = 0.083333 and 712a/Lc = 0.001254, respectively.

Here, ri2a is the sway excitation amplitude and the tank length L is 0609600m. 97

Ti

Transient downflooding flow through a small square opening (10m X10m) at

the compartment ceiling at the time instants O.Os, O.5s, lOs and 2.Os from top left to down right, (a), (b), (c) and (d), respectively. The reflective boundary conditions at the basin walls cause the piling up water at the walls in subfigure

(d). The constant wave elevation Ç_ at the opening is 25m 99

7.8 The dam-breaking convergence tests in a domain (100m long): water

eleva-tion. In the SWS simulation, 100, 200, 1000, and 4000 cells are used,

respec-tively, from top left to down right. The solid line and are initial conditions.

The dashed, dash-dot, dotted, dash-dot-dot are the SWS results against the

analytical solution , A, V, and at the time instants, lOs, 2.Os, 3.Os, and

List of Figures

7.9 The dam-breaking convergence tests in a domain (100m long): water velocity.

In the SWS simulation, 100, 200, 1000, and 4000 cells are used, respectively,

from top left to down right, (see Figure 7.8 for legend definitions) 101

7.10 Analytical and numerical roll angles of the 'PrrOl' ship in beam sea waves with

frequency ,j,₌ nO = 0.5l6rad/s and amplitude 05m. The superscripts 'exa'

and 'num' indicate the analytical and numerical solution, respectively. The subscript 'H' and 'P' indicate the homogenous and particular solution of eq.

(7.2), respectively. 103

7.11 Analytical and numerical roll velocities in waves with = Um0 = 0.5l6rad/s

(see Figure 7.10 for further explanations). 104

7.12 Analytical and numerical coupled sway-roll displacements in waves with w, =

w7 = 0.7O6rad/s (see Figure 7.10 for further explanations). 107

7.13 Analytical and numerical coupled sway-roll velocities in waves with w,L, =

UmnO = U.7O6rad/s (see Figure 7.10 for further explanations). 108

7.14 GZ curve for ITTC ship 'PrrOl' calculated by integrating the hydrostatic pressure over the wetted 3D geometry and the numerical results from the

benchmark test participants (Papanikolaou, 2001) for the intact condition. . 110

7.15 GZ curve for YFTC ship 'PrrOl' calculated by integrating the hydrostatic pres-sure over the wetted 3D geometry generated by the HRM and the numerical

results from the benchmark test participants (Papanikolaou. 2001) for the

damaged condition 111

7.16 GZ curves for ITTC ship 'PrrOl' obtained with the present method by heeling

the ship from port 50° to starboard +50° in the intact (solid line) and

damaged (dash-dot line) conditions. '1he GZ calculation for the damaged condition is based on the geometry model processed by the Hull Reshaped

Method. 112

7.17 Free decay simulations in intact and damaged conditions for the model 'PrrOl'

(Papanikolaou, 2001) with the initial heeling angles of 6.763° and +1.778°,

respectively. Intact: experiment and simulation marked with U and V,

re-spectively. Damaged: experiment and simulation marked with A and Q,

List of Figures

7.18 The arrangement of the tanker ship (model 'TNK') (cf. Papanikolaou and

Spanos, 2004b) and the floodable compartment used in the ITTC benchmark

test C. The unit in the figure is meter. 117

7.19 The measured (solid line) and simulated (dash-dot line) roll motions in the free decay test Cl (intact tanker ship 'TNK' from Papanikolaou and Spanos,

2004b). 118

T.20 The measured and simulated roll motions in the free decay test C2 with the water depth 100m in the tanker ship 'TNK', (see Papanikolaou and Spanos, 2004b). The dash-dot line (Q) and dotted line (A) represent the results from

the simulations using the analytical linear sloshing loads as extra added mass terms and unified approach (cf. Newman, 2005), respectively. The solid line

represents the experimental data. 119

7.21 The measured and simulated roll motions in the free decay test C2 with the water depth 100m in the tanker ship 'TNK' (see Papanikolaou and Spanos, 2004b). The results from the experimental data, SWS and analytical ap-proaches are marked with I. 9 and A, respectively. 120

7.22 The measured and simulated roll motions in the free decay test C3

(Papaniko-laou and Spanos, 2004b) with the full scale water depth 400m in the tanker ship 'TNK'. Experiment: solid line with ., simulation: solid line (roll only)

and dash-dot line (sway and roll) 122

v.23 The measured and simulated roll motions in the free decay test C4 (Papaniko-laou and Spanos, 2004b) with the full scale water depth 16.00m in the tanker

ship 'TNK'. Experiment: solid line with .. The solid and dashed lines

rep-resent the results from the simulation using MMS for single DoF in roll and

2DoFs in coupled sway-roll, respectively. 123

7.24 The Fourier analysis of the roll motions from the experimental data (a) and

simulation (b) for test C3 on the tanker ship 'TNK' (Papanikolaou and Spanos,

List of Figures

1 lie nondimensional moments and roll angle in the coupled sway-roll

sim-ulation of the free decay test C3 (Papanikolaou and Spanos, 2004b) with the water depth 4.0Dm in the tanker ship 'TNK'. Solid line: loo times the

nondimensional interior moment lOOF.("/(MgB), dashed line: 100 times

the nondimensional restoring moment

1O0F/(M8gB),

and dash-dot-dot

line: roll angle in degree 124

7.26 The Fourier analysis of the roll motions from the experimental data (a) and simulation (b) test C4 on the tanker ship 'TNK' (Papanikolaou and Spanos,

2004b). 125

7.27 The nondimensional moments and roll angle in the coupled sway-roll simula-tion of the free decay test C4 (Papanikolaou and Spanos, 2004b) with the full scale water depth 1600m for the tanker ship 'TNK'. Solid line: 100 times the

nondimensional interior moment

i00F/(M5gB),

dashed line: 100 times the

nondimensional restoring moment 100F4/ (M8gB), and dash-dot-dot line:

roll angle in degree 126

7.28 The studied RORO passenger ship (intact) and interior compartment (vehicle deck) in model scale, which are digitized from the paper by Fujiwara (2005), the model scale is 1: 48.571. Subflgures: (a) the RORO passenger ship with 3.135m long vehicle tank inside (green) and the freeboard curve (red); (b) the

top view of the ship and vehicle deck compartment 127

7.29 The model scale GZ curve of the studied RORO passenger ship (Fujiwara,

2005). 128

7.30 The studied RORO passenger ship (damaged) in full scale: (a) overview; (b)

local view of the damaged compartment with opening; (c) and (d) with a shallow water sloshing tank on the top of the damaged compartment. All

figures are obtained through digitization from the paper by Fujiwara (2005). 129

7.31 The experimental and simulated results for the roll decay test on the model

ship (Fujiwara, 2005) with only the midship damaged (without water on

ve-hicle deck). The different DoF combinations are used in the equations of

motion. The numbers 2, 3, 4, 5, 6 represent the sway, heave, roll, pitch and yaw modes, respectively. The experimental data is marked by., and the

nu-merical results from only roll (DoF4) mode are given by the dashed line with

List of Figures

7.32 The experimental (solid line with.) and simulated (dashed line) results for the roll decay test on the model ship (Fujiwara, 2005) with the midship damaged

and 10% ship weight floodwater on the vehicle deck. 131

7.33 The experimental and numerical results for the roll transfer function (or

called Response Amplitude Operator, RAO) of the ITTC benchmart test ship ('PrrOl' model), referring to Papanikolaou (2001) or ITTC (2003) for details. The numerical results are from the five benchmark study participating

insti-tutes coded from 'Pl' to 'PS' in the intact condition. The full scale values are used. The experimental results for the wave heights II = 12m and 24m are shown in solid and dashed lines without symbols and the investigated frequency range is 0.3rad/s ' 1.2rad/s. 133

7.34 The experimental (Papanikolaou, 2001; ITTC, 2003) and the presently

calcu-lated roll transfer functions for the ITTC ship ('PrrOl' model) in the intact

condition. The solid and dashed lines are the experimental results at the wave

heights H = 12m and H = 24m, respectively, while the dash-dot and

dot-ted lines are the corresponding numerical results using six degrees-of-freedom. 134

7.35 The numerical results for the roll Response Amplitude Operator (RAO) ob-tained by the five benchmark participating institutes coded from 'Pl' to 'PS' in damaged conditions for the ITTC ship ('PrrOl' model). The experimen-tal results (Papanikolaou, 2001) are also plotted (see Figure 7.33 for more

explanations) 135

7.36 The panelization of the damaged ITTC ship ('PrrOl' model) from

Papaniko-laou (2001). Subfigures (a), (b), (c) are the panels at the bow/astern part,

outer and inner structure in the damaged midship, respectively, with the

nor-mal vector (green) in (b) and (c) pointing into water. Subfigures (d) and (e)

are the top view and bottom view of the whole midship panels. 136

7.37 The full scale added mass coefficients (a): Alk, k = 1. . .6 and (b): A2k, k =

2. . . 6 and A33 for the ITTC ship ('PrrOl' model) from Papanikolaou (2001)

in the damaged condition as a function of frequency. 140

7.38 The full scale added mass coefficients (a): A3, k = 4. . . 6 and A4k, k = 4. .. 6,

(b): A55, A55, A66 and damping coefficients Blk, k = i . .. 3 for the ITTC

ship ('PrrOl' model) from Papanikolaou (2001) in the damaged condition as

List of Figures

7.39 The full scale damping coefficients (a): Bik, k = 4 ... 6 and B2k, k = 2.. . 4, (b): B25, B26 and B3k, k = 3.. .6 for the ITTC ship ('PrrOl' model) from

Papanikolaou (2001) in the damaged condition as a function of frequency. . 141

7.40 The full scale damping coefficients (a): B4k, k = 4. .. 6 and B55, B56, B66,

(b): excitation loads FX, j = 1

.. .6for the ITTC ship ('PrrOl' model) from

Papanikolaou (2001) in the damaged condition as a function of frequency. . . 141

7.41 A partially operi rectangular harbor with constant depth h and lengths L1 and L2 of the rectangular shaped waterplane with the opening width C. . . . 142

7.42 The transfer functions, RAOs:

(a) for llka/(a, k = i

.. . 6, and the phase

angles (b) for Ck, k = 1. . . 6 in six degrees-of-freedom for the damaged ITTC ship ('PrrOl' model) from Papanikolaou (2001) and the incident wave height

FI,1, = 12m was used in the simulations. The phase angles of the response

motions relative to the phase of the incident wave amidships are corresponding

to the transfer functions in (a). 144

7.43 The experimental and calculated roll response amplitude operators (RAOs)

for the ITTC ship ('PrrOl' model) from Papanikolaou (2001) in the damaged

condition with the wave heights I-Im = 12m and 24m within the frequency range 0.3rad/s «. 1.2rad/s. The solid and dashed lines are the experimental results at the wave heights I-L, = 12m and 24m, respectively. The dash-dot

and dotted lines are the corresponding numerical results using six

degrees-of-freedom 146

7.44 The cross-flow velocities at the center of the damaged opening in the ITTC

ship 'PrrOl' (Papanikolaou, 2001). The seven field points named by v» to

on the free surface in the middle of the opening are investigated. The points

have perpendicular distances -37248m, 0.000m, 12752m, 62752m, 112752m,

162752m and 212752m relative to the middle of the hull opening along the

dotted line in Figure 7.41. 147

7.45 The cross-flow velocity within the damaged opening in the ITTC ship 'PrrOl'

(Papanikolaou, 2001) at the piston resonance frequency w1 = 0.33rad/s (a)

and sloshing resonance frequency w2 = 0.90rad/s (h) with the incident wave

List of Figures

7.46 Resonant wave amplitudes in the damaged compartment of the ITTC ship

('PrrOl' model) from Papanikolaou (2001) at the frequencies: (a) w1 = 0.33rad/s

and (b) w2 = 0.90rad/s. In the calculations, the incident wave height H =

1.2m was used. 150

7.47 The phase angles of the response wave motions in the damaged compartment of the ITTC ship ('PrrOl' model) froni Papanikolaou (2001) at the frequencies:

(a) w = 0.33rad/s and (b) w2 = 0.90rad/s. The phase angles are relative

to the phase of the incident wave amidships. The incident wave height H0, =

12m was used. 151

7. 18 'l'he experimental and calculated roll response amplitude operators (RAOs)

for the RORO ship ('Prrjp') from Fujiwara (2005) in the damaged condition: (a) based on the general procedure in estimating the roll viscous damping; (b)

based on the selective experimental data 152

7.49 The relative wave elevation (including combined incident, radiation and

diffrac-tion effects) with respect to the vertical modiffrac-tion of the ship 'Prrjp' (Fujiwara,

2005). 153

7.50 The time loop to search for 'fTC. M is the sum mass of the ship and

the floodwater pAid f vi d(t)dt. The 6DOF 'ShipMotïon' module tracks the

ship's position and the underwater surfaces. 155

7.51 The time-histories of the exterior wave elevation ( (dashed line), opening position H0 (dash-dot line), vertical position Center Of Gravity Z9 (solid line), and the floodwater volume s (actually VId 0). 'I'he wave elevation,

opening position and vertical COG are relative to the mean sea water. The

wave period T = 10.00s and amplitude (a = 15m. The nìodel ship (Fujiwara,

2005) was used in the simulation. 156

7.52 Roll motion simulations of the model ship (Fujiwara, 2005) for the different

DoFs combinations. The numbers 1, 2, 3, 4, 5, 6 are used for surge, sway,

heave, roll, pitch and yaw modes, respectively. Subfigures (a), (b), (c) and (d), are the roll motion in time for coupled sway-heave-roll, sway-heave-roll-pitch, sway-heave-roll-pitch-yaw, and all coupled modes, respectively. 'l'he incident

List of Figures

7.53 Floodwater volume time series of the model ship (Fujiwara, 2005) for the dif-ferent DoFs simulations and assuming a small circular opening with diameter 05m on the port midship hull. The subscripts stand for the DoF combinations

explained in Figure 7.52 159

C. i The CFL definition and the upwind scheme for positive wave speed \. The

information propagates from the left. C/i is the ratio of the wave propagation

speed À and the numerical grid speed zx/t. 171

D.1 Wave speeds and intercell flux differences where i is the averaged Jacobian

matrix. 189

F.1 Analytical and numerical added mass coefficients A11, A22, A33 and A44. . . . 203

Chapter 1

Introduction

The flooding and damage on ships have caused great loss of life and property. In the wake of well-known marine disasters, the subject of safety has been forced to the forefront of marine technology development. In this framework, it was acknowledged that the ship design and operation strategy should use the state-of-the-art technology to ensure owners and operators

to have economic safety. Typically ship capsizing accidents are connected with flooding

caused by either a human error or a rough sea state. Therefore, not only from a design point of view but also for assessment of operational safety, a clear understanding of flooding and damage effects on a ship becomes indispensable to marine safety and ship stability in waves.

1.1

Background and Motivation

The maritime accidents that occurred along the years motivate suitable and effective inves-tigations. Table 1.1 shows the capsizing events for Roll-On/Roll-Off (RORO) vessels likely caused by flooding water effects. A RORO vessel is a type of vessel designed to carry wheeled cargo such as automobiles, trucks, semi-trailer trucks, trailers or railroad cars. The load and

unload operations are performed by means of large external doors close to the waterline and open vehicle decks with limited internal bulkheads. The RORO vessel is one of the most successful types operating today from the efficiency point of view, in contrast to the Lift-On/Lift-Off (LOLO) vessel which uses a crane to load and unload cargo. Despite the

comniercial success gained by the RORO vessels, the RORO concept received its continuous critics and even gained the reputation of an easy-to-capsize design due to public media about

Chapter 1. Introduction

Table 1.1: The RORO vessel accidents from 1953 to 2006. Most of the RORO vessel tragedies

are caused by water-on-deck or sloshing loads of floodwater through the loading doors or

vehicle decks. The FSE is the acronym for Free Surface Effect.

borne out by practical statistics. The Lloyd's World Casualty Statistics (World Casualty Statistics 2000, Lloyd's Register, London) reported that the annual chance of death from

ship accidents for general cargo crew is estimated as i in 2700, which is 50% higher than on a bulk carrier, twice as high as on an oil tanker, and 10 times as high as on a RORO passenger

ferry. Therefore, it is not the RORO ship stability that should be only concerned for, and

the knowledge about the RORO ship safety and its vulnerability should not be neglected by the ship designers and operators for the other ship types.

rrhe numbers of people lost in the accidents (see Table 1.1) look disappointing for re-searchers. Although many outstanding scientists worked on this area during the last fifty years, the casualties did not decrease but actually increased. These maritime disasters

ex-cited endless debates on the existing ship safety criteria. Many questions about the accidents do not have convincing answers yet. For instance, it is still not known why the Estonia and

al-Salam Boccaccio 98 sank so rapidly that over thousand passengers and crew members

died.

Roughly speaking, besides others, any of the following factors can lead a ship to capsize ship riding on a wave crest with quasi-steady reduction of the metacentric height, parametrically excited roll ¡notion,

broaching in aft or quartering waves, cargo shifting or loosening,

MV Princess Victoria Jan. 31, 1953 North Channel Flooding 133

SS Heraklion Dec. 08, 1966 Aegean Sea Flooding >200 TEV Wahine Apr. 10, 1968 Barrett Reef Flooding/FSE 52

MS Herald of Free Enterprise Mar. 06, 1987 Zeebrugge Flooding/FSE 193 MS Jan Hewelïusz Jan. 14, 1993 Baltic Sea Flooding/design 55 MS Estonia Sep. 28, 1994 Baltic Sea Flooding/FSE 852 MS Express Samina Sep. 26, 2000 Paros Flooding/FSE 82 MV al-Salam Boccaccio 98 Feb. 03, 2006 Red Sea FSE/sea/design 1020

I. I. Background and Motivation

collision

grounding,

flooding through openings or freeboard, floodwater-on-deck or sloshing.

Note that different type of ships may have different resistance to the above factors. As we know, it is virtually rare that the large ships were capsized without suffering flooding from hull damage, but there exists a serious hazard to smaller vessels that can experience

safety problems under normal operating conditions. For instance, the most capsized fishing vessel accidents are reported to be initiated by the water flooding down through open doors,

hatches, and/or flush deck fittings, and by the subsequent free movement of water on the deck, in the fish hold and the drinking water tank that are partially filled. Further, the

high-speed vessels may have more dynamic instability problems which are not included here. The hydrodynamics of the high-speed vehicles including the stability analysis can be found in the book by Faltinsen (2005).

Documented catastrophes show that it is not easy to make a right strategical decision

based on the regulated stability analysis for damaged ships. Empirical estimation based on rules is neither easy nor accurate. That is why so many people have to think that 'The Rules do not always reflect experience and the safety level in the Rules is still unknown!'. Therefore

one must rely on theoretical, numerical and experimental analyses. Ideally, all the static and dynamic properties of the ship, its various compartment configurations, surrounding environment and involved interactions should he taken into stability consideration even at

the initial design stage. Additionally, all kinds of damage scenarios should be modeled and

simulated. The knowledge database based on the theoretical analysis should be available onhoard to be referred to in case of emergency. However, in practice, there exist many

unpredictable natural and human factors influencing the ship-environment system.

The most likely cause of capsizing accident would be an opening of the hull and

free-board submergence. A large amount of water can thus run into compartments or on decks. This is called 'flooding damage'. The consequences of flooding through an opening can be minimized by subdividing the hull into compartments by watertight bulkheads. To minimize the flooding due to the freeboard suhniergence, one can design higher freeboard. However, it is impractical to build such a passenger ship that would survive all possible flooding since the fine subdivision will preclude effective use of the interior space and the high freeboard can significantly increase the ship's weight and reduce its metacentric height CAÍ. In reality,

Chapter 1. Introduction

flooding can not be avoided and the study on a damaged ship and its subsequent flooding

through an opening is necessary and significant for human lives and the marine safety. In this framework, by floodwater it is meant that the water is ingressed either from sea or from other compartments. While 'flooding water' indicates the change in time of the volume

of floodwater due to 'ingress' (flowing inwards) or 'egress' (flowing outwards) through an

opening or freeboard.

To build up proper design and safety criteria, one must try to answer the following major questions

Is it possible to identify a typical damage scenario for a given ship? Is it possible

to predict whether a damaged ship can survive or capsize? How much time is available after the damage for evacuating the people onboard if the damage is

going to cause capsizing?

Here, the word 'damage' can refer to any kind of adverse event, like flooding, collision, strong wind loads, grounding, etc. The first question can be possibly answered from damage statistics: damage scenarios generally vary from an accident to the following one, and for a given accident the damage features for two identical vessels can be different. Unfortunately,

neither the actual area nor the geometry of the damaged openings can be determined from the damage statistics since only the damage scale, i.e. the length, penetration and height

are usually recorded or reported. Therefore, the experimental and numerical investigations of a damaged ship are generally suggested to follow the SOLAS midship damage condition 1MO (1997), i.e., the longitudinal length of the damaged opening is assumed extending over

a length of 3m plus 3% of the ship length and a substantial penetration (one-fifth of the

ship beam) in the transverse direction. The second question is about the safety level and

the last question is about the value of Time To Capsizing (TTC) for a given damaged ship. The answer to the last two questions in the case of generic damage scenarios represents the objective of the present study.

1.2

Problem Description

The study of a (damaged or intact) ship motion problem at sea is to solve dynamic equilib-rium equations of forces and moments. The elastic deformations of the vehicle are neglected and the ship is assumed as a rigid body free to move in its six Degrees-of-Freedom (6DoF) in

I..l. Problem Description

waves. The influence of viscous effects and surface tension is of minor importance compared

to pressure effects when using a potential theory for the exterior flow problem. But the

viscous effects have to be included in the roll motion additionally.

For an intact ship in many practical cases, the diffraction and radiation phenomena are dominated by linear effects. The hydrodynamic problem of ship motions in waves can be solved through studying the diffraction and radiation problems separately in the exterior

fluid domain. (a)

Figure 1.1: The flooding phenomena are distinguished into: (a) water run up the edge and

freeboard exceedance; (b) water shipping on-/off-deck; (c) water sloshing within deck area. Note these phenomena are analogous to the cases of flooding through an opening in Figures 1.2 and 1.3 for 2D and 3D problems respectively.

For a damaged ship motion problem, things become more complicated, especially when

flooding is present. We distinguish three main stages that can occur to a ship, see Figure 1.1, i.e., 'water run-up and freeboard exceedance' (Figure 1.1(a), 'water shipping

on-/off-deck' (Figure 1.1(b) and 'water sloshing inside' (Figure 1.1(c). Water run-up and freeboard exceedance means that the incoming wave front causes a deck/opening edge exceedance, and the involved internal portion of the ship starts to be immersed in water. After the freeboard exceedance, a horizontal flow can lead to inrush of water over the deck, called water-on-deck. Part of onboard water flowing off the deck is called water-off-deck. Sloshing is excited if the flooding water runs into a compartment with walls so that the confined or partially confined water can be excited by ship motions to move back and forth inside the compartment. When the compartment is excited at or close to one of its natural frequencies, significant loads can result. Hence, the traditional mathematic models concerning the damaged ship behavior are challenged to correctly represent the static and dynamic effects due to the floodwater.

Ship capsizing due to flooding is characterized by many parameters that may affect the choice of solution model. Relevant but not exhaustive lists of such parameters for flooding

and opening are summarized in Tables 1.2 and 1.3, respectively. For instance, in terms of time scale, the flooding can develop slowly when the water flows through open doors,

Chapter 1. Introduction

Table 1.2: Relevant flooding parameters that influence damaged ship motions and capsizing. The studied terms by the present work are in italic and the regimes of water depth (*) refer to Faltinsen and Timokha (2009).

hatches, pipes, ventilation ducts. The flooding can also fill a damaged compartment (tank)

within a few seconds. This occurs especially when the opening is caused by a contact impact like collision or grounding.

The physical models adopted to handle flooding can be classified and outlined in Figures 1.2 and 1.3 for 2D and 3D cases, respectively.

A Flooding Model is the model designed to simulate how a fluid flows through an opening. An example is the Hydraulic Model which is based on the Bernoulli's equation

and assumes that the flow through the opening is quasi-steady (Turan, 1993).

A Floodwater, Flooding/Flooded water Model is the model used to simulate how a

fluid flows inside an interior domain of a compartment (tank). If the flow inside the interior

domain can be assumed quasi-static, a Rigid-lump Mass Model (Papanikolaou et al.,

2000) can be used for this purpose. While, a hydrodynamic approach is required in general conditions to capture the realistic features of fluid motions inside a compartment.

Flooding Parameters

Factor Category Comment

Time Scale fast slow

suddenly caused and massive flooding quasi-steady flooding

Developing

transient

progressive

steady-state

initial phase as shown in Figure 1.1(a) water accumulating as shown in Figure 1.1(b) equilibrium as shown in Figure 1.1(c) Symmetry symmetric

asymmetric

same flooding at port and starboard

different about central plane or flooding at one side Position over freeboard

through opening

due to water-shipping, large heel angle or in high wave on sidewall, bulkhead (cross-flooding), bottom (down-flooding) Flow Domain around edge

interior domain

including flow through opening and over freeboard edges interior flooding water flow

3D Effects 2D model 3D model

in multimodal solver and fully nonlinear wave tank in shallow water solver and exterior wave loads Air Effects trapped air

free air

air is compressed and may resist further flooding

air is not trapped and thus not affecting flooding

Water Depth1

shallow

intermediate finite

deep

small filling ratio of flooding water intermediate filling ratio of flooding water finite filling ratio of flooding water large filling ratio of flooding water

Problem Description

Table 1.3: Relevant opening parameters that influence damaged ship motions and capsizing.

The studied terms by the present work are in italic.

Flooding water models applied to Interio, domain

Figure 1.2: A Flooding Model is applied to the flow near an opening boundary and a Floodwater Model is applied to the interior flow (interior domain is marked by a circled I and exterior by a circled O). A Rigid-lump Mass Model is the floodwater model which

assumes that the interior water surface is horizontal (dashed) versus the curved free surface (solid) described by a hydrodynamic model.

Opening Parameters

Factor

Category

Comment

Size

small intermediate

large

an order smaller than ship's beam order of ship's beam

larger relative to ship's beam

Shape regular

irregular

rectangular, triangular, etc. arbitrary boundaries

Bending Feature radius

angle

opening plate deformation magnitude opening plate deformation direction

Symmetry symmetric

asymmetric

about ship's central plane

different between port and starboard or at one side

Position sidewall

top/bottom

of the ship or compartment's bulkhead of the ship or compartment (down-flooding)

N umber single several

only one opening caused

Chapter 1. Introduction

1.3

Previous Studies

Over the last fifty years, notable achievements have been made by the research. It is not

practical to report the exhaustive studies concerning the stability, damage and capsizing of

all types of vehicles. Here the focus is given to the ship stability and the major steps of the research development in the area of flooding and its interaction effects with the damaged

ship.

Ship stability can be, in general, divided into intact stability and damaged stability.

However, there is no clear definition of the damaged stability. If one follows Papanikolaou (2001), the damaged stability can be generalized to include any undesired event, like flooding, sloshing, collision, etc.

Either intact or damaged stability consists of static stability and dynamic stability. The static stability refers to the transverse or longitudinal stability which is associated with hydrostatics. The static stability of a ship reflect the ship's capability to maintain some

equilibrium or resume its original upright position after a finite transverse inclination (heel)

or longitudinal inclination (trim) due to waves or other external loads. A ship can have static instability when it experiences a reduced righting arm for a sufficient long time, in

case of a stationary wave crest amidship. The dynamic stability is mainly associated with the kinetic energy redistributing over the hull surface, for instance directional stability, broaching and porpoising (see Faltinsen, 2005). Flooding (including water-on-deck for the intact case

and water ingress for the holed-hull case), floodwater motion (including sloshing), strong

wind, cargo shifting and uneven-loaded cargo can also cause severe dynamic instability. In practical application, static and dynamic stability or effects of course change are considered in a synthesis way.

Nowadays, ship stability is an area of naval architecture and ship design that deals with how a ship behaves in calm water and in sea waves, as well as concerning intact and damaged conditions. The study on ship stability generally needs to solve the equations of motion of a ship in waves. The study on ship motion problem for the intact case has achieved significant

success. Sen (2002) performed a review of a three-dimensional (3D) ship ¡notion problem especially with large amplitude motions. Static intact stability calculations are relatively

straightforward (see Biran, 2003). The cargo arrangements and loadings, crane operations,

and the design sea states are usually taken into account.

I. Previous Studies

Water accumulated onboard cari run into the compartment or bilges, which generally lowers the Center Of Gravity (COG) and increases the metacentric height (GM) by assuming that the ship remains completely stationary and upright. On the other hand, flooding (as well as wind loads) can easily result in a ship inclination in roll, which can cause the liquid to move from one compartment to another. This can eventually lead to a permanent heel angle.

Stability can also be lost due to the dynamic flooding or dynamic free surface effects in the case of a compartment (with or without opening) partially filled with a liquid. The free surface effects have been noticed since Watts (1883) modeled the water motion in a tank by means of moving men (men walking on deck) to study the interior water motion effects on the roll stability. Nowadays, it is acknowledged that the flooding induced free surface effects have significant influence on the ship roll motion by changing the ship's roll damping and its GZ curve. A positive use of the dynamic free surface effects is the application of anti-rolling tanks.

For the stability calculation in a damaged condition, two methods are traditionally used: the Lost Buoyancy Method and the Added Weight Method. Both of them take into account, in general, only the hydrostatic forces. The strategy is to find an equilibrium position

accord-ing to the floodwater accumulation based on the calm water and static floodwater surface

assumptions. However, the hydrodynamic forces can have crucial effects on a damaged ship and affect the occurrence and features of capsizing. In reality, the hydrodynamic analysis of-ten has to be employed because the flooding area and floodwater volume can quickly become more complex than those in the intact case.

The studies on a damaged ship motion problem have been carried out since the last decade. The Rigid-lump Mass Model has been widely used to consider the floodwater

effect, for instance by Papanikolaou et al. (2000) and Ruponen (2007). In this approach, the

flooding effects can be simply accounted for by including the hydrostatic loads due to the floodwater which is assumed to behave as quasi-static, i.e., the floodwater is treated as a

rigid-lump mass. It is assumed that the mass of floodwater is concentrated in its centroid as

a rigid-lump mass. The centroid can be further assumed moving over a predefined surface

domain. This requires the assumption of the surface domain and this is not straightforward. For simplicity, the interior floodwater surface is usually set rigidly horizontal to the calm sea

s u r face.

Letizia (1996) and Jasionowski (2001) provided a six Degrees-of-Freedom (6DoF)

Chapter 1. Introduction

uses a linear approach to modify the general ship motion equations by including some terms

accounting for the mass variation effects. The ingress/egress of the floodwater through an

opening is also considered by using a hydraulic (flooding) model. The nonlinearities are taken into account by a DATABASE approach whereby a set of hydrodynamic forces and coeffi-cients are pre-calculated from a linear potential theory. I'he corresponding values are then interpolated from the stored values in the DATABASE. This is usually called the geometry

nonlinear approach since the nonlinearities are due to the change of underwater geometry.

Letizia (1996) and Jasionowski (2001) adopted a 'free mass' model, which is more realistic

than the rigid-lump mass model but involves still strong assumptions about the flooding water. The 'free mass' formulation indicates that the mass of the system includes both the

ship and floodwater mass. The floodwater mass is then decomposed into small rigid bodies

within the ship. The interaction between the floodwater and the ship is handled by the kinematics and dynamics of the multi-body system. But the interaction among the small rigid bodies (floodwater) can not be captured since such bodies are summed up into one inertia contribution to the mass of the system. This approach can be proved to be suitable

for the cargo-shifting problem but not for the floodwater flow.

Söding (2002) proposed that it is riot appropriate to simulate capsizing through a linear motion calculation in frequency domain followed by a statistical superposition in a stationary seaway. The accurate solution of the Time Tb Capsizing (TTC) for a damaged RORO ship

requires: 1) the simulation of the ship motions in 6DoF; 2) the simulation of the flow

through openings; 3) the simulation of the motions of water-on-deck of the ship. Söding (2002) provided some suggestions on how to deal with ship safety problems. Instead of

performing CFD calculations, relatively simple approaches were recommended to calculate the cross-flooding of damaged compartments, time of evacuation of persons on board, sinking of a damaged ship in still water, accelerations and loads on free-falling lifeboats and damaged ship survival time in a seaway.

Kim (2002) used the LAMP code to solve the exterior problem and a finite difference method for the interior floodwater in a tank. The developed numerical tool was used in studying the interaction of ship roll motion and sloshing. The method was also tested with the application of an anti-rolling tank in a container ship. Whereas, the finite difference

scheme can not be used to solve the shallow water flow when a hydraulic jump presents.

Schreuder (2005) incorporated the asymmetric effects of the underwater geometry in his numerical analysis based on a strip theory. The pressure loss around the opening area

1.3. Previous Studies

is taken into account as an additional load into the equations of motion for the exterior

hydrodynamic problem which is solved formally on the intact hull without opening.

Ruponen (2007) assumed the floodwater surface to he a constant and horizontal plane

without any wavy free surface and built up a floodwater accumulating equation for the mass balance of water. These assumptions can work well for a short-time and stationary flooding simulation but they are not valid for long-time or significant flooding cases.

The preliminary hydrostatic analysis by Valarito and Friesch (2008) showed that the water on a vehicle deck has been an important factor in the sinking sequence of the MV Estonia. 'the hydrostatic analysis showed that a mass of more than 1500 tonnes of water on the vehicle deck caused a hydrostatic heel of about 30°. The simulation of the sinking sequence of the MV

Estonia is started from the quasi-static equilibrium floating condition at this heel angle by

using the Added Weight Method. In their analysis, the exterior and interior flow domains are

separately solved. The exterior problem is solved by a 2D program based on a strip theory and the interior flow is solved by the Random Choice Method (RCM) by Glimm (1965). The one-way communication is performed to modify the opening boundary conditions for

the information to go from the exterior to the interior, i.e., the interior boundary conditions

(such as water depth and velocity) are taken from the (exterior) incident wave elevation.

However, for a damaged ship with an opening, the exterior flow around the opening can not be represented by a strip theory approach.

In case of flooding, a compartment can be partially ingressed and experience violent flow motion (sloshing) especially when the ship motions are in the vicinity of the highest natural period for the floodwater inside the compartment. This has been reported by Verhagen and Wijngaarden (1965). In their experiments, the wave elevation inside a compartment reached the same magnitude as the water depth and a hydraulic jump (or a bore) has been observed. Therefore severe loads on a damaged ship can be caused and affect the ship motions and the capsizing process.

Analytical solution for a linear sloshing problem can be obtained (see Abramson, 1966; Faltinsen and Timokha, 2009). But the linear solution fails to capture the resonant behavior. Further, in general, the nonlinear hydrodynamics of the floodwater plays an important role in the motions of a damaged ship with flooding flow present. In particular, when the floodwater

motion is significant, the ship can capsize with much less volume of floodwater than the

volume predicted by assuming that the floodwater behaves quasi-steady and using a

Chapter 1. Introduction

variation of floodwater, which is called the effects of mass changing dynamics (see Wynn,

2000).

It is highly rìecessary that the state-of-the-art models are built up to solve a damaged ship motion problem with flooding, green water-on/off-deck, sloshing and interior/exterior com-munication taken into account and their effects are incorporated into the dynamic equations of motion.

1.4

Present Study

The present study investigates damaged ship motions in regular beam sea waves associated

with flooding and floodwater. Both water-on/off-deck and sloshing are considered. Some

relevant physical parameters given in italic in Tables 1.2 and 1.3 are analyzed.

A mathematic model able to calculate all possible flooding scenarios is not available,

so different models are adopted according to different flooding scenarios. In particular, the

level of the flooding water accumulation is separated into four different regimes, shallow, intermediate, finite and deep levels (Faltinsen and Timokha, 2009). The corresponding models are adopted.

The shallow water equations are only relevant when the depth of the fluid is smaller

compared with the horizontal scale of the flow, i.e., dispersion effects are negligible. At the early flooding phase, the water depth is usually small with respect to the horizontal dimension

of the flow domain. The transient effects are important, and the interface of wet and dry

domains develops over the bottom of the deck or compartment. At this stage, the damaged

opening (or freeboard edge) is accessible for the sea water, and the horizontal acceleration component of a water particle is dominant over the vertical one near the opening area and the interior deck area. Further, the conservation of mass implies that the vertical velocity

of the fluid is small. Consequently, the shallow water assumption is appropriate and can be

used in predicting the flooding. From the momentum conservation equations, the vertical

pressure gradient can be approximated as pure hydrostatic relative to the instantaneous free surface elevation and the velocity field is nearly constant throughout the depth of the fluid. By neglecting the small vertical velocity and variations throughout the depth of the fluid in the Euler equations (Landau and Lifshitz, 1987), the shallow water equations are derived.

For the later stages, due to large heel angle or large filling ratio, the water can be

1.4. Present Study

dispersive. Thus the shallow water assuniption becomes inappropriate. The case of

non-shallow water condition can be solved by a nonlinear multimodal method by Faltinsen and

Tirnokha (2009). It is also possible to apply the multimodal method for a domain with a small opening. This idea is based on the fact that the small variation of the domain does

not change the sloshing eigenmodes significantly.

Comparing against the rigid-lump mass model, both the shallow water and the

multi-modal approach can capture more accurately the floodwater free surface and describe more realistic opening boundary conditions for the flooding calculation since the boundary condi-tions at the opening are used.

If the opening is large relative to the compartment dimension, the damaged body is modeled hydrodynamically as a complex concave geometry around the damage area. The

'interior' fluid in the floodable domain is treated as a part of the exterior sea water and the in-terior fluid motions as the partial sea wave motions. Thus all the hydrodynamic/hydrostatic

coefficients are calculated based on the new complex geometry. This kind of approach is

called the Hull Reshaped Method (HRM) by Kong and Faltinsen (2008).

1.4.1

Structure of the Thesis

The present thesis is organized as follows.

The equations of motion system for a damaged ship and flooding water is built up in

Chapter 2 based on the rigid-body theory.

The exterior and interior loads acting on a damaged ship are discussed in Chapter 3. Relevant flooding scenarios are modeled numerically and analyzed in Chapter 4.

To solve the problem concerning the coupled interior and exterior domains, two kinds of solvers are developed for the interior flow and the exterior flow. One is the interior

floodwater solver (FWS) which includes the shallow water solver (SWS) given in Chapter

5 and the multimodal solver (MMS) stated in Chapter 6. The FWS solves the interior

problem associated with the ingress/egress flooding. In addition, the exterior 6DoF ship

motion solver is needed for the exterior problem associated with the ship motions to handle the interaction with the flooding water.

Numerical applications of the above models, as well as the verification and validation are arranged in Chapter 7.

Chapter 1. Introduction

1.4.2

Major Contributions

The challenging problem of predicting the Time To Capsizing (TTC) of a given damaged

ship is tentatively solved. Some contributions to this objective are a.ssumably made in this

thesis.

Shallow water equation solver: the fully nonlinear shallow water equations built up in

a body-fixed accelerated coordinate system are accurately and robustly solved by the developed Shallow Water Solver (SWS).

Multimodal Solver: the adaptive nonlinear MultiModal Solver (MMS) is efficiently ap-plied to the sloshing problem to obtain the sloshing loads. Due to its analytical na-ture, the accuracy is relatively higher than other numerical methods as long as the

theoretical assumptions are appropriate.

Hull Reshaped Method: if a damaged ship contains a large opening (in 2D or 3D), the Hull Reshaped Method (HRM) can be efficiently used to consider the complicated

communication effects between the interior and exterior domains. The applied HRM

is an extended application of a 3D linear potential flow method for unconventional

geometry.

Important Resonances: the roll resonance of damaged ship motions is well known to us, but the piston mode and sloshing resonance are seldom noticed for a damaged ship

with openings. Through numerical observation and the simplified formulas, these two additional resonances are confirmed and the structural failure problem for a damaged ship is postulated.

F!OodTh

'flOd

aPP1,d_(Q

1..

Present Study

Figure 1.3: Three Dimensional Flooding Model and Floodwater Model. The Flooding Model

deals with different opening conditions through which the fluid can ingress or egress the