Inviscid and Viscous 2D Unsteady Flow Solvers Applied to ...

Applied to

Inviscid and Viscous 2D Unsteady Flow Solvers

ENVIRONMENTAL AND WATER RESOURCES ENGINEERING

DEPARTMENT OF CIVIL ENGINEERING

Austin, TX 78712

THE UNIVERSITY OF TEXAS AT AUSTIN

Report No. 04−7

FPSO Hull Roll Motions

Bharani Kacham

O EC NA

G

December 2004

ROUP

RENGINE EGNI

Copyright

by

Bharani Kacham

2004

Inviscid and Viscous 2D Unsteady Flow SolversApplied to


by

Bharani Kacham, B.Tech.

Thesis

Presented to the Faculty of the Graduate School

of The University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science in Engineering

The University of Texas at Austin

December 2004

Inviscid and Viscous 2D Unsteady Flow SolversApplied to


APPROVED BYSUPERVISING COMMITTEE:

Supervisor:Spyros A. Kinnas

Reader:Kamy Sepehrnoori

To family and friends

Acknowledgements

At the outset, I would like to express my gratitude to my advisor, Dr. Spyros A.

Kinnas for his unending support, encouragement and valuable advice which kept

me going for the entire duration of my masters’ program. His understanding nature

and fatherly concern are worth mentioning and thanking.

I would also like to thank Dr. Kamy Sepehrnoori for agreeing to be the reader

of my thesis in spite of his busy schedule. His comments and suggestions were of

immense help in giving this thesis a final shape.

It is with great pleasure that I mention the names of my CHL buddies; Dr.

Lee, Shreenaath, Hua, Vimal, Yi-Hsiang, Apurva, Bikash, Hong and Yumin, from

whom I have learnt a great deal. They have always been more than willing to help

and were fun to work with. Special thanks to Yi-Hsiang, without whose help the

thesis progress would have been very slow.

I am indebted to my parents and my brother Shravan for all the support and

freedom they have given me. It is very difficult not to list my friends Shilpa, Swapna,

Kranthi, Jeetain and Gopal who were of great support and strength during the in-

evitable tough times. I wish them all great health and prosperity.

Finally, I would like to thank the Offshore Technology Research Center for

providing financial support through their Cooperative Agreement with the Minerals

v

Management Service (MMS) and its Industry Consortium�

, and also the faculty of

UT for the superior quality of education they imparted.

�

Disclaimer: “The views and conclusions contained in this document are those of the authors andshould not be interpreted as representing the opinions or policies of the U.S. Government. Mention oftrade names or commercial products does not constitute their endorsement by the U.S. Government”.

vi

Inviscid and Viscous 2D Unsteady Flow Solvers

Applied to


by

Bharani Kacham, M.S.E.

The University of Texas at Austin, 2004

SUPERVISOR: Spyros A. Kinnas

The roll dynamics of a Floating, Production, Storage and Offloading (FPSO) hull

are of special interest in the present offshore industry. The FPSOs, while on duty

need to be stationary for long periods of time in order to enable smooth drilling and

oil transfer to the shuttle tankers. The present research is aimed at providing insights

into the effectiveness of using anti-roll appendages, like bilge keels, in mitigating

roll motion of FPSOs operating in mid-seas. Numerical modeling is a tool that

can be extensively used to simulate and investigate real ship motions. The present

work details a 2D unsteady Boundary Element Method and Navier-Stokes solver

based on Finite Volume Method and their application to modeling roll motions of

an FPSO hull. The Navier Stokes solver is a viscous solver and is advantageous

when compared to the traditional potential flow solvers due to its ability to capture

the effects of viscosity and separation past the bilge keel on the motion of the hull.

vii

The method could be applied to three dimensional hulls by using either strip theory

or by including the third dimension in the formulation.

viii

Table of Contents

Acknowledgements v

Abstract vii

List of Tables xii

List of Figures xiii

Nomenclature xix

Chapter 1. Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Chapter 2. Literature Review 102.1 Hull Motion Prediction . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Vortex Tracking Method . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 3. 2D Boundary Element Method and Its Applications 173.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Numerical Formulation . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.1 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.2 Application of Green’s Formula for a two-dimensional body . 19

3.2.3 Numerical Implementation . . . . . . . . . . . . . . . . . . . 24

3.3 Validations of the method . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 Prismatic cylinder of circular cross-section . . . . . . . . . . 25

3.3.2 Prismatic cylinder of elliptic cross-section . . . . . . . . . . 28

3.3.3 Prismatic cylinder of square cross-section . . . . . . . . . . . 30

ix

3.3.4 Prismatic cylinder of cross shaped cross-section . . . . . . . 33

3.4 Roll motions of a submerged body . . . . . . . . . . . . . . . . . . 35

3.4.1 Forces and added mass coefficient . . . . . . . . . . . . . . . 36

3.4.2 2D submerged hull without bilge keels . . . . . . . . . . . . 40

3.4.3 2D submerged hull with bilge keels . . . . . . . . . . . . . . 43

3.5 Oscillating hull at free surface . . . . . . . . . . . . . . . . . . . . . 44

3.5.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 46

3.5.2 Numerical Implementation . . . . . . . . . . . . . . . . . . . 50

3.5.3 Time-marching . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5.4 Forces and Hydrodynamic coefficients . . . . . . . . . . . . 53

3.6 Tip Vortex Tracking Method . . . . . . . . . . . . . . . . . . . . . . 55

3.6.1 Numerical Formulation and Implementation . . . . . . . . . 58

3.6.2 Application to flow over a foil . . . . . . . . . . . . . . . . . 60

Chapter 4. Numerical Formulation of 2D Viscous Solver 644.1 Non-dimensional governing equation . . . . . . . . . . . . . . . . . 64

4.2 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.3 Upwind scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Time Marching . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5 Pressure Correction Scheme . . . . . . . . . . . . . . . . . . . . . . 69

Chapter 5. Applications of 2D Navier-Stokes solver 725.1 2D Channel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Numerical Wavemaker . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.1 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 77

5.3 Heave and Roll Motions . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3.2 Coordinate System and Grid details . . . . . . . . . . . . . . 83

5.3.3 Froude number and Reynolds number . . . . . . . . . . . . . 85

5.3.4 Heave Motion . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.5 Roll Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3.7 Roll motion of a semi-circular hull . . . . . . . . . . . . . . 106

x

5.4 Submerged hull motions . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4.1 Fixed coordinate system and fixed grid . . . . . . . . . . . . 107

5.4.2 Fixed coordinate system and moving grid . . . . . . . . . . . 114

5.4.3 Convergence Studies . . . . . . . . . . . . . . . . . . . . . . 120

5.4.4 Hull with bilge keels . . . . . . . . . . . . . . . . . . . . . . 124

Chapter 6. Conclusions and Recommendations 1326.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

Bibliography 136

Vita 143

xi

List of Tables

5.1 Comparison of roll added mass coefficients obtained from viscousand potential solvers for a submerged hull without bilge keels un-dergoing roll motion . . . . . . . . . . . . . . . . . . . . . . . . . 120

xii

List of Figures

1.1 Terra Nova FPSO (source: www.provair.com/ IcebergNet/gallery.htm) 2

1.2 Description of motion under six degrees of freedom for a ship . . . 4

3.1 Volume � confined by a surface�

. . . . . . . . . . . . . . . . . . 19

3.2 Body B and a unit source P confined in a finite domain��

. . . . . . 21

3.3 Figure showing the discretized body surface and corresponding in-dex notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 An infinitely long cylinder of circular cross-section subjected to asinusoidal inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.5 Comparison of analytical and numerical values of perturbation po-tential on the circle . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.6 Time history of the force on the circle in the x-direction . . . . . . . 28

3.7 An infinitely long cylinder of elliptic cross-section subjected to asinusoidal inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.8 Time history of the force on the ellipse in the x-direction . . . . . . 30

3.9 An infinitely long cylinder of square cross-section subjected to rollmotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.10 Time history of the moment on the square in the z-direction . . . . . 32

3.11 An infinitely long cylinder of cross shaped cross-section subjectedto roll motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.12 Time history of the moment on the cross in the z-direction . . . . . 34

3.13 Figure showing cross-section of submerged hull without bilge keels 35

3.14 Figure showing cross-section of submerged hull with bilge keels . . 36

3.15 Comparison between numerical (BEM) and analytical pressure on aheaving circle at �� . . . . . . . . . . . . . . . . . . . . . . 38

3.16 Comparison between numerical (BEM) and analytical pressure on aheaving circle at �� . . . . . . . . . . . . . . . . . . . . . . 39

3.17 Geometry details and boundary conditions for a submerged hull with-out bilge keels undergoing roll motion . . . . . . . . . . . . . . . . 41

3.18 Time history of the moment on the hull without bilge keels under-going roll motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xiii

3.19 Convergence of the roll added mass coefficient �� with respect tonumber of panels on the hull (without bilge keels) surface . . . . . . 43

3.20 Error convergence plot for the roll added mass coefficient obtainedfor a submerged hull without bilge keels . . . . . . . . . . . . . . . 44

3.21 Geometry details and boundary conditions for a submerged hull withbilge keels undergoing roll motion . . . . . . . . . . . . . . . . . . 45

3.22 Time history of the moment on the hull with bilge keels undergoingroll motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.23 Geometry details and boundary conditions for a floating hull under-going harmonic heave motion . . . . . . . . . . . . . . . . . . . . . 49

3.24 Force history for a hull undergoing heave motion for�� . . 54

3.25 Comparison of heave added mass coefficients obtained from theBEM solver [Vinayan 2004] with those presented in [Newman 1977]and obtained from Euler solver [Kakar 2002] . . . . . . . . . . . . 56

3.26 Comparison of heave damping coefficients obtained from the BEMsolver [Vinayan 2004] with those presented in [Newman 1977] andobtained from Euler solver [Kakar 2002] . . . . . . . . . . . . . . . 57

3.27 A bilge keel with a trailing wake and a tip vortex . . . . . . . . . . 58

3.28 A 2D foil subjected to a uniform inflow with a lateral sinusoidal gust 61

3.29 Description of the initial wake and tip vortex geometry . . . . . . . 62

3.30 Figure showing trailing wake for a foil subject to a uniform inflowand a lateral sinusoidal gust . . . . . . . . . . . . . . . . . . . . . . 62

3.31 Vorticity being shed tangentially into the shear layer . . . . . . . . 63

3.32 Time history of the lift force on the foil . . . . . . . . . . . . . . . 63

4.1 Geometry details of the cell based scheme . . . . . . . . . . . . . . 67

5.1 Description of the boundary conditions applied for a 2D channel flow 74

5.2 The velocity and pressure contours for the fully developed flow in a2D channel obtained from the viscous solver . . . . . . . . . . . . . 75

5.3 Comparison of horizontal velocity profile obtained from the viscoussolver at the outflow boundary with analytical solution . . . . . . . 76

5.4 Description of the boundary conditions applied for a numerical wave-maker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Pressure contours under a wave and the corresponding wave eleva-tion at �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.6 Pressure contours under a wave and the corresponding wave eleva-tion at �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

xiv

5.7 Depiction of coordinate system and domain for a floating body un-dergoing harmonic motions . . . . . . . . . . . . . . . . . . . . . . 85

5.8 Grid details for a rectangular hull without bilge keels . . . . . . . . 86

5.9 Grid details for a rectangular hull without bilge keels . . . . . . . . 86

5.10 Comparison of hydrodynamic coefficients for a 2D hull undergoingheave motion obtained from the present solver with those measuredby Vugts [1968] as given in [Newman 1977] and Euler solver [Kakar2002] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.11 Force history for a heaving rectangular hull over one time period andfor

�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.12 Pressure contours at different time steps for a 2D rectangular hullundergoing heave motion . . . . . . . . . . . . . . . . . . . . . . . 91

5.13 Wave profiles at different time steps for a 2D rectangular hull under-going heave motion . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.14 Bilge and keel geometry details . . . . . . . . . . . . . . . . . . . . 94

5.15 Boundary conditions applied for a body undergoing forced harmonicroll motion at the free surface . . . . . . . . . . . . . . . . . . . . . 95

5.16 Figure explaining how to evaluate roll added mass and damping co-efficients from the moment history plot itself . . . . . . . . . . . . . 97

5.17 Moment history of a hull without bilge keels undergoing harmonicroll motions for

�� = 0.8 . . . . . . . . . . . . . . . . . . . . . . 98

5.18 Pressure contour plots at various time instants for a hull withoutbilge keels undergoing roll motion . . . . . . . . . . . . . . . . . . 99

5.19 Comparison of roll added mass coefficients from the present solverwith those obtained from the BEM solver, the Euler solver [Kakar2002] and Vugts [1968] for a hull without bilge keels . . . . . . . . 100

5.20 Comparison of roll damping coefficients from the present solverwith those obtained from the BEM solver, the Euler solver [Kakar2002] and Vugts [1968] for a hull without bilge keels . . . . . . . . 100

5.21 Wave profiles at various time instants for a hull without bilge keelsundergoing roll motion . . . . . . . . . . . . . . . . . . . . . . . . 101

5.22 Moment history of a hull with 4�

bilge keels undergoing harmonicroll motions for

�� = 0.8 . . . . . . . . . . . . . . . . . . . . . . 102

5.23 Comparison of roll added mass coefficients from the present solverwith those obtained from the BEM solver, the Euler solver [Kakar2002] and Yeung et al. [2000] for a hull with 4

�bilge keels . . . . 103

5.24 Comparison of roll damping coefficients from the present solverwith those obtained from the BEM solver, the Euler solver [Kakar2002] and Yeung et al. [2000] for a hull with 4

�bilge keels . . . . 103

xv

5.25 Wave profiles at various time instants for a hull with 4�

bilge keelsundergoing roll motion; The vertical axis represents the wave eleva-tion, � scaled by the beam length, � . . . . . . . . . . . . . . . . . 104

5.26 Pressure on the hull with 4�

bilge keels for� � � = 0.8 at �� = 0.8

(The discrepancies between the pressures from the current viscous solverand other solvers shown in the figure led to investigation and changes in theformulation of the solver, which are presented in the succeeding sectionsof the chapter.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.27 A close-up view of the grid near the semi-circular hull . . . . . . . 108

5.28 Plot of pressure on the semi-circular hull �� curve length at varioustime instants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.29 Description of the main length parameters for a submerged hull un-dergoing roll motions . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.30 a typical grid used for forced harmonic motions of a submerged hull 110

5.31 Description of boundary conditions applied for the submerged rollproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.32 Pressure on the submerged hull without bilge keels at �� for anon-moving grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.33 Pressure on the submerged hull without bilge keels at �� fora non-moving grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.34 Pressure contours around the submerged hull without bilge keels at�� for a non-moving grid . . . . . . . . . . . . . . . . . . . . 113

5.35 Pressure contours around the submerged hull without bilge keels at�� for a non-moving grid . . . . . . . . . . . . . . . . . . . . 114

5.36 Figure explaining the terms used in transformation of the unsteadyterm in the Navier-Stokes equations for a moving grid in a fixedinertial coordinate system . . . . . . . . . . . . . . . . . . . . . . . 116

5.37 Grid orientation for a submerged hull without bilge keels at �� and �� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.38 Pressure evaluated on the submerged hull without bilge keels at �� using a fixed coordinate system and a moving grid in the case ofviscous solver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.39 Pressure evaluated on the submerged hull without bilge keels at �� using a fixed coordinate system and a moving grid in the caseof viscous solver . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

5.40 Pressure contours around the submerged hull without bilge keels at�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

xvi

5.41 Pressure contours around the submerged hull without bilge keels at�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.42 Comparison between hydrodynamic moment obtained from viscousand potential solvers for a submerged hull without bilge keels un-dergoing roll motion . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.43 Flow field with respect to the submerged moving hull without bilgekeels, shown around the hull at time instant � �� . . . . . . . . . . 121

5.44 Flow field with respect to the submerged moving hull without bilgekeels, shown around the hull at time instant � �� . . . . . . . . 122



5.47 Comparison of the grid densities around the submerged hull withoutbilge keels used in the convergence study . . . . . . . . . . . . . . 125

5.48 Comparison of the pressure on the submerged hull without bilgekeels for increasing number of cells at �� . . . . . . . . . . 126



5.51 Comparison of the hydrodynamic moment on the submerged hullwithout bilge keels between three different grids for the first timeperiod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.52 a typical grid used for computation of forced harmonic motions of asubmerged hull with bilge keels . . . . . . . . . . . . . . . . . . . 128

5.53 Comparison of the pressure on the submerged hull with bilge keelsundergoing roll motion for varying Reynolds number at �� . 128



5.56 Comparison of the pressure on the submerged hull with bilge keelsundergoing roll motion between viscous and potential solvers at�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.57 Comparison of the pressure on the submerged hull with bilge keelsundergoing roll motion between viscous and potential solvers at�� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

xvii

5.58 Comparison of the hydrodynamic moment on the submerged hullwith bilge keels undergoing roll motion between viscous and poten-tial solvers for the first time period . . . . . . . . . . . . . . . . . . 131

xviii

Nomenclature

Latin Symbols

� wave amplitude

�� added-mass coefficient� �� damping coefficient

� beam of the ship� wave celerity, � � � �� (in deep water)�

water depth�propeller diameter,

� � �� , or

draft of the ship�� , body force per unit mass

F column matrix for the derivative terms��

Froude number based on propeller diameter�

,�� "!$#

��&%

Froude number based on beam � ,� �'% �)( � % �

�� Froude number based on draft

�,�� *( �

� ��

non-dimensionalized total - and + -direction force, gravitational acceleration

G column matrix for the + derivative terms

xix

�wave number,

� � ��

Keulegan-Carpenter number� �

non-dimensional bilge-keel depth�

reference length used in non-dimensionalization, or

wavelength, or length of 2-D channel��

deep-water wavelength� �

moment about the � axis�� , normal vector� pressure�� atmospheric pressure�� , body velocity

Q column matrix containing the source terms ��

residual of the continuity equation

�� Reynolds number based on reference length�

, ��

�� area of cell in two-dimensional formulation

� non-dimensional time�

time period of motion

U column matrix for time derivative terms��

flow velocity at infinity� amplitude of oscillating velocity function

��amplitude of heave velocity

� � � � � � + and � -direction velocities�� , total velocity vector!#"

ship speed$% �computational cell volume� � � � + � � � , location vector on the ship fixed

coordinate system

� � + � � � downstream, upward and port side coordinates respectively

xx

Greek Symbols� angle of roll for FPSO hull, � � � �

� � � ( �� amplitude of roll motion� � pressure difference�

� time step size

� � � � + � cell size in and + direction

� vertical coordinate of free surface� dynamic viscosity of water

� kinematic viscosity of water

( frequency of periodic heave and roll forcing function�( � � ( �� ( �� ( �� , vorticity vector��

perturbation potential� fluid density�

phase of the wave,� � � ( �

Subscripts

� � �� node numbers �

�� cell indices

� � �� node or cell indices in each direction;�

is axial,�

is radial, and�

is circumferential.� ��

face (in two-dimensions) indices

at north, west, south and east of a cell

xxi

Superscripts� intermediate velocity or pressure�

velocity or pressure correction� � �� time step indices

Acronyms

BEM Boundary Element Method

CFD Computational Fluid Dynamics

CPU Central Processing Unit (time)

FPSO Floating, Production, Storage and Offloading (vessels)

FVM Finite Volume Method

MIT Massachusetts Institute of Technology

RANS Reynolds Averaged Navier-Stokes (equations)

SIMPLE Semi-Implicit Method for Pressure Linked Equations

Computer Program Names

FLUENT commercial CFD software

WAMIT panel method based wave-structure interaction analyzer

xxii

Chapter 1

Introduction

1.1 Background

Oil in various forms has become an indispensable commodity in present day lives.

The oil basins or reserves in shallow waters have long been dried up or reduced to

non-profitable resources. The ever increasing demand for oil has forced explorers

to look towards deeper seas for newer and better opportunities. Deeper seas were

explored as early as 1950s. But, exploring, drilling and production in deeper seas

are not without their share of problems. It is an arduous and expensive task to set up

drilling and production units in deep seas mainly due to the depth of the sea bed and

the prevailing harsh environmental conditions.

The option of fixed structures is hence limited in deep waters. The best and obvious

alternative for this purpose is the use of floating structures. Floating structures are

used in all fields of marine technology, particularly in exploration work. Typical

floating structures used for offshore operations are semi-submersibles and drill ships.

Semi-submersibles are mainly floating drilling platforms which use pontoons and

columns flooded with sea water to stay afloat. Floating, Production, Storage and

Offloading (FPSO) vessels and Floating, Storage and Offloading (FSO) vessels are

the common drilling vessels. FPSO vessels are nowadays extensively being used

1

for oil extraction. Both FPSOs and Semi-submersibles are usually anchored with

mooring lines which in some cases can be assisted by dynamic positioning thrusters.

A typical FPSO operating in mid-seas is shown in figure 1.1.

Figure 1.1: Terra Nova FPSO (source: www.provair.com/ IcebergNet/gallery.htm)

1.2 Motivation

Floating structures, together with moorings, risers and other equipment can be con-

sidered as a single system. Such systems usually have a low stiffness and hence have

a low natural frequency. This in turn can cause the ship to move in all six degrees

of freedom when the structure is subject to three-dimensional loads due to various

factors like random waves, currents and winds. The translatory motions that occur

along the three axes and the rotational motions that occur about the axes form the

six degrees of motion. Translatory motions along the X-, Y- and Z-axes are called

surge, sway and heave respectively. The three rotational motions in the same order

2

are roll, pitch and yaw respectively. The six degrees of freedom are depicted in figure

1.2. One can represent the energy carried by the waves as an area of spectral den-

sity in the frequency domain. From the wave energy spectrum one can observe that

the energy intensive range of the spectrum is concentrated in the low frequencies.

Hence, resonance phenomenon can be very problematical for floating structures as

their natural frequency is low. Of the possible motions, pitch, roll and heave are

significant and need to be studied carefully. The prime origin of roll motions of FP-

SOs’ is the non-collinearity of wind, current, wind driven seas and swell. In storm

conditions the wind driven seas are normally collinear with the wind and dominant

over currents. Therefore, in extreme conditions FPSOs usually encounter seas and

wind head on or at a small angle. It should be noted that the wind driven seas exhibit

a directional spreading and that the FPSOs oscillate around the mean heading at the

same time. Both the phenomena contribute to transverse wave loading, sway, yaw

and roll motions. Swells originating from remote storms may arrive from the beam

direction. In cases where the currents govern the heading of the vessel, vessels have

to cope with the onslaught of beam seas. Other sources of transverse excitation are

the variations in wind direction and current direction. During wind shifts or change

of wind direction or tidal change of current, the FPSO may turn and experience bow

quartering or beam waves for a certain period. The main focus of the present work

is roll motion because the possibility of extremely large motions and even capsizing

make roll one of the most critical aspects of ship motions and sea keeping.

The FPSOs that are operating in deep waters need to be stationary for long periods

of time in order to facilitate smooth drilling operations. Hence, the mitigation of roll

motion acquires significant importance and needs to be studied extensively. The roll

motion plays an important role in determining the loads on deck cargo of an offshore

3

Figure 1.2: Description of motion under six degrees of freedom for a ship

vessel. The range of operability of the ship also can be predicted accurately if the

roll motion near resonance can be estimated correctly. Field observations indicate

that FPSOs roll more than expected based on their design and model tests. This can

lead to riser fatigue, loads on mooring system, turntable and turret and operational

difficulties (degraded process performance, operational limits on material transfer,

helicopter operations, crew comfort and safety and effectiveness). FPSOs consist of

both ballast tanks and cargo tanks which have a free surface at all times. Cargo tanks

are basically used to store oil till the oil shuttle tankers make their trip. Sloshing in

these tanks could pose a major problem due to roll motions at resonance. Hence, it

becomes imperative that roll motions be avoided as much as possible for the FPSOs.

Many types of devices and methods have been designed by sailors and naval archi-

tects to reduce the roll motions. The means to control the roll motion can be divided

into the following groups:

4

� Hull design (main dimensions, distribution of displacement, cross section

shape);

� Passive devices such as bilge keels, skegs and fins;

� Active systems based on moving weights and stabilizer fins;

� Active and passive anti-rolling tanks;

� Rudder-roll control and heading control;

In hull design, distribution of displacement mainly involves distributing the weight

on to either side of the vessel away from the centreline. Making the topside of the

vessel lighter also helps in increased stability by lowering the center of gravity, min-

imizing the moment of inertia and thus reducing the roll moments. But this approach

might not be feasible for an FPSO which houses a drilling unit on its deck. Accord-

ing to [Kasten 2002], active stabilizers can cause up to 90�

roll reduction but they

are most effective only when the ship is moving at its maximum speed. Stabilizer

fins and rudder-roll control are based on lift force generation with forward speed and

therefore not applicable for stationary vessels such as FPSOs. They are also rela-

tively expensive and complex to install. Using anti-roll tanks, roll reductions in both

amplitude and acceleration to the order of 50�

to 60�

have been possible. Vessel

speed is not an issue in this case. The main disadvantage is the added displacement

required to carry the extra deadweight of the tank contents. Space provision for anti-

roll tanks can lead to compromises in spaces for interior and storage. Another major

disadvantage is the possible effects on stability of the vessel due to the large free

surface effect in the tank. Orienting the ship into the wave direction using thrusters

5

is one desirable way of reducing ship motions, but, in rough weather the waves are

random both in direction and individual characteristics.

Bilge keels are appendages that form an obstruction to roll motion. A bilge keel

generally runs over the midship portion of the hull, perpendicular to the turn of the

bilge. According to [Kasten 2002], for sailing vessels, long, low aspect ratio bilge

keels offer roll reduction of the order of 35�

to 55�

and their efficacy is independent

of the vessel speed. There is some added frictional resistance due to increased wetted

surface area. Bilge keels are relatively inexpensive and also simple to build and are

hence widely preferred. They also have the advantage of having no moving parts

and require no more maintenance than that devoted to the hull surface. Properly

designed bilge keels create minimal drag and increase roll period while reducing

roll amplitude. The effectiveness of using bilge keels in FPSO hull roll mitigations

needs to be studied and hence, is the main focus of this thesis.

Accuracy in the prediction of ship motions under extreme conditions and the re-

sulting hydrodynamic loads is of great importance to the ship design process and

is a challenging task. Accurate predictions of the motion are also necessary for

the development of control methods. Early predictions of the ship motions were

based on scale model tests in given wave conditions in wave basins. Though these

tests yield fairly good results, it is cumbersome and expensive to model these tests.

These model tests are still being used, but, are limited by the time and efforts taken

in conducting them. The scale effects also pose a substantial problem in the ex-

periments. It is usually difficult for researchers to get a good correlation between

Reynolds number and Froude number for models of practical hull forms. In con-

trast, theoretical and numerical methods offer greater ease of use and are relatively

6

very inexpensive. A number of techniques were developed for the estimation of the

roll damping moment. Among these are empirical and semi-empirical formulae that

are derived from experimental data. Two types of experiments are generally used

for the estimation of roll damping; free decay and forced roll tests. Development

of numerical methods made ship motion prediction easier and faster. Continuous

advancements in technology has only helped increase the usage and effectiveness

of these numerical methods. Most of the numerical methods till now have been

potential based methods due to the simplicity in modeling the problem and cost ef-

fectiveness in terms of computer time and storage. The lack of proper computational

methods for prediction of ship motions also arose mainly due to the complexity of

the problem and limited knowledge of the actual governing physics. The main cause

of roll damping in the case of a hull, fitted with bilge keels, is the vortex created due

to the separated flow past the bilge keels in addition to the outwardly radiating free

surface waves. Accurately modeling the complex flow around the bilge keels in the

presence of the hull geometry and a free surface acquires great importance in the

process of determining the effectiveness of the bilge keels in roll attenuation.

There are existing potential solvers such as WAMIT (Waves at MIT)�

which are used

to determine the hydrodynamic coefficients; added mass and damping coefficients

for ships undergoing motions under six degrees of freedom. Potential solvers have

been proved fairly accurate in predictions of heave and sway motions. But, potential

solvers fail to predict the roll motion accurately because the flow around the hull

can neither be assumed to be inviscid nor irrotational in the roll case. Viscous ef-

fects dominate and separation of flow past sharp edges play a major role when a ship

�

WAMIT is a registered trademark of WAMIT, Inc. (www.wamit.com)

7

is undergoing roll motion, but, potential solvers can solve only for attached flows.

Hence, there arises a need for solvers which can take into account viscous and sep-

aration effects and predict the flow accurately. A Reynolds Averaged Navier-Stokes

(RANS)�

solver provides a good alternative to potential solvers. The whole motion

problem can be split up into the sum of harmonic oscillations of the ship in still wa-

ter and waves coming in on the restrained ship and the two fields can be investigated

entirely separately [Vugts, 1968]. In this thesis, the aim is to deal with the harmonic

oscillations of the ship hull. For all practical purposes the problem can be assumed

to be two-dimensional and solved for a general rectangular cross-section of a hull.

Strip theory then can be used to integrate the solutions of all the cross-sections along

the ship’s length and an approximate 3D solution can be obtained.

1.3 Objective

The objective of the work presented in this thesis is to develop and validate a two-

dimensional Navier-Stokes solver to solve the problem of radiation due to the roll

motions of a 2D FPSO hull at the free surface. Hulls with and without bilge keels

are considered. The free surface effects and the viscous effects are decoupled and

studied separately to get a better understanding of both phenomena. The present

method aims at capturing the separated flow and the radiated wave profile and at

predicting the roll hydrodynamic coefficients. The ultimate objective of the present

work is to develop a 3D solver which can simulate roll motions of 3D ship hulls.

�

In the present work, investigations have shown that the solution is not affected much by a changein the Reynolds number and hence, the present solver is based on laminar flow alone.

8

1.4 Overview

� Chapter 2 presents a literature review of the past work done on the prediction

of hull motions using various schemes. Also, a brief review of vortex tracking

methods is given.

� Chapter 3 discusses the detailed formulation of the boundary element method.

The chapter then presents results for various applications of the method. Com-

parisons with existing solutions and convergence studies are provided. It also

provides a brief outline of the vortex tracking method.

� Chapter 4 describes the numerical formulation of 2D unsteady Navier-Stokes

solver. The Crank-Nicolson scheme and SIMPLE pressure correction method

are explained.

� Chapter 5 presents the application of the 2D finite volume method to the prob-

lem of a floating hull undergoing forced harmonic motions. It presents the

comparisons between the present solver results and results from theory or ex-

periments. It also presents the formulation and results for the problem of

submerged body undergoing forced harmonic motions.

� Chapter 6 includes conclusions on the present work done and recommenda-

tions for the work to be done in the future.�

�

A copy of this thesis may also be downloaded from the following website:http://cavity.ce.utexas.edu/kinnas/oetheses.html

9

Chapter 2

Literature Review

This chapter reviews literature related to different methods and approaches to study

the roll motion of hull forms. The first section discusses previous literature which

deals with the problem of 2D and 3D hull forms undergoing roll motions in the

presence as well as the absence of a free surface. The second section discusses

literature that deals with vortex tracking methods based on potential theory.

2.1 Hull Motion Prediction

Prediction of roll motion is very important in ship dynamics and has interested re-

searchers for long. Most of the currently available techniques for the analysis of

ship motions and sea loads are based on potential flow assumptions. These potential

solvers have proven adequate in the analysis of sway, pitch and heave motions. But,

these solvers fail to predict roll motion accurately due to their fundamental assump-

tion of irrotationality and absence of viscous effects. Vugts [1968] was probably one

of the first to do a comprehensive study of ship motions and observe the importance

of viscous effects in the case of rolling bodies. Yeung et al. [1996] states that viscous

effects are known to have significant influence on hydrodynamic forces on bluff-

10

shape bodies. Ocean structures in long waves and roll damping arising from bilges

of a ship hull are important example. In the potential methods viscous effects may

be accounted for by empirical, semi-empirical formulations (Tanaka [1960], Ikeda

et al. [1977], Himeno [1981]). The empirical and semi-empirical formulations de-

pend mostly on various model tests and are hence used only on a trial and error basis.

They are also incapable of dealing with motions of bodies with complex geometries.

Another component of damping is the free surface waves which are well predicted

by potential theory. There have been efforts by Fink and Soh [1974], Brown and Pa-

tel [1985], Braathen and Faltinsen [1988a], Cozens [1987] and Downie et al. [1974]

to predict viscous damping without relying on empiricism. But, none of them were

able to model accurately the interaction of hull geometry, vorticity generation and

free surface simultaneously until Yeung and Vaidhyanathan [1994] who developed a

Free-Surface Random Vortex Method . On the other hand, a RANS equations based

technique, naturally incorporates the effect of viscosity and hence, produces better

results in cases where viscosity plays an important role [Sarkar and Vassalos 2000].

They can easily be extended to 3D practical ship forms and the creation of vorticity

in the boundary layer and vortex shedding during separation can be readily tackled.

Among the available techniques to predict vessel motions, the strip theory based

”Seakeeper”�

, or the panel method diffraction codes such as WAMIT (Waves at

MIT) assume inviscid flow and operate in the frequency domain. Klaka [2001]

observes that viscous forces are important and the non-linear nature of roll response

requires time domain modeling. According to Gentaz et al. [1997], viscous effects

are important for rectangular bodies in sway or roll motion. Therefore numerical

�

developed by Formation Design Systems Pvt. Ltd

11

simulations based on inviscid flow theory cannot give satisfactory results. It has

been shown in Yeung and Ananthakrishnan [1992] that for strongly separated flow,

the shear stress is of secondary importance. This is illustrated in Kakar [2002] and

Kinnas et al. [2003], where the flow past a flat plate is determined using both an

Euler and a Navier-Stokes solver. The values of the drag and inertia coefficients

from both the solvers compare very well with each other as well as with experiments

(experimental data presented in Sarpkaya and O’Keefe [1995]).

Some of the past work done on the subject of roll motions includes an investigation

into the eddy-making damping in slow-drift motions performed by Faltinsen and

Sortland [1987]. The authors showed the importance of bilge-keel depth, especially

for low Keulegan-Carpenter numbers. [Sarpkaya, 1995] presented experimental re-

sults for two- and three-dimensional bilge keels subject to an oscillating flow. The

authors conclude that bilge keel damping is affected by the vortex shedding from

the edge of the bilge keel and the use of damping coefficients from flat plates in a

free stream are not necessarily accurate for wall bounded bilge keels. Korpus and

Falzarano [1997] were the one of the first researchers to use a RANS solver to tackle

the problem of ship roll motion. Their work aimed at studying the viscous and vor-

tical flows around the hull corners and appendages in the absence of a free surface.

They performed a series of parametric studies in order to identify the individual

contributions of viscosity, vorticity, and pressure.

Yeung et al. [1998] applied the Free-Surface Random Vortex Method (FSRVM) to

a rectangular ship-like section oscillating in roll motion and compared the hydro-

dynamic coefficients obtained from the method with those obtained from their ex-

periments. Their study shows that the added mass coefficients are not affected by

12

further increase in the amplitude of roll beyond five degrees. A composite model

representing the effect of flow separation on the hydrodynamic moment is also de-

veloped. The moment is expressed as the sum of the added mass inertia, a linear

damping associated with surface wave generation and a quadratic damping associ-

ated with vortex generation. In Yeung et al. [2000], the authors extended the work

to include modeling of the complex flow around the bilge keels. In the FSRVM, the

flow-field is solved by decomposing it into irrotational and vortical parts. The irro-

tational part is solved using a complex-valued boundary-integral method, utilizing

Cauchy’s integral theorem for a region bounded by the body, the free surface and

the open boundary. The rotational part is solved by solving the vorticity equation

using the fractional step method. Results obtained using the solver are compared

to experimental data as well as results obtained by Alessandrini and Delhommeau

[1995] for various bilge keel depths and forcing function amplitudes. The increase

in size of the keels increased the added inertia and the damping coefficients.

Miller et al. [2002] was one of the first to use three-dimensional RANS calculations

to simulate roll motions of a circular cylinder with bilge keels. The numerical results

are compared with experiments performed at the Circulating Water Channel at the

Naval Surface Warfare Center, Carderock Division. The results compared well for

immersed body computations but emerged body results need to be improved further.

These calculations demonstrate that RANS can play an important role in variety of

hull motions in the near future. At the same time Wilson and Stern [2002] presented

results for unsteady simulation of a surface combatant under roll motion. Though the

authors did not have experimental data to validate their results, their efforts showed

the efficacy of a RANS solver in naval architecture applications. Other works in this

area include Sturova and Motygin [2002], where the authors solve, using a multi pole

13

expansion method, a system of boundary integral equations describing the linear

two-dimensional water-wave problem, for a horizontal cylinder undergoing small

oscillations at the interface of two layers of different densities.

Most recently, Felli et al. [2004] conducted free decay roll experiments on a DDG551

ship model with forward sped at the INSEAN facility in order to study the 3D flow

field around the hull. The flow field is resolved in phase with the roll motion using

Laser Doppler Velocimetry (LDV). The study is performed for a bare hull as well as

a fully appended hull (rudder, brackets and bilge keels). Bilge keels are found to be

the major contributors towards roll damping. The authors observe that LDV results

could be improved significantly by using Particle Image Velocimetry (PIV). Bishop

et al. [2004] conducted experiments at the Naval Surface Warfare Center, Carderock

Division to explore the viscous flow field in the region of the bilge keels while the

ship is undergoing roll motions.The model used in the experiments is DTMB model�

5415. Irvine et al. [2004] also conducted towing tank experiments for an advanc-

ing surface combatant (DTMB model 5512) in free roll decay. For free roll decay

experiments, results are presented for all motions under all the six degrees of free-

dom. All the studies conclude that with increasing forward speed, the roll damping

increases. This is attributed to the lift effect caused by the bilge keels. These stud-

ies could be useful for validation tests when the present solver is made capable of

handling 3D flows.

2.2 Vortex Tracking Method

This section does a review of some of the past work done in the field of prediction

and tracking of vortices that are shed from edges using inviscid flow theory. Rott

14

[1956] was one of the first among to consider the effects of viscous separation and

include it into calculations of fluid flow past sharp edges. In the problem of diffrac-

tion of shock waves he modeled the separation of flow by replacing the vortex region

by a single concentrated vortex. Assuming that the flow was irrotational, he argued

that neglecting viscosity would cause only a small deviation from real flow pattern

and solved the problem using dimensional analysis.

Researchers later tried to study problems involving unsteady motion of 2D vortex

sheets past wedges. According to Pullin [1978], the appropriate similarity law for

the wedge starting flow appears to have been originally discovered by Prandtl. Fink

and Soh [1974] later made an attempt to model impulsive flat-plate (zero wedge an-

gle) flow by a finite number of point vortices whose initial strengths and positions

represent a discretized model of the disturbed sheet circulation. Pullin [1978] ap-

plied a model consisting of a vortex sheet, a cut and an isolated vortex developed

by Smith [1968] to the impulsive starting flow past an infinite wedge. In Pullin

[1978] a similarity solution is used to transform the time-dependent problem for the

sheet motion into an integro-differential equation and finite difference solutions to

the same are obtained.

Two-dimensional methods based on a discrete vortex approach were used by Clements

and Maull [1975] and Bearman and Graham [1980] to model vortex sheets. These

methods were later applied to the problem of prediction of ship roll damping by

Standing et al. [1988] based on the method developed by Bearman et al. [1982] and

Cozens [1987]. Later this method was extended to three-dimensions and applied to

ship roll damping problem by Downie et al. [1991]. Graham and Cozens [1988]

adapted the Cloud-in-cell method (Christiansen [1973]) to model the vortex sheet

15

which is shed and rolls up from a single sharp edge. The method is a mesh method

in which a discrete moving point vortex representation of the vorticity field is trans-

ferred to a fixed mesh. Numerical approximation to the velocity field is carried out

on the mesh and transferred back to the moving points as a convection velocity.

Another approach used in modeling vortex sheets was developed by Faltinsen and

Pettersen [1987]. It was based on distributing sources and dipoles over boundaries

and free shear layers. It was applied for oscillating flow over bodies with either

curved surfaces or sharp edges. It was later extended to include the free surface

effects and applied to a 2D floating body with sharp as well as round corners under-

going forced harmonic roll motion by Braathen and Faltinsen [1988b].

16

Chapter 3

2D Boundary Element Method and Its Applications

In this chapter, the first section presents the Boundary Element Method (BEM) or the

Panel Method and its detailed formulation in two dimensions. Next, its application

to a few standard problems are presented for validation purposes. A problem of a

submerged hull undergoing forced harmonic roll motions is solved. The method

is extended to include trailing vortex prediction for flow past a bilge keel (wedge).

The ultimate objective of the present work with 2D BEM is to be able to predict the

vortex shedding past a bilge keel for a 2D hull section thats rolling at a free surface.

The motivation for using BEM is that it requires less computational time and storage

to solve a problem when compared to viscous solvers.

3.1 Background

Boundary Element Method is based on integral equations. Boundary value prob-

lems can be represented mathematically in terms of integral equations by transform-

ing the governing partial differential equations into integral equations relating only

boundary values. The integral representation of a problem relates the main variables

(velocity potential in fluid flows, temperature in heat transfer problems, etc) with

17

functions of their derivatives (velocities and heat flux respectively).

The advantages of using BEM are:

1) Only boundaries need to be discretized, hence, minimal computational storage

and time are used

2) Problems involving infinite or semi-infinite domains can be easily solved since

the boundaries at infinity need not be created

3) Problems involving some kind of singularity or discontinuity can be dealt with

effectively

4) One need not perform any discretization in the plane of symmetry in case of

problems involving symmetry

3.2 Numerical Formulation

This section presents the numerical formulation and implementation of 2D Bound-

ary Element Method�

.

3.2.1 Green’s Theorem

Consider a volume � surrounded by a surface�

as shown in Figure 3.1. Suppose�

and�

are two functions that satisfy the Laplace equation inside � , i.e, ��

and �� inside the volume, then, according to Green’s second identity the

following equation holds:

�

The formulation and numerical implementation of the method is based on the course work of-fered by Dr. Spyros Kinnas in CE 380 P.4, Boundary Element Methods, 2003.

18

n

ν

S

Figure 3.1: Volume � confined by a surface�

�� (3.1)

where,��

is the unit vector normal to the surface�

pointing out of the domain � as

shown in the Figure 3.1.

3.2.2 Application of Green’s Formula for a two-dimensional body

Consider a body � surrounded by a surface� %

in the two-dimensional space, as

shown in Figure 3.2. Consider a unit source at a point � outside � . The potential�

associated with the unit source is given by:

� �� ,� �� (3.2)

where, is the position vector of the point P. Assume a potential�

which satisfies

the Laplace equation outside B.

19

�� (3.3)

Consider a circle��

of radius � � surrounding the point P and a surface��

sur-

rounding the body and the source. Applying the Green’s theorem inside the volume

surrounded by� %

,��

and��

, and considering the limits��

0 and��

0, we

obtain the following equation:

� � � �� ,� � � � ,� � � � �

� (3.4)

The above equation shows that value of the potential�

at any point depends only on

the values of�

and� ! on the body boundary. It can also be seen that the potential

can be expressed as a superposition of the potentials due to distributions of sources

and normal dipoles. The integral equals� �

for a point outside the body, �� on the

body and 0 inside the body. Following the same approach for a function� � that is

harmonic inside the body, we obtain an integral that equals 0 outside the body,��

on the body and� � � inside the body. Adding the two integrals provides an integral

equation for the value of�

on a general 2D body which forms the governing equation

of 2D Boundary Element Method. The governing equations is as follows:

� � � ��

� � � � � �� ,� � � � � � � � ,� � � � � �(3.5)

Consider a body subject to an inflow of a velocity equal to�� ! . If � � ! is the velocity

potential of the inflow and � is the total potential of the resultant flow, then,

20

SP

SB∇2φi = 0

SC

y

x

ni

B

P

∇2φ = 0

n

rP

Figure 3.2: Body B and a unit source P confined in a finite domain��

� � � � � ! (3.6)

where,�

is the perturbation potential or the potential due to the body.

Perturbation potential on the body is normally solved for when the flows involved

are rotational. Choosing� � to be equal to � � ! , the governing equation under total

potential formulation is obtained. Choosing� � to be equal to zero, the governing

equation under perturbation potential is obtained and is as follows:

�

� � �� ,� � � � ,� � � � � � (3.7)

The above equation is a Fredholm integral equation of the second kind for the un-

21

known�

.

The boundary conditions that are required when the body is subject to an inflow are

kinematic boundary conditions applied on the body. The kinematic body bound-

ary condition states that the flow cannot penetrate the body and hence the flow is

tangential to the body at its boundaries.

�� (3.8)

�� ! �� (3.9)

hence, � �� !

��(3.10)

The body boundary condition that is applied when the body itself is under motion is

� �� (3.11)

where,�� is the velocity of the body.

! is substituted for in the governing equation

and a new form of governing equation is obtained.

In a numerical method the governing equation needs to be discretized so that it

can be applied on the discretized domain (discretized boundary in this case). The

boundary is first discretized into a number of straight panels. The integral equation

for perturbation potential formulation can be written in the following discretized

form (the corresponding geometry is shown in Figure 3.3):

22

j

i

i+1

ω

l

X o

Y o

rj i

ri

ri+1

(control point)

i-1

indexing direction

Figure 3.3: Figure showing the discretized body surface and corresponding indexnotation

� ��

��

�� ,� ��

��

�� ,� �� (3.12)

where, j and i are the indices representing the panels on the boundary. The above

discretized equation is to solve for the velocity potential of the� �� panel. The first

term represents the influence of the strength of the source located on the� �� panel

while the second term represents the influence of the strength of the dipole located

on the� �� panel. Source and dipole influence coefficients are defined as

��

�� ,� � �� (3.13)

� ��

�� ,� �� (3.14)

The resulting linear system of equations in terms of�

is given by:

23

�

� � � � � � �

��

�� ! ��

��

� �� (3.15)

Performing some algebra upon the expressions for influence coefficients we obtain:

� � � ( ��

(3.16)

where, ( � � � �� ( � � ) and � � � ��

� ��

� � , � �� (3.17)

� �� , � � ��

� � � �� ,� � � � �� #� �� (3.18)

where, �

and �

are distances as shown in Figure 3.3.

3.2.3 Numerical Implementation

1) The surface of the body is discretized into panels.

2) Large number of panels are concentrated in areas where changes in geometry are

abrupt or large gradients in solution are expected.

3) Straight panels are used.

4) Constant strength dipoles and discrete sources are used.

5) Collocation method is used.

24

6) Influence coefficients due to all the panels are computed for each panel at its

control point and the resulting linear system of equations in�

are solved.

7) Pressure distributions, forces and moments acting on the body are evaluated from�

, based on Bernoulli’s equation.

3.3 Validations of the method

The method is applied numerically to a few problems that have analytical solutions

so that the correctness of the method is verified before it can be applied to the desired

problem.

3.3.1 Prismatic cylinder of circular cross-section

The case of a circle in two dimensions subjected to an inflow in the x - direction is

the simplest of the problems that can be used for validation of the present method.

The details are shown in the Figure 3.4. The number of panels affects the solution of

a numerical scheme but since the present case is a test case, convergence studies have

not been performed and to obtain a good solution the numerical scheme is applied

to the problem with a large number of panels on the circle. The circle is subject to

a sinusoidal inflow,� � ! �

� � � � ( ��. The kinematic boundary condition is applied

where� ! is defined in terms of the inflow velocity. The perturbation potential

�is

solved for and checked against the available analytical solution. Also, the force on

the circle is evaluated based on the added mass of the circle [Newman, 1977] and

then compared with the force obtained from the numerical scheme.

The amplitude of the sinusoidal inflow is taken to be equal to 2 units while the cir-

25

x

y

Uin = 2cos(πt)

R

Figure 3.4: An infinitely long cylinder of circular cross-section subjected to a sinu-soidal inflow

cular frequency, ( is taken to be equal to � units. The radius of the circle is taken

equal to 0.5 units. Since we are considering the circle in an infinite domain only the

surface of the circle is discretized. First, the source and dipole influence coefficients

are found at each control point due to all the panels and then, the perturbation poten-

tial is solved for by using a standard matrix solver. The solution is moved forward in

time and a new�

is solved for by applying a new set of the time dependent boundary

conditions. The analytical formula for�

at a point (x,y) at a certain instant of time

is given as:

� � � � !� � � � � + � � � �� (3.19)

where, R is the radius of the circle and� � ! is the magnitude of the inflow at that

instant of time. Hence,�

on the circle is equal to� � ! . The perturbation potential

26

on the body obtained from the numerical scheme is compared with the analytical

solution in Figure 3.5. The comparison as seen is exact and this is an indication of

the correctness of the method in evaluating�

.

X --->

φ b,pe

rtur

batio

npo

tent

ialo

nth

eci

rcle

-0.25 0 0.25 0.5

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

φ numericalφ analytical

Figure 3.5: Comparison of analytical and numerical values of perturbation potentialon the circle

In the case of an oscillating circle, the force acting on the circle is given by��

��

��, where, � � � is the added mass coefficient in the x-direction and

��is the

acceleration of the circle. The added mass coefficient in the x-direction for an os-

cillating circle is � � � � � � � � [Newman, 1977], where, � is the radius of the circle.

The velocity of the body with respect to the flow is ��

�. Hence, we

obtain the following expression for the force in the x-direction.

��

(3.20)

� 27

� ��

�(3.21)

The amplitude of the term��

� is equal to 4.9348 units.

t/T --->

FX/ρ

onth

eci

rcle

2 2.25 2.5 2.75 3

-4

-3

-2

-1

0

1

2

3

4

Potential flow solver

4.9384

Figure 3.6: Time history of the force on the circle in the x-direction

In the numerical scheme, the pressure is calculated from the relation � � � � .

And the force� �

is evaluated from� � � � " � � � �

� which implies that� �

� � � " � � � + . Force history is plotted for one time period in Figure 3.6. The amplitude

of the force is found equal to 4.9384 units which is very close to the analytical value.

The difference will be further reduced if the number of panels is increased on the

circle.

3.3.2 Prismatic cylinder of elliptic cross-section

An elliptical cylinder subjected to a sinusoidal inflow of velocity � � � � � � ��

is con-

sidered. The details are given in Figure 3.7 The added mass of the ellipse in the

28

x

y

Uin = 2cos(πt)

Figure 3.7: An infinitely long cylinder of elliptic cross-section subjected to a sinu-soidal inflow

x-direction is given by � � � � � � � � , where, b is the minor axis of the ellipse. The

major axis is taken equal to 0.5 while the minor axis is taken equal to 0.25 units.

Following the procedure described in the case of the circle, we obtain the following

expression for the force on the ellipse;

� ��

�(3.22)

� ��

�(3.23)

The force history is plotted in Figure 3.8 and it can be seen that the amplitude of the

force obtained numerically is in good comparison with the analytical value.

29

t/T --->

FX/ρ

onth

eel

lipse

2 2.25 2.5 2.75 3

-1

-0.5

0

0.5

1


1.2345

Figure 3.8: Time history of the force on the ellipse in the x-direction

3.3.3 Prismatic cylinder of square cross-section

This section deals with a infinitely long cylinder of square cross-section subject to

forced harmonic roll motion about the axis of the cylinder. As seen in the previous

two sections, this problem is also treated in an infinite domain. Only the surface

of the square is discretized. The corner portions of a square assume importance

due to sudden changes in the geometry and also due to changes in the fluid flow in

the vicinity of the corner. Hence, the grid in the corner region is refined and large

number of panels are concentrated into that area. The details are shown in Figure

3.9. The boundary condition is applied on ! which is written in terms of the body

velocity. The circular frequency ( of the square is taken equal to � units. The

velocity of a point on the body is:

30

2a

U = rαωcos(ωt)

n

.

∂φ/∂n = U. n

r

Figure 3.9: An infinitely long cylinder of square cross-section subjected to roll mo-tion

! � � � � �� (3.24)

where,�� is the angular velocity of the body,

� is the position vector of the point

being considered and�

its index.

�� ( � � � � ( ��

(3.25)

where, � � is the angular amplitude and is taken equal to 0.05 units. The components

of the velocity are given by:

! � � � � � + � � � � � ( � � � � ( ��

(3.26)

!�� ( � � � � ( ��

(3.27)

Added mass moment of inertia for a square undergoing harmonic angular oscilla-

tions is � �� , where, � is the density of water and � is equal to half the

31

t/T --->

MX

Y/ρ

onth

esq

uare

2 2.25 2.5 2.75 3

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

Potential flow solver0.022377

Figure 3.10: Time history of the moment on the square in the z-direction

length of a side of the square. The length of each side of the square is taken equal to

1 unit. The moment on the body is obtained from the relation:

� � � � �� ( � � � � � ( �

�(3.28)

� � � ��

�(3.29)

In the numerical scheme the moment on the body is evaluated from the force com-

ponents through the relation:

� � � � � � �� " � + � � + � � � � (3.30)

Time history of the moment is plotted in Figure 3.10 and shows good comparison

with the theoretical moment.

32

3.3.4 Prismatic cylinder of cross shaped cross-section

An infinitely long cylinder of a cross shaped cross-section is subject to harmonic roll

motion about the axis of the cylinder. The boundary conditions applied are similar to

the boundary conditions applied in the case of a square cylinder. The cross consists

of four arms in total with each arm making an angle of �� with its neighboring

arm. Each arm is considered to be infinitely thin with thickness tending to zero. A

description of the cross and its discretization is shown in Figure 3.11

a

900

Figure 3.11: An infinitely long cylinder of cross shaped cross-section subjected toroll motion

The components of the velocity on the body at� �� panel are given by:

! � � � � � + � � � � � ( � � � � ( ��

(3.31)

! � � � � � � � � � � ( � � � � ( ��

(3.32)

33

t/T --->

MX

Y/ρ

onth

ecr

oss

2 2.25 2.5 2.75 3-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Potential flow solver0.312

Figure 3.12: Time history of the moment on the cross in the z-direction

Added mass moment of inertia for a cross undergoing harmonic angular oscillations

is � �� , where, � is the density of water and � is equal to the length of an

arm of the cross. The length of each arm of the cross is taken equal to 1 unit. The

moment on the body is obtained from the relation:

� � � � � �� ( � � � � � ( �

�(3.33)

� � � ��

�(3.34)

Time history of the moment is plotted in Figure 3.12 and the amplitude of the nu-

merical moment shows good comparison with the theoretical value.

34

3.4 Roll motions of a submerged body

The objective of the present section is to present results for a submerged body under-

going roll motion using the panel method. The body is considered to be an infinitely

long cylinder with a constant cross-section and hence, the problem can be treated in

two dimensions. Two types of cross-sections are considered and are shown in Fig-

ures 3.13 and 3.14. From the Figure 3.13 it can be seen that the body has rounded

corners. These rounded corners are called bilges. In the Figure 3.14 it can be seen

that the body has sharp fin like projections at its corners. These projections are called

bilge keels. These corners are akin to the bottom corners of a ship hull. For this rea-

son, henceforth, the submerged body is going to be referred to as a submerged hull.

The main parameter that is solved for in the current problem is the added mass coef-

ficient. Once the results are obtained for the 2D problem, strip theory can be applied

to them and the results can be extended into three dimensions. The following section

discusses how the roll added mass coefficient for the hull is evaluated from the force

history.

BILGE

Figure 3.13: Figure showing cross-section of submerged hull without bilge keels

35

BILGE KEEL

Figure 3.14: Figure showing cross-section of submerged hull with bilge keels

3.4.1 Forces and added mass coefficient

� Force: The hull forces are evaluated from the pressure integrated over the

surface area of the hull. The pressure on the hull is obtained from the velocity

potential and the velocities using the Bernoulli’s equation in the following

way:

� � ��

� � � � ��

� � ��

(3.35)

The values at infinity are all assumed to be zero. After non-dimensionalization

of pressure with respect to � � �� (��

%� ), the following expression is ob-

tained for the pressure:

� � � ��

� �� (3.36)

The parameter � denotes the change of potential with time at a fixed point

in space and is evaluated with respect to the inertial system. But in the cur-

rent case, the body undergoes an unsteady motion and hence, the point under

consideration is not fixed in space. A transformation needs to be done on� �

36

in order to account for the change in the location of the point. The following

equation gives the transformation for� � :

� ��

� � � �� + (3.37)

In the above expression, � implies the same as a material derivative does

for a fluid particle.�

� � represents the total change in the value of�

with both

increment in time and the corresponding change in the location of the point. � ��and � ��

denote the x- and y-velocities of the body at the previous time step.� �

and � denote the x- and y-velocities of the fluid particle at the point. The total

velocity � of the fluid particle is obtained from the normal component ! , and

the tangential component " . Since

�is known at every point on the body,

it is easy to calculate� " using either central, backward or forward second

order differences depending on the position of the point. If the control point

is located immediately before a corner, backward difference is used and if it is

located immediately after a corner, forward difference is used. The derivative ! is obtained directly from the boundary condition. Once the pressure is

evaluated, the forces can then be obtained by integrating pressure as given in

Section 3.3.1. A simple check is performed on the pressure evaluation method

by comparing the potential solver results for pressure with analytical values

in the case of a heaving circle in infinite fluid domain. The analytical pressure

on the circle is given by:

� � ! � � � � � � � � � � � � �! + (3.38)

where,!

is the heave velocity of the circle,�

is the angle made with the

37

positive y-axis and + is the y-coordinate of the point and�!

is the acceleration

of the circle in the y-direction. The numerical pressure is plotted against the

analytical pressure at two time instants and the comparison is shown in Figures

3.15 and 3.16. The numerical pressure matches exactly with the analytical

pressure at both the time instants thus proving the validity of the pressure

evaluation method.

S (arc length/B) -->

P/(

ρU2 )

1 2 3

-1

-0.5

0

0.5

1

numerical pressureanalytical pressure

Figure 3.15: Comparison between numerical (BEM) and analytical pressure on aheaving circle at ��

� Added mass: For roll, according to linear potential theory, the hydrodynamic

moment can be written as a linear combination of the inertia and damping

terms.

� � � � � � � �� (3.39)

38


P/(

ρU2 )

1 2 3

-0.1

-0.05

0

0.05

numerical pressureanalytical pressure

Figure 3.16: Comparison between numerical (BEM) and analytical pressure on aheaving circle at ��

where, � �� is the roll added-mass coefficient;� �� is the roll damping coeffi-

cient;�� and

�� are the angular acceleration and velocity. These can be ob-

tained by differentiating the expression of the roll angle, � , with respect to

time. Expanding the angular acceleration and velocity terms we obtain:

$� � � � � � � � ( � $� �� ( $�� ( $� �� ( $

��

(3.40)

The above expression can be identified as a Fourier series. The added mass

coefficient can be calculated by extracting the Fourier coefficient of the pri-

mary frequency over a period�

. The following expression is obtained for the

coefficient:

39

� � ( � $� ��

$� � � � � � � � � ( $�� $

� (3.41)

where, the variables$� � � � � and

$� are dimensional variables. In order to make

the coefficients non-dimensional, a normalization similar to [Yeung et al. 2000]

is performed.

� �� $� ��

� � � � � (3.42)

where,�

is the half-beam of the body and�

is the submerged sectional area

and is equal to ��

units for the present problem.

Introducing the non-dimensional variables in equation 3.41, the following ex-

pression is obtained for the normalized coefficient:

� ��

�

� � � � � � � � � � � ��

� (3.43)

Once the time history of the hydrodynamic moment is obtained, the coeffi-

cients can be obtained by numerically integrating the moment, according to

equation 5.30, using the trapezoidal rule.

3.4.2 2D submerged hull without bilge keels

The 2D hull is subjected to forced harmonic roll motion with the roll angle, � �� ( $

��, where, � � is the roll amplitude and ( is the circular frequency of the roll

motion. Velocity boundary condition is applied on the hull in terms of ! as shown

in the Figure 3.17. All the parameters in the BEM solver are non-dimensionalized

with respect to the corresponding characteristic scales.

40

� Lengths are non-dimensionalized with respect to the characteristic length� �

which is the beam length � of the hull in the current problem

� Time is non-dimensionalized with respect to the time period�

of the roll

motion

� Velocity is non-dimensionalized with respect to the characteristic velocity��

%�

B

U = rαωcos(ωt)

n

.

∂φ/∂n = U. n

r

D

D: Draft

B: Beam

U: Velocity

n: Normal

r: Position vector

bilge

Figure 3.17: Geometry details and boundary conditions for a submerged hull withoutbilge keels undergoing roll motion

Since the time quantity is non-dimensionalized with respect to the time period of

the roll motion, the value of the non-dimensional time period is equal to 1 unit.

Hence, the circular frequency ( is equal to � � units. Similarly the length of the

beam is equal to 1 unit. The amplitude of roll angle is taken equal to 0.05 units.

The radius of the bilge is taken equal to 2�

of the beam length � . The roll angle in

41

non-dimensional form is given by

� � � � � � � � ��

(3.44)

while the angular velocity of the hull is given by

��

(3.45)

The panel method is applied to the problem and the moment history over one time

period is obtained. The moment history for the hull without bilge keels is shown in

Figure 3.18.

t/T --->

Mxy

/ρU

2 B2

2 2.25 2.5 2.75 3

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Figure 3.18: Time history of the moment on the hull without bilge keels undergoingroll motion

Applying Fourier transformation to the moment history we obtain the added mass

coefficient � �� . The value of the non-dimensional added mass coefficient obtained

is found equal to 4.31152E-02 for 300 panels. The convergence of the added mass

42

coefficient with increasing number of panels is shown in Figure 3.19. The log plot

of error of the added mass coefficients �� square of number of panels is plotted in

Figure 3.20. The solution from the finest grid is assumed to be the best solution or

the exact solution.

N (number of panels) -->

a 66

100 200 300

0.04285

0.0429

0.04295

0.043

0.04305

0.0431

Figure 3.19: Convergence of the roll added mass coefficient � �� with respect tonumber of panels on the hull (without bilge keels) surface

3.4.3 2D submerged hull with bilge keels

This section presents the results for a 2D submerged hull with bilge keels undergo-

ing roll motion. The boundary conditions applied in the present problem are same

as those applied for the hull without bilge keels. The corner regions of the hull as-

sume importance due to the presence of the bilge keels where sudden changes in

the geometry occur. Hence, these regions need to be concentrated with a lot of grid

43

N2 -->

Err

or

50000 100000

10-5

10-4

Figure 3.20: Error convergence plot for the roll added mass coefficient obtained fora submerged hull without bilge keels

points. Full-cosine spacing is applied on all the sides of the hull. Geometry details

and boundary conditions are shown in Figure 3.21. The length of the bilge keel is

taken equal to 4�

of the beam length � . The moment history for one time period

is shown in Figure 3.22. The non-dimensional added mass coefficient obtained by

applying Fourier transformation is found equal to 6.5816E-02.

3.5 Oscillating hull at free surface

This section presents the radiation problem due to a hull undergoing heave motion

at free surface. Modeling and tracking the movement of a free surface due to a

disturbance is both interesting and challenging. In the present problem to reduce

the difficulty in modeling the free surface, linear wave theory is applied. Basically,

44

B

U = rαωcos(ωt)

n

.

∂φ/∂n = U. n

r

D

bilge keel

Figure 3.21: Geometry details and boundary conditions for a submerged hull withbilge keels undergoing roll motion

t/T --->

Mxy

/ρU

2 B2

2 2.25 2.5 2.75 3

-0.1

-0.05

0

0.05

0.1

Figure 3.22: Time history of the moment on the hull with bilge keels undergoingroll motion

45

linear wave theory assumes that the amplitude of the waves is very small compared

to their wave length. This assumption is valid for the present problem as long as the

heave motion of the hull is small in magnitude. The problem is solved for a range of

frequencies.

3.5.1 Boundary Conditions

A graphical description of the boundary conditions that are applied for the heaving

hull is given in Figure 3.23. The coordinate system is summed up by the location

of the origin and the orientation of coordinate axes. As seen in the Figure 3.23, the

origin is located at the center of floatation of the hull. The positive x-axis points

to the right side of the domain and the y-axis is positive in the upward direction.

The method is fully non-dimensional and the equations are non-dimensionalized

with draft�

, time period�

and velocity� �

�� as the length, time and velocity

reference scales. Since the heave motion is symmetric with respect to the y-axis,

only half the domain is considered for solving the problem. The domain consists of

four boundaries�; hull, free surface, far domain and the symmetry boundary. The

conditions that are applied on the boundaries are as follows:

� Hull boundary condition -

In 2D panel methods the boundary conditions are applied either in terms of�

or ! . On the hull, the kinematic boundary condition is applied. The kinematic

boundary condition ensures that the fluid particles on the surface of the body

do not have a velocity component normal to the body, i.e, the fluid particles

�

The solver is capable of handling irregular boundaries and hence, the boundaries need not benecessarily flat.

46

can only have components tangential to the surface of the body or the fluid

particles can only slide along the surface of the body.

� �� (3.46)

where,��

is the unit normal vector at a point on the surface of the body and��

is the velocity of the body. The velocity at a point on the body is given by

�� ( ��

(3.47)

where,��

is the amplitude of the heave velocity and ( is the circular frequency

of the heave motion. The amplitude is taken equal to 0.05 units and the non-

dimensional ( is equal to �� units. The equilibrium position of the motion is

assumed to be at the mean free surface level.

� Free surface boundary condition -

Linear wave theory is assumed to govern the waves that are radiated due to the

hull motion. There are two boundary conditions that need to be satisfied on the

free surface. The first boundary condition known as the Kinematic Boundary

Condition stipulates that the velocity of the fluid particles at the free surface is

equal to the velocity of the free surface. The Kinematic Boundary Condition

in terms of the velocity potential is given by:

� �� (3.48)

where, � denotes the wave elevation,�

the velocity potential of the fluid par-

ticles at the free surface and�

the normal to the linearized free surface. The

47

other boundary condition is known as the Dynamic boundary condition and it

requires that the pressure on the free surface is equal to the atmospheric pres-

sure. The Dynamic Boundary Condition in terms of the velocity potential is

given by:

� �� , � (3.49)

where, , represents the acceleration due to gravity. The above relation is ob-

tained from applying the Bernoulli’s equation on the free surface.

� Bottom boundary condition -

The bottom boundary of the computational domain represents the sea bed. At

the bottom boundary the vertical velocity is denoted by� ! and is equal to zero

because the bed is assumed to be impermeable.� �� (3.50)

� Far field boundary condition -

The far boundary in the problem consists of an extremely far vertical boundary

which represents a virtual boundary situated to the right of the hull at a loca-

tion where we assume that the waves radiated by the hull have not reached

yet. Hence, there is no disturbance felt, or, in other terms there is no velocity

potential at this location. For convenience, at the far right boundary we im-

pose� ! equal to zero since the velocity potential is expected to be zero in the

vicinity�

. Hence, the far boundary condition can be written as

�

The computation is carried out before the waves reach the far right boundary in order that reflec-tion of waves does not occur.

48

O

Y

X

b

D

d

b: half beamD: draftd: depth

hull∂φ/∂n = qn

free surface

∂φ/∂n = ∂η/∂t∂φ/∂t = -gη

∂φ/∂n = 0

φ = 0

symmetryboundary

sea bed

farboundary

Figure 3.23: Geometry details and boundary conditions for a floating hull undergo-ing harmonic heave motion

� �� (3.51)

� Symmetry boundary condition -

Since the heave motion is symmetric about the y-axis of the domain we can

assume a symmetry boundary along the y-axis. Along this boundary there

is no fluid passing across it due to the symmetry and hence the following

condition on�

is applicable,

� �� (3.52)

49

3.5.2 Numerical Implementation

The boundary value problem is solved as an internal flow problem using the panel

method. In an internal flow problem the fluid is bounded by boundaries, solid or

otherwise. In the present problem the hull, the free surface, the virtual symmetry

line under the hull, the sea bed and the far boundary form the boundaries that bound

the fluid.

The problem is considered in deep waters. Since the draft of the hull is chosen as

the characteristic length its length is taken as 1 unit. A hull with a ratio of%� � �

is considered. Hence, the half beam length�

as shown in the Figure 3.23 is equal

to 2 units. The length of the free surface or the extent of the domain is based on

the wavelength of the waves radiated due to the heave motion of the hull. The

wavelength is decided from the frequency of the motion and the extent is taken

such that the wave travels at least 2-3 wavelengths of distance. The bottom of the

domain representing the sea bed is chosen such that it’s depth is greater than half the

wavelength satisfying the deep water conditions.

The boundaries are discretized into a number of straight panels. Each panel consists

of a discrete source and dipole of constant strengths. The end points of each panel

are called grid points and the panel midpoints are called control points. Wherever

there is a sharp change in the shape of the boundary a large number of grid points

are concentrated into that area. In the present problem such instances occur at bilge

region of the hull, the intersection of the hull with the free surface and intersection

of the hull with the symmetry boundary. Full cosine spacing is used on the hull

side and bottom. Expansion ratios are used on the free surface and the symmetry

boundary concentrating more points towards the hull. Care is taken to provide at

50

least 15 panels per wavelength to capture the wave profile correctly near the hull.

The absence of lot of panels away from the hull numerically dampens the wave and

thus avoids the reflection problem, if any.

The source and dipole influence coefficients are calculated at each panel due to all

the panels and stored in separate arrays. The discretized governing equation is ap-

plied on each panel at its control point. At each control point either�

or ! is known

depending on the boundary condition applied. Thus, a linear system of equations in

the unknowns ! and

�is obtained. For the heave problem, the linear system of

equation is as follows:

��

�� % ��

�� %

� � ��

% � % � % % % � � % � �� % ��

� � % � � � � �

��

��

� ��

� %� �� ! �

��

��

��

�� % �

��

� ��

��

� % �

� � ��

� % � � %�

� % % � % � % ��

��

�� % �

��

�� % � � � � �

��

��

! � ! � ! %� ! ��

��

The above system of linear equations were solved using a matrix solver based on

LU Decomposition � . This system of linear equations is solved every time step and

LUD solver is useful because the coefficient matrix does not change with time.

LU Decomposition (LUD) solver denotes Lower Upper Decomposition matrix solver based on

inversion of the matrix.

51

3.5.3 Time-marching

The velocity potential�

is assumed to be known on the free surface at the first time

step of the problem. The motion of the hull is assumed to start from zero and hence

the potential on the free surface is assumed to be zero. At the other boundaries ! is known and the resulting linear system of equations is solved as discussed in

the previous section. On the free surface ! is solved for. Once the value of

! is

obtained the kinematic free surface boundary condition is applied to find the wave

elevation � ;

� ��

� ! � � � � ! � � � � �� (3.53)

Once the elevation at the new time step is obtained the Dynamic free surface bound-

ary condition can be applied to obtain the�

at the new time step;

� �� , � � � ! � � � � ! � � , � ! � �

(3.54)

A non-dimensional number called Froude number denoted by��

is obtained when

an expression containing , is non-dimensionalized. It is based on the frequency, ,and the hull draft

�. It is also called reduced frequency and is defined as follows:

� � �)(�� , (3.55)

After non-dimensionalization,�

at the new time step is obtained from the following

expression:

52

� ! � � � � ! � � � � �� ! � �

(3.56)

At the first time step first order backward Euler method is used and at later time steps

a second order backward method is used. The whole set of operations discussed

above is repeated at the next time step.

3.5.4 Forces and Hydrodynamic coefficients

The forces and moment are calculated from the pressure on the hull as explained

in Section 3.3.1. Since only half the hull is used for calculation a zero force in the

x-direction is not obtained as expected. To get the total force in the y-direction the

force has to be multiplied by a factor of two thus accounting for the other half of the

hull. The force history is plotted versus the non-dimensional time in Figure 3.24.

Applying Fourier transform to the force history in the y-direction, we obtain the

added mass coefficient � � � and the damping coefficient� � � respectively. For heave,

according to linear potential theory, the hydrodynamic moment can be written as a

linear combination of the inertia and damping terms.

� � � � � � � � � �� (3.57)

Expanding the velocity terms as a function of time, we obtain:

$� � � � � � �� ( $� � � � � � � ( $�� $� � � � � � � ( $

��

(3.58)

53

t/T --->

Fy/

(ρU

2 D)

0 1 2 3

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Figure 3.24: Force history for a hull undergoing heave motion for� ��

The above expression can be identified as a Fourier series. The added mass and

damping coefficients can be calculated by extracting the Fourier coefficients of the

primary and secondary frequencies over a period�

. The following expressions are

obtained for the coefficients:

�� ( $� � � � ��

$� � � � � � � � � ( $�� $

� (3.59)

�� $� � � � ��

$� � � � � � � � � ( $�� $

� (3.60)

where, the variables$� � � � � and

$� are dimensional variables. In order to make the co-

efficients non-dimensional, a normalization similar to [Newman 1977] is performed.

54

� � � �$� � ��

� (3.61)

� � � � $� � �

� � � ( (3.62)

Introducing the non-dimensional variables in equation 3.25, the following expres-

sions are obtained for the normalized coefficients:

� � � � � � ��

� �

�

� � � � � � � � � ��

� (3.63)

� � � �

� � ��

�

� � � � � � � � � ��

� (3.64)

Vinayan [2004] independently worked on the scheme and presented results for heave

motion. The non-dimensional added mass and damping coefficients are compared

with results presented in Newman [1977] in Figures 3.25 and 3.26. As seen in

the figures, the coefficients obtained from the potential flow solver compare well

with the coefficients presented in Newman [1977]. The deviation of Euler solver

results from the results of other methods at lower Froude numbers is attributed to

the discretization error. At lower Froude numbers larger domains need to be used,

and to save computational time coarse grids are used in the Euler solver.

3.6 Tip Vortex Tracking Method

Potential method, when applied to the problem of floating bodies undergoing har-

monic heave motion, proved satisfactory and the results compare well with theory

55

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

w(D/g)1/2

a 22

BEMNEWMANEULER

Figure 3.25: Comparison of heave added mass coefficients obtained from the BEMsolver [Vinayan 2004] with those presented in [Newman 1977] and obtained fromEuler solver [Kakar 2002]

and experiments. But, when it is applied to the problem of a floating body under-

going harmonic roll motion at the free surface Vinayan [2004], the method fails to

reproduce the coefficients that match with the experiment results. Potential method

tends to over predict the hydrodynamic coefficients and this discrepancy is mainly

due to the assumptions that the fluid is inviscid and the flow is irrotational around

the hull. Viscous separation and shedding of vortices past corners of the hull play

an important role in roll damping. Hence, there arises a need to incorporate these

viscous effects in order to predict roll damping more accurately. The following sec-

56

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

w(D/g)1/2

b22

BEMNEWMANEULER

Figure 3.26: Comparison of heave damping coefficients obtained from the BEMsolver [Vinayan 2004] with those presented in [Newman 1977] and obtained fromEuler solver [Kakar 2002]

tion discusses an attempt to develop a vortex tracking method � that can model the

separation of flow past the bilge keel using a shear layer. After the inclusion of the

shear layer, the flow is still assumed to be irrotational and the fluid, inviscid. Hence

the problem can be treated using potential method by distributing source and dipoles

over boundaries and dipoles over shear layers.

�The present method is based on a method developed by Lee [2002] to predict vortex roll-up

motion in 2D.

57

3.6.1 Numerical Formulation and Implementation

The governing integral equation is as follows:

� � � ��

�� ,� � � � � � � � , � � � � � �� , � � � , � � � � � (3.65)

where,� %

is the surface of the body and tip vortex while�� is the boundary of the

wake sheet. This equation is the same as the governing equation used to model flow

over a hydrofoil with wake panels included at the trailing edge. A bilge keel and a

shear layer attached at its tip are shown in Figure 3.27

n--

n+

Wake

Bilge Keel Tip Vortex

φN

φ1

Figure 3.27: A bilge keel with a trailing wake and a tip vortex

The second integral in the above governing equation can be written as:

�� ,� � � � � � � � � ,� � � � � � � (3.66)

58

where,� � and

� �denote the unit normal vectors on the positive side and negative

sides of the shear layer respectively. The normal components of the total fluid ve-

locities on the two sides of the wake sheet (or shear layer),�! �! � � � �! �

! have to have

the same magnitude in order that the lower and the upper layers of the wake,�� stay

together. The difference in the tangential velocities is what causes the shear layers

to curl as vortex is shed each time step. Hence, �� ! � � �� ! � and the first term of the

above integral goes to zero.

The final equation:

�

� � �� $� � !

$� � � ,� � � � ,� � � � � � ��

�� ,� � � � � �

(3.67)

NUMERICAL TREATMENT:

� At the first time step,� �

at all the panels is assumed to be��

� (Kutta

Condition).

� At the nth time step,� �

at the first wake control point is equal to��

�

while the� �

at the second wake control point is equal to��

� at the� �� time step and so on.

� At each time step the induced velocities due to the body and the wake itself

are found at each of the wake control points and then, on the tip vortex

� The new locations of the panel mid-points at time ��

� are computed using

an Euler scheme

� � � � � ��

� � � � � � �

59

� Once the control points are moved to new locations, coordinates and dipole

strengths of the new panel end points are obtained by interpolating the control

point coordinates and then, the dipole strengths.

� The problem is solved for the new wake and tip vortex geometry and the above

numerical scheme is repeated in the next time step.

3.6.2 Application to flow over a foil

The tip vortex tracking solver is applied to the problem of a 2D foil subjected to

a steady uniform flow with a lateral sinusoidal gust. Description of the problem is

given in the Figure 3.28. The amplitude � �, circular frequency ( , of the gust, and the

magnitude of the uniform inflow are taken equal to 0.025, 3 and 1 units respectively.

Initially a wake consisting of 2 to 3 panels and of certain length (equal to a user-

input percentage of the foil chord length) is considered (Figure 3.29). The tip vortex

radius is also taken from user given input. The trailing wake is expected to grow

in length as vorticity is shed from the trailing edge of the foil each time step and is

expected to exhibit a wavy pattern due to the lateral gust. The results for the wake

are shown in Figure 3.30 for two cases; one without a sinusoidal gust and another

with the sinusoidal gust. As expected, the trailing wake in the case without a lateral

gust is straight, while the wake in the lateral gust case progresses like a wave. The

vorticity is always shed from the trailing edge in the tangential direction (depending

on the direction of the gust) according to the Mangler condition � [Mangler and

Smith 1970] and as seen in Figure 3.31, the present method imposes the condition

well. Note that the method starts with a wake that is non-tangential to both sides

�

The Mangler condition ensures that the velocity at the trailing edge is finite.

60

of the trailing edge and no explicit tangency condition is imposed on the first wake

panel. In the figure, the x-axis and y-axis scales are made independent in order to

show the geometry of the wake and the trailing edge clearly. The lift force on the

foil is plotted in Figure 3.32 against the run time and is sinusoidal with time.

u = αosin(ωt)(sinusoidal gust)

v (uniform inflow)

Figure 3.28: A 2D foil subjected to a uniform inflow with a lateral sinusoidal gust

61

-1.02

-1.01

-1

Trailing edge of the foil

Wake panels

Tip vortex

node

Figure 3.29: Description of the initial wake and tip vortex geometry

u = αosin(ωt)(sinusoidal gust)

v = -1 (uniform inflow)

Figure 3.30: Figure showing trailing wake for a foil subject to a uniform inflow anda lateral sinusoidal gust

62

φN φ1

∆φw = φ1 - φN

+ - Trailing edge

wake

Figure 3.31: Vorticity being shed tangentially into the shear layer

t/T --->

Lif

ting

forc

eon

the

foil

1 2 3

-0.2

-0.1

0

0.1

0.2

Figure 3.32: Time history of the lift force on the foil

63

Chapter 4

Numerical Formulation of 2D Viscous Solver

This chapter discusses the formulation of the two-dimensional unsteady Navier-

Stokes solver�

based on finite volume method. The topics that the chapter covers

include the choice of the computational nodes, method of stepping the solution for-

ward in time, upwinding schemes and the pressure correction method.

4.1 Non-dimensional governing equation

Any fluid flow in nature is governed by some laws of physics. The underlying

physics can be approximated by a set of partial differential equations. A two di-

mensional, unsteady, incompressible and laminar flow is governed by Navier-Stokes

equations. Navier-Stokes equations consist of the continuity equation and the mo-

mentum equations in the x- and y-directions. The vector form of the continuity and

momentum equations for an incompressible and viscous fluid can be written in the

following way:

�

The new solver is being developed by Yi-Hsiang Yu, a doctoral graduate student in the OceanEng. group at UT Austin. The solver is based on the Euler solver developed by Kakar [2002], whichin turn was based on Choi [2000], Choi and Kinnas [2000] and Choi and Kinnas [2003], and recentlymodified by Dr. Hanseong Lee, a Research Associate in the Ocean Eng. group at UT Austin.

64

��

� $� � � (4.1)� � $��

� $� �� $� �

��

� $� � � $� � ��

�� $�

(4.2)

Where� $�

represents the total velocity,� $�

the body force per unit mass,$� the density

of the fluid,$� the pressure,

$� the time and � the kinematic viscosity. In the above

equations, the hat ($) implies a dimensional variable. The dimensional variables

in the governing equation are non-dimensionalized with respect to the following

characteristic variables;

� Characteristic length L, which typically is a main length dimension specific to

each problem

� Time period T

� Velocity��

, which is equal to��

After non-dimensionalization, the Navier-Stokes equations can be rewritten as

� ��

��

(4.3)

where the column matrices��

,�

and�� represent the following:

��

� � � � ��

��

� �� (4.4)

65

where �� denotes the Reynolds number for the flow and is defined as �� &�� .

Since the numerical scheme is split into solving the momentum equation and the

pressure correction equation, the pressure terms and the body force terms are grouped

and put on the right hand side of the equations. The above equations represent the

conservative form of the equations.

4.2 Finite Volume Method

In the finite volume method, the solution domain is subdivided into a finite num-

ber of small control volumes or cells by a 2D mesh (the present solver is two-

dimensional). The mesh could be made up of either quadrilateral or triangular cells,

but, the present method is capable of solving only for quadrilateral cells (structured

grids only). The finite volume method uses the integral form of the governing con-

servation equation. After applying the Gauss divergence theorem to the volume

integrals we obtain:

��

��

��

��

�� (4.5)

The integral conservation equation applies to each cell, as well as the whole domain.

If equations for all the cells are summed up, the global conservation equation is

obtained, as surface integrals over the inner cell faces cancel out. The above set of

integral equations can be written in the following semi-discrete integral form:

� �� " � �� !

! ��

� �� " � � � � � � � �� (4.6)

66

Figure 4.1: Geometry details of the cell based scheme

In the above equations�� . � � � + , when solving the x-momentum

equation and� � � , when solving the y-momentum equation.

�� denotes the

length of each face of the cell.!! is the velocity on a face normal to it in direction.

� � is the area of the cell represented by a set of indices:�

and�. From coordinate

geometry, given the end points (Figure 4.1), the area of a quadrilateral is defined as:

��

��

� � �� + � + �

��

� � + � + ��

��

(4.7)

In every numerical scheme a set of points have to be chosen at which the values

of unknown dependent variables are to be computed. The present scheme is based

on colocated grid arrangement. In the colocated arrangement all the variables are

stored at the same set of grid points. In the present solver, the cell centers, instead of

the cell corners are chosen to be the computational nodes and hence, the scheme is

a cell based scheme. The colocated arrangement is advantageous to use in compli-

cated solution domains, especially when the boundaries have slope discontinuities or

the boundary conditions are discontinuous (Ferziger and Peric [2002]). It provides

better accuracy for non-orthogonal grids and specification of boundary conditions at

singular points on the boundary can be avoided. The Figure 4.1 shows cell center

67

based geometry. While calculating the flux, the value on the cell face (at point D) is

needed, and it can be obtained from Taylor series expansion about the point C. The

first derivative of the velocities with respect to space�� and

�� also can be found at

D in a similar way using the values at point C.

4.3 Upwind scheme

When a problem is dominated by convection, the flow direction assumes importance

and the influence of the flow properties at upstream nodes on the downstream flow

field needs to be captured. Upwind schemes help in this regard. The current solver

uses a popular scheme called QUICK (Quadratic Upwind Interpolation for Convec-

tive Kinematics) by Leonard [1979] to evaluate velocity at a cell face. Depending

on the direction of the flow, a parabola is fit to the data at two nodes upstream of the

face center and one node downstream of the face center. The scheme is third order

accurate. Since the solver uses normal velocities at the face centers in the momen-

tum equations, the upstream and downstream data points are located on the normal,

equidistant from each other. If�

,�

, and� �

denote the downstream, the first up-

stream, and the second upstream nodes respectively, then, the normal velocity at the

face center is obtained by,

!! � � ��

�

�!! ��

��

!! �

�

!! �� (4.8)

The above equation is valid only for uniformly spaced nodes.

68

4.4 Time Marching

Crank-Nicolson scheme, a second order accurate method is applied for discretiza-

tion of the unsteady term in the semi-discrete integral equation (equation 4.6). The

scheme is an implicit scheme and unconditionally stable. Crank-Nicolson scheme is

of second order accuracy and is useful when time accuracy is of importance. when

applied to the equation 4.6, we have:

�� ! � �� !��

��

�� " �� !

!�

�� !�� " �� !

!�

�� ! � ��

� � �� " � � � � �� ! � �

(4.9)

where� � represents the present time step and

�� ! � �� represents the unknown

variable that is being evaluated. An iterative scheme is applied to the above equation

to obtain a converged value of the unknown dependent variable. An iterative scheme

is necessitated by the presence of non-linear terms at the present time step on the

right hand side.

4.5 Pressure Correction Scheme

Since there is no independent equation for the pressure in Navier-Stokes equations,

there is a difficulty in solving for the unknowns in the equations. The continuity

equation can be used to obtain a solution to the pressure. A pressure correction

method, SIMPLE (Semi-Implicit Method for Pressure Linked Equations) developed

by Patankar [1980] is applied to the current numerical scheme to solve for the pres-

sure field. In the scheme a correction is applied to both pressure and velocities:

69

� � �� (4.10)

� � �� (4.11)

� � � ��

��

(4.12)

where � � , � � and ��

denote the pressure correction, x-velocity correction and the

y-velocity correction respectively. For mass conservation to be satisfied on a cell,

cumulative flux in the cell has to be equal to zero. Cumulative flux is obtained from

adding the fluxes through all the faces of the cell. Hence, the continuity equation is

enforced on the faces of the cell. Therefore, a correction is applied to the velocities

at the cell face centers:

!! � � �� !

! �� !!� � � �� (4.13)

where!! � � � � is the normal velocity at the center of the cell face;

!! �� is the in-

termediate velocity in the iterations between momentum and pressure correction

equations and!!� � � �� is the correction to the velocity at the face center. The veloc-

ity at the face center is interpolated from its two neighboring cell center velocities

predicted from the momentum equations.

The main velocity variable!! in the normal direction momentum equation can be

substituted with the above expression. Neglecting the convective terms, we obtain

the following relation between the velocity correction and the pressure correction:

70

!!� � � ��

�� (4.14)

� !! � � �� !

! �� (4.15)

Since the corrected velocity satisfies the continuity equation, we can write the satis-

fied continuity equation in terms of face velocities as:

�� " � �� " ! �� (4.16)

As a non-staggered grid arrangement is used, the scheme has to be modified to avoid

the checkerboard oscillation problem:

!! �� !

! ��

� ��

� �� (4.17)

where�!! �� is the interpolated velocity;

! represents the derivative in the normal

direction to the cell face; � � denotes that the derivative is obtained by averaging the

adjacent cell center values and�

denotes that the derivative is obtained directly by a

differencing scheme. The above scheme helps avoid the checkerboard distribution of

pressure. Once the pressure correction is obtained from the equation 4.16, the face

velocities can be calculated from the equation 4.14. The velocities at the cell center

are then calculated from the momentum equations using the face velocities. The

new face velocities are determined by interpolating the adjacent cell center values.

And the whole process is repeated till the corrections reach the value of zero within

a tolerance limit.

71

Chapter 5

Applications of 2D Navier-Stokes solver

This chapter discusses the application of the 2D unsteady Navier-Stokes solver to

a few standard CFD problems and to the problem of a 2D body undergoing forced

harmonic motions. The solver is first applied to a simple problem of channel flow. It

is also applied to capture the wave profile of 2D progressive waves. The method is

then applied to the problem of a 2D floating body undergoing harmonic heave and

roll motions in the presence of a free surface. Modeling the interaction between the

free surface and the body is always a challenging problem and difficult to deal with.

When the solver is applied to the problem of a 2D hull undergoing forced harmonic

roll motions, the pressure evaluated on the hull is found to be erroneous. This forms

the motivation behind uncoupling the viscous and free surface effects and studying

them independently. The viscous solver is then applied to the problem of a 2D

submerged body undergoing forced harmonic roll motions to isolate viscous effects

on the pressure. The problem is solved with respect to an inertial fixed coordinate

system and the results are presented. The results for pressure are compared against

results from the potential solver.

72

5.1 2D Channel Flow

Simulating flow in a 2D channel is a simple test that the present solver can be put

to. The results obtained from the viscous solver can be compared with the analytical

solution available for fully developed flow in a 2D channel to validate the scheme.

A rectangular domain is considered. At the inlet boundary, the x-velocity is given a

uniform value of unity, the vertical velocity is taken equal to zero and the pressure

is extrapolated from inner nodes using a second derivative in x-direction. A no-

slip boundary condition is imposed for the velocities on the top and bottom wall

boundaries of the channel. At the outlet boundary first derivative of the horizontal

and vertical velocities, and the second derivative of the pressure with respect to are

taken equal to zero. The boundary conditions are shown in detail in Figure 5.1. The

results are presented for the horizontal velocity and the pressure inside the domain

in Figure 5.2. As seen from the figure the flow is fully developed and the pressure is

linear downstream of the inlet boundary as expected. Figure 5.3 shows comparison

of the velocity profile at the outflow boundary obtained from the viscous solver

with the classical analytical solution of the parabolic velocity profile in laminar flow

(Couette flow). As seen from the figure, the viscous solver solution for velocity

compares well with the analytical solution.

5.2 Numerical Wavemaker

A 2D hull undergoing harmonic motions at a free surface is bound to radiate waves in

the outward direction or away from the body. These waves play an important role in

damping the motion of the body. Therefore, it becomes necessary for the numerical

scheme/solver to be able to capture the wave profile of the radiated waves well in

73

u = 1

v = 0

∂2p/∂x2 = 0

u, v, ∂p/∂y = 0

u, v, ∂p/∂y = 0

∂(u, v)/∂x = 0

∂2p/∂x2 = 0

no-slip condition

no-slip condition

wall

wall

Inflow Outflow

Figure 5.1: Description of the boundary conditions applied for a 2D channel flow

order to predict the motions accurately. To check the ability of the scheme in this

regard, it is proposed to capture the wave profile of a developed set of progressive

waves. The waves are assumed to be governed by linear wave theory. Under linear

wave theory assumption, the amplitude of the wave is considered to be very small

compared to its wavelength; �� . The analytical expressions for the particle

velocities and the pressure are as follows:

� � � ( � �� (5.1)

� � � ( � �� (5.2)

� � ��, � � � �� (5.3)

where, � � � and � are the horizontal velocity component, vertical velocity compo-

74

Figure 5.2: The velocity and pressure contours for the fully developed flow in a 2Dchannel obtained from the viscous solver

nent and the pressure respectively. In the above expressions, � is the amplitude of the

wave; ( is the angular frequency, and is related to the time period�

by ( � �� ;�

is

the wave number and related to the wavelength� �

as� � �� ; and

� � � ( � . is

the horizontal coordinate, and � , the vertical coordinate ( � � � on the free surface);

� is the time; � is the density of the fluid, and � , the wave elevation, is a function of

both and � and is related to them as � � � � � � � . It should be noted here that all

the formulation in the scheme has been expressed in terms of the vertical coordinate

+ while � is the usual notation used to denote the vertical coordinate in wave theory.

From our definitions, we have � � + , i.e, if � � �, then, + � �

. It can

be seen from the above equations that, for negative � , the velocity components die

down exponentially. Also, for deep water (�� ), the particles move in circular

paths with radius � � � � ��.

75

Figure 5.3: Comparison of horizontal velocity profile obtained from the viscoussolver at the outflow boundary with analytical solution

The domain used for computation is as shown in Figure 5.4. The domain is chosen

such that the boundary at the center of the free surface is shaped like a hull. The

boundary conditions applied at this hull boundary are not of wall but of through-

flow type. Two types of input conditions can be given to the solver. In the first type,

the analytical values for particle velocities are given at the left extreme boundary

and in the second, the analytical values for particle velocities are given at the hull

shaped boundary. The boundary where analytical values of velocities are given,

is considered as the inflow boundary. At least fifteen grid points are necessary to

capture one wavelength numerically and hence, while assigning the number of grid

points on the free surface, care is taken to satisfy this criterion. The bottom of the

domain is kept at a distance from the free surface such that deep water conditions

are satisfied. The length dimensions are non-dimensionalized with respect to the

76

wavelength,� �

, and the time dimensions are non-dimensionalized with respect to

the time period,�

, of the wave. The velocity dimensions are non-dimensionalized

with respect to��

( =�� ) as defined previously. Since wavelength is chosen as the

characteristic length, its value is always equal to unity . The boundary conditions

are summarized in the following section and are also illustrated in Figure 5.4..

∂2(u,v,p)

∂x2 = -k2 (u,v,p)= -k2 p

∂2p

∂x2

∂(u,v,p)

∂y= k(u,v,p)

∂P/∂t = ρgv ∇ Free surface

v = aωe(ky)sin(kx-ωt)u = aωe(ky)cos(kx-ωt)

INFLOW OUTFLOW

Figure 5.4: Description of the boundary conditions applied for a numerical wave-maker

5.2.1 Boundary Conditions

� Inflow Boundary: At the inflow boundary the analytical values of velocity

components and pressure derivative are assigned explicitly;

77

� � �� #��

�

� � �� #��

�(5.4)� � ��

where, the frequency ( is replaced by �� and wave number�

is replaced by�� .� Outflow or Far Boundary: At the far boundaries analytical values of the deriva-

tives of velocity components and pressure are assigned;

� � �� (5.5)� � ��

� Bottom Boundary: The bottom boundary is assumed to be far enough for the

waves not to disturb the fluid particles at the boundary.

� � �

� � � (5.6)� �� + � � ��

� Free Surface Boundary: There are two boundary conditions that need to be

satisfied on the free surface. The first boundary condition known as the Kine-

matic Boundary Condition stipulates that the velocity of the fluid particles at

78

the free surface is equal to the velocity of the free surface. The Kinematic

Boundary Condition is given by:

� �� (5.7)

where, � denotes the wave elevation, � , the vertical velocity of the fluid par-

ticles at the free surface and�

, the normal to the linearized free surface. The

other boundary condition is known as the Dynamic boundary condition and it

requires that the pressure on the free surface is equal to the atmospheric pres-

sure. The Dynamic Boundary Condition in terms of the velocity potential�

is

given by:

� � �� , � � � (5.8)

where, , represents the acceleration due to gravity. The above relation is ob-

tained from applying the Bernoulli’s equation on the free surface.

In the viscous solver, the boundary conditions need to be specified in terms of

� , � and � . Hence, using the above two free surface conditions, an expression

for each of the above variables needs to be derived. Differentiating the equa-

tion 5.8 with respect to time and applying the Kinematic Boundary Condition,

we obtain,

� � �� , � � � (5.9)

Differentiating the above equation with respect to we obtain,

79

� �� , � �� (5.10)

Using the irrotationality condition on velocities, � � � �� , we obtain,

� � �� , � �� + � � (5.11)

Knowing that � � � � � � + � � �� , we can replace # � � # by ( � � and obtain the

following expression for the horizontal particle velocity � ,

( � � � , � �� + � �� + � ( �, � (5.12)

The Bernoulli equation is given by,

� � � � �� , � � � (5.13)

Differentiating the above with respect to time, we obtain,

� ��

� � � (5.14)

Using equation 5.9, we obtain the following expression for the pressure � on

the free surface,

� �� , � (5.15)

80

Differentiating the above equation with respect to + and interchanging the

derivatives for � , we get,

�� + � � ��, � �� + (5.16)

Differentiating the Bernoulli’s equation once with respect to + and once with

respect to time, we obtain,

�� + � � � � ��

� � � �� + � � � (5.17)

Modifying the above equation using equation 5.16, and replacing # � � # by ( � � ,

we obtain the following expression for the vertical velocity component � ,

� �� + � ( �, � (5.18)

The above condition is imposed explicitly, i.e, the vertical velocity on the right

hand side is taken from the previous time step.

Wave elevation and pressure contours at two different time instants are shown in

Figures 5.5 and 5.6. From the figures it can be seen that the scheme evaluates the

pressure correctly and captures the wave elevation very well. The wavelength ob-

tained is equal to unity as observed in the figures. The wave has traversed a distance

of half the wavelength in one half of a time period as expected.

81

X;V1-2 -1 0 1 2

P0.06340.05490.04630.03770.02920.02060.01210.0035

-0.0050-0.0136-0.0221-0.0307-0.0393-0.0478-0.0564

Figure 5.5: Pressure contours under a wave and the corresponding wave elevation at��

X;V1-2 -1 0 1 2

P0.06340.05490.04630.03770.02920.02060.01210.0035

-0.0050-0.0136-0.0221-0.0307-0.0393-0.0478-0.0564

Figure 5.6: Pressure contours under a wave and the corresponding wave elevation at��

82

5.3 Heave and Roll Motions

This section presents the results for the application of viscous solver to 2D har-

monic motions of a floating body. The hydrodynamic coefficients obtained for both

heave and roll motions are compared with results from theory, experiments and other

solvers.

5.3.1 Assumptions

In order to simplify the complex problem of harmonic motions of a floating body

certain assumptions are made. The assumptions made in the present problem are

taken from previous work done by [Kakar 2002], except the inviscid fluid assump-

tion. The present solver is a laminar flow solver and hence, the fluid is assumed to be

viscous. The hull is subjected to forced harmonic motions and decay in its motion

is assumed to be absent.

� Linear wave theory is assumed

� The motions of the hull are assumed to be small; non-moving grid is used

� Forward speed of the ship is assumed to be zero

� Motions are assumed to be uncoupled

� The center of floatation is assumed to be the roll center

5.3.2 Coordinate System and Grid details

The coordinate system used for simulations of hull motions is shown in Figure 5.7.

Its origin coincides with the center of floatation of the hull. The x-axis is positive

83

to the right and y-axis is positive in the upward direction. In the Naval Architecture

convention, the x-axis is positive towards the bow of the ship and the y-axis is posi-

tive to the port side of the ship. The y-axis in the present problem is the same as the

z-axis in Naval Architecture convention.

A C-type structured grid shown in Figures 5.8 and 5.9 is used for the present prob-

lem. One set of grid lines follow the shape of the hull and another set of grid lines

originate from the hull. The index representing the grid lines following the hull is

called�

index and the index representing the other set of grid lines is called�

in-

dex. The C-type grid is advantageous to use due to the nearly orthogonal cells that

can be created near the intersection of the hull and the free surface. An H-type grid

[Kakar 2002] if used, has highly non-orthogonal and coarse cells near the hull�

free surface intersection.

The geometry details of the hull, the extent of the domain, the depth of the domain,

and the number of grid points on each of the boundaries total up as the inputs for the

grid generation code. The beam, draft, bilge radius and bilge keel length (if keels

are present) are the geometry details of the hull. The extent of the domain on either

side of the hull is chosen such that it is equal to at least two wavelengths. In deep

water a wave group travels half its wavelength in one time period and the simulation

is typically run for 3-4 time periods. The number of grid points on the free surface

boundary is given such that there are at least 15 grid points per wavelength. The

study is done for non-dimensional frequency� � � ranging from 0.4 to 1.4. Hence,

the extent of the domain varies from Froude number to Froude number depending on

the wavelength� � � ��

� ! � # . The depth of the domain is chosen such that it satisfies

the deep water criterion,�� .

84

X

Y

∇ SWL Free surface0

Hull

Center of floatation

Far boundary Far boundary

Bottom boundary (sea bed)

Figure 5.7: Depiction of coordinate system and domain for a floating body undergo-ing harmonic motions

5.3.3 Froude number and Reynolds number

The non-dimensional frequency� � � is similar in form to the Froude number used

in typical Naval Architecture problems and is hence called Froude number for con-

venience. It is defined in terms of the circular frequency ( and the characteristic

length�

(draft�

for heave motion and half-beam�

for roll motion) in the following

manner:

� � � �*( � �,

The dimensional value of ( and hence, the time period are obtained from the above

expression for a particular value of the Froude number. The Reynolds number for

the problem is then calculated from the following relation between the characteristic

85

B/D = 2

B

D

Bilge

Figure 5.8: Grid details for a rectangular hull without bilge keels

B/D = 2

B

D

Bilge keel

Figure 5.9: Grid details for a rectangular hull without bilge keels

86

length and the time period:

��

� ��

5.3.4 Heave Motion

The viscous solver is applied to the standard problem of a 2D hull undergoing forced

harmonic heave motions at a free surface. Heave problem is relatively an easier

problem to solve when compared to the roll motion problem. Hence, the heave

problem is used as a validation test for the viscous solver. Also, many researchers

have successfully solved the heave problem and the results are standardized. A

2D rectangular hull, with a Beam/Draft ratio equal to 2 and without bilge keels is

considered for the problem. All the length dimensions are non-dimensionalized with

respect to the draft�

of the hull. Time and velocity are non-dimensionalized as done

previously in Section 3.4.2. Non-dimensional time period and frequency ( are equal

to 1 and �� units respectively. The velocity amplitude is taken equal to 0.05 units.

The following boundary conditions are applied for the problem:

� Hull boundary:

A no-slip boundary condition is applied for velocities on the hull. The pressure

on the hull is calculated from the momentum equation.

� � �

87

� � !�� ( � (5.19)� �� !� �

� � !� � !� � � � " � � !� � �

� Free surface: � �� + � ( �, �� + � ( �, � (5.20)� �� , �

� Far boundaries

� � �

� � � (5.21)� � ��

� Bottom boundary

� � �

� � � (5.22)� ��

Vugts [1968] presented results for a family of cylinders heaving at the free sur-

face. The results have been revised and presented in Newman [1977]. Kakar [2002]

successfully employed the Euler solver to solve the heave problem. Figure 5.10

compares the added mass and damping coefficients obtained from the present vis-

cous solver with the coefficients obtained from Euler solver and Newman [1977].

Method to evaluate the hydrodynamic coefficients is presented in Section 3.5.4. The

88

problem was solved for frequencies ranging from 0.4 to 1.4. As mentioned earlier,

the domain size varies with each frequency and depends on the wavelength. The re-

sults compare well with Newman [1977] for the whole range of frequencies. Since

the heave problem is studied solely to validate the method, stringent convergence

studies have not been performed. Hence, slight deviations from the theoretical val-

ues are acceptable and are within tolerable limits.

ω √ D/g

a 22/(

ρB

2),

b 22/(

ρB

2ω

)

0.5 1 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9a22 (Euler)a22 (Viscous)a22 (measured)b22 (Euler)b22 (Viscous)b22 (measured)

Figure 5.10: Comparison of hydrodynamic coefficients for a 2D hull undergoingheave motion obtained from the present solver with those measured by Vugts [1968]as given in [Newman 1977] and Euler solver [Kakar 2002]

Figure 5.12 shows the plots for pressure contours at various time instants ( �� ,�� and � � ) over a time period. It can be observed from the contours in the plots

that the present solver captures the symmetry in the heave problem. The pressure

on the hull is integrated to obtain the forces and moment on the hull as explained

in Section 3.3.1. It is to be noted here that the solver plots the solution at the cell

centers instead of the cell face centers and hence, the boundaries shown in each

89

time/T --->

Non

-dim

ensi

onal

ized

forc

es,F

X,F

Yan

dM

XY

0 0.25 0.5 0.75 1

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6FxFyM

Figure 5.11: Force history for a heaving rectangular hull over one time period andfor

��

plot do not represent the actual boundaries. The force in the x-direction and the

moment about the z-axis (pointing out of the plane) are zero on the hull due to the

symmetry involved in the problem. Forces on the hull in the x- and y- directions and

the moment are plotted in Figure 5.11 for a duration of one time period. The wave

elevation � is calculated from its relation with the pressure on the free surface. Wave

profiles at various time steps are shown in Figure 5.13. It is to be noted here that,

since linear wave theory is assumed, the grid points on the free surface boundary

are not moved at each time step in the actual computation. The wave profiles are

captured well and the wave elevation at the intersection point between the hull and

the free surface appears to be smooth without any visible numerical errors.

90

P0.150.130.110.090.060.040.020.00

-0.02-0.04-0.06-0.09-0.11-0.13-0.15

t/T=1.00

P0.150.130.110.090.060.040.020.00

-0.02-0.04-0.06-0.09-0.11-0.13-0.15

t/T=0.75X

-6 -4 -2 0 2 4 6

P0.150.130.110.090.060.040.020.00

-0.02-0.04-0.06-0.09-0.11-0.13-0.15

t/T=0.5

Figure 5.12: Pressure contours at different time steps for a 2D rectangular hull un-dergoing heave motion

91

η/B

-5 0 5-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0t/T = 0.5

η/B

-5 0 5-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

t/T = 1.0

η/B

-5 0 5-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

t/T = 0.75

Figure 5.13: Wave profiles at different time steps for a 2D rectangular hull undergo-ing heave motion

92

5.3.5 Roll Motion

The main objective of the work presented in the current section is to apply the vis-

cous solver to the radiation problem of a 2D floating hull undergoing forced har-

monic roll motion, improve upon the results presented by Kakar [2002] and obtain

hydrodynamic coefficients which compare well with the available results from ex-

periments and other solvers. Two kinds of hulls were considered for the problem;

one without bilge keels and the other with bilge keels attached. The length dimen-

sions are non-dimensionalized with respect to the beam � . Time and velocity terms

are non-dimensionalized as described previously in Section 3.4.2. Non-dimensional

time period and frequency ( are equal to 1 and �� units respectively.

The bilge is a rounded bilge and its radius is taken equal to 0.02 units (2�

of B).

When present, the bilge keel’s length is taken equal to 0.04 units (4�

of B). The

bisector axis of the bilge keel makes an angle of � � � with the vertical. Description

of the bilge corner and the bilge keel is given in Figure 5.14 [Kakar 2002]. The work

presented in [Kakar 2002] included results for additional lengths equal to 0.02 and

0.06 units. In the present work, only the 4�

bilge keel is considered for the sake of

thorough validation. The solver, once successful in the case of 4�

bilge keel, can

easily be extended to solve problems with other bilge keel lengths. The boundary

conditions applied at the free surface, far and bottom boundaries are the same as

those applied in the heave motion problem and are presented in the previous section.

The roll angle and angular velocity are defined as follows:

� � � � � � � � ( ��

(5.23)

�� ( � � � � ( ��

(5.24)

� � � � ( � � � � ( ��

(5.25)

93

Computational node

KD : Bilge keel depth

Bilgeradius

Bilge Keel

2D Rectangular Hull Cross Section

Figure 5.14: Bilge and keel geometry details

where, � is the roll angle,�� is the angular velocity, � is the total velocity of the

point being considered on the body, is the magnitude of the position vector, � �

is the amplitude of the roll angle and ( is the circular frequency of the harmonic

motion. An amplitude of 0.05 radians is considered for the roll angle. Since the

present solver is a viscous solver, a no-slip boundary condition is applied on the hull

and the corresponding boundary condition is as follows:

� Hull boundary: The boundary conditions are applied at the centers of the faces

that form the boundary;

� � + � � ( � � � � ( ��

� � � � ( � � � � ( ��

(5.26)� �� !� �� !

� � !� � � � " � � !� � �

94

The boundary conditions are also described in the Figure 5.15.

∇ Free Surface

-∂p/∂n = ∂u/∂t + u∂u/∂x + v∂u/∂yu =q

u = 0, ∂p/∂n = 0

Bottom boundary condition

∂2(u,v,p)/∂x2 = -k2(u,v,p)Far boundary condition

∂2(u,v,p)/∂x2 = -k2(u,v,p)

Far boundary condition

∂P/∂t = ρgv , ∂(u,v)/∂y = ω2(u,v)/g

Figure 5.15: Boundary conditions applied for a body undergoing forced harmonicroll motion at the free surface

5.3.6 Results

The intersection of free surface and the hull is a difficult point to deal with due to

the sudden changes in the boundaries and boundary conditions. Hence, a special

treatment is needed while evaluating flow properties at that point. But, in the present

solver treating the intersection point is avoided since the boundary conditions are

applied only at the face centers of the boundary cells and not at the nodal points.

A C-type grid as described in the previous section and shown in Figures 5.8 and

5.9 is used for the problem. The roll problem is also solved for the same range

of frequencies as the heave problem, i.e, 0.4 - 1.4. For the present problem, the

frequency or the Froude number is defined as�� ( � �� , where,

�represents the

95

half-beam of the hull. The extent of the domain is chosen such that its magnitude

is greater than at least two wavelengths. The simulations are usually run for a total

of three time periods. The simulation is started with a ramp function to start the

solution smoothly;

� � � � � � � � ( ��

(5.27)

� � � � � � � � � � � ( �� ( �

�(5.28)

�� ( � � � � � ( �� ( �

� � � � � � ( �� ( �

� �(5.29)

The hyperbolic tangent ramp function ramps up to a value of unity within half a time

period. Once the force history for the last time period is obtained, it is used obtain

the added mass and damping coefficients. The non-dimensional added mass and

damping coefficients are determined from the following expressions [Kakar 2002]:

� ��

�

� � � � � � � � � ��

� (5.30)

� �� %

� ��

�

� � � � � � � � � � � ��

�

where, � �� and� �� are the added mass and damping coefficients respectively.

� � � � �represents the non-dimensional moment as a function of time. The added mass and

damping coefficients can also be evaluated directly from the moment history plot

over one time period. The added mass coefficient can be found from the value of the

moment at �� ( � � � � or �� ( � � �& � � , while the damping

coefficient can be found from the value of moment at �� ( � � � � or

�� ( � � � � � using the following expressions:

96

� �� (5.31)

� �� (5.32)

The Figure 5.16 also explains how to evaluate � �� and� �� from the moment history

plot.

t/T -->

M/(

ρU2 B

2 )

0 0.25 0.5 0.75 1

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

a66 = M75/(2π2αo)

b66 = -(FnbM100)/(2π2αo)

a66 = M25/(2π2αo)

b66 = -(FnbM50)/(2π2αo)

Figure 5.16: Figure explaining how to evaluate roll added mass and damping coeffi-cients from the moment history plot itself

Hull with no bilge keels:

The pressure contour plots for a hull without bilge keels at various time steps are

presented in Figure 5.18. The solver seems to capture the anti-symmetry involved

in the roll motion well. The time history for hydrodynamic moment (�� ) is

97

shown in Figure 5.17. It can be seen from the figure that the force history converges

within 2-3 time periods. As mentioned earlier, the hydrodynamic coefficients are

evaluated using the time history of the converged moment over the last time period.

The coefficients obtained from the present solver are compared with the coefficients

from Euler solver [Kakar 2002] and experimental results from Vugts [1968] in Fig-

ures 5.19 and 5.20. The added mass coefficients are over-predicted by the present

solver. The difference is larger at lower Froude numbers as seen in the figure. And

the values do not differ much from those obtained from both the Euler solver [Kakar

2002] and the BEM solver. There are some discrepancies in the damping coefficient

values as well.

The wave profile and the hull geometry are plotted together for different instants of

time in Figure 5.21.

Time

Non

-dim

ensi

onal

mom

entM

xy/ρ

U2 B

2

1 2 3 4

-0.05

0

0.05

Figure 5.17: Moment history of a hull without bilge keels undergoing harmonic rollmotions for

�� = 0.8

98

X-4 -2 0 2 4

P0.150.130.110.090.060.040.020.00

-0.02-0.04-0.06-0.09-0.11-0.13-0.15

t/T=0.5

P0.150.130.110.090.060.040.020.00

-0.02-0.04-0.06-0.09-0.11-0.13-0.15

t/T=1.00

P0.150.130.110.090.060.040.020.00

-0.02-0.04-0.06-0.09-0.11-0.13-0.15

t/T=0.75

Figure 5.18: Pressure contour plots at various time instants for a hull without bilgekeels undergoing roll motion

99

ω√b/g

a 66

0.25 0.5 0.75 1 1.25 1.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

BEMEULERVISCOUSVUGTS’S EXPT

Figure 5.19: Comparison of roll added mass coefficients from the present solverwith those obtained from the BEM solver, the Euler solver [Kakar 2002] and Vugts[1968] for a hull without bilge keels

ω√b/g

b66

0.25 0.5 0.75 1 1.25 1.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

BEMEULERVISCOUSVUGTS’S EXPT

Figure 5.20: Comparison of roll damping coefficients from the present solver withthose obtained from the BEM solver, the Euler solver [Kakar 2002] and Vugts [1968]for a hull without bilge keels

100

η/B

-4 -2 0 2 4-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

t/T=0.5η/

B

-4 -2 0 2 4-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

t/T=0.75

η/B

-4 -2 0 2 4-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

t/T=1.00

Figure 5.21: Wave profiles at various time instants for a hull without bilge keelsundergoing roll motion

101

Hull with bilge keels:

A typical grid used for computations is shown in Figure 5.52. The time history for

hydrodynamic moment (� � � � � ) for a hull with bilge keels is shown in Figure

5.22. The roll added mass and damping coefficients are obtained using the equation

5.30. The coefficients obtained from the present viscous solver are compared with

the results from the Euler solver [Kakar 2002] and the experimental results from

Yeung et al. [2000] in Figures 5.23 and 5.24. As seen from the comparison plots,

the present solver over-predicts the added mass coefficients for lower values of the

Froude number and under-predicts the damping coefficients for higher values of

the Froude number, when compared to the experimental values. And, as observed

earlier in the case of a hull without bilge keels, the coefficient values obtained from

the present solver do not differ much from those obtained from both the BEM solver

and the Euler solver. The damping coefficients are under-predicted by all the solvers

compared to the experimental values. The wave profiles obtained at different instants

of time are plotted along with the hull geometry in Figure 5.25.

Time

Non

-dim

ensi

onal

mom

entM

xy/ρ

U2 B

2

1 2 3 4

-0.1

-0.05

0

0.05

Figure 5.22: Moment history of a hull with 4�

bilge keels undergoing harmonic rollmotions for

�� = 0.8

102

ω√b/g

a 66

0.25 0.5 0.75 1 1.25 1.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

BEMEULERVISCOUSYEUNG’S EXPT

Figure 5.23: Comparison of roll added mass coefficients from the present solverwith those obtained from the BEM solver, the Euler solver [Kakar 2002] and Yeunget al. [2000] for a hull with 4

�bilge keels

ω√b/g

b 66

0.5 0.75 1 1.25 1.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14 BEMEULERVISCOUSYEUNG’S EXPT

Figure 5.24: Comparison of roll damping coefficients from the present solver withthose obtained from the BEM solver, the Euler solver [Kakar 2002] and Yeung et al.[2000] for a hull with 4

�bilge keels

103

-4 -2 0 2 4

-0.4

-0.3

-0.2

-0.1

0

0.1

t/T=0.5

-4 -2 0 2 4-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

t/T=0.75

-4 -2 0 2 4

-0.4

-0.3

-0.2

-0.1

0

0.1t/T=1.00

Figure 5.25: Wave profiles at various time instants for a hull with 4�

bilge keelsundergoing roll motion; The vertical axis represents the wave elevation, � scaled bythe beam length, �

104

Analysis:

The hydrodynamic coefficients for roll are derived from the time history of the mo-

ment over one time period. The moment is calculated from the forces which are in

turn calculated from the pressure on the hull. Hence, accurate prediction of pressure

acting on the hull assumes importance in regard to obtaining coefficients within rea-

sonable limits of tolerance. Also, the wave elevation is evaluated from the pressure

on the free surface. The presence of the free surface also affects the evaluation of

pressure on the hull and in turn the damping of the roll motion. But, the wave profiles

shown in Figures 5.21 and 5.25 for a hull without and with bilge keels respectively

are clearly not smooth and exhibit a discontinuity at the free surface-hull intersec-

tion. Figure 5.26 shows the pressure on the hull with bilge keels for��

plotted versus curve length of the hull at �� . The plot shows comparison

between pressure from the viscous solver and pressure from the BEM solver. For

the pressure obtained from the viscous solver, there is a marked deviation from the

pressures obtained from BEM solver and Yeung et al. [2000] (not shown in the fig-

ure) near the free surface. This can be attributed to the inaccurate prediction of the

pressure on the free surface at each time instant. Also, the pressure has remarkable

oscillations near the bilge keel region. Viscous separation is known to play a major

role in damping the roll motion of a hull, with and without bilge keels. But, it can

be seen from the Figures 5.19 and 5.23 that the present solver is not able to capture

the viscous effects well and hence, does not perform better than the inviscid solvers.

These discrepancies highlight the need for further investigation, both into the free

surface modeling and the formulation of the viscous solver. It is therefore, decided to

uncouple both and investigate each independently so that viscous and vortical flow

effects can be isolated from the free surface effects. The following sections present

105

the work done in order to find the shortcomings in the solver and formulation of the

roll problem.

Figure 5.26: Pressure on the hull with 4�

bilge keels for�� = 0.8 at �� = 0.8

(The discrepancies between the pressures from the current viscous solver and other solversshown in the figure led to investigation and changes in the formulation of the solver, whichare presented in the succeeding sections of the chapter.)

5.3.7 Roll motion of a semi-circular hull

A simple test to check the pressure evaluation on the hull is to consider a semi-

circular hull undergoing forced harmonic roll motion in the presence of a free sur-

face. If the fluid is considered inviscid and flow is considered irrotational, then, the

pressure on the hull should be equal to zero at any instant of time. The boundary

conditions applied on the free surface, the far boundaries and the bottom boundary

are the same as the conditions used in the case of a rectangular hull undergoing roll

106

motion at the free surface. The kinematic boundary condition on the hull imposes

that the fluid particles on the hull do not have a normal (to the hull surface) compo-

nent of velocity. The irrotationality condition imposes that vorticity is equal to zero.

The hull boundary condition is applied as follows:

�! � � $� � �$�

(5.33)�! �� (5.34)� � "� � � � (5.35)

If the velocity at a point on the normal in the tangential direction is denoted by!!"

and�� is denoted by � ! , then the x-velocity, y-velocity and the pressure on the hull

are given by:

� � !!" � � � � !

� �(5.36)

� � !!" � � � � !

� �(5.37)� �� !� �

� � !� � !� � � � " � � !� � �

(5.38)

A close-up view of the grid near the hull is shown in Figure 5.27. The pressure on

the hull is plotted versus the curve length in the Figure 5.28 at various time instants.

It can be seen that the pressure is zero at all instants of time as expected.

5.4 Submerged hull motions

5.4.1 Fixed coordinate system and fixed grid

To isolate and uncouple the viscous effects from the free surface effects, the motions

of a submerged body (or hull) in infinite fluid are intended to be studied. Hence, an

107

semi circular hull

Figure 5.27: A close-up view of the grid near the semi-circular hull

S (arc length/R)-->

P/(

ρU2)

1 2 3

-4E-08

-3E-08

-2E-08

-1E-08

0

1E-08

2E-08

3E-08

4E-08

t= 0.005t= 0.100t= 0.125t= 0.250

Figure 5.28: Plot of pressure on the semi-circular hull �� curve length at varioustime instants

108

image of the hull is taken about the line of floatation and the resultant hull, shown in

Figure 5.29, is subject to forced harmonic roll motions surrounded by infinite fluid.

.

Beam, BDraft, D

Bilge Keel

Center of rotation: O

rBLBR

Figure 5.29: Description of the main length parameters for a submerged hull under-going roll motions

At first, a hull without bilge keels is considered. All the length dimensions are non-

dimensionalized with respect to the beam B as in the roll motion of a floating hull.

The%� ratio, the bilge radius and the bilge keel length are equal to the values used

for hull undergoing roll motions at the free surface. The distance between the finite

far boundary and the body is arbitrary. The only criterion in choosing the distance is,

the far boundary should be at least a minimum distance away from the hull such that

the velocities and pressure at the far boundary are not affected by the hull motions.

Figure 5.30 gives the details of the grid used in the present problem. A structured

O-type grid is used and is similar to the grid used in previous study with regard to

the indices. The origin is located at the centroid of the hull and the orientation of the

coordinate axes is the same as that of the axes used in the study of roll motion of a

109

Close up view ofthe grid near the hull

Figure 5.30: a typical grid used for forced harmonic motions of a submerged hull

floating body. The circular shape of the far boundary helps in creating a grid which

is less non-uniform and less non-orthogonal than a square shaped far boundary.

� Boundary Conditions: Exact boundary conditions are applied on the hull, but

the grid and the body are kept fixed. The boundary conditions applied in the

case of roll motion of a floating hull are applied in the current problem. The

only difference is made by the boundaries that act as the free surface in the

floating hull problem, which now coincide with each other and act as a contin-

uum boundary in the present problem. The boundary conditions are described

diagrammatically in Figure 5.31. The values for velocities and pressure on

the continuum boundary are obtained by interpolating the adjacent cell values

on both the sides of the boundary. It is thus ensured that the velocities and

pressure are equal on both the boundaries forming the continuum and the flow

110

is continuous across the boundaries.

U = 0, V = 0

∂P/∂n = 0

Far boundary condition

Continuum Boundary Condition(UA = UB

VA = VB

PA = PB)

U = -yαoωcos(ωt), V = xαoωcos(ωt)

-∂P/∂n = ∂Un/∂t + Un(∂Un/∂n) + Us(∂Un/∂s)

Body boundary condition

A

B

Figure 5.31: Description of boundary conditions applied for the submerged rollproblem

When compared to a 2D Cartesian grid, the body boundary is equivalent to the bot-

tom boundary, the far boundary is equivalent to the top boundary and the continuum

boundary is a coupling between the inflow and the outflow boundaries.

The Reynolds number is evaluated in the similar manner as in the floating body case.

The Reynolds number considered presently is equal to 69503 (= the �� for��

�� in floating body case). The roll amplitude is taken equal to 0.05 units. The

forces and moment on the hull are calculated as detailed in Section 3.3.1. Figures

5.32 and 5.33 show the pressure evaluated on the hull at the time instants �� and �� respectively. As can be seen from the Figure 5.32, there are large

111

S (Arc length/B) -->

P/(

ρU2 )

1 2 3

-0.2

-0.15

-0.1

-0.05

0

Viscous solver (non-moving grid)Potential solvert/T = 2.00

Figure 5.32: Pressure on the submerged hull without bilge keels at �� for anon-moving grid

oscillations in pressure in addition to the single peak in the regions near the bilges.

It is to be noted here that the a sudden substantial rise/drop in the pressure in the

bilge region is expected due to the sudden change in the geometry and nature of

the flow, but, the smaller oscillations are unacceptable. The flow speeds up at the

corner and a peak in the pressure is obtained due to the large magnitude of velocities.

The pressure at �� is smooth overall when compared to the pressure at

�� , but has small local jump in its value at the bilge regions. The occurrence

of these irregularities in the pressure on the hull can be reasoned from the pressure

contour plots at both instants of time. In Figure 5.34, it can be seen that there

pressure oscillations that are spread over reasonable length of the hull near the bilges.

Keeping the body and grid stationary through the duration of the simulation could

be the cause of these pressure patches as the solver might be experiencing memory

or history effects.

112


P/(

ρU2 )

1 2 3

-0.1

-0.05

0

0.05

0.1

Viscous solver (non-moving grid)Potential solvert/T = 2.25

Figure 5.33: Pressure on the submerged hull without bilge keels at �� for anon-moving grid

X

Y

-1 0 1

-1

-0.5

0

0.5

1

P/ρU2

0.00000-0.00750-0.01500-0.02250-0.03000-0.03750-0.04500-0.05250-0.06000-0.06750-0.07500-0.08250-0.09000-0.09750-0.10500

Figure 5.34: Pressure contours around the submerged hull without bilge keels at�� for a non-moving grid

113

X

Y

-1 0 1

-1

-0.5

0

0.5

1

P/ρU2

0.100000.085710.071430.057140.042860.028570.014290.00000

-0.01429-0.02857-0.04286-0.05714-0.07143-0.08571-0.10000

Figure 5.35: Pressure contours around the submerged hull without bilge keels at�� for a non-moving grid

5.4.2 Fixed coordinate system and moving grid

An alternative approach to the problem of submerged roll motion is to move the

grid along with the body while keeping the inertial coordinate system fixed at each

instant of time. This approach is the correct approach and is without any small mo-

tion assumptions. It is expected to remove the memory effects experienced in the

previous approach. It is advantageous while applying the solver to large motions of

the hull as well. Keeping the coordinate system fixed helps in doing without a coor-

dinate transformation every time for the velocities and the grid locations. But, when

the grid is moved every time step, the location of each computational node changes

with time. This is not the case in an Eulerian approach to solving fluid flow prob-

lems, where the attention is focused at a point fixed in space. Hence, a Lagrangian

approach has to be incorporated in time to modify the way time dependent variables

114

are calculated at each node in the current scheme. At a particular node, the unsteady

term in the x-momentum is treated in the following fashion.

� ��

� � � !

� � � � � �� !� � � � � �� + � (5.39)

The � operator is the same operator used in the potential solver for calculating pres-

sure in the case of a moving submerged body. The unsteady term in the y-momentum

equation is treated in the same manner. When the above expression is substituted

for the unsteady term in the momentum equation, a new set of momentum equations

are obtained:

� � ��

� � � � !� � � � � � � �� !

� � � � � � � �� + � � � ��

� � � � (5.40)

where, � � denotes either � or � depending on whether the momentum equation is

written in the x-direction or the y-direction. Since the velocity of a fluid particle

on the body is equal to the velocity of the body at that point (no-slip boundary

condition), the convective terms in the above momentum equations vanish on the

body boundary. A special care needs to be taken while using the Crank-Nicolson

scheme for solving the problem. While evaluating the derivatives at the� �� time

step, old locations of the corresponding points need to be considered and hence, it is

necessary for the scheme to store the geometry data of the previous time step.

The modification is explained diagrammatically in Figure 5.36. In the figure the

body is shown at two successive time steps and the movement of an arbitrary node is

followed. The movement of the hull is exaggerated in order to present the changes

115

t

t + ∆t

1

2∆Y

∆X

∇U/∇t = (U(2,t+∆t) - U(1,t))/∆t

unode = ∆X/∆t

vnode = ∆Y/∆t

Figure 5.36: Figure explaining the terms used in transformation of the unsteady termin the Navier-Stokes equations for a moving grid in a fixed inertial coordinate system

clearly. Figure 5.37 shows the positions of the body and the grid at two time instants;

�� and �� , with one overlapped over the other. It is to be noted here that

only cell centers are plotted and hence the boundaries do not represent the bound-

aries of the actual geometry. Figures 5.38 and 5.39 show the pressure on the hull

at �� and �� plotted versus the hull arc length. The plots compare the

pressure obtained from viscous solver with that obtained from the potential solver. It

is clear from the Figure 5.38 that the spikes observed in the pressure in Figure 5.32

are smoothed out and what have been understood as memory effects are not present

anymore. Also, the pressure from the viscous solver compares very well with the

pressure obtained from the potential solver. The pressure evaluation method in the

potential solver has been validated in Section 3.4.1 and hence, the potential solver

results can be used as a benchmark though they are purely inviscid results. The pres-

116

X

Y

0.3 0.4 0.5 0.6 0.70.3

0.4

0.5

0.6

______ t/T = 2.00

........ t/T = 2.25

Figure 5.37: Grid orientation for a submerged hull without bilge keels at �� and ��

sures from both the solvers compare well at both �� and �� . This can

be attributed to the fact that the effect of viscosity and vorticity are minimal in the

present problem. This can be observed clearly from the vorticity plots shown later

in the section. These results need to be validated by running the solver with inviscid

fluid assumption and comparing the resulting pressure with the pressure obtained

from the potential solver.

Added Mass coefficient

The forces are obtained from pressure as explained in Section 3.3.1. And the added

mass coefficient for roll motion of the submerged hull is evaluated from the expres-

sion obtained in Section 3.4.1. The added mass coefficient can also be evaluated

from the moment history plot using the equation 5.31 by multiplying the resultant

added mass coefficient with �� to account for the image of the hull in the submerged

117


P/(

ρU2 )

1 2 3

-0.2

-0.15

-0.1

-0.05

0

Viscous Solver (moving grid)Potential Solver

Figure 5.38: Pressure evaluated on the submerged hull without bilge keels at �� using a fixed coordinate system and a moving grid in the case of viscous solver


P/(

ρU2 )

1 2 3

-0.1

-0.05

0

0.05

0.1

Viscous Solver (moving grid)Potential Solver

Figure 5.39: Pressure evaluated on the submerged hull without bilge keels at �� using a fixed coordinate system and a moving grid in the case of viscous solver

118

X

Y

-1 0 1-1

-0.5

0

0.5

1

P/ρU2

0.0000-0.0075-0.0150-0.0225-0.0300-0.0375-0.0450-0.0525-0.0600-0.0675-0.0750-0.0825-0.0900-0.0975-0.1050

Figure 5.40: Pressure contours around the submerged hull without bilge keels at��

X

Y

-1 0 1-1

-0.5

0

0.5

1

P/ρU2

0.10000.08570.07140.05710.04290.02860.01430.0000

-0.0143-0.0286-0.0429-0.0571-0.0714-0.0857-0.1000

Figure 5.41: Pressure contours around the submerged hull without bilge keels at��

119

case;

� �� (5.41)

The hydrodynamic moments from both viscous and potential solvers are compared

with each other in Figure 5.42 for one time period. The added mass coefficients are

tabulated and compared with each other in Table 5.1. As observed from the moment

comparison plot and Table 5.1, the difference in the added mass coefficients obtained

from the two solvers is very small.

Solver Viscous Potential� �� 4.22524E-02 4.312122E-02

Table 5.1: Comparison of roll added mass coefficients obtained from viscous andpotential solvers for a submerged hull without bilge keels undergoing roll motion

Figures 5.43, 5.44, 5.45 and 5.46 show the converged flow field at four different

time instants over the last period of the simulation for the hull without bilge keels.

Since separation effects are minimal for a rounder bilge corner, strong vortices are

not created. It can be seen in the figures that the vorticity created is small and is very

local in the bilge region.

5.4.3 Convergence Studies

To validate the results obtained for a submerged hull without bilge keels undergo-

ing a forced harmonic roll motion, a grid study is performed. The convergence of

pressure and hydrodynamic moment on the hull is checked for increasing number of

cells keeping the domain size constant. Simulations are performed for only one time

120

t/T -->

M/ρ

U2 B

2

0 0.25 0.5 0.75 1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

VISCOUSBEM

Figure 5.42: Comparison between hydrodynamic moment obtained from viscousand potential solvers for a submerged hull without bilge keels undergoing roll mo-tion

X

Y

-0.5 0 0.5

-0.6

-0.4

-0.2

0

0.2

0.4

0.6Z Vorticity(Normalized)

0.17860.10000.0214

-0.0572-0.1357-0.2143-0.2929-0.3714-0.4500-0.5286-0.6071-0.6857-0.7643-0.8429-0.9214

t = 0

Figure 5.43: Flow field with respect to the submerged moving hull without bilgekeels, shown around the hull at time instant � ��

121

X

Y

-0.5 0 0.5

-0.6

-0.4

-0.2

0

0.2

0.4


0.26620.18180.09740.0130

-0.0714-0.1558-0.2403-0.3247-0.4091-0.4935-0.5779-0.6623-0.7468-0.8312-0.9156

t = T/4


X

Y

-0.5 0 0.5

-0.6

-0.4

-0.2

0

0.2

0.4


0.92060.84120.76180.68240.60300.52360.44420.36480.28540.20600.12660.0472

-0.0322-0.1116-0.1910

t = T/2


122

X

Y

-0.5 0 0.5

-0.6

-0.4

-0.2

0

0.2

0.4


0.91350.82710.74060.65420.56770.48120.39480.30830.22180.13540.0489

-0.0375-0.1240-0.2105-0.2969

t = 3T/4


period to save computational time. The convergence study is performed on grids of

three different sizes; 224 x 78, 272 x 78 and 320 x 78. While denoting the size of the

grid, number of grid points along the hull or the�

index is given by the first number

and the number of grid points along the�

index is given by the second. The number

of grid points along the�

index is kept constant but the expansion ratio is varied

such that an increase in the number of grid points on the hull is complimented by

an increase in the expansion ratio. An increase in the expansion ratio creates larger

cells near the far boundary and smaller cells near the hull. Figure 5.47 compares

the densities of the grids close to the hull. Figures 5.48, 5.49 and 5.50 compare the

pressure on the hull obtained for the three different grids at three instants of time.

The hydrodynamic moment on the hull for one time period is also compared for the

three grids and is shown in Figure 5.51. All the comparison plots show that the solu-

tion converges to the expected solution with increasing number of cells. Numerical

123

accuracy can be compromised to some extent by performing the simulations with a

grid coarser than the finest grid in order to save computational time.

5.4.4 Hull with bilge keels

The solver is also applied to model the roll motions of a hull with bilge keels. The

simulations are run for just one time period for different Reynolds numbers and

the results are compared for varying Reynolds number to investigate the effect of

Reynolds number on the solution. The pressures for varying Reynolds numbers

are compared at three different time instants; �� and are

plotted in Figures 5.53, 5.54 and 5.55. As can be seen from the figures, the solution

does not vary much when the Reynolds number is increased from � � �� to� � �� .

The solution diverges for all the Reynolds numbers near �� and this can

be seen in Figure 5.55. The viscous solver results for pressure are compared with

potential solver results in Figures 5.56 and 5.57. Results for the bilge keels are very

preliminary in nature and hence, the differences between the pressures from the

two solvers cannot be attributed to just the viscosity and separation effects without

stringent validation tests. The moment history for the first time period is compared

between the two solvers in Figure 5.58. The divergence of the solution in the viscous

solver case is clearly seen in the figure.

124

X

Y

0.25 0.5 0.75 1

-0.6

-0.5

-0.4

-0.3

272 x 78

X

Y

0.25 0.5 0.75 1

-0.6

-0.5

-0.4

-0.3

320 x 78

X

Y

0.25 0.5 0.75 1

-0.6

-0.5

-0.4

-0.3

224 x 78

Figure 5.47: Comparison of the grid densities around the submerged hull withoutbilge keels used in the convergence study

125


P/(

ρU2 )

1 2 3

-0.1

-0.05

0

0.05

0.1

224 x 78272 x 78320 x 78

t/T = 0.25

Figure 5.48: Comparison of the pressure on the submerged hull without bilge keelsfor increasing number of cells at ��


P/(

ρU2 )

1 2 3-0.2

-0.15

-0.1

-0.05

0

224 x 78272 x 78320 x 78

t/T = 0.50


126


P/(

ρU2 )

1 2 3

-0.1

-0.05

0

0.05

224 x 78272 x 78320 x 78

t/T = 0.90


t/T -->

M/(

ρU2 B

2)

0.25 0.5 0.75 1

-0.15

-0.1

-0.05

0

0.05

0.1224 x 78272 x 78320 x 78

Figure 5.51: Comparison of the hydrodynamic moment on the submerged hull with-out bilge keels between three different grids for the first time period

127

Close up view ofthe grid near the hull

Figure 5.52: a typical grid used for computation of forced harmonic motions of asubmerged hull with bilge keels


P/(

ρU2)

1 2 3 4

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Re 1000Re 10000Re 20000Re 69503

t/T = 0.25

Figure 5.53: Comparison of the pressure on the submerged hull with bilge keelsundergoing roll motion for varying Reynolds number at ��

128


P/(

ρU2)

1 2 3 4-0.2

-0.15

-0.1

-0.05

0

Re 1000Re 10000Re 20000Re 69503

t/T = 0.5



P/(

ρU2)

1 2 3 4

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Re 1000Re 10000Re 20000Re 69503

t/T = 1.00


129

S (arc length/B)-->

P/(

ρU2)

1 2 3 4

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

ViscousPotential

t/T = 0.25

Figure 5.56: Comparison of the pressure on the submerged hull with bilge keelsundergoing roll motion between viscous and potential solvers at ��

S (arc length/B)-->

P/(

ρU2)

1 2 3 4

-0.2

-0.15

-0.1

-0.05

0

ViscousPotential

t/T = 0.5

Figure 5.57: Comparison of the pressure on the submerged hull with bilge keelsundergoing roll motion between viscous and potential solvers at ��

130

time -->

M/(

ρU2B

2 )

0.25 0.5 0.75 1-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

ViscousPotential

Figure 5.58: Comparison of the hydrodynamic moment on the submerged hull withbilge keels undergoing roll motion between viscous and potential solvers for the firsttime period

131

Chapter 6

Conclusions and Recommendations

6.1 Conclusions

FPSOs operating in deep waters need to remain stationery for long durations of time.

But, they encounter various environmental loads which result in motions in all the

six degrees of freedom. Of these motions, roll motion is a cause of serious concern.

Bilge keels are one among the many anti-roll devices in use. The goal of the present

work is to study the roll motion and investigate the effectiveness of the bilge keels

in mitigating roll motions of FPSO hulls.

Conventional numerical solvers have been based on potential theory and have proved

adequate in studying most of the problems concerning sea keeping and ship manoeu-

vring. But, in the case of ship roll motions, the potential solvers have proved grossly

inadequate due to the major role the viscosity and separation effects play in damp-

ing of the motion. The empirical methods developed to improve upon the potential

solvers have proved satisfactory on conventional body forms but fail in case of ad-

vanced and complex hull designs. Hence, a robust numerical solver is the present

need in order to study and predict roll motions accurately. Accurate prediction of

roll motion can help in determining loads on the deck and also the range of operabil-

ity of the vessel. A solver which incorporates viscosity effects into its formulation

132

can fill in the gaps left by conventional solvers.

In the current work, a method based on 2D finite volume method has been developed

for solving the unsteady Navier-Stokes equations for laminar flows. Since the free

surface waves also play an important role in roll damping, free surface modeling

based on linear wave theory has been included in the solver. The method is based

on a structured, non-uniform, colocated grid scheme and incorporates a pressure

correction scheme based on SIMPLE method developed by Patankar [1980]. In the

colocation grid scheme, all the main variables are stored at the cell centers. This

approach deviates from the previous scheme developed by Kakar [2002], in which

the variables are stored at cell corners or nodes. Having the control volume centered

around the cell centers is advantageous over the nodal scheme as the need to treat

singularity points on the boundary is absent. The method is based on the Crank-

Nicolson scheme and hence, is implicit in nature.

In order to validate the method, the solver has been applied to a few standard prob-

lems which have analytical solutions. The solver has been applied to the channel

flow problem and is able to produce the expected parabolic profile in the fully devel-

oped flow region downstream of the inlet. It also proved successful in capturing the

wave profile of a given set of progressive waves in deep water conditions. The prob-

lem of a floating hull undergoing forced harmonic heave motion has been solved

and the hydrodynamic coefficients obtained from the solver compare well with the

values from other methods and experiments. Next, the solver has been applied to

the problem of forced harmonic roll motions of a floating hull, with and without

bilge keels. The hydrodynamic coefficients obtained from the present method over

predict the added mass coefficients and under predict the damping coefficients when

133

compared to the experimental values. Also, there has not been much improvement

over the results from the Euler solver. A thorough analysis has been launched to

look in to and find out the reasons for the inadequacies of the current method and

its formulation. An erroneous pressure distribution on the hull formed a motivation

behind uncoupling the free surface effects and viscous effects and investigating them

independently. The problem has been scaled down by considering the roll motions

of a submerged hull in the absence of the free surface. The assumption of small mo-

tions has been withdrawn and a moving grid with a fixed inertial coordinate system

was used to solve the problem. A Lagrangian transformation in time was imposed

and the results obtained for a hull without bilge keels compare well with the much

validated BEM solver results.

Finally, a tip-vortex tracking method based on potential flow theory has been devel-

oped and preliminary results have been obtained. Expected wake pattern is simu-

lated behind a 2D hydrofoil subjected to a uniform inflow and a sinusoidal lateral

gust.

6.2 Recommendations

The present solver has some distance to traverse before it can meet the ultimate

objective of modeling the viscous separated flow around a rolling FPSO vessel. The

following could be implemented in the future to achieve the final objective:

� In the case of hull roll motions at the free surface, the grid in the domain

needs to be re-meshed every time step according to the movement of the body.

Moving the body and the grid has proved to be successful in the submerged

134

body case and hence, it needs to be implemented in the floating body case too.

� A free surface tracking method based on the non-linear boundary condition is

more accurate than a method based on the linear boundary condition. And if

developed, it could prove better than the present method which is based on the

linear boundary conditions, in the case of FPSO hull roll motions, especially

at larger roll angles.

� The solver can be extended into three dimensions using two approaches; one is

to apply strip theory to the 2D hydrodynamic coefficients obtained for various

sections of the hull and the second is to develop a 3D Navier-Stokes solver by

including the z-momentum equation. Clearly the latter will be more accurate,

but also considerably more computer intensive than the former. The 3D NS

solver will also be able to predict lifting effects of the bilge keels, especially

when surge motion is coupled with the roll motion.

� The vortex tracking method needs to be developed successfully for an oscil-

lating flow past a flat plate or a bilge keel. Once successfully employed for

flow past the flat plate, the method can be applied to flows past the bilge keels

attached to hull forms in the presence of a free surface. The results from this

method should require less computer time to obtain. The current NS solver

can then be used to assess the accuracy of this simplified approach.

135

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Vita

Bharani Kumar Kacham was born to K. Prakash and K. Neeraja on December

18th, 1980 in the state of Andhra Pradesh, India. He finished his high schooling

at Vasavi Public School, Hyderabad in 1996 and after completing two years of pre-

professional schooling, joined the prestigious Indian Institute of Technology Madras

at Chennai. After receiving a Bachelor of Technology degree in Naval Architecture

and Ocean Engineering in 2002, he joined the University of Texas at Austin to pursue

a Masters degree in Civil Engineering with a focus on Ocean Engineering.

Permanent address: Flat-1, Shailaja Apartments, Tv TowerMalakpet, Hyderabad - 500036, India

This thesis was typeset with LATEX�

by the author.

�LATEX is a document preparation system developed by Leslie Lamport as a special version of

Donald Knuth’s TEX Program.

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