Hydrodynamics of High-Speed Vehicles

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Hydrodynamics of High-Speed Vehicles

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 empha-sized. 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 prob-lems and slamming. He has published more than 200 scientific papers, and his textbook Sea Loads on Ships and Offshore Struc-tures, published by Cambridge University Press in 1990, is used at universities worldwide.

Hydrodynamics of High-speed Marine Vehicles ODD M. FALTINSEN Norwegian University of Science and Technology CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Siio Paulo Cambridge University Press 40 West 20th Street, New York, NY 10011-4211, USA www.cambridge.org Information on this title: www.cambridge.org/9780521845687 O Cambridge University Press 2005 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without 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, 0. M. (Odd Magnus), 1944- Hydrodynamics of high-speed marine vehicles i 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 and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Preface List of symbols page xiii xv 1 INTRODUCTION 1.1 Operational limits 1.2 Hydrodynamic optimization 1.3 Summary of main chapters 2 RESISTANCE AND PROPULSION 2.1 Introduction 2.2 Viscous water resistance 2.2.1 Navier-Stokes equations 2.2.2 Reynolds-averaged Navier-Stokes (RANS) equations 2.2.3 Boundary-layer equations for 2D turbulent flow 2.2.4 Turbulent flow along a smooth flat plate. Frictional resistance component 2.2.5 Form resistance components 2.2.6 Effect of hull surface roughness on viscous resistance 2.2.7 Viscous foil resistance 2.3 Air resistance component 2.4 Spray and spray rail resistance components 2.5 Wave resistance component 2.6 Other resistance components 2.7 Model testing of ship resistance 2.7.1 Other scaling parameters 2.8 Resistance components for semi-displacement monohulls and catamarans 2.9 Wake flow 2.10 Propellers 2.10.1 Open-water propeller characteristics 2.10.2 Propellers for high-speed vessels 2.10.3 Hull-propeller interaction 2.11 Waterjet propulsion 2.11.1 Experimental determination of thrust and efficiency by model tests 2.1 1.2 Cavitation in the inlet area

vi Contents 2.12 Exercises 2.12.1 Scaling 2.12.2 Resistance by conservation of fluid momentum 2.12.3 Viscous flow around a strut 2.12.4 Thrust and efficiency of a waterjet system 2.12.5 Steering by means of waterjet 3 WAVES 3.1 Introduction 3.2 Harmonic waves in finite and infinite depth 3.2.1 Free-surface conditions 3.2.2 Linear long-crested propagating waves 3.2.3 Wave energy propagation velocity 3.2.4 Wave propagation from deep to shallow water 3.2.5 Wave refraction 3.2.6 Surface tension 3.3 Statistical description of waves in a sea state 3.4 Long-term predictions of sea states 3.5 Exercises 3.5.1 Fluid particle motion in regular waves 3.5.2 Sloshing modes 3.5.3 Second-order wave theory 3.5.4 Boussinesq equations 3.5.5 Gravity waves in a viscous fluid 4 WAVE RESISTANCE AND WASH 4.1 Introduction 4.1.1 Wave resistance 4.1.2 Wash 4.2 Ship waves in deep water 4.2.1 Simplified evaluation of Kelvin's angle 4.2.2 Far-field wave patterns 4.2.3 Transverse waves along the ship's track 4.2.4 Example 4.3 Wave resistance in deep water 4.3.1 Example: Wigley's wedge-shaped body 4.3.2 Example: Wigley ship model 4.3.3 Example: Tuck's parabolic strut 4.3.4 2.5D (2D;tt) theory 4.3.5 Multihull vessels 4.3.6 Wave resistance of SES and ACV 4.4 Ship in finite water depth 4.4.1 Wave patterns 4.5 Ship in shallow water 4.5.1 Near-field description 4.5.2 Far-field equations 4.5.3 Far-field description for supercritical speed

Contents vii 4.5.4 Far-field description for subcritical speed 4.5.5 Forces and moments 4.5.6 Trim and sinkage 4.6 Exercises 4.6.1 Thin ship theory 4.6.2 Two struts in tandem 4.6.3 Steady ship waves in a towing tank 4.6.4 Wash 4.6.5 Wave patterns for a ship on a circular course 4.6.6 Internal waves 5 SURFACE EFFECT SHIPS 5.1 Introduction 5.2 Water level inside the air cushion 5.3 Effect of air cushion on the metacentric height in roll 5.4 Characteristics of aft seal air bags 5.5 Characteristics of bow seal fingers 5.6 "Cobblestone" oscillations 5.6.1 Uniform pressure resonance in the air cushion 5.6.2 Acoustic wave resonance in the air cushion 5.6.3 Automatic control 5.7 Added resistance and speed loss in waves 5.8 Seakeeping characteristics 5.9 Exercises 5.9.1 Cushion support at zero speed 5.9.2 Steady airflow under an aft-seal air bag 5.9.3 Damping of cobblestone oscillations by T-foils 5.9.4 Wave equation 5.9.5 Speed of sound 5.9.6 Cobblestone oscillations with acoustic resonance 6 HYDROFOIL VESSELS AND FOIL THEORY 6.1 Introduction 6.2 Main particulars of hydrofoil vessels 6.3 Physical features 6.3.1 Static equilibrium in foilborne condition 6.3.2 Active control system 6.3.3 Cavitation 6.3.4 From hullborne to foilborne condition 6.3.5 Maneuvering 6.3.6 Seakeeping characteristics 6.4 Nonlinear hydrofoil theory 6.4.1 2D flow 6.4.2 3D flow 6.5 2D steady flow past a foil in infinite fluid. Forces 6.6 2D linear steady flow past a foil in infinite fluid 6.6.1 Flat plate

viii Contents 6.6.2 Foil with angle of attack and camber 6.6.3 Ideal angle of attack and angle of attack with zero lift 6.6.4 Weissinger's "quarter-three-quarter-chord" approximation 6.6.5 Foil with flap 6.7 3D linear steady flow past a foil in infinite fluid 6.7.1 Prandtl's lifting line theory 6.7.2 Drag force 6.8 Steady free-surface effects on a foil 6.8.1 2D flow 6.8.2 3D flow 6.9 Foil interaction 6.10 Ventilation and steady free-surface effects on a strut 6.11 Unsteady linear flow past a foil in infinite fluid 6.11.1 2D flow 6.11.2 2D flat foil oscillating harmonically in heave and pitch 6.11.3 3D flow 6.12 Wave-induced motions in foilborne conditions 6.12.1 Case study of vertical motions and accelerations in head and following waves 6.13 Exercises 6.13.1 Foil-strut intersection 6.13.2 Green's second identity 6.13.3 Linearized 2D flow 6.13.4 Far-field description of a high-aspect-ratio foil 6.13.5 Roll-up of vortices 6.13.6 Vertical wave-induced motions in regular waves 7 SEMI-DISPLACEMENT VESSELS 7.1 Introduction 7.1.1 Main characteristics of monohull vessels 7.1.2 Main characteristics of catamarans 7.1.3 Motion control 7.1.4 Single-degree mass-spring system with damping 7.2 Linear wave-induced motions in regular waves 7.2.1 The equations of motions 7.2.2 Simplified heave analysis in head sea for monohull at forward speed 7.2.3 Heave motion in beam seas of a monohull at zero speed 7.2.4 Ship-generated unsteady waves 7.2.5 Hydrodynamic hull interaction 7.2.6 Summary and concluding remarks on wave radiation damping 7.2.7 Hull-lift damping 7.2.8 Foil-lift damping 7.2.9 Example: Importance of hull- and foil-lift heave damping 7.2.10 Ride control of vertical motions by T-foils

Contents ix 7.2.11 Roll motion in beam sea of a catamaran at zero speed 7.2.12 Numerical predictions of unsteady flow at high speed 7.3 Linear time-domain response 7.4 Linear response in irregular waves 7.4.1 Short-term sea state response 7.4.2 Long-term predictions 7.5 Added resistance in waves 7.5.1 Added resistance in regular waves 7.5.2 Added resistance in a sea state 7.6 Seakeeping characteristics 7.7 Dynamic stability 7.7.1 Mathieu instability 7.8 Wave loads 7.8.1 Local pressures of non-impact type 7.8.2 Global wave loads on catamarans 7.9 Exercises 7.9.1 Mass matrix 7.9.2 2D heave-added mass and damping 7.9.3 Linear wavemaker solution 7.9.4 Foil-lift damping of vertical motions 7.9.5 Roll damping fins 7.9.6 Added mass and damping in roll 7.9.7 Global wave loads in the deck of a catamaran 8 SLAMMING, WHIPPING, AND SPRINGING 8.1 Introduction 8.2 Local hydroelastic slamming effects 8.2.1 Example: Local hydroelastic slamming on horizontal wetdeck 8.2.2 Relative importance of local hydroelasticity 8.3 Slamming on rigid bodies 8.3.1 Wagner's slamming model 8.3.2 Design pressure on rigid bodies 8.3.3 Example: Local slamming-induced stresses in longitudinal stiffener by quasi-steady beam theory 8.3.4 Effect of air cushions on slamming 8.3.5 Impact of a fluid wedge and green water 8.4 Global wetdeck slamming effects 8.4.1 Water entry and exit loads 8.4.2 Three-body model 8.5 Global hydroelastic effects on monohulls 8.5.1 Special case: Rigid body 8.5.2 Uniform beam 8.6 Global bow flare effects 8.7 Springing 8.7.1 Linear springing 8.8 Scaling of global hydroelastic effects

x Contents 8.9 Exercises 8.9.1 Probability of wetdeck slamming 8.9.2 Wave impact at the front of a wetdeck 8.9.3 Water entry of rigid wedge 8.9.4 Drop test of a wedge 8.8.5 Generalized Wagner method 8.9.6 3D flow effects during slamming 8.9.7 Whipping studies by a three-body model 8.9.8 Frequency-of-encounter wave spectrum in following sea 8.9.9 Springing 9 PLANING VESSELS 9.1 Introduction 9.2 Steady behavior of a planing vessel on a straight course 9.2.1 2.5D (2D+t) theory 9.2.2 Savitsky's formula 9.2.3 Stepped planing hull 9.2.4 High-aspect-ratio planing surfaces 9.3 Prediction of running attitude and resistance in calm water 9.3.1 Example: Forces act through COG 9.3.2 General case 9.4 Steady and dynamic stability 9.4.1 Porpoising 9.5 Wave-induced motions and loads 9.5.1 Wave excitation loads in heave and pitch in head sea 9.5.2 Frequency-domain solution of heave and pitch in head sea 9.5.3 Time-domain solution of heave and pitch in head sea 9.5.4 Example: Heave and pitch in regular head sea 9.6 Maneuvering 9.7 Exercises 9.7.1 2.5D theory for planing hulls 9.7.2 Minimalization of resistance by trim tabs 9.7.3 Steady heel restoring moment 9.7.4 Porpoising 9.7.5 Equation system of porpoising 9.7.6 Wave-induced vertical accelerations in head sea 10 MANEUVERING 10.1 Introduction 10.2 Traditional coordinate systems and notations in ship maneuvering 10.3 Linear ship maneuvering in deep water at moderate Froude number 10.3.1 Low-aspect-ratio lifting surface theory 10.3.2 Equations of sway and yaw velocities and accelerations 10.3.3 Directional stability 10.3.4 Example: Directional stability of a monohull

Contents xi 10.3.5 Steady-state turning 10.3.6 Multihull vessels 10.3.7 Automatic control 10.4 Linear ship maneuvering at moderate Froude number in finite water depth 10.5 Linear ship maneuvering in deep water at high Froude number 10.6 Nonlinear viscous effects for maneuvering in deep water at moderate speed 10.6.1 Cross-flow principle 10.6.2 2Di-t theory 10.6.3 Empirical nonlinear maneuvering models 10.7 Coupled surge, sway, and yaw motions of a monohull 10.7.1 Influence of course control on propulsion power 10.8 Control means 10.9 Maneuvering models in six degrees of freedom 10.9.1 Euler's equation of motion 10.9.2 Linearized equation system in six degrees of freedom 10.9.3 Coupled sway-roll-yaw of a monohull 10.10 Exercises 10.10.1 Course stability of a ship in a canal 10.10.2 Nonlinear, nonlifting and nonviscous hydrodynamic forces and moments on a maneuvering body 10.10.3 Maneuvering in waves and broaching 10.10.4 Linear coupled sway-yaw-roll motions of a monohull at moderate speed 10.10.5 High-speed motion in water of an accidentally dropped pipe APPENDIX: Units of Measurement and Physical Constants Index

Preface Writing a book on the hydrodynamics of high-speed marine vehicles was chal-lenging 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 lead-ership 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 super-vised. 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 Depart-ment of Marine Technology, Norwegian University of Science and Technology (NTNU). My philosophy in writing the book has been to start from basic fluid dynam-ics 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 nec-essary to read the book in detail. The reader should also be familiar with dynam-ics 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 experi-ments 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. xiii

xiv Preface I am thankful to Professor Newman for reading through the manuscript and offering suggestions for improvement. Dr. Svein Skjordal 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 con-firming 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 Flaeten, Pro-fessor Thor I. Fossen, Dr. Chunhua Ge, Dr. Marilena Greco, Dr. Martin Greenhow, Dr. Ole Hermundstad, Egil Jullumstro, Dr. Toru Katayama, Professor Katsuro Kijima, Professor Spyros A. Kinnas, Dr. Kourosh Koushan, David Kristiansen, Professor Claus Kruppa, Dr. Jan Kvaalsvold, Dr. Burkhard Miiller-Graf, Professor Dag Myrhaug, Professor Makoto Ohkusu, Professor Bjornar Pettersen, Dr. Olav Rognebakke, Renato Skejic, Dr. Nere Skomedal, Professor Sverre Steen, Gaute Storhaug, Professor Asgeir Sorensen, Professor Ernest 0. 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 FransvAg 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 area; planform area of foil developed area, propeller blades expanded area, propeller blades 3D added mass coefficient in the jth mode due to kth motion 2D added mass coefficient area of propeller disc after perpendicular rudder area waterplane area average hull roughness beam beam of section blade area ratio critical damping 3D damping coefficient in jth mode due to kth motion 2D damping coefficient chord length; half wetted length in 2D impact; speed of sound block coefficient, ship drag coefficient friction coefficient frictional force coefficient computational fluid dynamics head coefficient restoring force coefficient in jth mode due to kth motion lift coefficient lift coefficient for planing vessel CLP at zero deadrise angle Theodorsen function midship section coefficient; mass coefficient in Morison's equation center of gravity pressure coefficient minimum pressure coefficient longitudinal prismatic coefficient residual resistance coefficient propeller thrust-loading coefficient; total resistance coefficient wave-making resistance coefficient wave pattern resistance coefficient

xvi List of symbols 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/&L F~B beam Froude number F~D draft Froude number Fnh depth or submergence Froude number F~T transom draft Froude number FP forward perpendicular Fv volumetric Froude number g acceleration of gravity G(x,y,z;<J ,<)G reen function GM GM~ GZ h 4 H H1/3 i Ijk 4 j, k IVR J k KC k f KG KT KQ L LC LCB 1% LCG LK LOA Los transverse metacentric height longitudinal metacentric height moment arm in heel (roll) about COG water depth; submergence height of the center of the jet at station S7 (see Figure 2.54) above calm free surface wave height; head significant wave height imaginary unit moment or product of inertia unit vectors along x, y and z-axis, respectively inlet velocity ratio advance ratio of propeller wave number; roughness height; form factor Keulegan-Carpenter number reduced frequency height of COG above keel thrust coefficient torque coefficient length of ship; lift of a foil; hydrodynamic roll moment in maneuvering chine wetted length longitudinal center of buoyancy longitudinal center of gravity measured from the transom stern longitudinal center of gravity keel wetted length length, overall length, overall submerged

List of symbols xvii longitudinal position of the center of pressure measured along the keel from the transom stern length between perpendiculars length of the designer's load waterline mass; moment; hydrodynamic pitch moment in maneuvering fluid momentum vector mass per unit length components of mass matrix propeller revolutions per second surface normal vector positive into the fluid normal force; hydrodynamic yaw moment in maneuvering origin of coordinate system order of magnitude of E power; pitch of propeller; probability pressure; roll component of angular velocity; half of the distance between the center lines of the demihulls of a catamaran; stagger between foils atmospheric pressure delivered power ambient pressure; static excess pressure vapor pressure of water propeller torque; volume flux; source strength pitch component of angular velocity yaw component of angular velocity radius; resistance added resistance in air and wind added resistance in waves radius of gyration in rigid body mode j root mean square Reynolds number residual resistance spray resistance total resistance viscous resistance wave-making resistance span length of foil area of wetted surface; cross-sectional area body surface wave spectrum time; thrust-deduction coefficient; maximum foil thickness period; propeller thrust modal or peak period mean wave period mean wave period encounter period natural period surface tension

xviii List of symbols forward velocity of vessel mean velocity at the most narrow cross-section of the waterjet inlet propeller slip stream velocity x-component of vessel velocity y-component of vessel velocity vertical distance between COG and the keel group velocity phase velocity wall friction velocity water entry velocity weight wake fraction; z-component of vessel velocity; vertical deflection Weber number Cartesian coordinate system. Moving with the forward speed in seakeeping analysis. Body-fixed in maneuvering analysis. x-component of hydrodynamic force in maneuvering Earth-fixed coordinate system x-coordinate of transom LK- Lc y-component of hydrodynamic force in maneuvering z-component of hydrodynamic force in maneuvering Greek symbols a angle of attack a, Kelvin angle a€ flap angle ai ideal angle of attack ao angle of zero lift p wave propagation angle; deadrise angle; drift angle l- circulation; gamma function; dihedral angle Y vortex density; sweep angle; ratio of specific heat for air 6 boundary layer thickness; rudder angle; flap angle 6 * displacement thickness A vessel weight E angle r surface elevation Ta wave amplitude 'l overall propulsive efficiency 'l~ hull efficiency VJ jet efficiency 'l k wave-induced vessel motion response, where k = 1,2,3. . to surge, sway, heave, roll, pitch, and yaw, respectively VP propeller efficiency; pump efficiency 'l~ relative rotative efficiency 'ls sinkage VT thrust power efficiency 8 pitch angle; momentum thickness 6 refers

4.2 Ship waves in deep water 107 number. We explain this in more detail later. The conclusion of this discussion is that the divergent waves at the Kelvin angle are of primary concern from a wash point of view. It matters also from the seashore point of view that the wavelengths due to high-speed vessels are long. This can be illustrated ,..' ,_,'- by noting that the wavelength h, of the divergent __-'. ,/- waves at a = a, is according to eq. (4.1 I), given by A~ //- u (rad) Figure 4.9. Angles 01 and 02 of stationary phase as a function of the polar coordinate angle cr of a posi-tion inside the Kelvin angle a, = 19"28' '= 0.34 rad. 0 is defined in Figure 4.6. -HI and -02 are propagation angles for y > 0 of, respectively, transverse and diver-gent waves. We discussed in sections 3.2.4 and 3.2.5 how waves are modified as they come close to the seashore. We can find the wave crest positions inside the Kelvin angle as shown in Figure 4.3 by using eq. (4.22). If A(H,) are real, for the transverse waves they are given by and the Kelvin angle a,, that is, for positive y in (4.24) Figure 4.6. We note that 8, is always negative and that HI I < )&I. -8, and -Or are propagation angles (see Fig-ure 4.6) of respectively transverse and divergent waves. When a = a,, 81 = $ = -sin-'(l/&) = -35"16'. This is the angle illustrated in Figure 4.3 for the divergent waves at the Kelvin angle. When -a, < a < 0, we get similar solutions of Hi, but with opposite signs. Eq. (4.22) shows that the wave amplitude decays like r-lf2 when 0 5 la1 < ar. When la = a,, eq. (4.22) gives an infinite value for the amplitude. Actually, a separate analysis is needed in the close vicinity of la1 = ac. This will show that the wave amplitude decays like r-'fs. Here 8, is given by eq. (4.21). Eq. (4.24) can be written as in polar coordinates (r, a). Similarly for the diver-gent waves, that is, where HZ, is given by eq. (4.21). This decay factor or the fact that the wave ampli- 2.3 Transverse waves along the ship's track tude decays as a function of horizontal distance 1 y I from the ship's track is important from a coastal engineering point of view, that is, to assess the wash at the shore due to a passing ship. Because y = r sin a, the decay rate in terms of lyl is the same as that in terms of r. One aspect is the decay rate, as discussed. Another aspect is how large the amplitudes of the waves are. We will later see that when Fn > ~0.6th,e divergent waves will be dominant. The amplitude of the transverse waves is strongly dependent on the Froude number for Fn < =0.5. This has to do with phasing between transverse waves generated in the bow and at the stern. Because the phasing has to do with the wave-length of the transverse waves, we can understand from eq. (4.12) that this is related to the Froude We will now discuss how ship parameters influ-ence the transverse waves along the ship's track. However. our theoretical model of how the ship generates waves is very simplified at this stage. The problem is analyzed from the ship reference sys-tem. This means that the ship speed appears as an incident flow with velocity U. We assume that the ship has a long parallel midpart with submerged cross-sectional area S. It is only at a small part in the bow and the stern where the ship cross section changes. Because a high-speed semi-displacement vessel has a transom stern and the flow separates from the transom stern with a resulting hollow in the water behind the ship, we may in this case con-sider the body to consist of the ship and the hollow.

108 Wave Resistance and Wash This is based on the coordinate system (x, y, z) defined in Figure 4.6. (6, q, <) is the source point .- , and (x,y,z) IS a field point at which we want to evaluate the influence of the source (sink). El is the complex exponential integral as defined Figure 4.10. The ship moves the water The ship has in Abramowitz and Stegun (1964), and H 1s the a long parallel midpart wlth submerged cross-sectional ~~~step~ funcition ~id~ area S. The effect of the ship will be that it pushes the water out in the bow, that is, it acts like a source, whereas the ship attracts the water in the stern, that is, it acts locally like a sink. This is illus-trated from the Earth-fixed reference system in Figure 4.10. A source or a sink that properly satis-fies the boundary conditions for this problem is a complicated function. The linearized free-surface condition follows from eq. (3.10) by setting po and slat equal to zero. This means where G is the velocity potential due to the source (sink). This is also called a Green function. Fur-ther, we must require that there is no flow at large water depths. A radiation condition ensuring that the ship waves are downstream of the ship is also needed. Newman (1987) has derived the following velocity potential for the source (sink): where u = v[(z + <) sec2 N + i 1 y - sec2 H sin 8 Further, Q is the source (sink) strength. To be pre-cise, Q > 0 means a source and Q < 0 means a sink. The expression - Q/(4rr R) is a source (sink) in infinite fluid. The last integral term in eq. (4.28) represents the downstream waves, that is, for large positive values of (x - 6). It can be shown that the source strength in the bow can be expressed as US, whereas the sink strength in the stern is -US. Here S is the submerged midship cross-sectional area. Later we show that this is cor-rect. The source and sink coordinates are, respec-tively, 6 = -0.5L, q = 0, < = 0 and t = 05L, q = 0, C = 0. Here L is the ship length and the origin of the coordinate system is midships, and the x-axis points aft. Let us call the velocity potential due to the source-sink pair cp. The resulting free-surface ele-vation follows from the dynamic free-surface con-dition, that is, (see eq. (3.7) with po and tililt equal tozero). Let us start with the contribution CBS from the bow source to the downstream wave elevation. We then use eq. (4.31) in approximating the velocity potential. This gives By noting that it is the real part of eq. (4.33) that has physical meaning and introducing 0' = -0, we see that eq. (4.33) can be expressed in the form of

288 Slamming, Whipping, and Springing Figure 8.5. Drop tests a1 DNV of a wrdpc section of a typical sidehull design nl'an SES or a catamaran ?he top picture shows the secl picture illustrates the large spr fitted to reduce the three-dim lion at its initial drop height of about 9 m above water level. 7he lower ,ay created as a consequence of the impact. Note the circular end plates :nsional effect. - The hydrodynamicloading affects mestructural rmc. lrrc rvaumg is men appnea In a quasl-sreaoy elastic ribrations manner when the resulting static structural elastic ~ndst resses are calcu-cted and plastic deformations a The classical book on hydroelasticity of ships is by lated. The fluid flow is affe Bis.ho p and Price (1979). features such as compressi ~ ~~~~~ ~.. . . . . . . ... -- .. . .. by many physical bility and air cushions A convenrlonal srrunural analysls wtnout However. ~t 1s comDllcated to solve the comnlete hydroelasticity or dynamic effects considers the hydn 1st hydrodynamic loading by assuming a rigid struc- be m ns dynamic problem, and approximations mt ade. 'J3e guideline in making simplificatio

290 Slamming, Whipping, and Springing Fipune 8.8. Artist's impression of how slamming causin~gl obal elastic vibrations (whipping) of the ship's hull. (Artist: Biarnc Slenberel 8.2 Local hydmelastlc slamming effects Compressibility and the formation and collapse -..?.. . a . . .. .. . of nn air mehinn me irnnnrtrnt initi.1l.r rnA ---- "lIIerenI pnySlcal enects occur d"nng slamming, -- -.- .." '-'."". """""J' """ ""1- The effects of viscosity and surface tension are, in mally in a time scale, that is smaller than the general, negligible. When the local angle between time scale for local maximum stresses to occur. the water surface and the body surface is small at Hence, the effect of compressibility on maximum the impact position, an air cushion may be famed local stress is generally small. However, we can-me lmponanr rune scale rromastructuralpointof ~ ~ ~--- view is when maximum stresses occur. Thi- scale effects were demonstrated. The main parameters ~~ is given by the highest wet natural period (T,,) for for plates used in the drop tests are shown in the local structure. Table 8.1. The testse ctions were dividedI into three Table 8.1. Mnin pnrnmeters for pfntes used in the drop test results present"u ..A. , i.,n.. .F' i~l,.,. ~9-r., 0 ,n..nd r 8.10 Plate I1 Parameter Plate I (steel) (aluminum) Structural mass per unit length and breadth, ME 62 kgm? 21 kgm? Modulus of e1asticity.E 210 x lo9 Nm" 70 x lo9 Nm-2 Length of plate, L 0.50 m 0.50 m Breadth of plate, B 0.10 m 0.10 m Bending stiffness, EZ 8960 Nm2m-' 17060 Nmzm-' Structural mass parameter, Ms/pL 0.124 0.042 Connecting spring parameter, ksL/(ZEI) 2.85 1.50 Distance from neutral axis to strain measurements, z. 0.004 m 0.01375 m

312 Slamming, Whipping, and Springing 8a33 ~,$b-a)~ 8, 7 6 - - an SES. However, in the latter case, we accounted for leakage and inflow to the air cushion. Eqs. (8.58), (8.59), and (8.60) are now linearized by expressing p = pa + pl, where pl/pa <tl. It follows from eq. (8.60) by first writing p/pa = (p/pa)"' and then using a Taylor series expansion that Further, eq. (8.59) can be approximated as -dn = ~,,(b- a ) . dt This implies that constant U,, is assumed. The following linearized equation follows from eqs. (8.58), (8.61), and (8.62): mass a33 due to an oscillating alr cushlon of length (b-a) on the bottom of a 5 - semi-infin~tely long plate. a and b with b > a are d~stances from the leadlng 4 - edge of the plate (see Figure 8.32). The h~gh-frequency free-surface condition IS 3- used. p,, = mass density of the water. Here Qo is an average air cushion volume and (b - a) an average length of the air cushion. We still have two unknowns. that is, U,, and p,. How-ever, they can be related as follows. We consider the boundary-value problem in Figure 8.32. We could say this is the same as the problem for a heav-ing flat plate between x = a and b in combination with a free surface from x = -m to 0, and fixed flat plates from x = 0 to a and from x = b to m. We can solve this problem for unit U,, by a numer-ical method or conformal mapping. We will not show the details on how to solve the problem, but instead recall from Chapter 7 how added mass is defined. This means that forced oscillation of the Figure 8.33. Two-dimensional added plate with velocity U,, will cause a vertical force on the plate that can be expressed as a33dU,,/dt, where a33 is the two-dimensional added mass in heave. (Note that the sign of the force is consis-tent with the fact that positive U,, is in the negative z-direction.) This force comes from integrating a dynamic pressure. This pressure is the same as pl . which is approximated as a uniform pressure from the force, that is, We can express a33 in a nondimensional way as follows Here p, means mass density of the water. Calcu-lated values of K based on conformal mapping are presented in Figure 8.33. The added mass is The value when a = 0 is a33 = p,b22/n. When alb + 1, a33 + m because of the free surface,

8.3 Slammlng on rlgid bodies 313 Figure l deck ex deck an1 stages o ~sidear model test based on Froude 2,(b -a) /L,an d Qn/LZw ould be which is infinitely far away on the scale of (b -a). Let us then cot The .p roblem IS then mathematically similar scaling. U,I~ . .. .> . ... --.L,.- ..:.L <L., .L- :.. - to solving me auoeu mass proowrn wrrl ngru r~r3r auL~U ,,,ode1 and full scales Eq. (8.69) says free-surface condition. This gives infinite two- tl dimensional added mass in heave. f, We should note that there is a conflict between the full-scale lengm, me pressure In IUJI scale WIU assuming both U. and pl to be constant between be (Lf/Ld5 times the I e. x =a and b. So we .... must consider our analysis If Froude scaling of pres! le approx:m.ta f-"m thi. nnint ,," .... nf ". vipw nressure in full scale wol le We ;cal-assumf substitute eq. (8.64) into eq. (8.63) and : harmonic oscillations This gives the natu-ww K is defined by eq. (8.65). This equation also he time scaling; that is we should present in nondimensional form as a function of nensional time xessure in model scal lure had been used, tt ~lbde Lf/L, times tt r....-.. -~ ~.~-~ ~ pressure in model scale. This means Froude ! ing is clearly conservative when slamming p sures associated with air cushions are scaled. ' was also documented numerically by Greco e (2003), who also showed that the linear beha described in this section is appropriate in mt tests However, the oscillations of the air cusi had a strong nonlinear behavior in full scale. Figure 8.34 shows another case in whicl air cushion is created during impact. A ph ing breaker hits the top of a deck structure. can be an initial scenario for green water on I (Barcellona et al. 2003). The results in Figure are two-dimensional results by Greco (2001). -. Ires-ral fref Tbis :t al. nor ode1 hion where gives t I an results ung-nondi~ This ieck 8.34 Ihe where ...- "...-. ... . . . .. . . L1 LcLL6L.I lrOLLl YYY,. air cushion ~ ~ ~111l,n thls case, collapse lnto buohles words, we must introduce the finite dimensions of the body. pressu where tical results for slamming mis veloaty a proude scales. using eqs. ,a.w, r.L..ul... ".. ,... ., .,. .:a1 wall due to an impact-and (8.65), w ing fluid wedge with interior angle 6 and velocity V. The results are based on a similarity solution, ,e can now write PI In, fb-a'~* urn which neglects gravity. Thismeans that it does not -nI' - -- &FG=-J8 L Is2nJiE sinw,,!. need to be avertical wall but can be any flat surface (8.69) perpendicular to the impacting fluid wedge. When

Figure 8.35. Left: sketch of the equivalent problem of a Ruid (half) wedge impacting a flat wall at 90". Center: maximum pressure on a wall due to the water impact. Right: pressure distribution along the vertical wall for 5" 5 @ 5 75" with increment A@ = 10". The results are numerically obtained by neglecting gravity and using the similarity solution by Zhang et al. (1996) (Greco 2001). the interior angle p is close to 90", we could obtain similar results by.using a Wagner-type analysis. The results in Figure 8.35 are of relevance, for instance, in studying the impact on a deck house of green water on deck (Greco 2001). A scenario causing green water is shown in Figure 8.36. The relative vertical motions between the ship and the waves cause avertical wall of water around the bow. The behavior of the water later on is similar to the breaking of a dam. This causes water to flow with large velocities along the deck and to induce loading on the deck as well as on deck houses and equipment (Barcellona et al. 2003). The front of the water can locally be approximated as a fluid wedge with a small angle 0. There are, of course, three-dimensional effects modifying this picture. Ifthe relativeverticalvelocity isnot dominantin comparison with the relative longitudinal velocity, Figure 8.36. Illustration of green water when the tanker Sw! met vphoan Iudy southcast of Okinawa in 1963. Ihe relative vertical motions between the ship and the waves caused a vertical wall of water around the bow. The behavior of the water later on was similar to the breaking of a dam. nis caused water flowing with large velocity along the deck. Secondary slamming effects may occur when the water flows from the forecastle, shown on the picture, and hits the main deck (Photo: Per Meidel).

8.3 Slamming on rigid bodies 315 Figure 8.37. 2D and 3D cxperimcntsof green water on lhc deck ofa slalionar) ship that is rratraincc from oscillating (Greco 2001. Barcrllona ct al. 2003). the water may flow onto the deck in a manner similar to a plunging breaker (Greco 2001, see Figure 8.34). An extreme situation may be that a plunging breaker hits directly on a deck hous~ in the forward part of the ship. Figure 8.37 shows .. . 2D . and . 3D . .. experiments o f -, . -"L sults in Figure 8.35 are directly applicable in th, 1 case. i n. t he .i.n -itia l h e after the e r h i .p.as.w. at.has. .green water on me aecK or a starlonary sn~p.~ ne re e 21 t the vert~caw~al l. msn as been extensively stua~ed numerically and experimentally by Greco (2001), who also described the strong interaction between the flow on the deck and exterior to the ship hull. The water in her case came initially as a plung-ing breaker hitting the front of the deck. The sub-sequent fluid motion on the deck resembled but was not equal to the flow due to the breaking of a dam. Obviously, the results ir valid for the entire time alrrr urs urrpacr w~rnu x vertical wall, bec the vertically moving flub d is influenced by ( zravity. The water near the wa. I1 will at some stag, e overturn, as illustrated . ..... in Fig . ure 8.37. me overtummg . warer . WII rnen Impact on the underlying water, causing important pres-ause sure loading on the deck and the wall. Figure 8.37 shows the water after this impact of the overturn-ing water. It illustrates also that the 3D flow situ- Ition is only qualitatively the same as for the 2D low. The results in the figure are for a stationary and nonoscillating ship. However, large forward L 1 ~-~... . :-> .... >.L: ---. spew, wavc-muucrusn~pmu:.-u. u~-~-.s,,Lau ~urwu g currr- :try typical for a high-speed vessel are expected to have a clear influence on the results. -. """ . ~~A-~,~ ~-L.L.-~~. I rigure 8.36 snows expenmenral warenront velocity along the deck centerline for three sta-tionary ship models that are restrained from mov-ing in head sea. If the initial height of the water above the deck, H-f, is set equal to 10 m, the waterfront velocity along the deck centerline once the flow is almost fully developed varies between 11 ms-', for kn = 0.125, and 17 ms-', for ka = 0.225. Here ka means the incoming wave steep-ness These values can be used to get a rough esti-mate of the maximum pressure on a vertical wall, associated with an initial water-superstructure impact. At the beginning of the impact, the water-front velocity (impact velocity, V) and the angle 0 represent the impact parameters. Actually, if, as expected, is sufficiently small (less than W),

368 Planing Vessels Using Aw = 0.5 (LK + LC) /B and eqs. (9.50) and (9.51) gives - = -- v-cg-/B + z,,r l B lo COST sin2 r sm i (9.53) 0.25 tan B + (1 + zmxl Vfb2 Here T must be evaluated in radians. An impor-tant part of the static forces is Cu, (see eq. (9.5)). Differentiating CIB results in Here hn is the value of hw at the static equilibrium position. Using the expression for vertical static force given by eqs. (9.4) and (9.5) leads to The pitch moment about COG can be expressed as where e, is given by eq. (9.7). We can write It follows that The restoring coefficients for the case presented in Figure 9.26 are then These restoring coefficients show strong coupling between heave and pitch. This coupling effect is important for the occurrence of porpoising. Ikeda and Katayama (2000b) showed that porpoising did not occur if the coupled restoringcoefficients were set equal to zero in their analysis of a personal watercraft. Added mass in heave and pitch The added mass calculations will be based on a high-frequency free-surface condition and strip theory. The two-dimensional added mass coeffi-cient in heave a33 for a wedge is then an essen-tial part for a prismatic planing hull. An analyt-ical solution of has been presented by many researchers. One version is (Faltinsen 2000) - --[p-d2 n V1.5 - Bin) -I]. tan B sinp r2(1 - Bln)r(0.5 + Bin) (9.63) Here d is the draft, which is equal to 0.5btanB. Here b is the beam of the wedge. Further, r is the gamma function and K is by definition a33/pd2. Eq. (9.63) is graphically presented in Figure 9.27.

380 Planing Vessels Table 9.2. Calculated real (a) and imaginary (w) terms of eigenvalues s for coupled linear heave and pitch of prismatic planing hull with B = 20", hw = 4, leg/ B = 2.13, 7deg = 43, and ~/= 3.0m Eigenvalue no. A wJ? equations into explicit expressions for du3/dt and dus/dt. This gives, then, the following four first-order differential equations: -dv3 = Uj dt where Eq. (9.125) is now in a convenient form for stan-dard procedures for numerical time integration, but initial conditions have to be given. In our anal-ysis, we have used a Runge-Kutta method of fourth order. 9.5.4 Example: Heave and pitch in regular head sea A prismatic planing hull in regular head sea waves is considered, and the linear transfer functions of heave and pitch are calculated. The average wetted length-to-beam ratio hw = 4.0, the trim angle rdeg = 4', the deadrise angle Bdeg = 20°, the beam Froude number UI(~B)O' = 3.0, kg/B = 2.13, vcg/B = 0.25, the ship mass M = 1.28pB1, and the pitch radius of gyration rs5 with respect to COG is 1.3B. The theoretical procedure to find transfer functions is described in section 9.5.2. Three different methods are used to predict added mass, damping, and wave excitation loads. Transfer functions (or steady-state solutions) have no physical meaning if the system is dynami-cally unstable, that is, porpoising occurs. However, Figure 9.29 shows that the coupled heave and pitch motions will be stable when U/(gB)0.5 = 3.0. This is true both for our theoretical method and for Troesch's empirical method. The stability is deter-mined by the eigenvalues s (see eq. (9.81)). The four eigenvalues s calculated by our theory gives nondimensionalized real (a) and imaginary (w) parts, as presented in Table 9.2. We can combine the eigenvalues into two sets, in which the eigen-values are complex conjugates for each set. When studying the system response, it is sufficient to consider positive imaginary parts. Because all real parts are negative, Table 9.2 confirms that porpois-ing instability does not occur. If the ratio between the real and imaginary parts is small, strong ampli-fication of the transfer function occurs when the system is excited with a frequency we equal to w. The ratio between the absolute values of a and w can then be approximated as the ratio between damping and critical damping of the eigenmode. The transfer functions in heave and pitch are presented in Figures 9.34 and 9.35. The peak in the transfer functions predicted by our theory , . 5 - i 'i Empirical coefficients by ~rdescha nd -.---.- bd8 Figure 9.34. Transfer function for wave excited heave (11~) in regular head sea waves with incident wave amplitude to. h = wavelength, L= average wetted ship length, B = beam, prismatic planing hull with LIB = 4.0, sdeg = 4", p = 20", kg/ B = 2.13, and U/m = 3.0. G 7 6 Theory - - Theory with only -... - - .- - .; , F 'roude-Kriloff excitation

420 Maneuvering Figure 10.31. An all-movable rudder with a rudder angle 8 as defined in Figure 10.4. There is an incident flow velocity Jm to the rudder with an angle of attack 611. L and D are the rudder lift and drag force, respectively (see also Figure 2.16). The body-fixed coordinate system (x, y) is con-sistent with Figure 10.4. UR we can understand that the propeller slip stream is beneficial for the effectiveness of the rudder. However, the calculations in Figure 10.30 neglect the swirling flow in the propeller slip stream and do not consider the effect of the free surface and the vessel. Further, the analysis requires the rudder to be, let us say, at least the order of a propeller radius behind the propeller (Serding 1982). We illustrate how to evaluate forces and moments on the vessel due to an all-movable rud-der behind a propeller. As seen from the rudder, there is an incident flow velocity 4- with components UR and UR along the body-fixed neg-ative x- and y-axes, respectively (Figure 10.31). Here uR will vary along the rudder axis and is Us if we consider a rudder cross section within the pro-peller slip stream. Otherwise UR is U,. We intro-duce an averaged ui instead of using a varying ui. One way of doing this is by using the rudder plan-form area parts inside and outside the propeller slip stream as weighting factors, that is, where AR = rudder planform area ARS = rudder planform area within propeller slip stream The transverse (URi)n flow velocity component along the negative y-axis as observed from the rud-der can be expressed as where XR is the x-coordinate of the rudder centroid and y, and y, are flow rectification factors due to the hull and the propeller (Ankudinov et al. 1993). Neglecting this effect means that y, = y, = 1. The incident flow causes an angle of attack aH relative to the rudder (see Figure 2.16 for a definition of angle of attack). We can write (see Figure 10.31) The total or effective angle of attack 6, of an all-movable rudder must also include the rudder angle, that is, The resulting force on the rudder can be decom-posed into a lift (L) and a drag (D) component (see Figure 10.31). The lift is perpendicular to the inflow velocity direction and can be expressed as O.SpARCLR(~+% u i), where CI,R is the lift coef-ficient of the rudder. The drag force, which is in line with the inflow direction, can be expressed as O.~~ARC+~ vi~), (wUhe~re CDRi s the drag coefficient. The hydrodynamic longitudinal (XR) and trans-verse (YR) force components and yaw moment (NR) due to the rudder in the body-fixed coordi-nate system can then be expressed as XR P = - AR (u+~ 4~)( -c LR sin 6H - cDcRos SH) 2 P YR = -2 AR ([& -t u;) (CLRc os SW-k CDRs in SH) NR = YRxR. (10.96) Whicker and Fehlner (1958) did an extensive experimental study of the lift and drag CLR and CDR for rudders as a function of the angle of attack. A spatially uniform inflow velocity along the rudder span was considered. Investigations were made of the following: three aspect ratios. 1, 2, and 3; five section shapes; two tip shapes, faired and square; and three sweep angles (-8, 0,ll). Aspect ratios and sweep angles are defined by Figure 6.3. Tests were made in a low-speed wind tunnel, and the models were mounted on a ground board. If the rudder is free surface-piercing, then the ground board will simulate the effect of the free surface for small and moderate Froude numbers.

432 Maneuvering 10.10.2 Nonlinear, nonlifting, and nonviscous hydrodynamic forces and moments on a maneuvering body Kochin et al. (1964) presented expressions for the nonlinear, nonlifting, and nonviscous hydrody-namic force and moment on a maneuvering body in infinite fluid. A body-fixed Cartesian coordinate system with origin in 0 is introduced. The veloc-ity vector of an arbitrary point M of the body is expressed as where V is the velocity vector of the point 0, r is the radius vector from M to 0, and R is the vector of the angular velocity of rotation of the body. We express where Vl, V2, and V3 mean the components of V along the body-fixed coordinate axis. V4, V5, and V6 have similar meaning for R. We introduce Here Ajk means added mass coefficients. These can be calculated by, for instance, a 3D boundary element method (BEM). We define the vectors The hydrodynamic force vector F acting on the body is and the hydrodynamic moment vector M with respect to the body-fixed coordinate system is a) Consider a ship that maneuvers in the horizon-talplane. The water has infinite depth and horizon-tal extent. The Froude number is assumed mode-rate so that the rigid free-surface condition applies. Use eqs. (10.157) and (10.158) to express consis-tently with Figure 10.5 the yaw moment and the longitudinal and transverse forces on the ship in terms of the yaw angular velocity and the lon-gitudinal and lateral components of ship veloc-ity. The expressions should account for the fact that the hull is symmetric about the body-fixed xz-plane defined in Figure 10.4. You must explain why this causes some of the coupled added mass coefficients Ajk to be zero. (Hint: Start with how added mass was explained in section 7.2.1 and dis-cuss symmetry and antisymmetry of the flow with respect to the xz-plane due to forced surge, sway, and yaw velocity.) The answer is Here X, Y, N, u, v, and r = 4 are consistent with Figure 10.4 and Ajk are the low-frequency added mass coefficients for the ship. Modify eq. (10.159) by introducing lifting terms consistent with linear slender body theory. b) Why can we not apply the infinite fluid results by Kochin et al. (1964) to the roll of a ship moving at moderate Froude number? Consider now the free-surface condition that the velocity potential due to body motion is zero on the mean free surface. Use the infinite fluid results to derive hydrodynamic transverse force, roll, and yaw moment due to roll. 10.10.3 Maneuvering in waves and broaching We will consider the hydrodynamic loads during maneuvering of a monohull at moderate speed in linear regular deep-sea waves with a wavelength A that is long relative to the cross dimensions of the vessel. a) Show by making a coordinate transformation of the results in Table 3.1 that the incident wave potential can be represented as gt cpo = -ekzcos(oet -kucos,!l - kysin@ fe) on (10.160) in the body-fixed coordinate system defined in Figure 10.5 with z = 0 in the mean free surface and positive z upward. Here mi 2n we =wo+kucos,!l,k= - = --, (10.161) g A where u is the forward speed component of the vessel along the negative x-axis. Further, ,!l is the wave propagation direction measured relative to

434 Maneuvering Figure 10.37. Nomenclature and coordinate systems of motions of a slender pipe in the Earth-fixed X-Z-plane. The body-fixed coordinate system (x, z) has origin in the center of gravity of the pipe. U1 and U3 are translatory velocity components of the pipe along the x- and z-axes, respectively. S22 is the rate of turn of the pipe. of the pipe when it hits the water surface is an unknown parameter. The water entry of the pipe will subsequently change the orientation and velocity of the pipe. We will focus on the next phase, when the pipe is completely submerged and has no influence on the free surface. The situation is illustrated in Figure 10.37. We assume 2D flow in the global vertical X-2-plane. The X- and Z-coordinates of the center of gravity of the pipe are denoted XG and ZG. We introduce a body-fixed coordinate system (x, y, z) as illustrated in Figure 10.37. The origin is in the center of grav-ity of the pipe. The pipe has translatory velocity components Ul and U3 along the x- and z-axes, respectively. The angle of the pipe axis relative to the X-axis is denoted /?, and the angular velocity of the pipe about the y-axis is called 522. We can then set up the following relationships: -dXG = Ul cos ,!3 + U3 sin /? (10.166) dt dZG -- - U3 cos B - Ul sin fi (10.167) dt We will first assume the pipe has end caps. The following three equations follow from Newton's second law: dQ2 (15s + Ass) - = Ui (A33 f ~Ta33~) dt x Us - x~a~(~U~I( (Q10.217 1) + MD, + M,?. Here the longitudinal viscous force can be expressed as FD, = -0.5pCFnDLUl IUl(- P -~CD,PU~ lull, 8 (10.172) where The CF-value assumes turbulent axisymmetric flow along a smooth surface (White 1972). Rn means the Reynolds number. Further, Cox is a base drag coefficient that may be set equal to 0.65 (Hoerner 1965). Fn, and MD, in eqs. (10.170) and (10.171) also represent viscous loads. The mass of the pipe is called M, and we will assume uni-form mass distribution. In eqs. (10.169) through (10.171), we have not explicitly expressed all the mass and added mass terms. The latter terms are simply denoted F,.?, F,.?, M,,?. a) Explain the different terms in eqs. (10.169) through (10.171). Expressions for F,?, F,?, and M,) should be presented. Use the cross-flow princi-ple to formulate the viscous terms FD, and MD,. in eq. (10.170) and eq. (10.171), respectively. One should make a special effort to explain why IUl 1 appears in the equations. b) Assume now that the pipe has no end caps and there is a flow through the pipe. Consider the spe-cial case of steady incident flow with a constant small angle of attack relative to the cylinder axis and assume that the inner diameter of the pipe is equal to the outer diameter. Show by slender body theory that the lift force with interior flow is twice the lift force on the same pipe with end caps.

436 Appendix: Units of measurement and physical constants Table A.1. Conversion factors for different units of measurement Quantity SI unit Other unit Inverse factor Length Area Volume Velocity Mass Force Pressure Energy 3.281 feet (ft) 0.540 nautical miles 10.764 ft2 35.315 ft" 264.2 gallons (US) 220.0 gallons (UK) 3.281 fts-I 1.944 knots 2.205 pounds 0.984 tons (long) 1 tonne (metric) 0.225 pound-force 0.1020 kg-force (kgf) 102.0 tonne-force 100.4 ton-force 0.000145 psi (pounds per square inch) 1 0-5 bar 0.738 foot-pounds 0.00134 horsepower Table A.2. Mass density (p) and kinematic viscosity ( v ) of water and air Freshwater Saltwater (salinity 3.5%) Dry air Temperature p(kg m-3) v . 10h(rn2s-') ,o(kg m-') v . 106(m2s-') p(kg m-3) v . 106(m2s-') 0°C 999.8 1.79 1028.0 1.83 1.29 13.2 5°C 1000.0 1.52 1027.6 1.56 1.27 13.6 10°C 999.7 1.31 1026.9 1.35 1.25 14.1 15°C 999.1 1.14 1025.9 1.19 1.23 14.5 20°C 998.2 1 .00 1024.7 1.05 1.21 15.0 Table A.3. Vapor pressure of water for various temperatures (Breslin and Andersen 1994) Temperature Vapor pressure, " c OF PV(N~-~)

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