1. SEAKEEPING OF HIGH SPEED SHIPS WITH TRANSOM STERN AND
THE VALIDATION METHOD
WITH UNSTEADY WAVES AROUND SHIPS
A Dissertation
by
Muniyandy ELANGOVAN
Submitted in Partial Fulfilment of the Requirements for the Degree of
Doctor of Engineering
Graduate School of Engineering
December 2011
JAPAN

2. Abstract
Improvement in seakeeping qualities of the ship hull through numerical computation is
always a demand from shipping industry to have the better hull to reduce the resistance which
is related with the engine power requirement. So far, plenty of effort has been made to improve
the estimation accuracy of the seakeeping qualities that are the hydrodynamic forces, motions,
wave field around a ship and added wave resistance. Nowadays, Rankine Panel Method is popular
to carry out this numerical analysis. In this thesis, the Rankine panel method in the frequency
domain is developed and applied to several kinds of ships including the high speed transom
stern for the purpose of confirming its efficiency. The Neumann-Kelvin and Double-body flow
formulations are examined as a basis flow.
As numerical estimation methods become higher grade, more accurate and detailed exper-
imental data for their validation is required. An unsteady wave pattern is being used as one of
the methods for that purpose. To capture the wave pattern, Ohkusu, RIAM, Kyushu University,
Japan has developed a method to measure the waves and analysis the added wave resistance
by means of unsteady waves. In this thesis, the analysis method is improved by including the
interaction effect of steady wave and the incident wave in the original method. A Modified
Wigley hull is analyzed numerically for the unsteady waves and compared with experimental
data obtained by the present method. This interaction effect has been observed remarkably in
the comparison. In addition to the unsteady wave field, hydrodynamics forces, motions and
added wave resistance are also compared with experimental data.
To treat the high speed vessel which has a transom, a new boundary condition has been
introduced. This condition has been derived from the experimental observation which confirms
that the transom stern part is completely dry at the high forward speed. From this point of
view, the boundary condition is formulated at the transom stern just behind stern to implement
in the potential theory panel code. This condition corresponding to the Kutta condition in
the lifting body theory. High speed monohull and trimaran are taken for the analysis, and the
computed numerical results are compared with experimental data. Influence of a transom stern
is observed in hydrodynamic forces and moments, ship motions, unsteady waves and added
wave resistance. It is concluded that the new transom boundary condition can capture the
hydrodynamics phenomena around the transom and this can improve the estimation accuracy
of the seakeeping qualities in numerical computation for this kind of vessel.
ii

3. Acknowledgements
The completion of this thesis has been facilitated by several persons. I would like to thank
all of them for their help and cooperation during this research.
First, I would like to express my deepest gratitude to my academic supervisor Prof. Hidet-
sugu Iwashita, who has given me an opportunity to do research under his supervision. His
technical advice and guidance have significantly contributed to the success of this research and
also thank him for his valuable time to explain the critical research point which was raised during
the research period. He has helped me and my family in terms of advice and financial support
without which will be difficult to complete this research. Myself and my family members will
remember in our lifetime and thankful to him.
It is my great pleasure to thank Prof Yasuaki Doi, Prof Hironori Yasukawa and Prof. Hidemi
Mutsuda for evaluating this thesis and for their valuable suggestions. I take this opportunity to
thank Prof. Mikio Takaki, and we had a departmental party in the first year which is memorable
in my life.
I would like to thank students from Airworthiness and seakeeping for vehicles laboratory,
for their direct and indirect support. I also thanks to Mr. Tanabe and Mr. Ito for their support
in the final stage of thesis preparation. I very much acknowledge the help of the staff, Faculty
of Engineering Department and Graduate School of Engineering.
My deepest thanks go to my father, mother, wife, two daughters and father in-law whose
love, affection and encouragement during the period of course had been main back born to keep
me with more confident and motivation towards the completion of this research.
My stay and studies in Japan have been supported in-terms of scholarship of the Japanese
Government, Ministry of Education, Science, Sports and Culture, for which I am very thankful.
I would like to thank ”www.google.com” making the search part and translation of Japanese
document very easy and fast.
My last thanks but not least, will be for the citizens of Japan and especially Saijo city office
and people for their great cooperation for my family stay and me during our stay. I had a few
opportunities to participate in some of the Japanese cultural program in Japan, which made
unforgettable in my life and love their traditional culture. Therefore, once again, I am thankful
to the people in Japan.
Muniyandy Elangovan
December, 2011
Higashi-Hiroshima Shi
iii

4. Dedication
To All Citizens of Japan and My Family Members
iv

5. Nomenclature
Acronyms φj Radiation Potential
BEM Boundary Element Method Ψ Total Velocity Potential
BVP Boundary Value Problem ρ Fluid Density
CFD Computational Fluid Dynamics σ Source Strength
EUT Enhanced Unified Theory τ Reduced Frequency
GFM Green Function Method ξj Motion in j-th Direction
HSST High Speed Strip Theory ζ Free Surface Elevation
LES Large-Eddy Simulation ζ7 Diffraction wave
RANS Reynolds Averaged Navier-Stokes ζj Radiation wave in jth Direction
RPM Rankine Panel Method ζs Steady Wave Elevation
TSC Transom Stern Condition
Mathematical Symbols
Greek Symbols Two Dimmensional Laplacian with re-
spect to x and y
α Displacement Vector
GML Longitudinal Metacentric Height
χ Encounter Angle of Incident Waves
GMT Transverse Metacentric Height
λ Wave Length
mj jth Component of m Vector
ω0 Wave Circular Frequency
nj jth Component of n Vector
ωe Encounter Circular Frequency
n Unit Normal Vector
Φ Double body flow potential
A Wave Amplitude
φ Unsteady velocity Potential
Aij Added Mass acting in i-th direction due
φ0 Incident Wave Potential
to j-th Motion
φ7 Scattering Potential
v

6. vi
Bij Damping Coefficient acting in i-th di- Miscellaneous
rection due to j-th Motion
V Steady Velocity Vector
Cb Block Coefficient
B Breadth
Cij Matrix of Restoring Coefficients
d Draught
D/Dt Substantial Derivatives
g Gravitational Acceleration
Ej Exciting Force in j-th Direction
H Kochin Function
Fj Steady Force in j-th Direction
k Moments of Inertia and Centrifugal Mo-
Fn Froude Number ment
G Green Function k1 , k2 Elementary Wave
H Wave Height L Length Between Perpendiculars
K Wave Number
NF Number of Elements on Free Surface
K0 Steady Wave Number
NH Number of Elements on Hull Surface
Ke Encounter Wave Number
NT Total Number of Elements
Mij Mass Matrix Associated with Body
NF A No. of Elements on Transom Surface
p Unsteady Pressure
P (x, y, z) Field Point
ps Steady Pressure
Q(x, y, z) Source Point
U Forward Speed of the Ship
RAW Added Wave Resistance
xg , zg Coordinates of the Center of Gravity
SC Transom Surface
SF Free Surface
SH Hull Surface
Sw Waterline Area
xw Center of Water Line Area

7. Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
1 Introduction 1
1.1 Background of Theoretical Estimations of Seakeeping . . . . . . . . . . . . . . . . 1
1.2 Validation Methods of Theoretical Estimations . . . . . . . . . . . . . . . . . . . 7
1.3 Scope of Present Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2 Mathematical Formulation of Seakeeping 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Body Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Free Surface Boundary Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Radiation Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Proposed Transom Stern Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Formulation of Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . 27
2.6.1 Formulation for Double body flow potential . . . . . . . . . . . . . . . . . 29
2.6.2 Formulation for steady velocity potential . . . . . . . . . . . . . . . . . . 29
2.6.3 Formulation for unsteady velocity potential . . . . . . . . . . . . . . . . . 30
2.7 Hydrodynamic Forces and Exciting Forces . . . . . . . . . . . . . . . . . . . . . . 31
2.8 Ship Motions and Wave Elevation . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Numerical Method 35
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vii

8. Contents viii
3.2 Integral Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Formulation of Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Analytical Wave Source Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5 Rankine Panel Method based on the Panel Shift Method (PSM) . . . . . . . . . 38
3.6 Rankine Panel Method based on the Spline Interpolation Method (SIM) . . . . . 39
3.7 Comparison of Two Rankine Panel Methods in the Calculation of a Point Source 40
3.8 Treatment of Transom Stern Condition by Panel Shift Method . . . . . . . . . . 45
3.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Experiments for the Validation of Seakeeping 47
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Measurement of Ship Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Forced Motion Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 Mass estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.2 Restoring force coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.3 Forced heave motion test . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.3.4 Forced pitch motion test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.4 Measurement of Unsteady Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.5 Interaction Between Incident Wave and Generated Unsteady Waves . . . . . . . 57
4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Interaction Effect of Incident Wave in the Unsteady Wave Analysis 61
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Experiment for Modified Wigley Hull . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.4 Experimental Results on the Interaction Effect of Incident Wave . . . . . . . . . 64
5.4.1 Wave probes dependency study . . . . . . . . . . . . . . . . . . . . . . . . 64
5.4.2 Influence of 2nd order term of unsteady wave . . . . . . . . . . . . . . . . 65
5.4.3 Interaction effect between double body flow and incident wave . . . . . . 66
5.4.4 Interaction effect between Kelvin wave and incident wave . . . . . . . . . 66
5.5 Analysis of Hydrodynamic Forces, Motions and Added wave Resistance . . . . . 70
5.5.1 Steady wave field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.5.2 Added mass and damping coefficient . . . . . . . . . . . . . . . . . . . . . 71

9. Contents ix
5.5.3 Wave exciting forces and moment . . . . . . . . . . . . . . . . . . . . . . . 72
5.5.4 Ship motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.5.5 Pressure distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5.6 Unsteady wave field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.5.7 Added wave resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Seakeeping of High Speed Ships with Transom Stern 87
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.2 Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Experiment for Monohull with Transoms Stern . . . . . . . . . . . . . . . . . . . 90
6.4 Evaluation of Numerical and Experimental Results . . . . . . . . . . . . . . . . . 90
6.4.1 Steady wave field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.4.2 Added mass and damping coefficients . . . . . . . . . . . . . . . . . . . . 91
6.4.3 Wave exciting forces and moment . . . . . . . . . . . . . . . . . . . . . . . 93
6.4.4 Ship motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.4.5 Pressure distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.4.6 Unsteady wave field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.4.7 Added wave resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.5 Application in the Trimaran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.5.1 Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.5.2 Steady wave field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.5.3 Added mass and damping coefficients . . . . . . . . . . . . . . . . . . . . 106
6.5.4 Wave exciting forces and moment . . . . . . . . . . . . . . . . . . . . . . . 108
6.5.5 Unsteady wave field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.5.6 Analysis of sinkage and trim effect . . . . . . . . . . . . . . . . . . . . . . 109
6.5.7 Ship motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.5.8 Pressure distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.6 Comparison of pressure on monohull versus trimaran . . . . . . . . . . . . . . . . 112
6.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 Conclusions 115
Bibliography 117

10. List of Figures
2.1 Body boundary bondition - coordinate system . . . . . . . . . . . . . . . . . . . 13
2.2 Transom stern with steady wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Definition of problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4 Basis flow approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1 SPM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 5.0, τ = 0.447 . 41
3.2 SPM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 10, τ = 0.633 . 41
3.3 SPM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 20, τ = 0.894 . 42
3.4 SPM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 30, τ = 1.095 . 42
3.5 SIM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 5.0, τ = 0.447 . 43
3.6 SIM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 10, τ = 0.633 . . 43
3.7 SIM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 20, τ = 0.894 . . 44
3.8 SIM - Comp. of wave pattern and pers. view for Fn = 0.2, Ke = 30, τ = 1.095 . . 44
3.9 Numerical treatment of transom stern . . . . . . . . . . . . . . . . . . . . . . . . 45
3.10 Panel shift method for the transom stern problem . . . . . . . . . . . . . . . . . 45
4.1 Motion free experimental setup diagram . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Forced motion experimental setup diagram . . . . . . . . . . . . . . . . . . . . . 49
4.3 Schematic diagram for the wave measurement by multifold method . . . . . . . . 55
4.4 Wave propagation with respect to time . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Plans of the modified Wigley hull . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Computation grids for Rankine panel method . . . . . . . . . . . . . . . . . . . 63
5.3 Effect of number of wave-probe in accuracy of wave pattern analysis (Diffraction
wave at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.5, χ = π) . . . . . . . . . . . . . . . 65
x

11. List of Figures xi
5.4 2nd-order term of wave pattern analysis (Diffraction wave at y/(B/2) = 1.4 for
Fn = 0.2, λ/L = 0.5, χ = π) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5 Interaction effect between double-body flow and incident wave in wave pattern
analysis (Diffraction wave at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.5, χ = π) . . . 66
5.6 Kelvin wave at Fn = 0.2, y/(B/2) = 1.4 . . . . . . . . . . . . . . . . . . . . . . . 66
5.7 Interaction effect between Kelvin wave and incident wave in wave pattern analysis
(Diffraction wave at y/(B/2) = 1.4 for Fn = 0.2, χ = π) . . . . . . . . . . . . . . 67
5.8 Kochin functions computed with diffraction waves measured at Fn = 0.2, λ/L =
0.5, χ = π, y/(B/2) = 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.9 Steady Kelvin wave pattern of modified Wigley model (blunt) at Fn = 0.2 . . . . 70
5.10 Added mass and damping coefficients due to forced heave motion at Fn = 0.2 . . 71
5.11 Added mass and damping coefficients due to forced pitch motion at Fn = 0.2 . . 72
5.12 Wave exciting forces and moment at Fn = 0.2, χ = π . . . . . . . . . . . . . . . . 73
5.13 Ship motions at Fn = 0.2, χ = π . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.14 Steady pressure distribution of modified Wigley model (blunt) at Fn = 0.2 . . . . 74
5.15 Wave pressure (cos component) at Fn = 0.2, λ/L = 0.5, χ = π . . . . . . . . . . 75
5.16 Wave pressure (cos component) at Fn = 0.2, λ/L = 1.3, χ = π . . . . . . . . . . 75
5.17 Unsteady pressure (cos component) at Fn = 0.2, λ/L = 0.5, χ = π . . . . . . . . 76
5.18 Unsteady pressure (cos component) at Fn = 0.2, λ/L = 1.3, χ = π . . . . . . . . 76
5.19 Contour plots of heave radiation wave at Fn = 0.2, KL = 30 . . . . . . . . . . . . 78
5.20 Contour plots of diffraction wave at Fn = 0.2, λ/L = 0.5, χ = π . . . . . . . . . . 78
5.21 Contour plots of total wave at Fn = 0.2, λ/L = 0.5, χ = π . . . . . . . . . . . . . 79
5.22 Contour plots of total wave at Fn = 0.2, λ/L = 1.3, χ = π . . . . . . . . . . . . . 79
5.23 Heave radiation waves at y/(B/2) = 1.4 for Fn = 0.2, KL = 30, 35 . . . . . . . . 80
5.24 Diffraction waves at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.5, 0.7, χ = π . . . . . . 80
5.25 Wave profile for Wigley model at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.7, χ = π . 82
5.26 Wave profile for Wigley model at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 0.9, χ = π . 83
5.27 Wave profile for Wigley model at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 1.1, χ = π . 83
5.28 Wave profile for Wigley model at y/(B/2) = 1.4 for Fn = 0.2, λ/L = 1.4, χ = π . 84
5.29 Added wave resistance at Fn = 0.2, χ = π . . . . . . . . . . . . . . . . . . . . . . 84

12. List of Figures xii
6.1 Plans of the monohull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.2 Perspective view of the monohull with computation grids . . . . . . . . . . . . . 88
6.3 Computation grids (NH = 1480(74 × 20), NF = 3888(162 × 24), NF A = 297(99 × 3)) . 89
6.4 A snapshot of the transom stern in the motion measurement test . . . . . . . . . 90
6.5 Steady Kelvin wave pattern at Fn = 0.5 . . . . . . . . . . . . . . . . . . . . . . . 91
6.6 Measured steady resistance (total), sinkage and trim . . . . . . . . . . . . . . . . 91
6.7 Added mass and damping coefficients due to forced heave motion at Fn = 0.5 . . 92
6.8 Added mass and damping coefficients due to forced pitch motion at Fn = 0.5 . . 93
6.9 Wave exciting forces and moment at Fn = 0.5, χ = 180degs. . . . . . . . . . . . . 94
6.10 Ship motions at Fn = 0.5, χ = 180degs. . . . . . . . . . . . . . . . . . . . . . . . 94
6.11 Wave pressure on the hull at Fn = 0.5, λ/L = 1.1, χ = 180degs. . . . . . . . . . . 96
6.12 Total unsteady pressure on the hull at Fn = 0.5, λ/L = 1.1, χ = 180degs. . . . . 96
6.13 Wave pressure on the hull at Fn = 0.5, λ/L = 1.1, χ = 180degs. . . . . . . . . . . 97
6.14 Total unsteady pressure on the hull at Fn = 0.5, λ/L = 1.1, χ = 180degs. . . . . 97
6.15 Comparisons of measured and computed wave patterns . . . . . . . . . . . . . . . 98
6.16 Comparisons of measured and computed wave profiles along y/(B/2) = 1.52 . . . 98
6.17 Added wave resistance computed by the wave pattern analysis . . . . . . . . . . 101
6.18 Plans of the Trimaran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.19 Perspective view of the model with computation grids . . . . . . . . . . . . . . . 104
6.20 Computational grids used for Trimaran . . . . . . . . . . . . . . . . . . . . . . . 104
6.21 Steady Kelvin wave pattern at Fn = 0.5 . . . . . . . . . . . . . . . . . . . . . . . 105
6.22 Steady wave view at Fn = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.23 Steady pressure on hull at Fn = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.24 Added mass and damping coefficients due to forced heave motion at Fn = 0.5 . . 107
6.25 Added mass and damping coefficients due to forced pitch motion at Fn = 0.5 . . 107
6.26 Wave exciting forces and moment at Fn = 0.5, χ = 180degs. . . . . . . . . . . . . 108
6.27 Diffraction and Radiation Wave Pattern . . . . . . . . . . . . . . . . . . . . . . . 109
6.28 Perspective view of the model with sinkage and trim . . . . . . . . . . . . . . . . 109

13. List of Figures xiii
6.29 Computation grids which include sinkage and trim . . . . . . . . . . . . . . . . . 110
6.30 Ship motions at Fn = 0.5, χ = 180degs. . . . . . . . . . . . . . . . . . . . . . . . 111
6.31 Pressure p/ρ g A at Fn = 0.5, λ/L = 1.1χ = 180degs. (with TSC, sinkage & trim
effect) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.32 Comparison of wave pressure on monohull Vs trimaran . . . . . . . . . . . . . . . 112
6.33 Comparison of total unsteady pressure on monohull Vs trimaran . . . . . . . . . 112

14. List of Tables
5.1 Principal dimensions of the model (Modified Wigley Hull) . . . . . . . . . . . . . 62
6.1 Main particulars of monohull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Main particulars of trimaran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
xiv

15. Chapter 1
Introduction
1.1 Background of Theoretical Estimations of Seakeeping
A ship operated at sea is exposed to forces due to waves, current and winds. These forces
not only cause the motions of the ship, which can be very annoying for its passengers but also
account for the resistance of the ship or drift the ship away from its course. Resistance due to
wind is very low when compared to water resistance. Estimations of resistance are important for
deciding the power requirements of any ship. The resistance is decomposed into the resistance
originated in the viscosity and the resistance due to the wave making.
Before a ship is built, knowing the maximum data about the ship’s performance in calm
water, and in waves can lead to a better ship construction. Therefore, a scale model of the
ship is built and tested in a basin with and without incoming waves. These model tests are
very expensive, and it is time consuming. Making new computer simulation software, which can
predict well about the ship behavior can replace a model test. With the increase of computer
technology, it has become possible to simulate a ship’s behavior in waves numerically. These
simulations are based on mathematical descriptions of the physics of the ship and the sea which
are extremely complex. To understand the gross fluid motion and corresponding interaction with
the ship, one must understand and predict turbulence, wave breaking, water spray, non-linear
motion, slamming, green water on deck, sloshing, acoustics, etc.
Viscous fluid motion is governed by the continuity equation and the motion equations of the
rigid body must be coupled when we treat the fixed-body interaction problem. When we focus
on the ocean waves and the interaction between the ship and ocean waves, the gravity effect is
dominant and the viscous effect is negligible. Therefore, the fluid can be treated as an ideal fluid
and the potential theory can be applied. Potential flow solvers are usually based on boundary
1

16. 1.1. Background of Theoretical Estimations of Seakeeping 2
element methods and need only to be discretized the boundaries of the domain, not the whole
domain. This reduces the load in grid generation and less time computation. Potential flow
solver needs suitable boundary conditions that consist of the body boundary condition, the free
surface boundary condition and the radiation condition to satisfy the physical condition on the
free surface.
The well-known strip theory was the first numerical method used as a practical design
tool for predicting ship hydrodynamic forces and ship motions. This method solves the 2-D
flow problem for each strip of the ship and integrates the results over the ship to find out the
hydrodynamic forces and motions. This problem was solved by U rsell[1]for heaving motion of
a half-immersed circular cylinder. Other extensive works were done afterward by Korvin −
Kroukovski[2], T asai[3] and Chapman[4]. The rational foundations for the strip theory were
provided by W atanabe[5], T asai & T akagi[6] , Salvasan et al. [7] and Gerritsma[8]. Now also
it is popular in this field because of satisfactory performance and computational simplicity.
When the forward speed increases, the efficiency of the strip theory based on the 2D theory
got reduced, due to a strong 3D effect near the bow part and forward speed influence from the
steady flow to the unsteady wave field. It has been reported by T akaki & Iwashita[9]that the
applicable limitation of the strip theory is around Fn = 0.4 for the typical high speed vessel.
The high speed strip theory (HSST), so called 2.5D theory, can be applied effectively for the
high speed vessels. The theory originated in Chapman[10] and developed subsequently by many
researchers, Saito & T akagi[11], Y ueng & Kim[12][13], F altinsen[14] , Ohkusu & F altinsen[15]
can capture the forward speed effect within the framework of uniform flow approximation in the
free-surface condition.
The rational justification of strip theory, as a method valid for high frequencies and moderate
Froude numbers, was derived from systematic analysis based on the slender-body theory by
Ogilvie and T uck[16]. This theory was extended to the diffraction problem by F altinsen[17]
and was further refined by M aruo & Sasaki[18]. The high-frequency restriction in slender ship
theories was removed by the unified theory framework presented by N ewman [19]. Its extension
to the diffraction problem was derived by Sclavounos[20] and applied to the seakeeping of ships
by N ewman and Sclavounos[21] and Sclavounos[22].
The increasing accessibility of computers of high capacity led to the development of three-
dimensional theories that removed some of the deficiencies of strip theory. The choice of the

17. 1.1. Background of Theoretical Estimations of Seakeeping 3
elementary singularities leads to the classification of these methods into the Green function
method (GFM) and the Rankine panel method (RPM). In GFM, the wave Green function is
applied only at the ship surface and in the RPM, simple source will be applied to ship surface
and free surface.
The 3-D Green function method has been applied and succeded for the floating bodies
without forward speed and extended for the forward speed. In the Green Function method,
the unsteady wave source, which satisfies the radiation condition and the linearized free surface
condition based on the uniform flow, is chosen as the elementary singularity. Important devel-
opments for its fast and accurate evaluation were made by Iwashita & Ohkusu[23] based on
the single integral formulation derived by Bessho[24]. Iwashita et al.[25][26] have rigorously
examined the wave pressure distribution on a blunt VLCC advancing in oblique waves by ap-
plying the Green function method and demonstrated that the strip method practically used for
the estimation of ship motions is insufficient for this purpose.
They have also showed that a significant discrepancy of the wave pressure between numerical
results and experiments still remains at blunt bow part even if the three-dimensional method
is applied. It was decided to include the influence of the steady field in an unsteady wave field
in different boundary condition. Then some improvements have been reported by Iwashita &
Bertram[27], where the influence of the steady flow on the wave pressure is taken into account
through the body boundary condition. The problem is often formulated in the frequency domain,
which assumes that the body motions are strictly sinusoidal in time. First achievements were
reported by Chang[28], Kabayashi[29], Inglis & price[30], Guevel & Bougis[31] and they found
good agreement with experimental data.
The Ranking panel method was proposed by Gadd[32] and Dawson[33] for the steady
problem and extended by N akos & sclavounos[34] to the unsteady problem. Y asukawa [42]
and Iwashita et al.[35] suspected that alternative influence of the steady flow through the
free surface condition might affect more strongly the local wave pressure, especially at the bow
part. They used for the computation a Rankine panel method based on the double body flow
linearization for steady wave field, and its influence is taken into account in the unsteady problem
through the free surface condition and the body boundary condition. This formulation will be
called as double body flow formulation. It was considered that the influence of the steady double
body flow through the free surface condition is important for estimating hydrodynamic forces

18. 1.1. Background of Theoretical Estimations of Seakeeping 4
and local pressures.
In RPM, the radiation condition must be satisfied numerically, and this numerical radiation
can be solved in different method. The typical one is the finite difference method originated in
Dawson[33]. The upward differential operator is used to evaluate the partial derivative of the
velocity potential on the free surface and the radiation condition which proves non-radiating
waves in front of a ship is satisfied. Sclavounos & N akos[36] introduced the B-Spline function
to express the potential distribution on the free surface and satisfied the radiation condition
by adding a non wave condition at the leading edge of the computation domain of the free
surface. They also applied their method to seakeeping problem provided the high reduced
frequency τ (= U we /g) > 0.25 where any disturbed wave due to a ship does not propagate
upward. The alternative method is Jensen s[37] method so called collocation method. The
collocation points on the free surface are shifted just one panel upward to satisfy the radiation
condition numerically. This method has been applied to the seakeeping problem by Bertram[38],
and the analytical proof on the numerical radiation condition has been done by Seto[39] for the
2D problem.
For the investigation of fully non-linear steady kelvin wave field and its influence on the
unsteady wave field, Bertram[40] employed a desingularized Rankine panel method proposed
initially for steady problem by Jensen et al.[37]. The steady problem is solved so that the
fully nonlinear free surface condition is satisfied and the influence terms of the steady wave field
on the unsteady flow are evaluated by assuming the small amplitude of the incident wave and
small ship motions. The boundary conditions are satisfied on the steady free surface and wetted
surface of the body.
Rankine source method is a highly efficient method for seakeeping analysis. There are
many researchers working with RPM to satisfy the radiation condition in the frequency domain
in different ways. T akagi[41] and Y asukawa [42] have introduced Rayleigh’s viscosity numeri-
cally to satisfy the radiation condition numerically and getting the adequate value of Rayleigh’s
viscosity is not strait forward. Bertram [43] and Sclavounos et al. [44] has introduced desingu-
larized panel method to satisfy the radiation condition, and this is applicable when the reduced
frequency is τ > 0.25. Iwashita et al.[45], Lin et al.[46], T akagi et al.[47] proposed a hybrid
method named combined boundary integral equation method (CBIEM) which is the combina-
tion of Rankine Panel Method and Green Function Method. The method makes possible to

19. 1.1. Background of Theoretical Estimations of Seakeeping 5
satisfy the radiation condition accurately by introducing the Green function method in the far
field.
Three dimensional computation method was extended to the time domain by
Lin et al.[50],[51] had developed in Green function based 3D panel method satisfying the linear
free surface condition, and the ship surface is treated nonlinear boundary condition. The free
surface is to be re-meshed according to ship position. This program is called as LAMP (Large
Amplitude Motion Program), and it was made commercially available. Later, free surface has
been improved with partially nonlinear and further; there are many series as Lamp-1 to Lamp-4.
This software has been extended as a Non-linear Large Amplitude Motions and Loads method
by Shin et al. [52] used for offshore structure application. Using this program, many calculations
have been carried out by W eems et al. [53], [54] for the trimaran and wave piercer.
M askew [55] has developed Rankine source based 3D panel method where the three surface
treated as a fully nonlinear and the ship surface is treated as partially nonlinear. It has been
applied for Frigate ship with waves. This program also considered lift effect and made available
commercially. Though it is the first program based Rankine source in time domian but there are
some disadvantages in accuracy of the calculation. This has been applied for the high speed and
demonstrated few results for S-60, SAWTH, S175 and Frigate with waves by M askew [56], [57].
Beck.R.F [58], [59]. has developed Rankine source based fully nonlinear boundary condition for
free surface as well as ship surface and this method used the desingularized panel source. This
code has been applied for Wigley hull for ship motion problems. This has been extended by
Scorpio et al.[60] for the container ship.
N akos et al.[61] have developed the Rankine source based time domain program where
the double flow is introduced, and it has been applied for Wigley hull with linear free surface
condition. Later, it has been improved by Kring et al. [62] for the application of contained ship
motion and load estimation where the free surface and body surface have taken as nonlinear.
This program gives much higher accuracy without many panels. However, this software has
been made commercially for the ship motion and load estimation with the name called SWAN.
Adegeest et al[63] has improved in the Rankine based time domain program for the container
ship in unsteady waves.
Bunnik and Hermans [64] & Buunik [65]have developed code based Rankine source
method for the estimation added wave resistance and ship motion. It used the numerical beach

20. 1.1. Background of Theoretical Estimations of Seakeeping 6
to satisfy the radiation condition. However, it cannot be used for blunt ship and high speed
vessel. Rankine source based code is developed by Colagrossi et al.[66], [67] with non linear
free surface condition and applied for slender ship body. This method used the desingularized
method for free surface condition and numerical beach for radiation condition. It has been
applied for catamaran and trimaran. This method has been developed and applied for the S-
60(Cb=0.8) by Iwashita et al.[68]. T anizawa[69] and Shirokura[70] has developed the fully
nonlinear free surface and body boundary condition seakeeping code based on Rankine source
method. This code is applied for the Wigley hull and demonstrated high-accuracy results.
Y asukawa[71], [72] has developed nonlinear code for seakeeping problem based on Rankine
source. For the high speed calculation SOR method has been introduced where after the transom
is modelled with a dummy grid surface to get the smooth flow. It has been applied for the Wigley
hull and S175 by Y asukawa[73], [74] and validated with experimental results.
Green function based time domain code is developed by Kataoka and Iwashita [75], [76]
in which the fluid domain is decomposed into near field and far field, and the space fixed artificial
surface like side-walls of a towing tank is employed to separate them. The Rankine method and
the Green function method are applied to two domains respectively, and the solutions in both
domains are combined on the artificial surface. This method is called hybrid method. It has
been applied for the modified Wigley hull and S-60 to estimate the forces and motions. It has
been extended for the different kind of vessel by Kataoka & Iwashita [77] ∼ [81] for the S175,
S-60 (Cb=0.7, Cb=0.8) and modified Wigley hull. The added wave resistance and pressure are
calculated and compared with an experimental result for the validation.
In practical fluid motion problem, due to the presence of turbulence suggests using the
unsteady Reynolds Averaged Navier-Stokes (RANS) equations or Large-Eddy Simulations (LES)
to model the fluid motion or seakeeping problem. In recent years, the CFD simulation tools
are applied to seakeeping. In grid-based method, three-dimensional CFD simulations of vertical
plane motion in waves are reported by Sato et al.[82] and W eymouth et al.[83]. A non-
linear motion for a high speed catamaran vessel, and the ship resistance in 3D was presented
by P anahi et al.[84]. On the other hand, in particle based method, Shibata et al.[85]&[86]
computed a shipping water and pressure onto moving ship. Oger et al.[87] SPH method applied
to a ship motion at high Froude number. M utsuda.H. et al.[88] − [90], has validated for free
surface and impact pressure problems.

21. 1.2. Validation Methods of Theoretical Estimations 7
M utsuda.H.etal.[91], the seakeeping performance of a blunt ship in nonlinear waves studied
and validated. As an improvement, CFD has been applied by Suandar.B. et al.[92]. for fishing
boat and high speed ferry. Ship motions, pressure distribution on the hull and velocity field
have been validated with panel method and experimental results.
A practical computational method to analysis the transom stern flow is important in the
area of marine hydrodynamics. Research on the transom stern steady flow was conducted by
Saunders[93]. He provided advice that the speed of ventilation occurs at a FT ∼ 4.0 to 5.0
(where FT is the Froude number based on the transom draft). More recent observation on
naval combatants put the ventilation in the range of FT ∼ 3.0 to 3.5. V anden − Broeck and
T uck[94] describes a potential flow solution with series expansion in FT that is valid for low
speeds where the transom stagnation point travels vertically upward on the transom as the
FT increases. V anden − Broeck[95] offers a second solution applicable to the post ventilation
speeds. For a 3D full ship problem, Subramani[96] extended the fully nonlinear desingularized
potential code (DELTA) to analyze a body with a transom stern. Additionally, Doctors[97] has
strived to suitably model the transom hollow for use in linearized potential flow program. By
Doctors et al.[98] non linear effect has been introduced to analyze the transom stern vessel.
Because of its advantage, most of the high speed vessel and naval ships are built with transom.
It is also important to study hydrodynamic flow behavior at transom and behind the ships.
When the ship reaches to a sufficient speed, flow leaves at transom and the transom area is
exposed to air. The transom flow detaches smoothly from the underside of the transom, and
a depression is created on the free surface behind the transom. This creates influence in the
pressure reduction and resistance on the hull.
For unsteady flow, many researchers are working on this to get transom flow with different
approaches but there is no exact solution is arrived yet. Considering the need and a new
boundary condition has been proposed to treat the transom part and details will be covered
inside the thesis.
1.2 Validation Methods of Theoretical Estimations
A ship advancing in waves generate waves. The generated waves are classified into the time-
independent and time-dependent waves. The former is the steady wave called Kelvin’s wave and

22. 1.3. Scope of Present Research 8
corresponds to the wave that is generated by the ship advancing in calm water surface. The wave
is independent of time when we observe it from the body-fixed coordinate system. The second
one is the unsteady wave that consists of the radiation waves and the diffraction wave. The
radiation waves are the wave generated by the ship motions, and the diffraction wave is scatted
wave of the incident wave by the hull. It is understood that both the steady and unsteady waves
dissipate more energy, which leads to a power loss. The resistance of a ship in wave minus the
calm water resistance is called added wave resistance. Besides this power loss, a transverse force
will drift the ship from its course, and a rotating moment about the ship’s vertical axis leads to
a change in its course. Therefore, it is necessary to consider the above when designing the ship,
especially its hull.
The asymptotic unsteady wave patterns which are for away from the ship were studied
initially by Eggers[99]. Ohkusus.M [100] proposed a method for measuring ship generated
unsteady waves and then evaluating the wave amplitude function and the added wave resistance.
Iwashita[101] has made systematic work, including the analysis near the cusp (the caustics)
wave pattern. That was successful in treating with the surface elevation near the cusp for a
wave system of shorter wave components with introducing Hogner’s approach [102]. A 2nd order
unsteady wave pattern has been investigated by Ohkusu.M [103]. These unsteady wave patterns
physically show the pressure distributions over the free surface. Therefore, the comparison of
unsteady wave distribution between computed and measured corresponds to the comparison as
for the pressure distributions that can be considered as the local physical value. From these
reasons, it is very valuable to utilize the unsteady waves in order to validate the numerical
computation methods more precisely.
In this paper, the interaction effect between the incident wave and steady wave, which has
not been considered in the measurement analysis up to now, is investigated in order to make
the experimental unsteady waves more accurate.
1.3 Scope of Present Research
To the analysis of seakeeping problems, Rankine panel method based mathematical formulation
is derived which includes the body boundary condition, the free surface boundary condition and
the radiation condition. Computational code is developed to handle the conventional ship hull.

23. 1.4. Organization of the Thesis 9
As a basis flow, uniform flow approximation and the Neumann-Kelvin flow approximation are
considered in the formulation and applied for the modified Wigley hull to study the seakeeping
qualities.
Improve the measured unsteady wave taking into account of the interaction between the
incident wave and the steady wave. A new analysis method of the measured unsteady wave is
proposed and the interaction effect related above is confirmed by applying the method for two
modified Wigley hull models. Additionally, the RPM code is developed in the thesis is validated
through the comparisons with experimental data, that is, hydrodynamic forces, motions, added
wave resistance and wave fields.
To treat the transom stern condition, until now there is no proper boundary condition to
treat the transom stern by panel method. A new boundary condition is derived to treat the
transom effect mathematically in numerical method. Therefore, the developed code is extended
for the application of high speed monhull to predict the transom stern effect at the transom.
The RPM is applied to the high speed ships with the newly introduced transom stern condition
and validated through the comparison with experiments. Transom effect is compared between
by numerical method taking into account of transom condition and without transom condition.
This method is also applied to a trimaran and the seakeeping data is compared with experimental
data.
1.4 Organization of the Thesis
The general scope of the thesis is divided into five chapters excluding the first chapter is the
introduction which covers the previous research and the conclusion as a last chapter.
The thesis is outlined as follows:
Chapter 2: This chapter covers the mathematical formulation of a boundary value problem
(BVP) for seakeeping analysis. As a first step, to formulate BVP, appropriate bound-
ary conditions need to be derived for frequency-domain formulation. Derivation of body
boundary condition, free surface boundary condition and radiation condition derivation
are given. A new boundary condition to deal with transom stern is developed and the
details also covered. The final section is the formulation of the boundary value problem to

24. 1.4. Organization of the Thesis 10
solve the potential equation in the panel method. And also the formulation of estimating
the hydrodynamic forces, exciting forces and moments, motions and waves are derived.
Chapter 3: In panel method, it is very critical to select the suitable formulation to solve the
problem numerically. To calculate the potential in RPM, there are two methods available
i.e., (i) direct method and (ii) indirect method. Each method has been discussed. As a
next step, it is very important to select the appropriate method to satisfy the radiation
condition either panel shift method or spline interpolation method. Therefore, attempt
has been made to plot wave pattern by both the method and compared with an analytical
wave pattern. This chapter also covers explanation about the treatment of the transom
boundary conditions in panel method.
Chapter 4: In Engineering field, any numerical solution must be verified and compared with
experimental results. Therefore, required tests are identified based on requirements. Mea-
suring method for forces, motions and unsteady wave method is discussed towards im-
provement in capturing the waves around the hull. Formulation is derived to capture
interaction effect of incident wave and steady wave in an unsteady waves.
Chapter 5: To investigate the wave interaction effect, modified Wigley hulls have been numer-
ically analyzed and the results are compared with experimental results. Hydrodynamics
forces, exciting forces and moments are compared with experimental results. Ship motions
and added wave resistance are also compared with experimental results. This chapter is
to deal with the validation of incident wave and steady wave interaction effect with the
unsteady waves.
Chapter 6: High speed monohull has been taken for the analysis. Radiation forces and ex-
citing forces are calculated with transom and without transom stern condition to see the
effect of a newly introduced condition. All the results are compared with experimental
results. Unsteady waves and pressure plots are also validated with an experimental data.
Ship motion and added wave resistance are also compared with experimental results. It
has been extended for the trimaran application and numerical results are compared with
experimental data. This chapter is to deal with transom stern boundary condition with
an experimental validation.
Chapter 7: In this chapter, the thesis is concluded making clear the obtained results.

25. Chapter 2
Mathematical Formulation of Seakeeping
2.1 Introduction
Due to the development in computer technology and advanced applied mathematics, engineering
problems are solved by a numerical method, and being validated by experimental results. In
marine hydrodynamics, ship/offshore structure’s problems are formulated as a boundary value
problem where the physical parameters are to be defined well in the form of mathematical
expression to solve the problem numerically. It is very important that boundary conditions of
real physics are to be expressed mathematically, which contribute to the prediction accuracy of
the expected results.
In this research, boundary value problems are solved by boundary element method (BEM)
also referred as a panel method or boundary integral equation methods. BEM is more suitable
for the marine structure hydrodynamics problem which takes the surface of the fluid domain, and
it is the best numerical method when compare to other methods like finite difference method,
finite-element method, finite volume method, etc. in terms of computational time because it
solves for entire domain, which is not required for practical needs.
Consider a ship which is floating on water and the water surface which is in turn touch
with the air is called a free surface. The most exact description of the flow of water is given by
the Navier-Stokes equations, which take into account of the water viscosity. Viscosity in ship
hydrodynamics can be important in turbulent areas like, for example, near a rudder, propulsion
or sharp edges of hull, but none of these is considered in this research. Near the hull, a small
boundary layer exists in which viscous effects dominate, but this layer does not really affect the
large-scale interactions of ocean waves and ship motions.
Assuming that the flow is irrotational and incompressible, the flow can be described using
11

26. 2.2. Body Boundary Condition 12
potential theory. By defining the fluid velocity by a scalar potential, the velocity field of the
flow can be expressed as the gradient of a scalar function, namely the velocity potential,
u(x, t) = Ψ(x, t) (2.1)
. The continuity equation or conservation of mass reduces to Laplace’s equation because the
fluid is incompressible, which states that the divergence of the velocity field is equal to zero.
·V =0 (2.2)
and the velocity potential must be a harmonic function which satisfies the Laplace equation.
2 ∂2Ψ ∂2Ψ ∂2Ψ
Ψ= + + =0 (2.3)
∂x2 ∂y 2 ∂z 2
The conservation of momentum equation can be reduced to Bernouli’s equation.
−1 ∂Ψ 1 p − pa 1 2
z= + Ψ· Ψ+ − U (2.4)
g ∂t 2 ρ 2
where ρ is the fluid density, g is the gravitational constant and pa is the atmospheric pressure.
The total velocity potential can be decomposed into steady velocity potential and unsteady
velocity potential.
Ψ (x, y, z; t) = Ψs (x, y, z) + Ψt (x, y, z; t) (2.5)
For a real ship in a seaway, the fluid domain is effectively unbounded relative to the scale of the
ship. For the computational purpose, the fluid domain must be truncated. On the boundaries
of the truncated fluid domain, the hull of the ship SH , the free surface SF , the bottom of the
water SB and the control surface (truncation surface) SC are covered. The total surface can
be written as S = SH + SF + SB + SC . In addition to the above boundary condition, it is
necessary to satisfy the radiation mathematically to get unique results. The control surface and
the bottom surface need not be considered for analysis because of the simple source in RPM
method. We discuss on each boundary in details on further section.
2.2 Body Boundary Condition
The boundary condition on the hull should take into account the interaction between the
motion of the hull and the motion of the water at the hull. Just like the water, the
hull of the ship cannot be crossed by a fluid particle. The water should therefore have
the same normal velocity as the ship’s hull and the water does not penetrate the hull.

27. 2.2. Body Boundary Condition 13
iωt Z
r = r + αe Z
∂Ψ
= V SH · n on SH (2.6)
∂n Y
Where the normal vector n is from hull to-
Y
wards the fluid domain. This condition must
r
be simplified to the mean wetted surface from
r
an instantaneous surface. Let us consider a iωt
SH αe
three-dimensional object in a fluid with a free
SH
surface. The object, for instance, a ship, sail Instantaneous Wetted Surface
SH
through an incident wave field with a velocity SH Mean Wetted Surface
U (t) in the negative x direction; this is equiv- Figure 2.1: Body boundary bondition - coordinate
alent to an object with zero speed in current system
U (t) in positive x direction.
The object is free to translate in three
directions and to rotate around the three axes. Therefore, six motions are taken to represent
the body boundary condition. To derive the mathematical equation of the moving body, there
are two coordinate systems are followed. The first coordinate is the space coordinate system
which can be represented as,
r = (x, y, z) (2.7)
The second coordinate system is body fixed coordinate system is represented as
r = (x, y, z) (2.8)
Bring the relation between teh Space and body fixed coordinate system, the equation shall be
written as
r = r − αeiωt (2.9)
Where α is a displacement vector of a point on the body and ω is a circular frequency of
oscillations. α = ξj ; (j=1,2..6 surge, sway, heave, roll, pitch, yaw). α = iξ1 + jξ2 + kξ3 + (iξ4 +
jξ5 + kξ6 ) × r. The r coordinates are fixed with respect to a body which is defined by the
equation F (x, y, z) = 0, then the potential flow kinematic boundary condition to be satisfied on
DF (r)
the surface is 0 = .
Dt

28. 2.2. Body Boundary Condition 14
The velocity potential is decomposed into a steady mean potential and unsteady perturba-
tion potential. The Velocity of the fluid is represented by the vector Φ = V (r) + eiωt φ(r)
and V is the steady flow field due to forward motion of the body(negative x direction ) i.e.,
V (r) = U (−x + ϕ) and φ(r) is the potential of the oscillating velocity vector. The coordinate
system chosen such that undisturbed free surface coincides with the plane z=0 and the centre
of gravity of the object is on the z axis, with z pointing upwards. In this formulation, SH is the
mean wetted surface and S H is the instantaneous wetted surface, Fig. 2.1
The boundary condition on the body is
DF (r) ∂
0= = + Φ. F (r) (2.10)
Dt ∂t
DF (r) ∂F (r)
0 = = + V (r) + eiωt φ(r) . F (r) on S H (2.11)
Dt ∂t
The above equation can be written as
∂F ∂x ∂F ∂y ∂F ∂z
0 = + + + V (r) + eiωt φ(r) .
∂x ∂t ∂y ∂t ∂z ∂t
∂F ∂x ∂F ∂y ∂F ∂z ∂F ∂x ∂F ∂y ∂F ∂z
i + + +j + +
∂x ∂x ∂y ∂x ∂z ∂x ∂x ∂y ∂y ∂y ∂z ∂y
∂F ∂x ∂F ∂y ∂F ∂z
+k + + on S H (2.12)
∂x ∂z ∂y ∂z ∂z ∂z
In the above equation, equation can be simplied using the below relation as
 
iωt ∂ α . F (r) − jeiωt ∂ α . F (r)
 
 F (r) = F (r) − ie

 


 ∂x ∂y 



 

−ke iωt ∂ α . F (r)


 ∂z 



 

 
 ∂ ∂ ∂ 

(r ) = i ∂x + j ∂y + k ∂z 
Now the equation can be simplified as
∂F ∂x ∂F ∂y ∂F ∂z
0 = + + + (V + eiωt φ).
∂x ∂t ∂y ∂t ∂z ∂t
∂α ∂α
F (r) − ieiωt . F (r) − jeiωt . F (r)
∂x ∂y
∂α
− keiωt . F (r) on S H (2.13)
∂z

29. 2.2. Body Boundary Condition 15
Now the equatin must be simplied. Let us see each term individually
∂F ∂x ∂F ∂y ∂F ∂z
First Term : + + = −iωeiωt α. F (r) (2.14)
∂x ∂t ∂y ∂t ∂z ∂t
Now the equation can be rewritten as
0 = −iωeiωt α. F (r) + (V + eiωt φ).
∂α ∂α
F (r) − ieiωt . F (r) − jeiωt . F (r)
∂x ∂y
∂α
− keiωt . F (r) on S H (2.15)
∂z
The above formulation is satisfied on the instantaneous wetted surface (Exact body surface
F (x , y, z , t). This can be linearized under the assumption that unsteady displacement amplitude
small. So Taylor’s expansion can be applied for both steady and unsteady velocity field on the
mean wetted surface. Taylor’s expansion for V and φ
V (r) = V (r)mean + (α. )V (r) eiωt + 0(α2 ) (2.16)
mean
φ = φ(r)mean + (α. ) φ(r) eiωt + 0(α2 ) (2.17)
mean
Substituting the above formation in boundary condition ( 2.15 ), we can obtain the equation
on mean wetted surface
0 = −iωeiωt α. F (r) + V (r) + eiωt φ + eiωt (α. )V (r) .
∂α ∂α
F (r) − i eiωt . F (r) − j eiωt . F (r)
∂x ∂y
∂α
− k eiωt . F (r) on SH (2.18)
∂z
∂α ∂α
Last Term : F (r) − i eiωt . F (r) + j eiωt . F (r)
∂x ∂y
∂α
+k eiωt . F (r) (2.19)
∂z
Boundary Condition by Steady State Term for: V
V (r). F (r) = 0; or V . F = 0

30. 2.2. Body Boundary Condition 16
Boundary Condition by Oscillatory Function for: φ
eiωt φ. F (r) = iωeiωt α. F (r) − eiωt (α. )V . F r + (V )mean .
mean
∂α ∂α
i eiωt . F (r) + j eiωt . F (r)
∂x ∂y
∂α
+ k eiωt . F (r) on SH (2.20)
∂z
After removing the time part out of this equation, the equation becomes
φ. F (r) = iω (α. F (r)) − (α. )V . F (r) + (V )mean .
mean
∂α ∂α
i . F (r) + j . F (r)
∂x ∂y
∂α
+k . F (r) on SH (2.21)
∂z
or
φ. F (r) = iω α. F (r)
− (V . )α − (α. )V . F (r) on SH (2.22)
mean
All the terms are small, of the same order as α or φ. Thus to this order of approximation it
is no longer necessary to distinguish between the actual position and of the body and its mean
position, or between the co-ordinates r and r
φ· F = {iωα · F − (α · )V · F + (V · ) α} · F (2.23)
Vector Identity:
(V . )A − (A. )V = × (A × V ) − A .V + V .A (2.24)
.V = 0 for Incompressibility;
V. F =0 from the Steady Potential Condition;
so the unsteady condition shall be written from the equation ( 2.23 ) as
φ. F = iωα. F + × (α × V ) . F on SH (2.25)
∂
since: F ⇒ vector normal to the body surface and n. = ∂n
Boundary Condition for φ on the Body is :
∂φ
= iωα + × (α × V ) .n on SH (2.26)
∂n

31. 2.2. Body Boundary Condition 17
(i) Zero Speed Case
The derived body boundary condition shall be written as
∂φ
= iωα + × (α × V ) .n on SH (2.27)
∂n
The forward velocity is considered zero. So the second term will be neglected from equation (
2.27 ). So the boundary condition shall be,
∂φ
= iωα.n on SH (2.28)
∂n
Substituting the α in the above equation
∂φ
= iω iξ1 + jξ2 + kξ3 + (iξ4 + jξ5 + kξ6 ) × r .n on SH (2.29)
∂n
Vector relations and the normal vectors are written as
. .
 
 (ξ × r) n = ξ (n × r)
 


 

n = n1 , n 2 , n 3 (2.30)

 


 n×r =n , n , n 

4 5 6
Applying the above relation in equation ( 2.29 ), the equation shall be written as
3 6
∂φ
∂n
= iω ξj nj + iω .
ξ j (n × r)j on SH (2.31)
j=1 j=4
To consider the six degree of motion, boundary condition shall be written as
6
∂φ
= iω ξj n j on SH (2.32)
∂n
j=1
(ii) With Forward Speed
a) Translatory Motion (j = 1 · · · 3): α = ξ1 i + ξ2 j + ξ3 k ≡ A
Now the derived boundary condition shall be applied for the translatory motion as
∂φ
= iωα + × (α × V ) .n on SH (2.33)
∂n
Let us rewrite the vector identity and the motions are considered only three direction.
× (A × V ) = (V · )A + V ( · V )A − V ( · A) − (A · )V (2.34)

32. 2.2. Body Boundary Condition 18
Considering the translatory motion, the below mentioned relation can be used
·A=0
·V =0
Now in the vector identity, applying the above relations, the last term n (A · ) V can be
modified using given equation as
∂vi ∂ ∂φ ∂vj
= = (1 ≤ i, j ≤ 3) (2.35)
∂xj ∂xj ∂xi ∂xi
n (A · ) V = A (n · )V (2.36)
Now applying above relation in the body boundary condition, the final equation for translatory
motion with forward speed shall be written as
∂φ
= iωα · F+ × (α × V ) · n
∂n
= iωα · n − (n · )V · A
= iωn − (n · )V A (2.37)
b) Rotational Motion (j = 4 · · · 6): α = ξ4 i + ξ5 j + ξ6 k × r ≡ B × r
Body boundary condition shall be written as
× (A × V ) = (V · )A + V ( · V )A − V ( · A) = (A · )V (2.38)
Let us apply only the rotation motion in the condition as
iωn (B × r) = −iωB (n × r) = iωB (r × n) (2.39)
The above equation can be applied to the vector identity as
n· × {(B × r) × V } = n {(V · ) (B × r) + (B × r) ( ·V)−V [ · (B × r)] − [(B × r) · }
] V(2.40)
Using the below mentioned mathematical relation
∂u ∂v ∂w
(B × r) + + =0
∂x ∂y ∂z
(2.41)
∂ ∂ ∂
V + + (B × r) = 0
∂x ∂y ∂z
The vector identity term can be simplified as
n (V · ) (B × r) = n (B × V ) = −B (n × V )
(2.42)
n (B × r) · V = (B × r) · (n · ) V = − (n · ) × r B = r × (n · ) B

33. 2.3. Free Surface Boundary Condition 19
Applying to the boundary condition equation ( 2.26 ) as
∂φ
= iωα + × (α × V ) · n
∂n
= iω (r × n) − (n × ) − r × (n · )V B (2.43)
To simplify the above equation, tensor relation can be used
∂vj ∂
xi + vj = (xi · vj )
∂xi ∂xi
∂ ∂vk
(xj · vk ) = xj (1 ≤ i, j, k ≤ 3) (2.44)
∂xi ∂xi
Vector idetity which can be used
(n × ) + r × (n · ) V = (n · ) (r × V ) (2.45)
Applying the above mentioned relations, the final equation for body boundary condition shall
be written as
∂φj
= (iωnj + U mj ) ξj (j = 1 · · · 6) (2.46)
∂n
The full form of m and n vectors are written as
n = n1 i + n2 j + n3 k
r × n = n4 i + n5 j + n6 k
− (n · )V = m1 i + m2 j + m3 k
− (n · ) (r × V ) = m4 i + m5 j + m6 k (2.47)
2.3 Free Surface Boundary Condition
In the free surface, two boundary conditions are to be satisfied. The first one is (a) kinematic-free
surface boundary condition, and the second one is (b) dynamic free surface boundary condition.
The kinematic-free surface boundary condition states that the normal velocity of the fluid surface
& air surface must be equal, and the water surface does not penetrate the air surface. The fluid
particle on the fluid surface remains at the free surface. The exact free surface shall be written
as z = ζ (x, y; t)

34. 2.3. Free Surface Boundary Condition 20
Equation (2.5) is the total velocity potential. Now this can decomposed into steady wave
and unsteady wave field as
Ψ (x, y, z, t) = U ΦS (x, y, z) + φ (x, y, z) eiωe t
In the abve equation, steady flow can be further expanded as
Ψ (x, y, z, t) = U [Φ (x, y, z) + ϕ (x, y, z)] + φ (x, y, z) eiωe t (2.48)
Where Φ = −x + ΦD The unsteady velocity potential equation shall be written as,
6
gA
φ= (φ0 + φ7 ) + iωe ξj φ j (2.49)
ω0
j=1
where φ0 = ieKz−iK(x cos χ+y sin χ)
Φ means the double body flow, ϕ the steady wave field and φ the unsteady wave field which
consists of the incident wave velocity potential, the radiation potentials φj (j = 1 ∼ 6) and
the scattering potential φ7 that represents the disturbance of the incident waves by the fixed
ship. The radiation potentials represent the velocity potentials of a rigid body motion with unit
amplitude, in the absence of the incident waves. The total velocity potential must be applied
over the fluid domian and potential equation has been derived based Green’s theorem as follows,
∂ ∂φj (Q)
φj (P ) = − φj (Q) − G(P, Q)dS (2.50)
SH ∂n ∂n
1
where G(P, Q) = and r = (x − x)2 + (y − y)2 + (z − z)2 , P (x, y, z) is the field point
4πr
and Q(x, y, z) is the source point.
Now let us derive the free surface boundary condition.
Kinematic Free surface Boundary Condition: The substantial derivative D/Dt of a function
express that the rate of change with time of the function, if we follow a fluid particle in the free
surface. This can be applied in our exact free surface as,
D ∂
(z − ζ) = + Ψ· z − ζ (x, y; t) = 0 on z = ζ (x, y; t) (2.51)
Dt ∂t
D/Dt means the substantial derivative and is defined as a two-dimensional Laplacian with
respect to x and y on the free surface. z = ζ (x, y; t) is the wave elevation around a ship and

35. 2.3. Free Surface Boundary Condition 21
considered to be expressed by the summation of the steady wave ζs and the unsteady wave ζt
as follows
ζ (x, y; t) = ζs (x, y) + ζt (x, y; t) (2.52)
Dynamic Free surface Boundary Condition: The dynamics-free surface boundary condition is
that the pressure on the water surface is equal to the constant atmospheric pressure pa and this
can be obtained from Bernoulli’s equation.
1 p pa 1 2
Ψt + Ψ· Ψ + gz + = + U (2.53)
2 ρ ρ 2
where U is the forward speed of the vessel. Now, shifting the pressure to one side, the equation
shall be written as
1 1
p − pa = −ρ Ψt + Ψ· Ψ − U 2 + gz (2.54)
2 2
This pressure equation is applied on the exact free surface z = ζ (x, y; t). Equation shall be
rewritten considering the fluid pressure is equal to atmospheric pressure as,
∂Ψ 1 1
pa − pa = −ρ + Ψ· Ψ − U 2 + gζ (x, y; t) (2.55)
∂t 2 2
Keeping the free surface elevation in one side and the free surface is written as
1 ∂Ψ 1 1
ζ (x, y; t) = − + Ψ· Ψ − U2 on z = ζ (x, y; t) (2.56)
g ∂t 2 2
Now substituting the dynamic free surface boundary equation ( 2.56 )in the kinematic-free
surface boundary equation eq.(2.51), the free surface boundary condition is written as,
∂ 1 1 1
0= + Ψ· z+ Ψt + Ψ· Ψ − U2
∂t g 2 2
1 1
= Ψ· z+ Ψtt + Ψ· Ψt + Ψ · Ψt + Ψ· ( Ψ· Ψ)
g 2
1 1
= Ψz + Ψtt + 2 Ψ · Ψt + Ψ· ( Ψ· Ψ) on z = ζ (x, y; t) (2.57)
g 2
Now the exact free surface boundary condition shall be written as
1
Ψtt + 2 Ψ · Ψt + Ψ· ( Ψ· Ψ) + gΨz = 0 on z = ζ (x, y; t) (2.58)
2
Total velocity potential equation (2.48) shall be substituted in the free surface bounndary
condition eq.(2.58) as
φtt + 2 (Φ + ϕ + φ) · φt
1
+ (Φ + ϕ + φ) · (Φ + ϕ + φ) · (Φ + ϕ + φ)
2
+ g (Φz + ϕz + φz ) = 0 (2.59)

36. 2.3. Free Surface Boundary Condition 22
φtt + 2 Φ · φt + Φ· Φ· (ϕ + φ)
1
+ ( Φ· Φ) · (ϕ + φ) + g (ϕz + φz )
2
1
= −gΦz − ( Φ· Φ) · Φ on z = ζ (x, y; t) (2.60)
2
Similarly, total velocity potential shall be applied to the dynamic free surface as
1 1 1
ζ (x, y; t) = − φt + Ψ· Ψ − U2 on z = ζ (x, y; t) (2.61)
g 2 2
1 1 1
ζ (x, y; t) = − φt + (Φ + ϕ + φ) · (Φ + ϕ + φ) − U 2 on z = ζ (x, y; t) (2.62)
g 2 2
1 1 1
ζ (x, y; t) = − Φ· Φ − U2 − ( Φ· ϕ) − (φt + Φ· φ) on z = ζ (x, y; t) (2.63)
2g g g
In the above wave equation, the first part of equation is the wave which is the function of only
forward speed. The double body potential satisfies the Miller condition on z = 0, which do not
generate waves and the double body flow is satisfied on the free surface i.e, ∂Φ/∂z = 0 on z = 0.
Considering the above condition, the forward speed wave shall be written as,
1
ζ (x, y) = − Φ· Φ − U 2 on z = ζ (x, y) (2.64)
2g
Now the free surface equation eq.(2.63) shall be applied to the forward speed wave z = ζ by
applying Tailor series
1 1
ζ (x, y; t) = ζ − ( Φ· ϕ) − (φt + Φ· φ)
g g
∂ 1
+ ζ −ζ − Φ· Φ on z = ζ (x, y) (2.65)
∂z 2g
( Φ· ϕ) + (φt + Φ · φ)
ζ −ζ = − on z = ζ (x, y) (2.66)
g + Φ · Φz
Similarly the free surface equation equation (2.60) shall be applied on the double body flow free
surface by applying Tailor series
φtt + 2 Φ · φt + Φ· Φ· (ϕ + φ)
1
+ ( Φ · Φ) · (ϕ + φ) + g (ϕz + φz )
2
∂ 1 φt + Φ · (ϕ + φ)
− Φ · ( Φ · Φ) + gΦz
∂z 2 g + Φ · Φz
1
= −gΦz − ( Φ · Φ) · Φ on z = ζ (x, y) (2.67)
2

37. 2.3. Free Surface Boundary Condition 23
Presently, the free surface boundary condition is satisfied on the double body flow free surface.
Now we need to linearize the free surface i.e., z = 0. In this formulation, the steady flow
disturbance potential ϕ is very small when compare to other potential and the higher order can
be neglected. Applying the Tailor series for the above free surface equation with respect to z = 0
φtt + 2 Φ · φt + Φ· Φ· (ϕ + φ)
1
+ ( Φ · Φ) · (ϕ + φ) + g (ϕz + φz )
2
∂ 1 φt + Φ · (ϕ + φ)
− Φ · ( Φ · Φ) + gΦz
∂z 2 g + Φ · Φz
∂ 1 1
− Φ · ( Φ · Φ) + gΦz Φ · Φ − U2
∂z 2 2g
1
= −gΦz − ( Φ · Φ) · Φ on z = 0 (2.68)
2
On the free surface, the double body flow potential ∂Φ/∂z = 0 on z = 0. Considering that we
can bring the summarization as
Φ· Φz = 0 on z = 0 (2.69)
∂
Φ· ( Φ· Φ) on z = 0 (2.70)
∂z
Now subsituting the above relation and the free surface equation equation (2.68)
φtt + 2 Φ · φt + Φ· Φ· (ϕ + φ)
1
+ ( Φ· Φ) · (ϕ + φ) + g (ϕz + φz )
2
1
− Φzz φt + Φ· (ϕ + φ) − Φzz Φ· Φ − U2
2
1
=− ( Φ· Φ) · Φ on z = 0 (2.71)
2
This is the linearixed free surface boundary condition which must be statifed on z = 0 and this
can be separated as a steady part and unsteady part. The steady part potential free surface
shall be written as
1
Φ· ( Φ· ϕ) + ( Φ· Φ) · ϕ + gϕz − Φzz ( Φ · ϕ)
2
1 1
=− ( Φ· Φ) · Φ− U2 − Φ· Φ Φzz on z = 0 (2.72)
2 2

38. 2.4. Radiation Condition 24
For unsteady velocity potential, the free suface condition shall be written as
1
φtt + 2 Φ · φt + Φ· ( Φ· φ) + ( Φ· Φ) · φ
2
+ gφz − Φzz (φt + Φ· φ) = 0 on z = 0 (2.73)
The steady wave elevation shall be written as
1 1 1
ζ (x, y) = − Φ· Φ − U2 + Φ· ϕ on z = 0 (2.74)
g 2 2
The unsteady wave elevation shallbe written as
1
ζ (x, y; t) = − (φt + Φ· φ) on z = 0 (2.75)
g
In the free surface boundary condition ( 2.72 ), the boundary condition shall be written ne-
glecting the higher order term as
1 1
Φ· ( Φ· ϕ) + ( Φ· Φ) · ϕ + gϕz + ( Φ· Φ) · Φ = 0 on z = 0 (2.76)
2 2
The unsteady potential free suface shall be written as
1
φtt + 2 Φ · φt + Φ· ( Φ· φ) + ( Φ· Φ) · φ + gφz = 0 on z = 0 (2.77)
2
Taking the account of double body flow velocity potential Φ = U [Φ], the steady disturbance
potential ϕ = U [ϕ] and the unsteady velocity potential φ = φ (x, y, z) eiωe t , steady and
unsteady part shall be written. The steady potential free suface shall be written as
1 1 1 ∂ϕ
Φ· ( Φ· ϕ)+ ( Φ· Φ)· ϕ+ ( Φ· Φ)· Φ+ = 0 on z = 0 (2.78)
K0 2K0 2K0 ∂z
The unsteady potential free suface shall be written as
1 1 ∂φj
−Ke φj +2iτ Φ· φj + Φ· ( Φ· φj )+ ( Φ· Φ)· φj + = 0 on z = 0 (2.79)
K0 2K0 ∂z
where K0 = g/U 2 ; Ke = ωe /g and τ = U ωe /g
2
2.4 Radiation Condition
Radiation condition indicates that the generated waves by the body propagate to infinity. But
it is difficult to express this condition by mathematical equations. This condition is satisfied
numerically when we introduce the RPM.