Cambridge University Press
0521845688 - Hydrodynamics of High-Speed Marine Vehicles - by Odd M. Faltinsen
Frontmatter/Prelims



HYDRODYNAMICS OF HIGH-SPEED MARINE VEHICLES

Hydrodynamics of High-Speed Vehicles discusses the three main categories of high-speed marine vehicles, vessels supported by submerged hulls, air cushions, or foils. The wave environment, resistance, propulsion, seakeeping, sea loads, and maneuvering are extensively covered based on rational and simplified methods. Links to automatic control and structural mechanics are emphasized. A detailed description of waterjet propulsion is given, and the effect of water depth on wash, resistance, sinkage, and trim is discussed. Chapter topics include resistance and wash; slamming; air cushion–supported vessels, including a detailed discussion of wave-excited resonant oscillations in air cushion; and hydrofoil vessels. The book contains numerous illustrations, examples, and exercises.

Odd M. Faltinsen received his Ph.D. in naval architecture and marine engineering from the University of Michigan in 1971 and has been a Professor of Marine Hydrodynamics at the Norwegian University of Science and Technology since 1974. Dr. Faltinsen has experience with a broad spectrum of hydrodynamically related problems for ships and sea structures, including hydroelastic problems and slamming. He has published more than 200 scientific papers, and his textbook Sea Loads on Ships and Offshore Structures, published by Cambridge University Press in 1990, is used at universities worldwide.




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Hydrodynamics of High-Speed
Marine Vehicles

ODD M. FALTINSEN
Norwegian University of Science and Technology




CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
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© Cambridge University Press 2005

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the written permission of Cambridge University Press.

First published 2005

Printed in the United States of America

A catalog record for this publication is available from the British Library.

Library of Congress Cataloging in Publication Data

Faltinsen, O. M. (Odd Magnus), 1944–
Hydrodynamics of high-speed marine vehicles / Odd M. Faltinsen.
p. cm.
Includes bibliographical references and index.
ISBN 0-521-84568-8 (hardback)
1. Motorboats. 2. Ships – Hydrodynamics. 3. Hydrodynamics. 4. Hydrofoil boats. I. Title.
VM341.F35    2005
623.8′1231 – dc22    2005006328

ISBN-13 978-0-521-84568-7 hardback
ISBN-10 0-521-84568-8 hardback

Cambridge University Press has no responsibility for
the persistence or accuracy of URLs for external or
third-party Internet Web sites referred to in this publication
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Contents

Preface page xiii
List of symbols xv
1   INTRODUCTION 1
    1.1  Operational limits 6
    1.2  Hydrodynamic optimization 10
    1.3  Summary of main chapters 10
2   RESISTANCE AND PROPULSION 12
    2.1  Introduction 12
    2.2  Viscous water resistance 13
       2.2.1  Navier-Stokes equations 16
       2.2.2  Reynolds-averaged Navier-Stokes (RANS) equations 18
       2.2.3  Boundary-layer equations for 2D turbulent flow 19
       2.2.4  Turbulent flow along a smooth flat plate. Frictional resistance component 20
       2.2.5  Form resistance components 25
       2.2.6  Effect of hull surface roughness on viscous resistance 28
       2.2.7  Viscous foil resistance 31
    2.3  Air resistance component 35
    2.4  Spray and spray rail resistance components 36
    2.5  Wave resistance component 38
    2.6  Other resistance components 38
    2.7  Model testing of ship resistance 39
       2.7.1  Other scaling parameters 42
    2.8  Resistance components for semi-displacement monohulls and catamarans 42
    2.9  Wake flow 45
    2.10  Propellers 47
       2.10.1  Open-water propeller characteristics 53
       2.10.2  Propellers for high-speed vessels 55
       2.10.3  Hull-propeller interaction 60
    2.11  Waterjet propulsion 61
       2.11.1  Experimental determination of thrust and efficiency by model tests 63
       2.11.2  Cavitation in the inlet area 70
    2.12  Exercises 73
       2.12.1  Scaling 73
       2.12.2  Resistance by conservation of fluid momentum 74
       2.12.3  Viscous flow around a strut 75
       2.12.4  Thrust and efficiency of a waterjet system 75
       2.12.5  Steering by means of waterjet 77
3   WAVES 78
    3.1  Introduction 78
    3.2  Harmonic waves in finite and infinite depth 78
       3.2.1  Free-surface conditions 78
       3.2.2  Linear long-crested propagating waves 81
       3.2.3  Wave energy propagation velocity 84
       3.2.4  Wave propagation from deep to shallow water 86
       3.2.5  Wave refraction 87
       3.2.6  Surface tension 90
    3.3  Statistical description of waves in a sea state 91
    3.4  Long-term predictions of sea states 94
    3.5  Exercises 95
       3.5.1  Fluid particle motion in regular waves 95
       3.5.2  Sloshing modes 97
       3.5.3  Second-order wave theory 97
       3.5.4  Boussinesq equations 98
       3.5.5  Gravity waves in a viscous fluid 98
4   WAVE RESISTANCE AND WASH 99
    4.1  Introduction 99
       4.1.1  Wave resistance 99
       4.1.2  Wash 101
    4.2  Ship waves in deep water 103
       4.2.1  Simplified evaluation of Kelvin’s angle 105
       4.2.2  Far-field wave patterns 105
       4.2.3  Transverse waves along the ship’s track 107
       4.2.4  Example 110
    4.3  Wave resistance in deep water 110
       4.3.1  Example: Wigley’s wedge-shaped body 112
       4.3.2  Example: Wigley ship model 112
       4.3.3  Example: Tuck’s parabolic strut 114
       4.3.4  2.5D (2D+t) theory 115
       4.3.5  Multihull vessels 120
       4.3.6  Wave resistance of SES and ACV 122
    4.4  Ship in finite water depth 123
       4.4.1  Wave patterns 126
    4.5  Ship in shallow water 128
       4.5.1  Near-field description 128
       4.5.2  Far-field equations 129
       4.5.3  Far-field description for supercritical speed 130
       4.5.4  Far-field description for subcritical speed 131
       4.5.5  Forces and moments 132
       4.5.6  Trim and sinkage 134
    4.6  Exercises 135
       4.6.1  Thin ship theory 135
       4.6.2  Two struts in tandem 136
       4.6.3  Steady ship waves in a towing tank 136
       4.6.4  Wash 137
       4.6.5  Wave patterns for a ship on a circular course 138
       4.6.6  Internal waves 138
5   SURFACE EFFECT SHIPS 141
    5.1  Introduction 141
    5.2  Water level inside the air cushion 141
    5.3  Effect of air cushion on the metacentric height in roll 143
    5.4  Characteristics of aft seal air bags 145
    5.5  Characteristics of bow seal fingers 147
    5.6  “Cobblestone” oscillations 149
       5.6.1  Uniform pressure resonance in the air cushion 150
       5.6.2  Acoustic wave resonance in the air cushion 154
       5.6.3  Automatic control 158
    5.7  Added resistance and speed loss in waves 159
    5.8  Seakeeping characteristics 161
    5.9  Exercises 163
       5.9.1  Cushion support at zero speed 163
       5.9.2  Steady airflow under an aft-seal air bag 163
       5.9.3  Damping of cobblestone oscillations by T-foils 163
       5.9.4  Wave equation 164
       5.9.5  Speed of sound 164
       5.9.6  Cobblestone oscillations with acoustic resonance 164
6   HYDROFOIL VESSELS AND FOIL THEORY 165
    6.1  Introduction 165
    6.2  Main particulars of hydrofoil vessels 166
    6.3  Physical features 166
       6.3.1  Static equilibrium in foilborne condition 166
       6.3.2  Active control system 169
       6.3.3  Cavitation 169
       6.3.4  From hullborne to foilborne condition 173
       6.3.5  Maneuvering 176
       6.3.6  Seakeeping characteristics 178
    6.4  Nonlinear hydrofoil theory 178
       6.4.1  2D flow 178
       6.4.2  3D flow 184
    6.5  2D steady flow past a foil in infinite fluid. Forces 187
    6.6  2D linear steady flow past a foil in infinite fluid 188
       6.6.1  Flat plate 192
       6.6.2  Foil with angle of attack and camber 193
       6.6.3  Ideal angle of attack and angle of attack with zero lift 193
       6.6.4  Weissinger’s “quarter-three-quarter-chord” approximation 193
       6.6.5  Foil with flap 194
    6.7  3D linear steady flow past a foil in infinite fluid 195
       6.7.1  Prandtl’s lifting line theory 195
       6.7.2  Drag force 197
    6.8  Steady free-surface effects on a foil 199
       6.8.1  2D flow 199
       6.8.2  3D flow 202
    6.9  Foil interaction 205
    6.10  Ventilation and steady free-surface effects on a strut 208
    6.11  Unsteady linear flow past a foil in infinite fluid 209
       6.11.1  2D flow 209
       6.11.2  2D flat foil oscillating harmonically in heave and pitch 210
       6.11.3  3D flow 212
    6.12  Wave-induced motions in foilborne conditions 212
       6.12.1  Case study of vertical motions and accelerations in head and following waves 216
    6.13  Exercises 219
       6.13.1  Foil-strut intersection 219
       6.13.2  Green’s second identity 219
       6.13.3  Linearized 2D flow 219
       6.13.4  Far-field description of a high-aspect–ratio foil 219
       6.13.5  Roll-up of vortices 219
       6.13.6  Vertical wave-induced motions in regular waves 220
7   SEMI-DISPLACEMENT VESSELS 221
    7.1  Introduction 221
       7.1.1  Main characteristics of monohull vessels 221
       7.1.2  Main characteristics of catamarans 221
       7.1.3  Motion control 224
       7.1.4  Single-degree mass-spring system with damping 226
    7.2  Linear wave-induced motions in regular waves 229
       7.2.1  The equations of motions 233
       7.2.2  Simplified heave analysis in head sea for monohull at forward speed 236
       7.2.3  Heave motion in beam seas of a monohull at zero speed 237
       7.2.4  Ship-generated unsteady waves 238
       7.2.5  Hydrodynamic hull interaction 240
       7.2.6  Summary and concluding remarks on wave radiation damping 246
       7.2.7  Hull-lift damping 246
       7.2.8  Foil-lift damping 247
       7.2.9  Example: Importance of hull- and foil-lift heave damping 249
       7.2.10  Ride control of vertical motions by T-foils 249
       7.2.11  Roll motion in beam sea of a catamaran at zero speed 250
       7.2.12  Numerical predictions of unsteady flow at high speed 253
    7.3  Linear time-domain response 257
    7.4  Linear response in irregular waves 259
       7.4.1  Short-term sea state response 259
       7.4.2  Long-term predictions 260
    7.5  Added resistance in waves 261
       7.5.1  Added resistance in regular waves 261
       7.5.2  Added resistance in a sea state 263
    7.6  Seakeeping characteristics 263
    7.7  Dynamic stability 266
       7.7.1  Mathieu instability 268
    7.8  Wave loads 270
       7.8.1  Local pressures of non-impact type 271
       7.8.2  Global wave loads on catamarans 273
    7.9  Exercises 282
       7.9.1  Mass matrix 282
       7.9.2  2D heave-added mass and damping 282
       7.9.3  Linear wavemaker solution 283
       7.9.4  Foil-lift damping of vertical motions 284
       7.9.5  Roll damping fins 285
       7.9.6  Added mass and damping in roll 285
       7.9.7  Global wave loads in the deck of a catamaran 285
8   SLAMMING, WHIPPING, AND SPRINGING 286
    8.1  Introduction 286
    8.2  Local hydroelastic slamming effects 290
       8.2.1  Example: Local hydroelastic slamming on horizontal wetdeck 298
       8.2.2  Relative importance of local hydroelasticity 299
    8.3  Slamming on rigid bodies 301
       8.3.1  Wagner’s slamming model 305
       8.3.2  Design pressure on rigid bodies 309
       8.3.3  Example: Local slamming-induced stresses in longitudinal stiffener by quasi-steady beam theory 310
       8.3.4  Effect of air cushions on slamming 310
       8.3.5  Impact of a fluid wedge and green water 313
    8.4  Global wetdeck slamming effects 317
       8.4.1  Water entry and exit loads 319
       8.4.2  Three-body model 321
    8.5  Global hydroelastic effects on monohulls 325
       8.5.1  Special case: Rigid body 328
       8.5.2  Uniform beam 329
    8.6  Global bow flare effects 330
    8.7  Springing 334
       8.7.1  Linear springing 336
    8.8  Scaling of global hydroelastic effects 338
    8.9  Exercises 338
       8.9.1  Probability of wetdeck slamming 338
       8.9.2  Wave impact at the front of a wetdeck 339
       8.9.3  Water entry of rigid wedge 339
       8.9.4  Drop test of a wedge 340
       8.8.5  Generalized Wagner method 340
       8.9.6  3D flow effects during slamming 340
       8.9.7  Whipping studies by a three-body model 341
       8.9.8  Frequency-of-encounter wave spectrum in following sea 341
       8.9.9  Springing 341
9   PLANING VESSELS 342
    9.1  Introduction 342
    9.2  Steady behavior of a planing vessel on a straight course 344
       9.2.1  2.5D (2D+t) theory 345
       9.2.2  Savitsky’s formula 349
       9.2.3  Stepped planing hull 355
       9.2.4  High-aspect–ratio planing surfaces 358
    9.3  Prediction of running attitude and resistance in calm water 360
       9.3.1  Example: Forces act through COG 360
       9.3.2  General case 362
    9.4  Steady and dynamic stability 363
       9.4.1  Porpoising 365
    9.5  Wave-induced motions and loads 373
       9.5.1  Wave excitation loads in heave and pitch in head sea 374
       9.5.2  Frequency-domain solution of heave and pitch in head sea 378
       9.5.3  Time-domain solution of heave and pitch in head sea 378
       9.5.4  Example: Heave and pitch in regular head sea 380
    9.6  Maneuvering 383
    9.7  Exercises 385
       9.7.1  2.5D theory for planing hulls 385
       9.7.2  Minimalization of resistance by trim tabs 386
       9.7.3  Steady heel restoring moment 386
       9.7.4  Porpoising 388
       9.7.5  Equation system of porpoising 388
       9.7.6  Wave-induced vertical accelerations in head sea 388
10   MANEUVERING 390
    10.1  Introduction 390
    10.2  Traditional coordinate systems and notations in ship maneuvering 393
    10.3  Linear ship maneuvering in deep water at moderate Froude number 395
       10.3.1  Low-aspect–ratio lifting surface theory 398
       10.3.2  Equations of sway and yaw velocities and accelerations 399
       10.3.3  Directional stability 400
       10.3.4  Example: Directional stability of a monohull 401
       10.3.5  Steady-state turning 401
       10.3.6  Multihull vessels 402
       10.3.7  Automatic control 403
    10.4  Linear ship maneuvering at moderate Froude number in finite water depth 403
    10.5  Linear ship maneuvering in deep water at high Froude number 403
    10.6  Nonlinear viscous effects for maneuvering in deep water at moderate speed 406
       10.6.1  Cross-flow principle 406
       10.6.2  2D+t theory 410
       10.6.3  Empirical nonlinear maneuvering models 415
    10.7  Coupled surge, sway, and yaw motions of a monohull 416
       10.7.1  Influence of course control on propulsion power 417
    10.8  Control means 419
    10.9  Maneuvering models in six degrees of freedom 421
       10.9.1  Euler’s equation of motion 421
       10.9.2  Linearized equation system in six degrees of freedom 425
       10.9.3  Coupled sway-roll-yaw of a monohull 426
    10.10  Exercises 431
       10.10.1  Course stability of a ship in a canal 431
       10.10.2  Nonlinear, nonlifting and nonviscous hydrodynamic forces and moments on a maneuvering body 432
       10.10.3  Maneuvering in waves and broaching 432
       10.10.4  Linear coupled sway-yaw-roll motions of a monohull at moderate speed 433
       10.10.5  High-speed motion in water of an accidentally dropped pipe 433
APPENDIX: Units of Measurement and Physical Constants 435
References 437
Index 451



Preface

Writing a book on the hydrodynamics of high-speed marine vehicles was challenging because I have had to cover all areas of traditional marine hydrodynamics, resistance, propulsion, seakeeping, and maneuvering. However, there is a need to combine all aspects of hydrodynamics in the design of which high-speed vessels are very different from conventional ships, depending on whether they are hull supported, air cushion supported, foil supported, or hybrids.

   High-speed vessels are a fascinating topic, and I have been deeply involved in research on high-speed vessels since a national research program under the leadership of Kjell Holden started in Norway in 1989. We also started the International Conference on Fast Sea Transportation (FAST), which has a much broader scope than marine hydrodynamics. I have also benefited from being the chairman of the Committee of High-Speed Marine Vehicles of the International Towing Tank Conference (ITTC) from 1990 to 1993. Further, this book would not have been possible without the work done by the many doctoral students who I have supervised. Their theses are referenced in the book. Parts of the book have been taught to the fourth year, master of science students and doctoral students at the Department of Marine Technology, Norwegian University of Science and Technology (NTNU).

   My philosophy in writing the book has been to start from basic fluid dynamics and to link this to practical issues for high-speed vessels. Mathematics is a necessity, but I have tried to avoid this when physical explanations can be given. Knowledge of calculus, including vector analysis and differential equations, is necessary to read the book in detail. The reader should also be familiar with dynamics and basic hydrodynamics of potential and viscous flow of an incompressible fluid.

   Computational fluid dynamics (CFD) are commonly used nowadays, but my emphasis is on giving simplified and rational explanations of fluid behavior and its interaction with the vessel. This is beneficial in planning and interpreting experiments and computations. I also believe that examples and exercises are important parts of the learning process.

   Automatic control and structural mechanics of high-speed marine vehicles are two disciplines that rely on hydrodynamics. These links are emphasized in the book and are also important aspects of the Centre for Ships and Ocean Structures, NTNU, where I participate.

   My presentation of the material is inspired by the book Marine Hydrodynamics by Professor J. N. Newman.

   I am thankful to Professor Newman for reading through the manuscript and offering suggestions for improvement. Dr. Svein Skj⊘rdal spent a lot of time giving detailed comments on different versions of the manuscript. He was also helpful in seeing the topics from a practical point of view. Sun Hui also did a great job in confirming all my calculations and providing solutions to all exercises. I have benefited from Professor K. J. Minsaas’ expertise in propulsion and hydrodynamic design of hydrofoil vessels. Many other people should be thanked for their critical reviews and contributions, including Dr. Tony Armstrong, Professor Tor Einar Berg, J. Bloch Helmers, Professor Lawrence Doctors, Dr. Svein Ersdal, Lars Flæten, Professor Thor I. Fossen, Dr. Chunhua Ge, Dr. Marilena Greco, Dr. Martin Greenhow, Dr. Ole Hermundstad, Egil Jullumstr⊘, Dr. Toru Katayama, Professor Katsuro Kijima, Professor Spyros A. Kinnas, Dr. Kourosh Koushan, David Kristiansen, Professor Claus Kruppa, Dr. Jan Kvaalsvold, Dr. Burkhard Müller-Graf, Professor Dag Myrhaug, Professor Makoto Ohkusu, Professor Bj⊘rnar Pettersen, Dr. Olav Rognebakke, Renato Skejic, Dr. Nere Skomedal, Professor Sverre Steen, Gaute Storhaug, Professor Asgeir S⊘rensen, Professor Ernest O. Tuck, and Dr. Frans van Walree.

   The artwork was done by Bjarne Stenberg. Anne-Irene Johannessen and Keivan Koushan were helpful in drawing figures. Jorunn Fransvåg organized and typed the many versions of the manuscript in an accurate and efficient way, which required a tremendous amount of work.

   The support from the Centre of Ships and Ocean Structures and the Department of Marine Technology at NTNU is appreciated.




List of symbols

A area; planform area of foil
AD developed area, propeller blades
AE expanded area, propeller blades
Ajk 3D added mass coefficient in the jth mode due to kth motion
ajk 2D added mass coefficient
AO area of propeller disc
AP after perpendicular
AR rudder area
AW waterplane area
AHR average hull roughness
B beam
b beam of section
BAR blade area ratio
Bcr, bcr critical damping
Bjk 3D damping coefficient in jth mode due to kth motion
bjk 2D damping coefficient
c chord length; half wetted length in 2D impact; speed of sound
CB block coefficient, ship
CD drag coefficient
Cf friction coefficient
CF frictional force coefficient
CFD computational fluid dynamics
CH head coefficient
Cjk restoring force coefficient in jth mode due to kth motion
CL lift coefficient
CLβ lift coefficient for planing vessel
CL0 CLβ at zero deadrise angle
C(kf) Theodorsen function
CM midship section coefficient; mass coefficient in Morison’s equation
COG center of gravity
Cp pressure coefficient
Cp min minimum pressure coefficient
CP longitudinal prismatic coefficient
CR residual resistance coefficient
CT propeller thrust-loading coefficient; total resistance coefficient
CW wave-making resistance coefficient
CWP wave pattern resistance coefficient
CQ capacity coefficient
D draft; drag force; propeller diameter
DNV Det Norske Veritas
DT transom draft
E Young’s modulus of elasticity
EI flexural rigidity of a beam
Ek kinetic fluid energy
E(t) energy
f frequency (Hz); maximum camber
F densimetric Froude number; fetch length
Fn Froude number U∕√gL
FnB beam Froude number
FnD draft Froude number
Fnh depth or submergence Froude number
FnT transom draft Froude number
FP forward perpendicular
Fv volumetric Froude number
g acceleration of gravity
G(x,y,z;ξ,η,ζ) Green function
□ transverse metacentric height
□L longitudinal metacentric height
□ moment arm in heel (roll) about COG
h water depth; submergence
hj height of the center of the jet at station S7 (see Figure 2.54) above calm free surface
H wave height; head
H1∕3 significant wave height
i imaginary unit
Ijk moment or product of inertia
i, j, k unit vectors along x, y and z-axis, respectively
IVR inlet velocity ratio
J advance ratio of propeller
k wave number; roughness height; form factor
KC Keulegan-Carpenter number
kf reduced frequency
□ height of COG above keel
KT thrust coefficient
KQ torque coefficient
L length of ship; lift of a foil; hydrodynamic roll moment in maneuvering
LC chine wetted length
LCB longitudinal center of buoyancy
lcg longitudinal center of gravity measured from the transom stern
LCG longitudinal center of gravity
LK keel wetted length
LOA length, overall
LOS length, overall submerged
lp longitudinal position of the center of pressure measured along the keel from the transom stern
LPP length between perpendiculars
LWL length of the designer’s load waterline
M mass; moment; hydrodynamic pitch moment in maneuvering
M fluid momentum vector
m mass per unit length
Mjk components of mass matrix
n propeller revolutions per second
n surface normal vector positive into the fluid
N normal force; hydrodynamic yaw moment in maneuvering
O origin of coordinate system
O(ε) order of magnitude of ε
P power; pitch of propeller; probability
p pressure; roll component of angular velocity; half of the distance between the center lines of the demihulls of a catamaran; stagger between foils
pa atmospheric pressure
PD delivered power
po ambient pressure; static excess pressure
pv vapor pressure of water
Q propeller torque; volume flux; source strength
q pitch component of angular velocity
r yaw component of angular velocity
R radius; resistance
RAA added resistance in air and wind
RAW added resistance in waves
rjj radius of gyration in rigid body mode j
RMS root mean square
Rn Reynolds number
RR residual resistance
RS spray resistance
RT total resistance
RV viscous resistance
RW wave-making resistance
s span length of foil
S area of wetted surface; cross-sectional area
SB body surface
S(ω) wave spectrum
t time; thrust-deduction coefficient; maximum foil thickness
T period; propeller thrust
T0 modal or peak period
T1 mean wave period
T2 mean wave period
Te encounter period
Tn natural period
TS surface tension
U forward velocity of vessel
UI mean velocity at the most narrow cross-section of the waterjet inlet
US propeller slip stream velocity
u x-component of vessel velocity
v y-component of vessel velocity
vcg vertical distance between COG and the keel
Vg group velocity
Vp phase velocity
v∗ wall friction velocity
V water entry velocity
W weight
w wake fraction; z-component of vessel velocity; vertical deflection
Wn Weber number
x, y, z Cartesian coordinate system. Moving with the forward speed in seakeeping analysis. Body-fixed in maneuvering analysis.
X x-component of hydrodynamic force in maneuvering
XE, YE, ZE Earth-fixed coordinate system
xT x-coordinate of transom
xs LK– LC
Y y-component of hydrodynamic force in maneuvering
Z z-component of hydrodynamic force in maneuvering
 
Greek symbols
 
α angle of attack
αc Kelvin angle
αf flap angle
αi ideal angle of attack
α0 angle of zero lift
β wave propagation angle; deadrise angle; drift angle
Γ circulation; gamma function; dihedral angle
γ vortex density; sweep angle; ratio of specific heat for air
δ boundary layer thickness; rudder angle; flap angle
δ∗ displacement thickness
Δ vessel weight
ε angle
ζ surface elevation
ζa wave amplitude
η overall propulsive efficiency
ηH hull efficiency
ηJ jet efficiency
ηk wave-induced vessel motion response, where k = 1, 2, 3....6 refers to surge, sway, heave, roll, pitch, and yaw, respectively
ηp propeller efficiency; pump efficiency
ηR relative rotative efficiency
ηS sinkage
ηT thrust power efficiency
θ pitch angle; momentum thickness
Λ aspect ratio of foil
ΛL ratio between full scale and model length
λ wavelength
λw mean wetted length-to-beam ratio
μ dynamic viscosity coefficient
ν kinematic viscosity coefficient
ξ ratio between damping and critical damping
ρ mass density of fluid (water)
ρa mass density of air
σ cavitation number; source density; standard deviation
σi cavitation inception index
σo propeller cavitation number
σ0.7 propeller cavitation number defined at 0.7 R
τ trim angle in radians; ωeU/g
τdeg trim angle in degrees
τij Newtonian stress relations
τw frictional stress on hull surface
ɸ heel (roll) angle
φ velocity potential
ψ yaw angle
ω circular frequency in radians per second
ω vorticity vector; vector of rotational vessel motion
ωn natural frequency
ωe frequency of encounter
ωo frequency of waves in an Earth-fixed coordinate system
Ω vector of rotational vessel velocity
Ω volume
 
Special symbols
 
∇ displaced volume of water; vector differential operator
∇2 ∂2∕∂x2+∂2∕∂y2+∂2∕∂z2



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